Research Article Sum of Bernoulli Mixtures: Beyond...

15
Research Article Sum of Bernoulli Mixtures: Beyond Conditional Independence Taehan Bae 1 and Ian Iscoe 2 1 Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada S4S 0A2 2 Quantitative Research, Risk Analytics, IBM Corporation, 185 Spadina Avenue, Toronto, ON, Canada M5T 2C6 Correspondence should be addressed to Taehan Bae; [email protected] Received 23 January 2014; Accepted 22 April 2014; Published 2 June 2014 Academic Editor: Aera avaneswaran Copyright © 2014 T. Bae and I. Iscoe. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the distribution of the sum of Bernoulli mixtures under a general dependence structure. e level of dependence is measured in terms of a limiting conditional correlation between two of the Bernoulli random variables. e conditioning event is that the mixing random variable is larger than a threshold and the limit is with respect to the threshold tending to one. e large-sample distribution of the empirical frequency and its use in approximating the risk measures, value at risk and conditional tail expectation, are presented for a new class of models which we call double mixtures. Several illustrative examples with a Beta mixing distribution, are given. As well, some data from the area of credit risk are fit with the models, and comparisons are made between the new models and also the classical Beta-binomial model. 1. Introduction Since the U.S. subprime mortgage crisis which led the world economy into the global recession in the late 2000s, there has been serious criticism about the use of inappropriate depen- dent default models for portfolio credit risk (see Donnelly and Embrechts [1]). Specifically, the Gaussian copula model (Basel II Accord [2] and Li [3]) had become the industry standard model and has been widely used for both the pricing of portfolio credit derivatives such as collateralized debt obligations (CDOs) or mortgage backed securities and the credit risk management. Its inability to incorporate extreme tail dependence, however, resulted in a serious lack of default clustering, especially under stressful economic situations. In the literature, several alternative approaches were pro- posed to cope with the issue of an insufficient level of depen- dency among defaults. As a simple extension of the Gaussian copula model, Andersen and Sidenius [4] and Burtschell et al. [5] considered stochastic asset correlation models. A more popular way of incorporating further dependence beyond the Gaussian copula model is the use of heavy tail copulas such as Archimedean copulas. Amongst many others, we mention Burtschell et al. [6] and Sch¨ onbucher and Schubert [7] for applications of copula models under the structural and the intensity-based credit risk model framework, respectively. It is important to note that factor copula models under the typical conditional independence assumption have a close link with Bernoulli mixture models. Specifically, by de Finettis theorem, there exists a common mixing random variable on which the default indicator random variables, in an exchangeable credit portfolio, are dependent. e mixing random variable is essentially the random probability of default which can be represented as a function of the common systematic factor in a factor copula model. Under the typical conditional independence assumption, however, the level of dependence between the default indicator random variables is controlled solely by the distributional property of the systematic factor, which can impose limitations for some applications where extreme levels of dependence are required. See Bluhm et al. [8], Cousin and Laurent [9], Frey and McNeil [10, 11], KMV [12], McNeil et al. [13], McNeil and Wendin [14], Moraux [15], and RiskMetrics Group [16] for more references on credit risk applications of the Bernoulli mixture models under the conditional independence model- ing framework. Formally, a Bernoulli mixture model is defined as follows. Let ( ) =1 be a sequence of identically distributed indicator random variables. We assume that each , = 1,...,, follows the Bernoulli distribution with the common default probability . e probability is randomly drawn from Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2014, Article ID 838625, 14 pages http://dx.doi.org/10.1155/2014/838625

Transcript of Research Article Sum of Bernoulli Mixtures: Beyond...

Page 1: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

Research ArticleSum of Bernoulli Mixtures Beyond Conditional Independence

Taehan Bae1 and Ian Iscoe2

1 Department of Mathematics and Statistics University of Regina Regina SK Canada S4S 0A22Quantitative Research Risk Analytics IBM Corporation 185 Spadina Avenue Toronto ON Canada M5T 2C6

Correspondence should be addressed to Taehan Bae taehanbaeureginaca

Received 23 January 2014 Accepted 22 April 2014 Published 2 June 2014

Academic Editor Aera Thavaneswaran

Copyright copy 2014 T Bae and I Iscoe This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider the distribution of the sum of Bernoulli mixtures under a general dependence structure The level of dependence ismeasured in terms of a limiting conditional correlation between two of the Bernoulli random variables The conditioning eventis that the mixing random variable is larger than a threshold and the limit is with respect to the threshold tending to one Thelarge-sample distribution of the empirical frequency and its use in approximating the risk measures value at risk and conditionaltail expectation are presented for a new class of models which we call double mixtures Several illustrative examples with a Betamixing distribution are given As well some data from the area of credit risk are fit with the models and comparisons are madebetween the new models and also the classical Beta-binomial model

1 Introduction

Since the US subprime mortgage crisis which led the worldeconomy into the global recession in the late 2000s there hasbeen serious criticism about the use of inappropriate depen-dent default models for portfolio credit risk (see Donnellyand Embrechts [1]) Specifically the Gaussian copula model(Basel II Accord [2] and Li [3]) had become the industrystandardmodel and has beenwidely used for both the pricingof portfolio credit derivatives such as collateralized debtobligations (CDOs) or mortgage backed securities and thecredit risk management Its inability to incorporate extremetail dependence however resulted in a serious lack of defaultclustering especially under stressful economic situations

In the literature several alternative approaches were pro-posed to cope with the issue of an insufficient level of depen-dency among defaults As a simple extension of the Gaussiancopulamodel Andersen and Sidenius [4] and Burtschell et al[5] considered stochastic asset correlation models A morepopular way of incorporating further dependence beyond theGaussian copula model is the use of heavy tail copulas suchas Archimedean copulas Amongst many others we mentionBurtschell et al [6] and Schonbucher and Schubert [7] forapplications of copula models under the structural and theintensity-based credit risk model framework respectively

It is important to note that factor copula models underthe typical conditional independence assumption have aclose link with Bernoulli mixture models Specifically byde Finettis theorem there exists a common mixing randomvariable on which the default indicator random variablesin an exchangeable credit portfolio are dependent Themixing random variable is essentially the random probabilityof default which can be represented as a function of thecommon systematic factor in a factor copula model Underthe typical conditional independence assumption howeverthe level of dependence between the default indicator randomvariables is controlled solely by the distributional propertyof the systematic factor which can impose limitations forsome applications where extreme levels of dependence arerequired See Bluhm et al [8] Cousin and Laurent [9] FreyandMcNeil [10 11] KMV [12] McNeil et al [13] McNeil andWendin [14] Moraux [15] and RiskMetrics Group [16] formore references on credit risk applications of the Bernoullimixture models under the conditional independence model-ing framework

Formally a Bernoulli mixturemodel is defined as followsLet (119883119895)

119899119895=1 be a sequence of 119899 identically distributed indicator

random variables We assume that each 119883119895 119895 = 1 119899follows the Bernoulli distribution with the common defaultprobability 119875 The probability 119875 is randomly drawn from

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 838625 14 pageshttpdxdoiorg1011552014838625

2 Journal of Probability and Statistics

a distribution with cumulative distribution function (cdf)119865119875(119901) 119901 isin (0 1) At this point the dependency structure for(119883119895)119899119895=1 is still arbitraryIn credit applications the statistical properties of the sum

of the 119899 Bernoulli random variables are the main interestand the following assumption plays an important role inalleviating computational difficulties 119883119895 are conditionallyindependent given 119875 In this case given 119875 the sum 119878119899 =

sum119899119895=1119883119895 follows the binomial distribution with mean 119899119875 and

variance 119899119875(1minus119875) Then the unconditional probability massfunction (pmf) is given as follows

120583119868119899 (119896) = Pr [119878119899 = 119896]

= int1

0

(119899

119896)119901119896(1 minus 119901)

119899minus119896119889119865119875 (119901) 119896 = 0 119899

(1)

where the superscript ldquo119868rdquo on 120583 indicates the assumed con-ditional independence The most commonly used mixingdistribution in modeling the credit risk of a homogeneousportfolio such as a mortgage or credit card portfolio is theBeta distribution This model is referred to as the Beta-binomial (120573-I) model

The present paper is concerned with the level of depen-dence incorporated in Bernoulli mixture models especiallyunder stress situations Together with the asset correlationthe default correlation is used as the standard measure ofdependence in portfolio credit risk models In particularwe consider the behavior of the correlation between twoarbitrary terms as the random default probability 119875 isconditioned to be larger than 119901 tending to 1 Specificallywe define the conditional default correlation between twoterms 119883119894 and 119883119895 given 119875 gt 119901 0 lt 119901 lt 1 as follows(Bae and Iscoe [17] used an analogous definition to studythe asset correlations under stress for various single-factorcredit risk modelsmdashsome dynamicmdashwhile Kalkbrener andPackham [18] also studied asset correlations under stress forstatic normal variance mixture models In both studies inplace of the conditioning on 119875 gt 119901 conditioning is doneon an auxiliary risk factor which for example is related to asystematic factor)

120588119894119895 (119901) = Corr [119883119894 119883119895 | 119875 gt 119901]

=Cov [119883119894 119883119895 | 119875 gt 119901]

radicVar [119883119894 | 119875 gt 119901]radicVar [119883119895 | 119875 gt 119901]

=Cov [119883119894 119883119895 | 119875 gt 119901]Var [119883119894 | 119875 gt 119901]

(2)

where

Cov [119883119894 119883119895 | 119875 gt 119901]

= 119864 [119883119894119883119895 | 119875 gt 119901] minus 119864 [119883119894 | 119875 gt 119901] 119864 [119883119895 | 119875 gt 119901]

= 119864 [119883119894119883119895 | 119875 gt 119901] minus 119864 [119883119894 | 119875 gt 119901]2

Var [119883119894 | 119875 gt 119901]

= 119864 [1198832119894 | 119875 gt 119901] minus 119864 [119883119894 | 119875 gt 119901]

2

= 119864 [119883119894 | 119875 gt 119901] (1 minus 119864 [119883119894 | 119875 gt 119901])

(3)Then the limit of the conditional correlation as 119901 tends

to 1 (referred to as the limiting correlation) is defined as120588119894119895 = lim

119901rarr1120588119894119895 (119901) (4)

provided the limit existsIt is intuitive to expect that the correlations in (2) increase

monotonically as the threshold 119901 approaches 1 Unfortu-nately this is not the case for the Beta-binomial model Infact the conditional correlation always converges to zeroregardless of the mixing distribution as long as conditionalindependence is postulated (Corollary 2) This type of modelfailure can introduce a significant bias in the measurementand management of risk in stress situations

The objectives of the present paper are threefold (i) toderive the relationship between the limiting correlation andthe probability structure (ii) to construct general Bernoullimixture models with nontrivial limiting correlations (iii)to demonstrate implications of the constructed models fortail risk measures such as value at risk and conditional tailexpectation

The theoretical results for these objectives are presentedin Section 2 We first examine the limiting behavior ofcorrelations for Bernoulli mixture models in Section 21Then Section 22 introduces a general model frameworkmdashthe class of double mixture modelsmdashwhich allows furtherpositive dependence between entities beyond conditionalindependence (Theorem 4) Section 23 is devoted to thethird objective and it considers the large-sample distributionof the sum of Bernoulli mixtures under the general modelframework and its application to the approximation of riskmeasures is discussed As a specific case Section 3 considersa Beta Bernoulli mixture and Section 4 provides the resultsfor an example with specific parameter values Section 5demonstrates the fitting results of the double mixture modelsto a dataset from the area of credit risk Finally Section 6concludes this paper with a summary Technical details ofsome proofs are given in the appendices

2 Main Results

21 Limiting Correlation for Bernoulli Mixtures We denoteby 119886119898(119901) the regular conditional probability given 119875 = 119901that all terms (119883119894)

119898119894=1 of a fixed sequence of Bernoulli random

variables (119883119894)119899119894=1 take on the value 1 (For this definition we

allow 119899 = infin) Formally119886119898 (119901) = Pr [119878119898 = 119898 | 119875 = 119901] (5)

where 119878119898 = sum119898119894=1119883119894 Note that under an assumption of

conditional independence 119886119898(119901) = 119901119898

The following hypothesis on the properties of 119865119875 ina deleted neighbourhood of 119901 = 1 will be part of theassumptions inTheorem 1 and its Corollary 2 below

Journal of Probability and Statistics 3

Hypothesis H 119865119875 is absolutely continuous in a neighbour-hood of 119901 = 1 with probability density function (pdf) 119891119875thereon such that

(i) 119891119875(119901) gt 0(ii) 119891119875 is continuous in a deleted neighbourhood of119901 = 1(iii) (1 minus 119901)119891119875(119901)(1 minus 119865119875(119901)) converges to a finite limit as

119901 tends to 1

The significance of 119867(i) is clear it is required for theconditioning on 119875 gt 119901 to be well defined Parts 119867(ii) and119867(iii) are technical assumptionswhich play a role in the proofof Theorem 1 Part (iii) of Hypothesis 119867 is satisfied in manycases of practical interest where 119891119875 has a simple asymptoticbehaviour near 119901 = 1 For example 119867(iii) is satisfied if forsome positive (finite) constant 119888 and some (finite) constants119887 gt minus1 and 119889 (the relation ldquo119909 sim 119910rdquo denotes asymptoticequivalence in the sense that the ratio 119909119910 tends to 1 as119901 rarr 1)

119891119875 (119901) sim 119888(1 minus 119901)119887(log [1 minus 119901])119889 as 119901 997888rarr 1 (6)

Here is the verification that (6) implies that119867(iii) holdsin the case that 119889 = 0 (The case 119889 = 0 is similar but easier sothe details will be omitted) Applying (6) in the first step andLrsquoHospitalrsquos rule in the second step as 119901 tends to 1 we have

(1 minus 119901) 119891119875 (119901)

1 minus 119865119875 (119901)

sim119888(1 minus 119901)

119887+1(log [1 minus 119901])119889

1 minus 119865119875 (119901)

sim (minus119888 (119887 + 1) (1 minus 119901)119887(log [1 minus 119901])119889

minus 119888(1 minus 119901)119887+1119889(log [1 minus 119901])119889minus1(1 minus 119901)minus1)

times (minus119888(1 minus 119901)119887(log [1 minus 119901])119889)

minus1

sim 119887 + 1 + 119889(log [1 minus 119901])minus1 sim 119887 + 1

(7)

so119867(iii) is satisfied with the value of the limit being 119887 + 1 Itwill be clear that the proof of Theorem 1 is easily generalizedto allow the limit in 119867(iii) to be infinite However it isnot apparent given the previous example whether such asituation can actually occur For this reason we have takenthe limit to be finite in the formulation of119867(iii)

Based on the conditional joint probability 1198862(119901) weobtain the following result for the limiting correlation

Theorem 1 Let 119883119894 and 119883119895 be two Bernoulli mixture randomvariables with correlation 120588119894119895(119901) as in (2) Suppose thatHypothesis119867 holds One further assumes that lim119901rarr11198862(119901) =1 1198862(119901) is differentiable for 119901 in a deleted neighbourhood of 1and lim119901rarr1119886

10158402(119901) exists Then the limiting correlation in (4)

exists and satisfies

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) (8)

Proof See the appendices

In the context of a sequence (119883119896) of Bernoulli mixturerandom variables the notation ldquo1198862rdquo suppresses the depen-dence on 119894 and 119895 more precisely it should be written as 1198862119894119895In order to have a result which does not actually depend on119894 or 119895 an extra hypothesis on the Bernoulli mixtures must beimposed We will return to this point in Section 22

An interesting example is the case that both terms takeon identical values (comonotonicity) (here comonotonicityrefers to the case that all the components (identically dis-tributed) of a Bernoulli random vector coincide in value withprobability one conditional on 119875) that is 1198862(119901) = 119901 UnderHypothesis119867Theorem 1 implies that the limiting correlationfor this case is 1 The following corollary another importantexample states that the limiting correlation between twoarbitrary terms from an identically distributed Bernoullimixture sequence converges to zero under the assumptionof conditional independence

Corollary 2 Suppose that two Bernoulli mixture randomvariables 119883119894 and 119883119895 are conditionally independent given thecommon default probability 119875 Then under Hypothesis119867 thelimiting correlation in (4) exists and satisfies

120588119894119895 = lim119901rarr1

120588119894119895 (119901) = 0 (9)

Proof Under the conditional independence assumption1198862(119901) = 1199012 The desired result follows immediately fromTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) = 2 minus 2 = 0 (10)

Remark 3 Results similar to those of Theorem 1 andCorollary 2 can also be given for the other extreme condi-tional on 119875 lt 119901 with 119901 tending to 0 One simply replacesin Hypothesis119867 andTheorem 1 the phrase ldquoneighbourhoodof 119901 = 1rdquo with ldquoneighbourhood of 119901 = 0rdquo and 119867(iii) withldquo119901119891119875(119901)119865119875(119901) converges to a finite limit as 119901 tends to 0rdquothen in Theorem 1 and Corollary 2 one replaces ldquolim119901rarr1rdquowith ldquolim119901rarr0rdquo (Note that lim119901rarr01198862(119901) is automatically 0because 1198862(119901) le Pr[119883119894 = 1 | 119875 = 119901] = 119901) However inour applications it is the case ldquo119901 rarr 1rdquo which is importantso we will not dwell further on the case ldquo119901 rarr 0rdquo

22 DoubleMixtureModels Wenow turn to the constructionof a more general probability structure beyond conditionalindependencemdasha structure which reveals the fundamentalrole of the quantities 119886119896 We require a stronger assumptionon the sequence (119883119895)

119899119895=1 than their being just conditionally

identically distributed which is a statement about the 1-dimensionalmarginalsWe require that (119883119895)

119899119895=1 be condition-

ally exchangeable meaning that under each Pr[sdot | 119875 = 119901]all permutations of (1198831 1198832 119883119899) have a common jointdistribution which may depend on 119901 (Note that a sequenceof 119899 conditionally iid random variables is conditionallyexchangeable as is a comonotonic sequence) In particular

4 Journal of Probability and Statistics

conditional exchangeability is sufficient to guarantee that1198862(119901) 120588119894119895(119901) and hence the limiting correlation inTheorem 1are independent of 119894 and 119895

We assume for the remainder of the paper that oursequence of 119899 Bernoulli mixture random variables is condi-tionally exchangeable The quantities 119886119896(119901) 1 le 119896 le 119899 thendo not depend on the choice of subset of 119896 variables Theprobability that any 119896 of the Bernoulli random variables takeon the value 1 is given in the following theorem which showsthat the pmf of 119878119899 is essentially determined by the quantities119886119896(119901) 119896 = 1 119899 which in turn can be specified quitegenerally as input to the model

Theorem 4 Let (119883119895)119899119895=1 be a sequence of conditionally

exchangeable Bernoulli mixture random variables Then for119896 = 0 119899

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901] = (119899

119896)119892119899 (119896 119901) (11)

119892119899 (119896 119901) = 119886119896 (119901) minus (119899 minus 119896

1) 119886119896+1 (119901)

+ (119899 minus 119896

2) 119886119896+2 (119901) + sdot sdot sdot + (minus1)

119899minus119896119886119899 (119901)

(12)

where 1198860(119901) equiv 1 and 1198861(119901) = 119901Conversely let ](sdot 119901) 119901 isin (0 1) be a family of proba-

bility measures on the interval [0 1] such that int10119903](119889119903 119901) =

119901 and ](119860 119901) is a Borel measurable function of 119901 for eachBorel set 119860 If 119886119896(119901) is defined to be int1

0119903119896](119889119903 119901) then 119891119899 as

defined by the right-hand sides of (11) and (12) is a valid pmfMoreover suppose that (119883119895)

119899119895=1 follows a Bernoulli mixtures

distribution with conditional distribution

Pr [(1198831 119883119899) = (1199091 119909119899) | 119875 = 119901] = 119892119899 (119896 119901) (13)

for all (1199091 119909119899) which have precisely 119896 components equal to1 and the remaining 119899minus119896 components equal to 0Then (119883119895)

119899119895=1

is conditionally exchangeable Pr [119878119896 = 119896 | 119875 = 119901] = 119886119896(119901)for all 119896 = 1 119899 hence 119878119899 has conditional pmf given by (11)and (12) Furthermore

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119896 = 0 119899 (14)

therefore for all (1199091 119909119899)which have precisely 119896 componentsequal to 1 and the remaining 119899 minus 119896 components equal to 0

Pr [(1198831 119883119899) = (1199091 119909119899)]

= ∬1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119889119865119875 (119901) (15)

and hence

Pr [119878119899 = 119896] = ∬1

0

(119899

119896) 119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119889119865119875 (119901)

119896 = 0 119899

(16)

Proof See the appendices

Remark 5 For a general characterization of multivariateBernoulli random variables see Sharakhmetov and Ibragi-mov [19] With the exchangeability assumption (12) can bededuced fromTheorem 3 of the aforementioned paper

Remark 6 Under the assumption of conditional indepen-dence 119891119899(119896 119901) reduces to the pmf of a binomial distribution(ie 119886119896(119901) = 119901

119896)

Remark 7 The converse part of the theorem is connectedwith the easy part of the classical Hausdorffmoment problemand its application to the proof of de Finettirsquos theoremfor sequences of exchangeable random variables (see egChapter VII in Feller [20])

Definition 8 Let ](sdot 119901) 0 lt 119901 lt 1 be as in Theorem 4 andlet 119886119896(119901) = int

1

0119903119896](119889119903 119901) for 119896 = 0 1 2 119899 A Bernoulli

mixture model with conditional distribution as specifiedin (14) (or equivalently (12)) and (13) is called a Bernoullidouble mixture model It is determined by the followingdistributions the family of measures ](sdot 119901) 0 lt 119901 lt 1 andthe mixing distribution 119865119875 and it can be formally denotedas a 119865119875-](sdot 119901) Bernoulli double mixture model Similarlymodel (16) for the sum 119878119899 of the 119865119875-](sdot 119901) Bernoulli doublemixture random variables can be formally referred to as a119865119875-](sdot 119901) binomial double mixturemodel

In concrete examples we will employ some meaningfulworded description or an acronym rather than through theformal notation of Definition 8 In addition for the remain-der of the paper all considered models will be Bernoulli (orbinomial) double mixture models therefore we may omitthose words from the names of models simply retaining thedescription of the building blocks ](sdot 119901) and 119865119875

The following simple example illustrates the converse ofthe theorem

Example 9 (ICM) We consider a weighted average of inde-pendence and comonotonicity Specifically for a specifiedweight 120596 isin [0 1] the random 119899-vector 119883 equiv (119883119895)

119899119895=1 is a two-

point mixture of a conditionally comonotonic 119899-vector ofBernoulli mixtures and a conditionally independent 119899-vectorof Bernoulli mixtures (with common mixing distribution119865119875) with respective weights 120596 and 1 minus 120596 For such a modelthe (conditional) joint probability 119886119896(119901) 119896 = 2 119899 isexpressed as the weighted average of the joint probability ofthe comonotonic case and that of independence case

119886119896 (119901) = 120596119901 + (1 minus 120596) 119901119896 119896 = 2 119899 (17)

In practice theweight parameter120596 isin [0 1]must be estimatedfrom data or be chosen exogenously

This model (with 119865119875 unspecified) will be denoted bythe acronym ICM standing for independent comonotonicmixture (with everything being implicitly conditional) It

Journal of Probability and Statistics 5

corresponds to the following choice for ](sdot 119901) which is adouble mixture of point masses

] (sdot 119901) = 120596 [1199011205751 + (1 minus 119901) 1205750] + (1 minus 120596) 120575119901 (18)

Note that int101199030](119889119903 119901) = 1 follows from the usual under-

standing that 00 = 1Then the limiting correlation is (cf (4))

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) = 120596 (19)

and the unconditional pmf of the sum 120583119899(119896) = Pr[119878119899 = 119896]is

120583119899 (119896)

=

120596 (1 minus 119864 [119875]) + (1 minus 120596) 120583119868119899 (0)

= 120596 (1 minus 119864 [119875])

+ (1 minus 120596) 119864 [(1 minus 119875)119899] 119896 = 0

(1 minus 120596) 120583119868119899 (119896) 119896 = 1 119899 minus 1

120596119864 [119875] + (1 minus 120596) 120583119868119899 (119896)

= 120596119864 [119875] + (1 minus 120596) 119864 [119875119899] 119896 = 119899

(20)

with 120583119868119899(119896) given in (1) The mean and variance of the sum 119878119899can be easily obtained as follows

119864 [119878119899] = 119899119864 [119875]

Var [119878119899] = 120596 1198992119864 [119875] minus 119899

2(119864 [119875])

2 + (1 minus 120596)

times 119899119864 [119875] minus 119899119864 [1198752] + 1198992119864 [1198752] minus 1198992(119864 [119875])

2

(21)

Note that both the mean and variance are the weightedaverages of those for the independent and comonotone cases(In general the 119896th moment is the weighted average of the119896th moment of the independent and comonotone cases)

23 Large-Sample Distribution of Empirical Frequency Theproportion of observed defaults to the total number of entitiesin a credit portfolio is of interest in practice For examplethe historical default rate for a homogenous credit portfoliois used to estimate the probability of default for a genericcounterparty in the portfolio

For a fixed 119899 let 119871119899 = 119878119899119899 denote the empiricalfrequency which can be considered as the percentage grossloss of a portfolio of 119899 loans in equal dollar amounts

The probability distribution of 119871119899 is

119865119871119899

(119909) = Pr [119871119899 le 119909] = Pr [119878119899 le 119899119909] 119909 ge 0 (22)

For the general binomial double mixture model basedon a family of probability measures ](sdot 119901) and mixingdistribution 119865119875 we have the following result

119865119871infin

(119909)

equiv lim119899rarrinfin

119865119871119899

(119909)

=

int1

0

] (0 119901) 119889119865119875 (119901) 119909 = 0

int1

0

[] ([0 119909) 119901) +1

2] (119909 119901)] 119889119865119875 (119901) 0 lt 119909 lt 1

1 119909 = 1

(23)

Here is the proof of (22) The case 119909 = 1 is trivial For0 le 119909 lt 1 first note that by (14)

Pr [119878119899 le 119899119909 | 119875 = 119901] =[119899119909]

sum119896=0

(119899

119896)119892119899 (119896 119901)

= int1

0

[119899119909]

sum119896=0

(119899

119896) 119903119896(1 minus 119903)

119899minus119896] (119889119903 119901)

(24)

where for a nonnegative number 119910 [119910] represents thegreatest integer less than or equal to 119910 The integrand is thecdf at 119909 of 1198781015840119899119899 where 1198781015840119899 sim Bin(119899 119903) so by the LLN as119899 rarr infin 1198781015840119899119899 rarr 119903 as Therefore for 0 lt 119909 lt 1

lim119899rarrinfin

Pr[1198781015840119899

119899le 119909] = 1[0119909) (119903) +

1

21119909 (119903) (25)

where 1119860 denotes the indicator function of a set 119860 (Thesecond term with the factor ldquo12rdquo comes from an applicationof the CLT to Pr[1198781015840119899 minus 119899119903 le 0] when 119909 = 119903) For 119909 = 0

Pr [119878119899 = 0 | 119875 = 119901]

= int1

0

(1 minus 119903)119899] (119889119903 119901) 997888rarr ] (0 119901) as 119899 997888rarr infin

(26)

Integrating these two limiting results with respect to](119889119903 119901)119889119865119875(119901) yields the first two cases of (22) The inter-change of limit and integrals is justified by the boundedconvergence theorem

Example 10 (ICM reprise) For the 120596-weighted average ofconditionally comonotonic and conditionally independentBernoulli mixtures described in Example 9 we can easilyobtain the following limiting distribution (this is theweightedaverage of the obvious limit for the conditionally comono-tonic component and the convergence result in Vasicek

6 Journal of Probability and Statistics

[21] for the conditionally independent component of themixture)

lim119899rarrinfin

119865119871119899

(119909)

equiv 119865119871infin

(119909) = 120596 (1 minus 119864 [119875]) + (1 minus 120596) 119865119875 (119909) 0 le 119909 lt 1

1 119909 = 1

(27)

Then the mean and variance of the limiting distribution are

119864 [119871infin] = 119864 [119875] = lim119899rarrinfin

119864 [119878119899

119899]

Var [119871infin] = 120596 119864 [119875] minus (119864 [119875])2 + (1 minus 120596)Var [119875]

= lim119899rarrinfin

Var [119878119899

119899]

(28)

In general the 119896th moment of the limiting distribution is

119864 [119871119896infin] = (1 minus 120596) 119864 [119875

119896] + 120596119864 [119875] 119896 = 1 2 (29)

The limiting result (27) can be applied to the area offinancial credit risk management The tail risk measuresof a large-sample portfolio credit loss distribution are ofparticular interest for financial risk managers The two well-known risk measures value at risk (VaR) and conditional tailexpectation (CTE) can be approximated by using the limitingdistribution (27) Formally for a loss randomvariable119883withthe cumulative distribution function 119865(119909) VaR and CTE atconfidence level 120572 isin [0 1] are defined as follows

VaR120572 (119883) = 119865minus1(120572) = inf 119909 119865 (119909) ge 120572

CTE120572 (119883) = VaR120572 (119883) +Pr [119883 gt VaR120572 (119883)]

(1 minus 120572)

times119864 [119883 minus VaR120572 (119883) | 119883 gt VaR120572 (119883)]

(30)

Note that this definition reduces to

CTE120572 (119883) = 119864 [119883 | 119883 gt VaR120572 (119883)] (31)

if 119883 is a continuous random variable (see Hardy [22] orAcerbi and Tasche [23])

Then by the scaling property of these risk measures andthe limiting result (27) for a large 119899 the VaR of the sum of 119899Bernoulli mixtures at level 120572 can be approximated as follows

VaR120572 (119878119899)

= 119899VaR120572 (119871119899) asymp 119899VaR120572 (119871infin) = 119899119865minus1119871infin

(120572)

=

0 0 le 120572 le 120596 (1 minus 119864 [119875])

119899119865minus1119875 (120572lowast) 120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(32)

where 120572lowast = 120572 minus 120596(1 minus 119864[119875])(1 minus 120596)

Similarly the CTE at level 120572 can be approximated as

CTE120572 (119878119899)

= 119899CTE120572 (119871119899)

asymp 119899CTE120572 (119871infin)

=

(119899

1 minus 120572)119864 [119875] 0 le 120572 le 120596 (1 minus 119864 [119875])

119899

1 minus 120572(1 minus 120572lowast) (1 minus 120596)CTE120572lowast (119875) + 120596119864 [119875]

120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(33)

3 Beta Mixing Distribution

We now specialize the results of the preceding section tothe case where the mixing distribution 119865119875 is the Betadistribution with parameters 119886 gt 0 and 119887 gt 0 Specificallythe density 119891119875(119901) is given by

119891119875 (119901) =Γ (119886 + 119887)

Γ (119886) Γ (119887)119901119886minus1(1 minus 119901)

119887minus1

=1

119861 (119886 119887)119901119886minus1(1 minus 119901)

119887minus1 0 lt 119901 lt 1

(34)

where Γ(sdot) and 119861(sdot sdot) are the Gamma and Beta functionsrespectively

For a Beta Bernoulli mixture several computationallyconvenient expressions are available We first introduce anotation which is convenient for the cumulative distributionfunction

119865119875 (119901) = 119868119901 (119886 119887) =119861119901 (119886 119887)

119861 (119886 119887) (35)

where 119861119909(119886 119887) = int119909

0119904119886minus1(1 minus 119904)

119887minus1119889119904 is the incomplete

Beta function Equation (1) the pmf of the sum under theconditional independence assumption can be evaluated as

120583119868119899 (119896) = (

119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) (36)

The 119896th moments for 119875 and (1 minus 119875) in this case are

119864 [119875119896] =

119861 (119886 + 119896 119887)

119861 (119886 119887)

119864 [(1 minus 119875)119896] =

119861 (119886 119887 + 119896)

119861 (119886 119887)

(37)

The following lists several results for the ICMmodel withthe Beta mixing distribution (120573-ICMmodel)

Journal of Probability and Statistics 7

Example 11 (120573-ICM) The pmf of the binomial double mix-ture is given from (20) as

120583119899 (119896)

=

120596(119887

119886 + 119887) + (1 minus 120596)

119861 (119886 119887 + 119899)

119861 (119886 119887) 119896 = 0

(1 minus 120596)(119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) 119896 = 1 119899 minus 1

120596(119886

119886 + 119887) + (1 minus 120596)

119861 (119886 + 119899 119887)

119861 (119886 119887) 119896 = 119899

(38)

The limiting distribution of the empirical frequency (27) is

119865119871infin

(119909) =

120596(119887

119886 + 119887) + (1 minus 120596) 119868119909 (119886 119887) 0 le 119909 lt 1

1 119909 = 1

(39)

Since the inverse cumulative Beta distribution function doesnot admit a closed-form expression a numerical method isrequired to approximate the VaR of a large-sample creditportfolio Given the VaR at level 120572 the CTE can be writtenin a concise form Specifically

VaR120572 (119878119899) asymp

0 0 le 120572 le120596119887

119886 + 119887

119899119868minus1120572lowast (119886 119887) 120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(40)

where 120572lowast = 120572 minus 120596(119887(119886 + 119887))(1 minus 120596) and

CTE120572 (119878119899)

asymp

(119899

1 minus 120572)(

119886

119886 + 119887) 0 le 120572 le

120596119887

119886 + 119887

(119899

1 minus 120572)(

119886

119886 + 119887) [1 minus (1 minus 119908) 119868119901lowast (119886 + 1 119887)]

120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(41)

where 119901lowast = VaR120572lowast(119875) = 119868minus1120572lowast (119886 119887)

As can be seen from (27) the limiting distribution ofthe empirical frequency under the 120573-ICM model has pointmasses at both end points 0 and 1 (when 120596 = 0) This mayrestrict the use of a certain parameter estimation methodin the case that all observations are strictly inside thedistributional range which is often the case In the followingexample we consider a Beta distribution for the measures](sdot 119901) 0 lt 119901 lt 1 as well as a Beta mixing distribution for119865119875 We call the model a double Beta model with acronym 120573-120573 The limiting empirical frequency for the 120573-120573 model doesnot have point masses at the boundary points 0 and 1

Example 12 (120573-120573) In this example the choice for ](sdot 119901) is theBeta distribution with the two shape parameters 120579119901 and 120579(1minus119901)

] (119889119903 119901)

= 119891] (119903 119901) 119889119903

=1

119861 (120579119901 120579 (1 minus 119901))119903120579119901minus1 (1 minus 119903)

120579(1minus119901)minus1119889119903 0 lt 119903 lt 1

(42)

where parameter 120579 satisfies 0 lt 120579 lt infin Then using therecursive property of the Beta function (119861(120572+1 120573) = [120572(120572+120573)]119861(120572 120573))

119886119896 (119901) = int1

0

119903119896] (119889119903 119901) =

119896minus1

prod119895=0

120579119901 + 119895

120579 + 119895 (43)

with the usual convention that a product over an empty set ofindices equals 1 In particular

1198862 (119901) =120579

1 + 1205791199012+

1

1 + 120579119901 (44)

and thus the limiting correlation for this model is byTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) =

1

1 + 120579 (45)

(Note that this is identical to the result for an ICM modelwith 120596 = 1(1 + 120579) cf (19) in Example 9)

The pmf of the binomial double mixture is given byTheorem 4 as

120583119899 (119896) = int1

0

119891119899 (119896 119901) 119889119865119875 (119901)

= (119899

119896)int1

0

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) 119889119865119875 (119901)

= (119899

119896)

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894

times int1

0

119896+119894minus1

prod119895=0

120579119901 + 119895

120579 + 119895119889119865119875 (119901) 119896 = 0 119899

(46)

By (22) the limiting distribution of the empirical fre-quency is absolutely continuous and its pdf is

119891119871infin

(119909) = int1

0

119891] (119909 119901) 119889119865119875 (119901) 0 lt 119909 lt 1 (47)

Since (47) does not admit a closed-form expression anumerical method is required to calculate the approximateVaR and CTE of a large-sample credit portfolio

8 Journal of Probability and Statistics

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 00120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 10

Figure 1The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-ICMmodel

4 Numerical Examples

As an illustrative example we set 119886 = 15 and 119887 = 135

for the Beta mixing distribution for both the 120573-ICM and120573-120573 models (These prescribed parameter values are chosenfrom the authorsrsquo previous experience in portfolio credit riskmodeling) This gives 119864[119875] = 119886(119886 + 119887) = 01 and Var[119875] =119886119887[(119886 + 119887)

2(119886 + 119887 + 1)] = 0005625

41 120573-ICMModel In Figure 1 we plot the cumulative distri-bution functions of the 120573-ICM binomial double mixture 11987820for several different values of120596 (the limiting correlation)Theplot shows that both the left and right tails get thicker as thelevel of limiting correlation 120596 increases to 1

In Figures 2 and 3 we display the approximate riskmeasures VaR and CTE for a scale of confidence levels120572 isin 08 085 090 095 099 and by the level of the limitingcorrelation120596The results show that both VaR120572 and CTE120572 areincreasing in the level of limiting correlation for 120572 ge 090This is not the case for VaR120572 when the confidence level is lessthan 1 minus 119864[119875] = 119887(119886 + 119887) because

120572lowast=

1

1 minus 120596120572 minus 120596(

119887

119886 + 119887)

=1

1 minus 120596120572 minus

119887

119886 + 119887 +

119887

119886 + 119887

(48)

is decreasing in 120596 when 120572 lt 119887(119886 + 119887)More specifically Table 1 compares the approximatedVaR

and CTE at the level 120572 = 095

42 120573-120573Model Recall that the limiting correlation of the 120573-120573 model is 1(1 + 120579) For the purpose of comparison withthe 120573-ICM model we reparametrize the 120573-120573 mixture modelas follows 120579 = (1 minus 120596)120596 such that the limiting correlationbecomes 120596

Figure 4 displays the cumulative distribution functions ofthe 120573-120573 binomial double mixture 11987820 for several different

Valu

e at r

isk

0

200

400

600

800

1000

00 02 04 06 08 10120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 2The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

00 02 04 06 08 10

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 3The approximatedCTEs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

Table 1The approximated VaR and CTE of 1198781000 at confidence level120572 = 095 under the 120573-ICMmodel

120596 VaR120572 CTE12057200 24751 3031401 25467 4477202 26483 5913503 28073 7333804 31100 8722305 100000 100000

values of 120596 (the limiting correlation) The plot shows that 11987820has a heavier left tail under the 120573-120573model than under the 120573-ICM model for each limiting correlation level Note that the120573-120573model does not admit the two extreme cases 120596 = 0 and120596 = 1

Figures 5 and 6 show the approximate risk measuresVaR and CTE for a scale of confidence levels 120572 isin

08 085 090 095 099 and by the level of the limiting

Journal of Probability and Statistics 9

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 01120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 09

Figure 4The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-120573model

Valu

e at r

isk

02 04 06 08

0

200

400

600

800

1000

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 5The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-120573model

correlation 120596 The approximate risk measures are smallerthan those of the 120573-ICMmodel for each confidence level andlimiting correlation compare for example Figures 2 and 3This indicates that the right tail of the 120573-120573 model is thinnerthan that of the 120573-ICMmodel for the same value of 120596

5 Models Fit to Real Data

In addition to the previous examples with prescribed param-eter values we illustrate the two binomial double mixturemodels with real data Bloomberg mortgage delinquencyrate index US residential mortgage loans are segmentedinto three buckets Prime Alt-A and Subprime based upon

02 04 06 08

0

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 6The approximatedCTEs of 1198781000 by confidence level120572 andthe limiting correlation 120596 under the 120573-120573model

Table 2Themethod ofmoments estimates of the shape parameters119886 and 119887 of Beta mixing distribution and the limiting correlation 120596

Model Parameter119886 119887 120596

120573-ICM 4038 14563 0009120573-I 3381 12191

the loan type Several delinquency rates for each bucketare calculated as the percentage of loans that are reportedto be delinquent beyond a certain number of days (3060 or 90 days) or are classified as real estate owned(REO) or foreclosure In this section we only use theAlt-A 60+ days (60+90+REO+foreclosure) delinquency rate(Ticker BBMDA60P) for the purpose of illustration Thesame approach can be taken for other delinquency indexessuch as the Prime (Tickers BBMDP60P BBMDPDLQ) orSubprime (Tickers BBMDS60P BBMDSDLQ) segmentsFigure 7 is taken from the Bloomberg interactive chart andit plots the monthly history of the delinquency rate duringMay 2007ndashMarch 2012

51 120573-ICMModel Note that none of the observations has thevalue 0 or 1 In this case the maximum likelihood estimateof the weight parameter 120596 is 0 because for 0 lt 119901 lt 1log119891119871

infin

(119901) = log(1 minus 120596) + log119891119875(119901) where 119891119875(119901) is thedensity of the Beta distribution (34) To avoid this issue weestimate the three parameters in the 120573-ICMmodel by meansof the method of moments (MMEs) which uses the first threemoments (29) and (37) of the limiting distribution of theempirical frequency

For comparison we also estimate the two shape parame-ters of the Beta distributionmdashthe limiting distribution of theclassical Beta-binomial (120573-I) modelmdashby matching the firstand second sample moments with the theoretical ones using(28) and (37) Table 2 gives the parameter estimates

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

2 Journal of Probability and Statistics

a distribution with cumulative distribution function (cdf)119865119875(119901) 119901 isin (0 1) At this point the dependency structure for(119883119895)119899119895=1 is still arbitraryIn credit applications the statistical properties of the sum

of the 119899 Bernoulli random variables are the main interestand the following assumption plays an important role inalleviating computational difficulties 119883119895 are conditionallyindependent given 119875 In this case given 119875 the sum 119878119899 =

sum119899119895=1119883119895 follows the binomial distribution with mean 119899119875 and

variance 119899119875(1minus119875) Then the unconditional probability massfunction (pmf) is given as follows

120583119868119899 (119896) = Pr [119878119899 = 119896]

= int1

0

(119899

119896)119901119896(1 minus 119901)

119899minus119896119889119865119875 (119901) 119896 = 0 119899

(1)

where the superscript ldquo119868rdquo on 120583 indicates the assumed con-ditional independence The most commonly used mixingdistribution in modeling the credit risk of a homogeneousportfolio such as a mortgage or credit card portfolio is theBeta distribution This model is referred to as the Beta-binomial (120573-I) model

The present paper is concerned with the level of depen-dence incorporated in Bernoulli mixture models especiallyunder stress situations Together with the asset correlationthe default correlation is used as the standard measure ofdependence in portfolio credit risk models In particularwe consider the behavior of the correlation between twoarbitrary terms as the random default probability 119875 isconditioned to be larger than 119901 tending to 1 Specificallywe define the conditional default correlation between twoterms 119883119894 and 119883119895 given 119875 gt 119901 0 lt 119901 lt 1 as follows(Bae and Iscoe [17] used an analogous definition to studythe asset correlations under stress for various single-factorcredit risk modelsmdashsome dynamicmdashwhile Kalkbrener andPackham [18] also studied asset correlations under stress forstatic normal variance mixture models In both studies inplace of the conditioning on 119875 gt 119901 conditioning is doneon an auxiliary risk factor which for example is related to asystematic factor)

120588119894119895 (119901) = Corr [119883119894 119883119895 | 119875 gt 119901]

=Cov [119883119894 119883119895 | 119875 gt 119901]

radicVar [119883119894 | 119875 gt 119901]radicVar [119883119895 | 119875 gt 119901]

=Cov [119883119894 119883119895 | 119875 gt 119901]Var [119883119894 | 119875 gt 119901]

(2)

where

Cov [119883119894 119883119895 | 119875 gt 119901]

= 119864 [119883119894119883119895 | 119875 gt 119901] minus 119864 [119883119894 | 119875 gt 119901] 119864 [119883119895 | 119875 gt 119901]

= 119864 [119883119894119883119895 | 119875 gt 119901] minus 119864 [119883119894 | 119875 gt 119901]2

Var [119883119894 | 119875 gt 119901]

= 119864 [1198832119894 | 119875 gt 119901] minus 119864 [119883119894 | 119875 gt 119901]

2

= 119864 [119883119894 | 119875 gt 119901] (1 minus 119864 [119883119894 | 119875 gt 119901])

(3)Then the limit of the conditional correlation as 119901 tends

to 1 (referred to as the limiting correlation) is defined as120588119894119895 = lim

119901rarr1120588119894119895 (119901) (4)

provided the limit existsIt is intuitive to expect that the correlations in (2) increase

monotonically as the threshold 119901 approaches 1 Unfortu-nately this is not the case for the Beta-binomial model Infact the conditional correlation always converges to zeroregardless of the mixing distribution as long as conditionalindependence is postulated (Corollary 2) This type of modelfailure can introduce a significant bias in the measurementand management of risk in stress situations

The objectives of the present paper are threefold (i) toderive the relationship between the limiting correlation andthe probability structure (ii) to construct general Bernoullimixture models with nontrivial limiting correlations (iii)to demonstrate implications of the constructed models fortail risk measures such as value at risk and conditional tailexpectation

The theoretical results for these objectives are presentedin Section 2 We first examine the limiting behavior ofcorrelations for Bernoulli mixture models in Section 21Then Section 22 introduces a general model frameworkmdashthe class of double mixture modelsmdashwhich allows furtherpositive dependence between entities beyond conditionalindependence (Theorem 4) Section 23 is devoted to thethird objective and it considers the large-sample distributionof the sum of Bernoulli mixtures under the general modelframework and its application to the approximation of riskmeasures is discussed As a specific case Section 3 considersa Beta Bernoulli mixture and Section 4 provides the resultsfor an example with specific parameter values Section 5demonstrates the fitting results of the double mixture modelsto a dataset from the area of credit risk Finally Section 6concludes this paper with a summary Technical details ofsome proofs are given in the appendices

2 Main Results

21 Limiting Correlation for Bernoulli Mixtures We denoteby 119886119898(119901) the regular conditional probability given 119875 = 119901that all terms (119883119894)

119898119894=1 of a fixed sequence of Bernoulli random

variables (119883119894)119899119894=1 take on the value 1 (For this definition we

allow 119899 = infin) Formally119886119898 (119901) = Pr [119878119898 = 119898 | 119875 = 119901] (5)

where 119878119898 = sum119898119894=1119883119894 Note that under an assumption of

conditional independence 119886119898(119901) = 119901119898

The following hypothesis on the properties of 119865119875 ina deleted neighbourhood of 119901 = 1 will be part of theassumptions inTheorem 1 and its Corollary 2 below

Journal of Probability and Statistics 3

Hypothesis H 119865119875 is absolutely continuous in a neighbour-hood of 119901 = 1 with probability density function (pdf) 119891119875thereon such that

(i) 119891119875(119901) gt 0(ii) 119891119875 is continuous in a deleted neighbourhood of119901 = 1(iii) (1 minus 119901)119891119875(119901)(1 minus 119865119875(119901)) converges to a finite limit as

119901 tends to 1

The significance of 119867(i) is clear it is required for theconditioning on 119875 gt 119901 to be well defined Parts 119867(ii) and119867(iii) are technical assumptionswhich play a role in the proofof Theorem 1 Part (iii) of Hypothesis 119867 is satisfied in manycases of practical interest where 119891119875 has a simple asymptoticbehaviour near 119901 = 1 For example 119867(iii) is satisfied if forsome positive (finite) constant 119888 and some (finite) constants119887 gt minus1 and 119889 (the relation ldquo119909 sim 119910rdquo denotes asymptoticequivalence in the sense that the ratio 119909119910 tends to 1 as119901 rarr 1)

119891119875 (119901) sim 119888(1 minus 119901)119887(log [1 minus 119901])119889 as 119901 997888rarr 1 (6)

Here is the verification that (6) implies that119867(iii) holdsin the case that 119889 = 0 (The case 119889 = 0 is similar but easier sothe details will be omitted) Applying (6) in the first step andLrsquoHospitalrsquos rule in the second step as 119901 tends to 1 we have

(1 minus 119901) 119891119875 (119901)

1 minus 119865119875 (119901)

sim119888(1 minus 119901)

119887+1(log [1 minus 119901])119889

1 minus 119865119875 (119901)

sim (minus119888 (119887 + 1) (1 minus 119901)119887(log [1 minus 119901])119889

minus 119888(1 minus 119901)119887+1119889(log [1 minus 119901])119889minus1(1 minus 119901)minus1)

times (minus119888(1 minus 119901)119887(log [1 minus 119901])119889)

minus1

sim 119887 + 1 + 119889(log [1 minus 119901])minus1 sim 119887 + 1

(7)

so119867(iii) is satisfied with the value of the limit being 119887 + 1 Itwill be clear that the proof of Theorem 1 is easily generalizedto allow the limit in 119867(iii) to be infinite However it isnot apparent given the previous example whether such asituation can actually occur For this reason we have takenthe limit to be finite in the formulation of119867(iii)

Based on the conditional joint probability 1198862(119901) weobtain the following result for the limiting correlation

Theorem 1 Let 119883119894 and 119883119895 be two Bernoulli mixture randomvariables with correlation 120588119894119895(119901) as in (2) Suppose thatHypothesis119867 holds One further assumes that lim119901rarr11198862(119901) =1 1198862(119901) is differentiable for 119901 in a deleted neighbourhood of 1and lim119901rarr1119886

10158402(119901) exists Then the limiting correlation in (4)

exists and satisfies

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) (8)

Proof See the appendices

In the context of a sequence (119883119896) of Bernoulli mixturerandom variables the notation ldquo1198862rdquo suppresses the depen-dence on 119894 and 119895 more precisely it should be written as 1198862119894119895In order to have a result which does not actually depend on119894 or 119895 an extra hypothesis on the Bernoulli mixtures must beimposed We will return to this point in Section 22

An interesting example is the case that both terms takeon identical values (comonotonicity) (here comonotonicityrefers to the case that all the components (identically dis-tributed) of a Bernoulli random vector coincide in value withprobability one conditional on 119875) that is 1198862(119901) = 119901 UnderHypothesis119867Theorem 1 implies that the limiting correlationfor this case is 1 The following corollary another importantexample states that the limiting correlation between twoarbitrary terms from an identically distributed Bernoullimixture sequence converges to zero under the assumptionof conditional independence

Corollary 2 Suppose that two Bernoulli mixture randomvariables 119883119894 and 119883119895 are conditionally independent given thecommon default probability 119875 Then under Hypothesis119867 thelimiting correlation in (4) exists and satisfies

120588119894119895 = lim119901rarr1

120588119894119895 (119901) = 0 (9)

Proof Under the conditional independence assumption1198862(119901) = 1199012 The desired result follows immediately fromTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) = 2 minus 2 = 0 (10)

Remark 3 Results similar to those of Theorem 1 andCorollary 2 can also be given for the other extreme condi-tional on 119875 lt 119901 with 119901 tending to 0 One simply replacesin Hypothesis119867 andTheorem 1 the phrase ldquoneighbourhoodof 119901 = 1rdquo with ldquoneighbourhood of 119901 = 0rdquo and 119867(iii) withldquo119901119891119875(119901)119865119875(119901) converges to a finite limit as 119901 tends to 0rdquothen in Theorem 1 and Corollary 2 one replaces ldquolim119901rarr1rdquowith ldquolim119901rarr0rdquo (Note that lim119901rarr01198862(119901) is automatically 0because 1198862(119901) le Pr[119883119894 = 1 | 119875 = 119901] = 119901) However inour applications it is the case ldquo119901 rarr 1rdquo which is importantso we will not dwell further on the case ldquo119901 rarr 0rdquo

22 DoubleMixtureModels Wenow turn to the constructionof a more general probability structure beyond conditionalindependencemdasha structure which reveals the fundamentalrole of the quantities 119886119896 We require a stronger assumptionon the sequence (119883119895)

119899119895=1 than their being just conditionally

identically distributed which is a statement about the 1-dimensionalmarginalsWe require that (119883119895)

119899119895=1 be condition-

ally exchangeable meaning that under each Pr[sdot | 119875 = 119901]all permutations of (1198831 1198832 119883119899) have a common jointdistribution which may depend on 119901 (Note that a sequenceof 119899 conditionally iid random variables is conditionallyexchangeable as is a comonotonic sequence) In particular

4 Journal of Probability and Statistics

conditional exchangeability is sufficient to guarantee that1198862(119901) 120588119894119895(119901) and hence the limiting correlation inTheorem 1are independent of 119894 and 119895

We assume for the remainder of the paper that oursequence of 119899 Bernoulli mixture random variables is condi-tionally exchangeable The quantities 119886119896(119901) 1 le 119896 le 119899 thendo not depend on the choice of subset of 119896 variables Theprobability that any 119896 of the Bernoulli random variables takeon the value 1 is given in the following theorem which showsthat the pmf of 119878119899 is essentially determined by the quantities119886119896(119901) 119896 = 1 119899 which in turn can be specified quitegenerally as input to the model

Theorem 4 Let (119883119895)119899119895=1 be a sequence of conditionally

exchangeable Bernoulli mixture random variables Then for119896 = 0 119899

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901] = (119899

119896)119892119899 (119896 119901) (11)

119892119899 (119896 119901) = 119886119896 (119901) minus (119899 minus 119896

1) 119886119896+1 (119901)

+ (119899 minus 119896

2) 119886119896+2 (119901) + sdot sdot sdot + (minus1)

119899minus119896119886119899 (119901)

(12)

where 1198860(119901) equiv 1 and 1198861(119901) = 119901Conversely let ](sdot 119901) 119901 isin (0 1) be a family of proba-

bility measures on the interval [0 1] such that int10119903](119889119903 119901) =

119901 and ](119860 119901) is a Borel measurable function of 119901 for eachBorel set 119860 If 119886119896(119901) is defined to be int1

0119903119896](119889119903 119901) then 119891119899 as

defined by the right-hand sides of (11) and (12) is a valid pmfMoreover suppose that (119883119895)

119899119895=1 follows a Bernoulli mixtures

distribution with conditional distribution

Pr [(1198831 119883119899) = (1199091 119909119899) | 119875 = 119901] = 119892119899 (119896 119901) (13)

for all (1199091 119909119899) which have precisely 119896 components equal to1 and the remaining 119899minus119896 components equal to 0Then (119883119895)

119899119895=1

is conditionally exchangeable Pr [119878119896 = 119896 | 119875 = 119901] = 119886119896(119901)for all 119896 = 1 119899 hence 119878119899 has conditional pmf given by (11)and (12) Furthermore

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119896 = 0 119899 (14)

therefore for all (1199091 119909119899)which have precisely 119896 componentsequal to 1 and the remaining 119899 minus 119896 components equal to 0

Pr [(1198831 119883119899) = (1199091 119909119899)]

= ∬1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119889119865119875 (119901) (15)

and hence

Pr [119878119899 = 119896] = ∬1

0

(119899

119896) 119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119889119865119875 (119901)

119896 = 0 119899

(16)

Proof See the appendices

Remark 5 For a general characterization of multivariateBernoulli random variables see Sharakhmetov and Ibragi-mov [19] With the exchangeability assumption (12) can bededuced fromTheorem 3 of the aforementioned paper

Remark 6 Under the assumption of conditional indepen-dence 119891119899(119896 119901) reduces to the pmf of a binomial distribution(ie 119886119896(119901) = 119901

119896)

Remark 7 The converse part of the theorem is connectedwith the easy part of the classical Hausdorffmoment problemand its application to the proof of de Finettirsquos theoremfor sequences of exchangeable random variables (see egChapter VII in Feller [20])

Definition 8 Let ](sdot 119901) 0 lt 119901 lt 1 be as in Theorem 4 andlet 119886119896(119901) = int

1

0119903119896](119889119903 119901) for 119896 = 0 1 2 119899 A Bernoulli

mixture model with conditional distribution as specifiedin (14) (or equivalently (12)) and (13) is called a Bernoullidouble mixture model It is determined by the followingdistributions the family of measures ](sdot 119901) 0 lt 119901 lt 1 andthe mixing distribution 119865119875 and it can be formally denotedas a 119865119875-](sdot 119901) Bernoulli double mixture model Similarlymodel (16) for the sum 119878119899 of the 119865119875-](sdot 119901) Bernoulli doublemixture random variables can be formally referred to as a119865119875-](sdot 119901) binomial double mixturemodel

In concrete examples we will employ some meaningfulworded description or an acronym rather than through theformal notation of Definition 8 In addition for the remain-der of the paper all considered models will be Bernoulli (orbinomial) double mixture models therefore we may omitthose words from the names of models simply retaining thedescription of the building blocks ](sdot 119901) and 119865119875

The following simple example illustrates the converse ofthe theorem

Example 9 (ICM) We consider a weighted average of inde-pendence and comonotonicity Specifically for a specifiedweight 120596 isin [0 1] the random 119899-vector 119883 equiv (119883119895)

119899119895=1 is a two-

point mixture of a conditionally comonotonic 119899-vector ofBernoulli mixtures and a conditionally independent 119899-vectorof Bernoulli mixtures (with common mixing distribution119865119875) with respective weights 120596 and 1 minus 120596 For such a modelthe (conditional) joint probability 119886119896(119901) 119896 = 2 119899 isexpressed as the weighted average of the joint probability ofthe comonotonic case and that of independence case

119886119896 (119901) = 120596119901 + (1 minus 120596) 119901119896 119896 = 2 119899 (17)

In practice theweight parameter120596 isin [0 1]must be estimatedfrom data or be chosen exogenously

This model (with 119865119875 unspecified) will be denoted bythe acronym ICM standing for independent comonotonicmixture (with everything being implicitly conditional) It

Journal of Probability and Statistics 5

corresponds to the following choice for ](sdot 119901) which is adouble mixture of point masses

] (sdot 119901) = 120596 [1199011205751 + (1 minus 119901) 1205750] + (1 minus 120596) 120575119901 (18)

Note that int101199030](119889119903 119901) = 1 follows from the usual under-

standing that 00 = 1Then the limiting correlation is (cf (4))

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) = 120596 (19)

and the unconditional pmf of the sum 120583119899(119896) = Pr[119878119899 = 119896]is

120583119899 (119896)

=

120596 (1 minus 119864 [119875]) + (1 minus 120596) 120583119868119899 (0)

= 120596 (1 minus 119864 [119875])

+ (1 minus 120596) 119864 [(1 minus 119875)119899] 119896 = 0

(1 minus 120596) 120583119868119899 (119896) 119896 = 1 119899 minus 1

120596119864 [119875] + (1 minus 120596) 120583119868119899 (119896)

= 120596119864 [119875] + (1 minus 120596) 119864 [119875119899] 119896 = 119899

(20)

with 120583119868119899(119896) given in (1) The mean and variance of the sum 119878119899can be easily obtained as follows

119864 [119878119899] = 119899119864 [119875]

Var [119878119899] = 120596 1198992119864 [119875] minus 119899

2(119864 [119875])

2 + (1 minus 120596)

times 119899119864 [119875] minus 119899119864 [1198752] + 1198992119864 [1198752] minus 1198992(119864 [119875])

2

(21)

Note that both the mean and variance are the weightedaverages of those for the independent and comonotone cases(In general the 119896th moment is the weighted average of the119896th moment of the independent and comonotone cases)

23 Large-Sample Distribution of Empirical Frequency Theproportion of observed defaults to the total number of entitiesin a credit portfolio is of interest in practice For examplethe historical default rate for a homogenous credit portfoliois used to estimate the probability of default for a genericcounterparty in the portfolio

For a fixed 119899 let 119871119899 = 119878119899119899 denote the empiricalfrequency which can be considered as the percentage grossloss of a portfolio of 119899 loans in equal dollar amounts

The probability distribution of 119871119899 is

119865119871119899

(119909) = Pr [119871119899 le 119909] = Pr [119878119899 le 119899119909] 119909 ge 0 (22)

For the general binomial double mixture model basedon a family of probability measures ](sdot 119901) and mixingdistribution 119865119875 we have the following result

119865119871infin

(119909)

equiv lim119899rarrinfin

119865119871119899

(119909)

=

int1

0

] (0 119901) 119889119865119875 (119901) 119909 = 0

int1

0

[] ([0 119909) 119901) +1

2] (119909 119901)] 119889119865119875 (119901) 0 lt 119909 lt 1

1 119909 = 1

(23)

Here is the proof of (22) The case 119909 = 1 is trivial For0 le 119909 lt 1 first note that by (14)

Pr [119878119899 le 119899119909 | 119875 = 119901] =[119899119909]

sum119896=0

(119899

119896)119892119899 (119896 119901)

= int1

0

[119899119909]

sum119896=0

(119899

119896) 119903119896(1 minus 119903)

119899minus119896] (119889119903 119901)

(24)

where for a nonnegative number 119910 [119910] represents thegreatest integer less than or equal to 119910 The integrand is thecdf at 119909 of 1198781015840119899119899 where 1198781015840119899 sim Bin(119899 119903) so by the LLN as119899 rarr infin 1198781015840119899119899 rarr 119903 as Therefore for 0 lt 119909 lt 1

lim119899rarrinfin

Pr[1198781015840119899

119899le 119909] = 1[0119909) (119903) +

1

21119909 (119903) (25)

where 1119860 denotes the indicator function of a set 119860 (Thesecond term with the factor ldquo12rdquo comes from an applicationof the CLT to Pr[1198781015840119899 minus 119899119903 le 0] when 119909 = 119903) For 119909 = 0

Pr [119878119899 = 0 | 119875 = 119901]

= int1

0

(1 minus 119903)119899] (119889119903 119901) 997888rarr ] (0 119901) as 119899 997888rarr infin

(26)

Integrating these two limiting results with respect to](119889119903 119901)119889119865119875(119901) yields the first two cases of (22) The inter-change of limit and integrals is justified by the boundedconvergence theorem

Example 10 (ICM reprise) For the 120596-weighted average ofconditionally comonotonic and conditionally independentBernoulli mixtures described in Example 9 we can easilyobtain the following limiting distribution (this is theweightedaverage of the obvious limit for the conditionally comono-tonic component and the convergence result in Vasicek

6 Journal of Probability and Statistics

[21] for the conditionally independent component of themixture)

lim119899rarrinfin

119865119871119899

(119909)

equiv 119865119871infin

(119909) = 120596 (1 minus 119864 [119875]) + (1 minus 120596) 119865119875 (119909) 0 le 119909 lt 1

1 119909 = 1

(27)

Then the mean and variance of the limiting distribution are

119864 [119871infin] = 119864 [119875] = lim119899rarrinfin

119864 [119878119899

119899]

Var [119871infin] = 120596 119864 [119875] minus (119864 [119875])2 + (1 minus 120596)Var [119875]

= lim119899rarrinfin

Var [119878119899

119899]

(28)

In general the 119896th moment of the limiting distribution is

119864 [119871119896infin] = (1 minus 120596) 119864 [119875

119896] + 120596119864 [119875] 119896 = 1 2 (29)

The limiting result (27) can be applied to the area offinancial credit risk management The tail risk measuresof a large-sample portfolio credit loss distribution are ofparticular interest for financial risk managers The two well-known risk measures value at risk (VaR) and conditional tailexpectation (CTE) can be approximated by using the limitingdistribution (27) Formally for a loss randomvariable119883withthe cumulative distribution function 119865(119909) VaR and CTE atconfidence level 120572 isin [0 1] are defined as follows

VaR120572 (119883) = 119865minus1(120572) = inf 119909 119865 (119909) ge 120572

CTE120572 (119883) = VaR120572 (119883) +Pr [119883 gt VaR120572 (119883)]

(1 minus 120572)

times119864 [119883 minus VaR120572 (119883) | 119883 gt VaR120572 (119883)]

(30)

Note that this definition reduces to

CTE120572 (119883) = 119864 [119883 | 119883 gt VaR120572 (119883)] (31)

if 119883 is a continuous random variable (see Hardy [22] orAcerbi and Tasche [23])

Then by the scaling property of these risk measures andthe limiting result (27) for a large 119899 the VaR of the sum of 119899Bernoulli mixtures at level 120572 can be approximated as follows

VaR120572 (119878119899)

= 119899VaR120572 (119871119899) asymp 119899VaR120572 (119871infin) = 119899119865minus1119871infin

(120572)

=

0 0 le 120572 le 120596 (1 minus 119864 [119875])

119899119865minus1119875 (120572lowast) 120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(32)

where 120572lowast = 120572 minus 120596(1 minus 119864[119875])(1 minus 120596)

Similarly the CTE at level 120572 can be approximated as

CTE120572 (119878119899)

= 119899CTE120572 (119871119899)

asymp 119899CTE120572 (119871infin)

=

(119899

1 minus 120572)119864 [119875] 0 le 120572 le 120596 (1 minus 119864 [119875])

119899

1 minus 120572(1 minus 120572lowast) (1 minus 120596)CTE120572lowast (119875) + 120596119864 [119875]

120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(33)

3 Beta Mixing Distribution

We now specialize the results of the preceding section tothe case where the mixing distribution 119865119875 is the Betadistribution with parameters 119886 gt 0 and 119887 gt 0 Specificallythe density 119891119875(119901) is given by

119891119875 (119901) =Γ (119886 + 119887)

Γ (119886) Γ (119887)119901119886minus1(1 minus 119901)

119887minus1

=1

119861 (119886 119887)119901119886minus1(1 minus 119901)

119887minus1 0 lt 119901 lt 1

(34)

where Γ(sdot) and 119861(sdot sdot) are the Gamma and Beta functionsrespectively

For a Beta Bernoulli mixture several computationallyconvenient expressions are available We first introduce anotation which is convenient for the cumulative distributionfunction

119865119875 (119901) = 119868119901 (119886 119887) =119861119901 (119886 119887)

119861 (119886 119887) (35)

where 119861119909(119886 119887) = int119909

0119904119886minus1(1 minus 119904)

119887minus1119889119904 is the incomplete

Beta function Equation (1) the pmf of the sum under theconditional independence assumption can be evaluated as

120583119868119899 (119896) = (

119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) (36)

The 119896th moments for 119875 and (1 minus 119875) in this case are

119864 [119875119896] =

119861 (119886 + 119896 119887)

119861 (119886 119887)

119864 [(1 minus 119875)119896] =

119861 (119886 119887 + 119896)

119861 (119886 119887)

(37)

The following lists several results for the ICMmodel withthe Beta mixing distribution (120573-ICMmodel)

Journal of Probability and Statistics 7

Example 11 (120573-ICM) The pmf of the binomial double mix-ture is given from (20) as

120583119899 (119896)

=

120596(119887

119886 + 119887) + (1 minus 120596)

119861 (119886 119887 + 119899)

119861 (119886 119887) 119896 = 0

(1 minus 120596)(119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) 119896 = 1 119899 minus 1

120596(119886

119886 + 119887) + (1 minus 120596)

119861 (119886 + 119899 119887)

119861 (119886 119887) 119896 = 119899

(38)

The limiting distribution of the empirical frequency (27) is

119865119871infin

(119909) =

120596(119887

119886 + 119887) + (1 minus 120596) 119868119909 (119886 119887) 0 le 119909 lt 1

1 119909 = 1

(39)

Since the inverse cumulative Beta distribution function doesnot admit a closed-form expression a numerical method isrequired to approximate the VaR of a large-sample creditportfolio Given the VaR at level 120572 the CTE can be writtenin a concise form Specifically

VaR120572 (119878119899) asymp

0 0 le 120572 le120596119887

119886 + 119887

119899119868minus1120572lowast (119886 119887) 120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(40)

where 120572lowast = 120572 minus 120596(119887(119886 + 119887))(1 minus 120596) and

CTE120572 (119878119899)

asymp

(119899

1 minus 120572)(

119886

119886 + 119887) 0 le 120572 le

120596119887

119886 + 119887

(119899

1 minus 120572)(

119886

119886 + 119887) [1 minus (1 minus 119908) 119868119901lowast (119886 + 1 119887)]

120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(41)

where 119901lowast = VaR120572lowast(119875) = 119868minus1120572lowast (119886 119887)

As can be seen from (27) the limiting distribution ofthe empirical frequency under the 120573-ICM model has pointmasses at both end points 0 and 1 (when 120596 = 0) This mayrestrict the use of a certain parameter estimation methodin the case that all observations are strictly inside thedistributional range which is often the case In the followingexample we consider a Beta distribution for the measures](sdot 119901) 0 lt 119901 lt 1 as well as a Beta mixing distribution for119865119875 We call the model a double Beta model with acronym 120573-120573 The limiting empirical frequency for the 120573-120573 model doesnot have point masses at the boundary points 0 and 1

Example 12 (120573-120573) In this example the choice for ](sdot 119901) is theBeta distribution with the two shape parameters 120579119901 and 120579(1minus119901)

] (119889119903 119901)

= 119891] (119903 119901) 119889119903

=1

119861 (120579119901 120579 (1 minus 119901))119903120579119901minus1 (1 minus 119903)

120579(1minus119901)minus1119889119903 0 lt 119903 lt 1

(42)

where parameter 120579 satisfies 0 lt 120579 lt infin Then using therecursive property of the Beta function (119861(120572+1 120573) = [120572(120572+120573)]119861(120572 120573))

119886119896 (119901) = int1

0

119903119896] (119889119903 119901) =

119896minus1

prod119895=0

120579119901 + 119895

120579 + 119895 (43)

with the usual convention that a product over an empty set ofindices equals 1 In particular

1198862 (119901) =120579

1 + 1205791199012+

1

1 + 120579119901 (44)

and thus the limiting correlation for this model is byTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) =

1

1 + 120579 (45)

(Note that this is identical to the result for an ICM modelwith 120596 = 1(1 + 120579) cf (19) in Example 9)

The pmf of the binomial double mixture is given byTheorem 4 as

120583119899 (119896) = int1

0

119891119899 (119896 119901) 119889119865119875 (119901)

= (119899

119896)int1

0

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) 119889119865119875 (119901)

= (119899

119896)

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894

times int1

0

119896+119894minus1

prod119895=0

120579119901 + 119895

120579 + 119895119889119865119875 (119901) 119896 = 0 119899

(46)

By (22) the limiting distribution of the empirical fre-quency is absolutely continuous and its pdf is

119891119871infin

(119909) = int1

0

119891] (119909 119901) 119889119865119875 (119901) 0 lt 119909 lt 1 (47)

Since (47) does not admit a closed-form expression anumerical method is required to calculate the approximateVaR and CTE of a large-sample credit portfolio

8 Journal of Probability and Statistics

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 00120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 10

Figure 1The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-ICMmodel

4 Numerical Examples

As an illustrative example we set 119886 = 15 and 119887 = 135

for the Beta mixing distribution for both the 120573-ICM and120573-120573 models (These prescribed parameter values are chosenfrom the authorsrsquo previous experience in portfolio credit riskmodeling) This gives 119864[119875] = 119886(119886 + 119887) = 01 and Var[119875] =119886119887[(119886 + 119887)

2(119886 + 119887 + 1)] = 0005625

41 120573-ICMModel In Figure 1 we plot the cumulative distri-bution functions of the 120573-ICM binomial double mixture 11987820for several different values of120596 (the limiting correlation)Theplot shows that both the left and right tails get thicker as thelevel of limiting correlation 120596 increases to 1

In Figures 2 and 3 we display the approximate riskmeasures VaR and CTE for a scale of confidence levels120572 isin 08 085 090 095 099 and by the level of the limitingcorrelation120596The results show that both VaR120572 and CTE120572 areincreasing in the level of limiting correlation for 120572 ge 090This is not the case for VaR120572 when the confidence level is lessthan 1 minus 119864[119875] = 119887(119886 + 119887) because

120572lowast=

1

1 minus 120596120572 minus 120596(

119887

119886 + 119887)

=1

1 minus 120596120572 minus

119887

119886 + 119887 +

119887

119886 + 119887

(48)

is decreasing in 120596 when 120572 lt 119887(119886 + 119887)More specifically Table 1 compares the approximatedVaR

and CTE at the level 120572 = 095

42 120573-120573Model Recall that the limiting correlation of the 120573-120573 model is 1(1 + 120579) For the purpose of comparison withthe 120573-ICM model we reparametrize the 120573-120573 mixture modelas follows 120579 = (1 minus 120596)120596 such that the limiting correlationbecomes 120596

Figure 4 displays the cumulative distribution functions ofthe 120573-120573 binomial double mixture 11987820 for several different

Valu

e at r

isk

0

200

400

600

800

1000

00 02 04 06 08 10120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 2The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

00 02 04 06 08 10

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 3The approximatedCTEs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

Table 1The approximated VaR and CTE of 1198781000 at confidence level120572 = 095 under the 120573-ICMmodel

120596 VaR120572 CTE12057200 24751 3031401 25467 4477202 26483 5913503 28073 7333804 31100 8722305 100000 100000

values of 120596 (the limiting correlation) The plot shows that 11987820has a heavier left tail under the 120573-120573model than under the 120573-ICM model for each limiting correlation level Note that the120573-120573model does not admit the two extreme cases 120596 = 0 and120596 = 1

Figures 5 and 6 show the approximate risk measuresVaR and CTE for a scale of confidence levels 120572 isin

08 085 090 095 099 and by the level of the limiting

Journal of Probability and Statistics 9

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 01120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 09

Figure 4The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-120573model

Valu

e at r

isk

02 04 06 08

0

200

400

600

800

1000

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 5The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-120573model

correlation 120596 The approximate risk measures are smallerthan those of the 120573-ICMmodel for each confidence level andlimiting correlation compare for example Figures 2 and 3This indicates that the right tail of the 120573-120573 model is thinnerthan that of the 120573-ICMmodel for the same value of 120596

5 Models Fit to Real Data

In addition to the previous examples with prescribed param-eter values we illustrate the two binomial double mixturemodels with real data Bloomberg mortgage delinquencyrate index US residential mortgage loans are segmentedinto three buckets Prime Alt-A and Subprime based upon

02 04 06 08

0

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 6The approximatedCTEs of 1198781000 by confidence level120572 andthe limiting correlation 120596 under the 120573-120573model

Table 2Themethod ofmoments estimates of the shape parameters119886 and 119887 of Beta mixing distribution and the limiting correlation 120596

Model Parameter119886 119887 120596

120573-ICM 4038 14563 0009120573-I 3381 12191

the loan type Several delinquency rates for each bucketare calculated as the percentage of loans that are reportedto be delinquent beyond a certain number of days (3060 or 90 days) or are classified as real estate owned(REO) or foreclosure In this section we only use theAlt-A 60+ days (60+90+REO+foreclosure) delinquency rate(Ticker BBMDA60P) for the purpose of illustration Thesame approach can be taken for other delinquency indexessuch as the Prime (Tickers BBMDP60P BBMDPDLQ) orSubprime (Tickers BBMDS60P BBMDSDLQ) segmentsFigure 7 is taken from the Bloomberg interactive chart andit plots the monthly history of the delinquency rate duringMay 2007ndashMarch 2012

51 120573-ICMModel Note that none of the observations has thevalue 0 or 1 In this case the maximum likelihood estimateof the weight parameter 120596 is 0 because for 0 lt 119901 lt 1log119891119871

infin

(119901) = log(1 minus 120596) + log119891119875(119901) where 119891119875(119901) is thedensity of the Beta distribution (34) To avoid this issue weestimate the three parameters in the 120573-ICMmodel by meansof the method of moments (MMEs) which uses the first threemoments (29) and (37) of the limiting distribution of theempirical frequency

For comparison we also estimate the two shape parame-ters of the Beta distributionmdashthe limiting distribution of theclassical Beta-binomial (120573-I) modelmdashby matching the firstand second sample moments with the theoretical ones using(28) and (37) Table 2 gives the parameter estimates

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

Journal of Probability and Statistics 3

Hypothesis H 119865119875 is absolutely continuous in a neighbour-hood of 119901 = 1 with probability density function (pdf) 119891119875thereon such that

(i) 119891119875(119901) gt 0(ii) 119891119875 is continuous in a deleted neighbourhood of119901 = 1(iii) (1 minus 119901)119891119875(119901)(1 minus 119865119875(119901)) converges to a finite limit as

119901 tends to 1

The significance of 119867(i) is clear it is required for theconditioning on 119875 gt 119901 to be well defined Parts 119867(ii) and119867(iii) are technical assumptionswhich play a role in the proofof Theorem 1 Part (iii) of Hypothesis 119867 is satisfied in manycases of practical interest where 119891119875 has a simple asymptoticbehaviour near 119901 = 1 For example 119867(iii) is satisfied if forsome positive (finite) constant 119888 and some (finite) constants119887 gt minus1 and 119889 (the relation ldquo119909 sim 119910rdquo denotes asymptoticequivalence in the sense that the ratio 119909119910 tends to 1 as119901 rarr 1)

119891119875 (119901) sim 119888(1 minus 119901)119887(log [1 minus 119901])119889 as 119901 997888rarr 1 (6)

Here is the verification that (6) implies that119867(iii) holdsin the case that 119889 = 0 (The case 119889 = 0 is similar but easier sothe details will be omitted) Applying (6) in the first step andLrsquoHospitalrsquos rule in the second step as 119901 tends to 1 we have

(1 minus 119901) 119891119875 (119901)

1 minus 119865119875 (119901)

sim119888(1 minus 119901)

119887+1(log [1 minus 119901])119889

1 minus 119865119875 (119901)

sim (minus119888 (119887 + 1) (1 minus 119901)119887(log [1 minus 119901])119889

minus 119888(1 minus 119901)119887+1119889(log [1 minus 119901])119889minus1(1 minus 119901)minus1)

times (minus119888(1 minus 119901)119887(log [1 minus 119901])119889)

minus1

sim 119887 + 1 + 119889(log [1 minus 119901])minus1 sim 119887 + 1

(7)

so119867(iii) is satisfied with the value of the limit being 119887 + 1 Itwill be clear that the proof of Theorem 1 is easily generalizedto allow the limit in 119867(iii) to be infinite However it isnot apparent given the previous example whether such asituation can actually occur For this reason we have takenthe limit to be finite in the formulation of119867(iii)

Based on the conditional joint probability 1198862(119901) weobtain the following result for the limiting correlation

Theorem 1 Let 119883119894 and 119883119895 be two Bernoulli mixture randomvariables with correlation 120588119894119895(119901) as in (2) Suppose thatHypothesis119867 holds One further assumes that lim119901rarr11198862(119901) =1 1198862(119901) is differentiable for 119901 in a deleted neighbourhood of 1and lim119901rarr1119886

10158402(119901) exists Then the limiting correlation in (4)

exists and satisfies

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) (8)

Proof See the appendices

In the context of a sequence (119883119896) of Bernoulli mixturerandom variables the notation ldquo1198862rdquo suppresses the depen-dence on 119894 and 119895 more precisely it should be written as 1198862119894119895In order to have a result which does not actually depend on119894 or 119895 an extra hypothesis on the Bernoulli mixtures must beimposed We will return to this point in Section 22

An interesting example is the case that both terms takeon identical values (comonotonicity) (here comonotonicityrefers to the case that all the components (identically dis-tributed) of a Bernoulli random vector coincide in value withprobability one conditional on 119875) that is 1198862(119901) = 119901 UnderHypothesis119867Theorem 1 implies that the limiting correlationfor this case is 1 The following corollary another importantexample states that the limiting correlation between twoarbitrary terms from an identically distributed Bernoullimixture sequence converges to zero under the assumptionof conditional independence

Corollary 2 Suppose that two Bernoulli mixture randomvariables 119883119894 and 119883119895 are conditionally independent given thecommon default probability 119875 Then under Hypothesis119867 thelimiting correlation in (4) exists and satisfies

120588119894119895 = lim119901rarr1

120588119894119895 (119901) = 0 (9)

Proof Under the conditional independence assumption1198862(119901) = 1199012 The desired result follows immediately fromTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) = 2 minus 2 = 0 (10)

Remark 3 Results similar to those of Theorem 1 andCorollary 2 can also be given for the other extreme condi-tional on 119875 lt 119901 with 119901 tending to 0 One simply replacesin Hypothesis119867 andTheorem 1 the phrase ldquoneighbourhoodof 119901 = 1rdquo with ldquoneighbourhood of 119901 = 0rdquo and 119867(iii) withldquo119901119891119875(119901)119865119875(119901) converges to a finite limit as 119901 tends to 0rdquothen in Theorem 1 and Corollary 2 one replaces ldquolim119901rarr1rdquowith ldquolim119901rarr0rdquo (Note that lim119901rarr01198862(119901) is automatically 0because 1198862(119901) le Pr[119883119894 = 1 | 119875 = 119901] = 119901) However inour applications it is the case ldquo119901 rarr 1rdquo which is importantso we will not dwell further on the case ldquo119901 rarr 0rdquo

22 DoubleMixtureModels Wenow turn to the constructionof a more general probability structure beyond conditionalindependencemdasha structure which reveals the fundamentalrole of the quantities 119886119896 We require a stronger assumptionon the sequence (119883119895)

119899119895=1 than their being just conditionally

identically distributed which is a statement about the 1-dimensionalmarginalsWe require that (119883119895)

119899119895=1 be condition-

ally exchangeable meaning that under each Pr[sdot | 119875 = 119901]all permutations of (1198831 1198832 119883119899) have a common jointdistribution which may depend on 119901 (Note that a sequenceof 119899 conditionally iid random variables is conditionallyexchangeable as is a comonotonic sequence) In particular

4 Journal of Probability and Statistics

conditional exchangeability is sufficient to guarantee that1198862(119901) 120588119894119895(119901) and hence the limiting correlation inTheorem 1are independent of 119894 and 119895

We assume for the remainder of the paper that oursequence of 119899 Bernoulli mixture random variables is condi-tionally exchangeable The quantities 119886119896(119901) 1 le 119896 le 119899 thendo not depend on the choice of subset of 119896 variables Theprobability that any 119896 of the Bernoulli random variables takeon the value 1 is given in the following theorem which showsthat the pmf of 119878119899 is essentially determined by the quantities119886119896(119901) 119896 = 1 119899 which in turn can be specified quitegenerally as input to the model

Theorem 4 Let (119883119895)119899119895=1 be a sequence of conditionally

exchangeable Bernoulli mixture random variables Then for119896 = 0 119899

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901] = (119899

119896)119892119899 (119896 119901) (11)

119892119899 (119896 119901) = 119886119896 (119901) minus (119899 minus 119896

1) 119886119896+1 (119901)

+ (119899 minus 119896

2) 119886119896+2 (119901) + sdot sdot sdot + (minus1)

119899minus119896119886119899 (119901)

(12)

where 1198860(119901) equiv 1 and 1198861(119901) = 119901Conversely let ](sdot 119901) 119901 isin (0 1) be a family of proba-

bility measures on the interval [0 1] such that int10119903](119889119903 119901) =

119901 and ](119860 119901) is a Borel measurable function of 119901 for eachBorel set 119860 If 119886119896(119901) is defined to be int1

0119903119896](119889119903 119901) then 119891119899 as

defined by the right-hand sides of (11) and (12) is a valid pmfMoreover suppose that (119883119895)

119899119895=1 follows a Bernoulli mixtures

distribution with conditional distribution

Pr [(1198831 119883119899) = (1199091 119909119899) | 119875 = 119901] = 119892119899 (119896 119901) (13)

for all (1199091 119909119899) which have precisely 119896 components equal to1 and the remaining 119899minus119896 components equal to 0Then (119883119895)

119899119895=1

is conditionally exchangeable Pr [119878119896 = 119896 | 119875 = 119901] = 119886119896(119901)for all 119896 = 1 119899 hence 119878119899 has conditional pmf given by (11)and (12) Furthermore

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119896 = 0 119899 (14)

therefore for all (1199091 119909119899)which have precisely 119896 componentsequal to 1 and the remaining 119899 minus 119896 components equal to 0

Pr [(1198831 119883119899) = (1199091 119909119899)]

= ∬1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119889119865119875 (119901) (15)

and hence

Pr [119878119899 = 119896] = ∬1

0

(119899

119896) 119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119889119865119875 (119901)

119896 = 0 119899

(16)

Proof See the appendices

Remark 5 For a general characterization of multivariateBernoulli random variables see Sharakhmetov and Ibragi-mov [19] With the exchangeability assumption (12) can bededuced fromTheorem 3 of the aforementioned paper

Remark 6 Under the assumption of conditional indepen-dence 119891119899(119896 119901) reduces to the pmf of a binomial distribution(ie 119886119896(119901) = 119901

119896)

Remark 7 The converse part of the theorem is connectedwith the easy part of the classical Hausdorffmoment problemand its application to the proof of de Finettirsquos theoremfor sequences of exchangeable random variables (see egChapter VII in Feller [20])

Definition 8 Let ](sdot 119901) 0 lt 119901 lt 1 be as in Theorem 4 andlet 119886119896(119901) = int

1

0119903119896](119889119903 119901) for 119896 = 0 1 2 119899 A Bernoulli

mixture model with conditional distribution as specifiedin (14) (or equivalently (12)) and (13) is called a Bernoullidouble mixture model It is determined by the followingdistributions the family of measures ](sdot 119901) 0 lt 119901 lt 1 andthe mixing distribution 119865119875 and it can be formally denotedas a 119865119875-](sdot 119901) Bernoulli double mixture model Similarlymodel (16) for the sum 119878119899 of the 119865119875-](sdot 119901) Bernoulli doublemixture random variables can be formally referred to as a119865119875-](sdot 119901) binomial double mixturemodel

In concrete examples we will employ some meaningfulworded description or an acronym rather than through theformal notation of Definition 8 In addition for the remain-der of the paper all considered models will be Bernoulli (orbinomial) double mixture models therefore we may omitthose words from the names of models simply retaining thedescription of the building blocks ](sdot 119901) and 119865119875

The following simple example illustrates the converse ofthe theorem

Example 9 (ICM) We consider a weighted average of inde-pendence and comonotonicity Specifically for a specifiedweight 120596 isin [0 1] the random 119899-vector 119883 equiv (119883119895)

119899119895=1 is a two-

point mixture of a conditionally comonotonic 119899-vector ofBernoulli mixtures and a conditionally independent 119899-vectorof Bernoulli mixtures (with common mixing distribution119865119875) with respective weights 120596 and 1 minus 120596 For such a modelthe (conditional) joint probability 119886119896(119901) 119896 = 2 119899 isexpressed as the weighted average of the joint probability ofthe comonotonic case and that of independence case

119886119896 (119901) = 120596119901 + (1 minus 120596) 119901119896 119896 = 2 119899 (17)

In practice theweight parameter120596 isin [0 1]must be estimatedfrom data or be chosen exogenously

This model (with 119865119875 unspecified) will be denoted bythe acronym ICM standing for independent comonotonicmixture (with everything being implicitly conditional) It

Journal of Probability and Statistics 5

corresponds to the following choice for ](sdot 119901) which is adouble mixture of point masses

] (sdot 119901) = 120596 [1199011205751 + (1 minus 119901) 1205750] + (1 minus 120596) 120575119901 (18)

Note that int101199030](119889119903 119901) = 1 follows from the usual under-

standing that 00 = 1Then the limiting correlation is (cf (4))

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) = 120596 (19)

and the unconditional pmf of the sum 120583119899(119896) = Pr[119878119899 = 119896]is

120583119899 (119896)

=

120596 (1 minus 119864 [119875]) + (1 minus 120596) 120583119868119899 (0)

= 120596 (1 minus 119864 [119875])

+ (1 minus 120596) 119864 [(1 minus 119875)119899] 119896 = 0

(1 minus 120596) 120583119868119899 (119896) 119896 = 1 119899 minus 1

120596119864 [119875] + (1 minus 120596) 120583119868119899 (119896)

= 120596119864 [119875] + (1 minus 120596) 119864 [119875119899] 119896 = 119899

(20)

with 120583119868119899(119896) given in (1) The mean and variance of the sum 119878119899can be easily obtained as follows

119864 [119878119899] = 119899119864 [119875]

Var [119878119899] = 120596 1198992119864 [119875] minus 119899

2(119864 [119875])

2 + (1 minus 120596)

times 119899119864 [119875] minus 119899119864 [1198752] + 1198992119864 [1198752] minus 1198992(119864 [119875])

2

(21)

Note that both the mean and variance are the weightedaverages of those for the independent and comonotone cases(In general the 119896th moment is the weighted average of the119896th moment of the independent and comonotone cases)

23 Large-Sample Distribution of Empirical Frequency Theproportion of observed defaults to the total number of entitiesin a credit portfolio is of interest in practice For examplethe historical default rate for a homogenous credit portfoliois used to estimate the probability of default for a genericcounterparty in the portfolio

For a fixed 119899 let 119871119899 = 119878119899119899 denote the empiricalfrequency which can be considered as the percentage grossloss of a portfolio of 119899 loans in equal dollar amounts

The probability distribution of 119871119899 is

119865119871119899

(119909) = Pr [119871119899 le 119909] = Pr [119878119899 le 119899119909] 119909 ge 0 (22)

For the general binomial double mixture model basedon a family of probability measures ](sdot 119901) and mixingdistribution 119865119875 we have the following result

119865119871infin

(119909)

equiv lim119899rarrinfin

119865119871119899

(119909)

=

int1

0

] (0 119901) 119889119865119875 (119901) 119909 = 0

int1

0

[] ([0 119909) 119901) +1

2] (119909 119901)] 119889119865119875 (119901) 0 lt 119909 lt 1

1 119909 = 1

(23)

Here is the proof of (22) The case 119909 = 1 is trivial For0 le 119909 lt 1 first note that by (14)

Pr [119878119899 le 119899119909 | 119875 = 119901] =[119899119909]

sum119896=0

(119899

119896)119892119899 (119896 119901)

= int1

0

[119899119909]

sum119896=0

(119899

119896) 119903119896(1 minus 119903)

119899minus119896] (119889119903 119901)

(24)

where for a nonnegative number 119910 [119910] represents thegreatest integer less than or equal to 119910 The integrand is thecdf at 119909 of 1198781015840119899119899 where 1198781015840119899 sim Bin(119899 119903) so by the LLN as119899 rarr infin 1198781015840119899119899 rarr 119903 as Therefore for 0 lt 119909 lt 1

lim119899rarrinfin

Pr[1198781015840119899

119899le 119909] = 1[0119909) (119903) +

1

21119909 (119903) (25)

where 1119860 denotes the indicator function of a set 119860 (Thesecond term with the factor ldquo12rdquo comes from an applicationof the CLT to Pr[1198781015840119899 minus 119899119903 le 0] when 119909 = 119903) For 119909 = 0

Pr [119878119899 = 0 | 119875 = 119901]

= int1

0

(1 minus 119903)119899] (119889119903 119901) 997888rarr ] (0 119901) as 119899 997888rarr infin

(26)

Integrating these two limiting results with respect to](119889119903 119901)119889119865119875(119901) yields the first two cases of (22) The inter-change of limit and integrals is justified by the boundedconvergence theorem

Example 10 (ICM reprise) For the 120596-weighted average ofconditionally comonotonic and conditionally independentBernoulli mixtures described in Example 9 we can easilyobtain the following limiting distribution (this is theweightedaverage of the obvious limit for the conditionally comono-tonic component and the convergence result in Vasicek

6 Journal of Probability and Statistics

[21] for the conditionally independent component of themixture)

lim119899rarrinfin

119865119871119899

(119909)

equiv 119865119871infin

(119909) = 120596 (1 minus 119864 [119875]) + (1 minus 120596) 119865119875 (119909) 0 le 119909 lt 1

1 119909 = 1

(27)

Then the mean and variance of the limiting distribution are

119864 [119871infin] = 119864 [119875] = lim119899rarrinfin

119864 [119878119899

119899]

Var [119871infin] = 120596 119864 [119875] minus (119864 [119875])2 + (1 minus 120596)Var [119875]

= lim119899rarrinfin

Var [119878119899

119899]

(28)

In general the 119896th moment of the limiting distribution is

119864 [119871119896infin] = (1 minus 120596) 119864 [119875

119896] + 120596119864 [119875] 119896 = 1 2 (29)

The limiting result (27) can be applied to the area offinancial credit risk management The tail risk measuresof a large-sample portfolio credit loss distribution are ofparticular interest for financial risk managers The two well-known risk measures value at risk (VaR) and conditional tailexpectation (CTE) can be approximated by using the limitingdistribution (27) Formally for a loss randomvariable119883withthe cumulative distribution function 119865(119909) VaR and CTE atconfidence level 120572 isin [0 1] are defined as follows

VaR120572 (119883) = 119865minus1(120572) = inf 119909 119865 (119909) ge 120572

CTE120572 (119883) = VaR120572 (119883) +Pr [119883 gt VaR120572 (119883)]

(1 minus 120572)

times119864 [119883 minus VaR120572 (119883) | 119883 gt VaR120572 (119883)]

(30)

Note that this definition reduces to

CTE120572 (119883) = 119864 [119883 | 119883 gt VaR120572 (119883)] (31)

if 119883 is a continuous random variable (see Hardy [22] orAcerbi and Tasche [23])

Then by the scaling property of these risk measures andthe limiting result (27) for a large 119899 the VaR of the sum of 119899Bernoulli mixtures at level 120572 can be approximated as follows

VaR120572 (119878119899)

= 119899VaR120572 (119871119899) asymp 119899VaR120572 (119871infin) = 119899119865minus1119871infin

(120572)

=

0 0 le 120572 le 120596 (1 minus 119864 [119875])

119899119865minus1119875 (120572lowast) 120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(32)

where 120572lowast = 120572 minus 120596(1 minus 119864[119875])(1 minus 120596)

Similarly the CTE at level 120572 can be approximated as

CTE120572 (119878119899)

= 119899CTE120572 (119871119899)

asymp 119899CTE120572 (119871infin)

=

(119899

1 minus 120572)119864 [119875] 0 le 120572 le 120596 (1 minus 119864 [119875])

119899

1 minus 120572(1 minus 120572lowast) (1 minus 120596)CTE120572lowast (119875) + 120596119864 [119875]

120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(33)

3 Beta Mixing Distribution

We now specialize the results of the preceding section tothe case where the mixing distribution 119865119875 is the Betadistribution with parameters 119886 gt 0 and 119887 gt 0 Specificallythe density 119891119875(119901) is given by

119891119875 (119901) =Γ (119886 + 119887)

Γ (119886) Γ (119887)119901119886minus1(1 minus 119901)

119887minus1

=1

119861 (119886 119887)119901119886minus1(1 minus 119901)

119887minus1 0 lt 119901 lt 1

(34)

where Γ(sdot) and 119861(sdot sdot) are the Gamma and Beta functionsrespectively

For a Beta Bernoulli mixture several computationallyconvenient expressions are available We first introduce anotation which is convenient for the cumulative distributionfunction

119865119875 (119901) = 119868119901 (119886 119887) =119861119901 (119886 119887)

119861 (119886 119887) (35)

where 119861119909(119886 119887) = int119909

0119904119886minus1(1 minus 119904)

119887minus1119889119904 is the incomplete

Beta function Equation (1) the pmf of the sum under theconditional independence assumption can be evaluated as

120583119868119899 (119896) = (

119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) (36)

The 119896th moments for 119875 and (1 minus 119875) in this case are

119864 [119875119896] =

119861 (119886 + 119896 119887)

119861 (119886 119887)

119864 [(1 minus 119875)119896] =

119861 (119886 119887 + 119896)

119861 (119886 119887)

(37)

The following lists several results for the ICMmodel withthe Beta mixing distribution (120573-ICMmodel)

Journal of Probability and Statistics 7

Example 11 (120573-ICM) The pmf of the binomial double mix-ture is given from (20) as

120583119899 (119896)

=

120596(119887

119886 + 119887) + (1 minus 120596)

119861 (119886 119887 + 119899)

119861 (119886 119887) 119896 = 0

(1 minus 120596)(119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) 119896 = 1 119899 minus 1

120596(119886

119886 + 119887) + (1 minus 120596)

119861 (119886 + 119899 119887)

119861 (119886 119887) 119896 = 119899

(38)

The limiting distribution of the empirical frequency (27) is

119865119871infin

(119909) =

120596(119887

119886 + 119887) + (1 minus 120596) 119868119909 (119886 119887) 0 le 119909 lt 1

1 119909 = 1

(39)

Since the inverse cumulative Beta distribution function doesnot admit a closed-form expression a numerical method isrequired to approximate the VaR of a large-sample creditportfolio Given the VaR at level 120572 the CTE can be writtenin a concise form Specifically

VaR120572 (119878119899) asymp

0 0 le 120572 le120596119887

119886 + 119887

119899119868minus1120572lowast (119886 119887) 120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(40)

where 120572lowast = 120572 minus 120596(119887(119886 + 119887))(1 minus 120596) and

CTE120572 (119878119899)

asymp

(119899

1 minus 120572)(

119886

119886 + 119887) 0 le 120572 le

120596119887

119886 + 119887

(119899

1 minus 120572)(

119886

119886 + 119887) [1 minus (1 minus 119908) 119868119901lowast (119886 + 1 119887)]

120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(41)

where 119901lowast = VaR120572lowast(119875) = 119868minus1120572lowast (119886 119887)

As can be seen from (27) the limiting distribution ofthe empirical frequency under the 120573-ICM model has pointmasses at both end points 0 and 1 (when 120596 = 0) This mayrestrict the use of a certain parameter estimation methodin the case that all observations are strictly inside thedistributional range which is often the case In the followingexample we consider a Beta distribution for the measures](sdot 119901) 0 lt 119901 lt 1 as well as a Beta mixing distribution for119865119875 We call the model a double Beta model with acronym 120573-120573 The limiting empirical frequency for the 120573-120573 model doesnot have point masses at the boundary points 0 and 1

Example 12 (120573-120573) In this example the choice for ](sdot 119901) is theBeta distribution with the two shape parameters 120579119901 and 120579(1minus119901)

] (119889119903 119901)

= 119891] (119903 119901) 119889119903

=1

119861 (120579119901 120579 (1 minus 119901))119903120579119901minus1 (1 minus 119903)

120579(1minus119901)minus1119889119903 0 lt 119903 lt 1

(42)

where parameter 120579 satisfies 0 lt 120579 lt infin Then using therecursive property of the Beta function (119861(120572+1 120573) = [120572(120572+120573)]119861(120572 120573))

119886119896 (119901) = int1

0

119903119896] (119889119903 119901) =

119896minus1

prod119895=0

120579119901 + 119895

120579 + 119895 (43)

with the usual convention that a product over an empty set ofindices equals 1 In particular

1198862 (119901) =120579

1 + 1205791199012+

1

1 + 120579119901 (44)

and thus the limiting correlation for this model is byTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) =

1

1 + 120579 (45)

(Note that this is identical to the result for an ICM modelwith 120596 = 1(1 + 120579) cf (19) in Example 9)

The pmf of the binomial double mixture is given byTheorem 4 as

120583119899 (119896) = int1

0

119891119899 (119896 119901) 119889119865119875 (119901)

= (119899

119896)int1

0

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) 119889119865119875 (119901)

= (119899

119896)

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894

times int1

0

119896+119894minus1

prod119895=0

120579119901 + 119895

120579 + 119895119889119865119875 (119901) 119896 = 0 119899

(46)

By (22) the limiting distribution of the empirical fre-quency is absolutely continuous and its pdf is

119891119871infin

(119909) = int1

0

119891] (119909 119901) 119889119865119875 (119901) 0 lt 119909 lt 1 (47)

Since (47) does not admit a closed-form expression anumerical method is required to calculate the approximateVaR and CTE of a large-sample credit portfolio

8 Journal of Probability and Statistics

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 00120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 10

Figure 1The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-ICMmodel

4 Numerical Examples

As an illustrative example we set 119886 = 15 and 119887 = 135

for the Beta mixing distribution for both the 120573-ICM and120573-120573 models (These prescribed parameter values are chosenfrom the authorsrsquo previous experience in portfolio credit riskmodeling) This gives 119864[119875] = 119886(119886 + 119887) = 01 and Var[119875] =119886119887[(119886 + 119887)

2(119886 + 119887 + 1)] = 0005625

41 120573-ICMModel In Figure 1 we plot the cumulative distri-bution functions of the 120573-ICM binomial double mixture 11987820for several different values of120596 (the limiting correlation)Theplot shows that both the left and right tails get thicker as thelevel of limiting correlation 120596 increases to 1

In Figures 2 and 3 we display the approximate riskmeasures VaR and CTE for a scale of confidence levels120572 isin 08 085 090 095 099 and by the level of the limitingcorrelation120596The results show that both VaR120572 and CTE120572 areincreasing in the level of limiting correlation for 120572 ge 090This is not the case for VaR120572 when the confidence level is lessthan 1 minus 119864[119875] = 119887(119886 + 119887) because

120572lowast=

1

1 minus 120596120572 minus 120596(

119887

119886 + 119887)

=1

1 minus 120596120572 minus

119887

119886 + 119887 +

119887

119886 + 119887

(48)

is decreasing in 120596 when 120572 lt 119887(119886 + 119887)More specifically Table 1 compares the approximatedVaR

and CTE at the level 120572 = 095

42 120573-120573Model Recall that the limiting correlation of the 120573-120573 model is 1(1 + 120579) For the purpose of comparison withthe 120573-ICM model we reparametrize the 120573-120573 mixture modelas follows 120579 = (1 minus 120596)120596 such that the limiting correlationbecomes 120596

Figure 4 displays the cumulative distribution functions ofthe 120573-120573 binomial double mixture 11987820 for several different

Valu

e at r

isk

0

200

400

600

800

1000

00 02 04 06 08 10120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 2The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

00 02 04 06 08 10

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 3The approximatedCTEs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

Table 1The approximated VaR and CTE of 1198781000 at confidence level120572 = 095 under the 120573-ICMmodel

120596 VaR120572 CTE12057200 24751 3031401 25467 4477202 26483 5913503 28073 7333804 31100 8722305 100000 100000

values of 120596 (the limiting correlation) The plot shows that 11987820has a heavier left tail under the 120573-120573model than under the 120573-ICM model for each limiting correlation level Note that the120573-120573model does not admit the two extreme cases 120596 = 0 and120596 = 1

Figures 5 and 6 show the approximate risk measuresVaR and CTE for a scale of confidence levels 120572 isin

08 085 090 095 099 and by the level of the limiting

Journal of Probability and Statistics 9

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 01120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 09

Figure 4The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-120573model

Valu

e at r

isk

02 04 06 08

0

200

400

600

800

1000

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 5The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-120573model

correlation 120596 The approximate risk measures are smallerthan those of the 120573-ICMmodel for each confidence level andlimiting correlation compare for example Figures 2 and 3This indicates that the right tail of the 120573-120573 model is thinnerthan that of the 120573-ICMmodel for the same value of 120596

5 Models Fit to Real Data

In addition to the previous examples with prescribed param-eter values we illustrate the two binomial double mixturemodels with real data Bloomberg mortgage delinquencyrate index US residential mortgage loans are segmentedinto three buckets Prime Alt-A and Subprime based upon

02 04 06 08

0

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 6The approximatedCTEs of 1198781000 by confidence level120572 andthe limiting correlation 120596 under the 120573-120573model

Table 2Themethod ofmoments estimates of the shape parameters119886 and 119887 of Beta mixing distribution and the limiting correlation 120596

Model Parameter119886 119887 120596

120573-ICM 4038 14563 0009120573-I 3381 12191

the loan type Several delinquency rates for each bucketare calculated as the percentage of loans that are reportedto be delinquent beyond a certain number of days (3060 or 90 days) or are classified as real estate owned(REO) or foreclosure In this section we only use theAlt-A 60+ days (60+90+REO+foreclosure) delinquency rate(Ticker BBMDA60P) for the purpose of illustration Thesame approach can be taken for other delinquency indexessuch as the Prime (Tickers BBMDP60P BBMDPDLQ) orSubprime (Tickers BBMDS60P BBMDSDLQ) segmentsFigure 7 is taken from the Bloomberg interactive chart andit plots the monthly history of the delinquency rate duringMay 2007ndashMarch 2012

51 120573-ICMModel Note that none of the observations has thevalue 0 or 1 In this case the maximum likelihood estimateof the weight parameter 120596 is 0 because for 0 lt 119901 lt 1log119891119871

infin

(119901) = log(1 minus 120596) + log119891119875(119901) where 119891119875(119901) is thedensity of the Beta distribution (34) To avoid this issue weestimate the three parameters in the 120573-ICMmodel by meansof the method of moments (MMEs) which uses the first threemoments (29) and (37) of the limiting distribution of theempirical frequency

For comparison we also estimate the two shape parame-ters of the Beta distributionmdashthe limiting distribution of theclassical Beta-binomial (120573-I) modelmdashby matching the firstand second sample moments with the theoretical ones using(28) and (37) Table 2 gives the parameter estimates

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

4 Journal of Probability and Statistics

conditional exchangeability is sufficient to guarantee that1198862(119901) 120588119894119895(119901) and hence the limiting correlation inTheorem 1are independent of 119894 and 119895

We assume for the remainder of the paper that oursequence of 119899 Bernoulli mixture random variables is condi-tionally exchangeable The quantities 119886119896(119901) 1 le 119896 le 119899 thendo not depend on the choice of subset of 119896 variables Theprobability that any 119896 of the Bernoulli random variables takeon the value 1 is given in the following theorem which showsthat the pmf of 119878119899 is essentially determined by the quantities119886119896(119901) 119896 = 1 119899 which in turn can be specified quitegenerally as input to the model

Theorem 4 Let (119883119895)119899119895=1 be a sequence of conditionally

exchangeable Bernoulli mixture random variables Then for119896 = 0 119899

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901] = (119899

119896)119892119899 (119896 119901) (11)

119892119899 (119896 119901) = 119886119896 (119901) minus (119899 minus 119896

1) 119886119896+1 (119901)

+ (119899 minus 119896

2) 119886119896+2 (119901) + sdot sdot sdot + (minus1)

119899minus119896119886119899 (119901)

(12)

where 1198860(119901) equiv 1 and 1198861(119901) = 119901Conversely let ](sdot 119901) 119901 isin (0 1) be a family of proba-

bility measures on the interval [0 1] such that int10119903](119889119903 119901) =

119901 and ](119860 119901) is a Borel measurable function of 119901 for eachBorel set 119860 If 119886119896(119901) is defined to be int1

0119903119896](119889119903 119901) then 119891119899 as

defined by the right-hand sides of (11) and (12) is a valid pmfMoreover suppose that (119883119895)

119899119895=1 follows a Bernoulli mixtures

distribution with conditional distribution

Pr [(1198831 119883119899) = (1199091 119909119899) | 119875 = 119901] = 119892119899 (119896 119901) (13)

for all (1199091 119909119899) which have precisely 119896 components equal to1 and the remaining 119899minus119896 components equal to 0Then (119883119895)

119899119895=1

is conditionally exchangeable Pr [119878119896 = 119896 | 119875 = 119901] = 119886119896(119901)for all 119896 = 1 119899 hence 119878119899 has conditional pmf given by (11)and (12) Furthermore

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119896 = 0 119899 (14)

therefore for all (1199091 119909119899)which have precisely 119896 componentsequal to 1 and the remaining 119899 minus 119896 components equal to 0

Pr [(1198831 119883119899) = (1199091 119909119899)]

= ∬1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119889119865119875 (119901) (15)

and hence

Pr [119878119899 = 119896] = ∬1

0

(119899

119896) 119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) 119889119865119875 (119901)

119896 = 0 119899

(16)

Proof See the appendices

Remark 5 For a general characterization of multivariateBernoulli random variables see Sharakhmetov and Ibragi-mov [19] With the exchangeability assumption (12) can bededuced fromTheorem 3 of the aforementioned paper

Remark 6 Under the assumption of conditional indepen-dence 119891119899(119896 119901) reduces to the pmf of a binomial distribution(ie 119886119896(119901) = 119901

119896)

Remark 7 The converse part of the theorem is connectedwith the easy part of the classical Hausdorffmoment problemand its application to the proof of de Finettirsquos theoremfor sequences of exchangeable random variables (see egChapter VII in Feller [20])

Definition 8 Let ](sdot 119901) 0 lt 119901 lt 1 be as in Theorem 4 andlet 119886119896(119901) = int

1

0119903119896](119889119903 119901) for 119896 = 0 1 2 119899 A Bernoulli

mixture model with conditional distribution as specifiedin (14) (or equivalently (12)) and (13) is called a Bernoullidouble mixture model It is determined by the followingdistributions the family of measures ](sdot 119901) 0 lt 119901 lt 1 andthe mixing distribution 119865119875 and it can be formally denotedas a 119865119875-](sdot 119901) Bernoulli double mixture model Similarlymodel (16) for the sum 119878119899 of the 119865119875-](sdot 119901) Bernoulli doublemixture random variables can be formally referred to as a119865119875-](sdot 119901) binomial double mixturemodel

In concrete examples we will employ some meaningfulworded description or an acronym rather than through theformal notation of Definition 8 In addition for the remain-der of the paper all considered models will be Bernoulli (orbinomial) double mixture models therefore we may omitthose words from the names of models simply retaining thedescription of the building blocks ](sdot 119901) and 119865119875

The following simple example illustrates the converse ofthe theorem

Example 9 (ICM) We consider a weighted average of inde-pendence and comonotonicity Specifically for a specifiedweight 120596 isin [0 1] the random 119899-vector 119883 equiv (119883119895)

119899119895=1 is a two-

point mixture of a conditionally comonotonic 119899-vector ofBernoulli mixtures and a conditionally independent 119899-vectorof Bernoulli mixtures (with common mixing distribution119865119875) with respective weights 120596 and 1 minus 120596 For such a modelthe (conditional) joint probability 119886119896(119901) 119896 = 2 119899 isexpressed as the weighted average of the joint probability ofthe comonotonic case and that of independence case

119886119896 (119901) = 120596119901 + (1 minus 120596) 119901119896 119896 = 2 119899 (17)

In practice theweight parameter120596 isin [0 1]must be estimatedfrom data or be chosen exogenously

This model (with 119865119875 unspecified) will be denoted bythe acronym ICM standing for independent comonotonicmixture (with everything being implicitly conditional) It

Journal of Probability and Statistics 5

corresponds to the following choice for ](sdot 119901) which is adouble mixture of point masses

] (sdot 119901) = 120596 [1199011205751 + (1 minus 119901) 1205750] + (1 minus 120596) 120575119901 (18)

Note that int101199030](119889119903 119901) = 1 follows from the usual under-

standing that 00 = 1Then the limiting correlation is (cf (4))

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) = 120596 (19)

and the unconditional pmf of the sum 120583119899(119896) = Pr[119878119899 = 119896]is

120583119899 (119896)

=

120596 (1 minus 119864 [119875]) + (1 minus 120596) 120583119868119899 (0)

= 120596 (1 minus 119864 [119875])

+ (1 minus 120596) 119864 [(1 minus 119875)119899] 119896 = 0

(1 minus 120596) 120583119868119899 (119896) 119896 = 1 119899 minus 1

120596119864 [119875] + (1 minus 120596) 120583119868119899 (119896)

= 120596119864 [119875] + (1 minus 120596) 119864 [119875119899] 119896 = 119899

(20)

with 120583119868119899(119896) given in (1) The mean and variance of the sum 119878119899can be easily obtained as follows

119864 [119878119899] = 119899119864 [119875]

Var [119878119899] = 120596 1198992119864 [119875] minus 119899

2(119864 [119875])

2 + (1 minus 120596)

times 119899119864 [119875] minus 119899119864 [1198752] + 1198992119864 [1198752] minus 1198992(119864 [119875])

2

(21)

Note that both the mean and variance are the weightedaverages of those for the independent and comonotone cases(In general the 119896th moment is the weighted average of the119896th moment of the independent and comonotone cases)

23 Large-Sample Distribution of Empirical Frequency Theproportion of observed defaults to the total number of entitiesin a credit portfolio is of interest in practice For examplethe historical default rate for a homogenous credit portfoliois used to estimate the probability of default for a genericcounterparty in the portfolio

For a fixed 119899 let 119871119899 = 119878119899119899 denote the empiricalfrequency which can be considered as the percentage grossloss of a portfolio of 119899 loans in equal dollar amounts

The probability distribution of 119871119899 is

119865119871119899

(119909) = Pr [119871119899 le 119909] = Pr [119878119899 le 119899119909] 119909 ge 0 (22)

For the general binomial double mixture model basedon a family of probability measures ](sdot 119901) and mixingdistribution 119865119875 we have the following result

119865119871infin

(119909)

equiv lim119899rarrinfin

119865119871119899

(119909)

=

int1

0

] (0 119901) 119889119865119875 (119901) 119909 = 0

int1

0

[] ([0 119909) 119901) +1

2] (119909 119901)] 119889119865119875 (119901) 0 lt 119909 lt 1

1 119909 = 1

(23)

Here is the proof of (22) The case 119909 = 1 is trivial For0 le 119909 lt 1 first note that by (14)

Pr [119878119899 le 119899119909 | 119875 = 119901] =[119899119909]

sum119896=0

(119899

119896)119892119899 (119896 119901)

= int1

0

[119899119909]

sum119896=0

(119899

119896) 119903119896(1 minus 119903)

119899minus119896] (119889119903 119901)

(24)

where for a nonnegative number 119910 [119910] represents thegreatest integer less than or equal to 119910 The integrand is thecdf at 119909 of 1198781015840119899119899 where 1198781015840119899 sim Bin(119899 119903) so by the LLN as119899 rarr infin 1198781015840119899119899 rarr 119903 as Therefore for 0 lt 119909 lt 1

lim119899rarrinfin

Pr[1198781015840119899

119899le 119909] = 1[0119909) (119903) +

1

21119909 (119903) (25)

where 1119860 denotes the indicator function of a set 119860 (Thesecond term with the factor ldquo12rdquo comes from an applicationof the CLT to Pr[1198781015840119899 minus 119899119903 le 0] when 119909 = 119903) For 119909 = 0

Pr [119878119899 = 0 | 119875 = 119901]

= int1

0

(1 minus 119903)119899] (119889119903 119901) 997888rarr ] (0 119901) as 119899 997888rarr infin

(26)

Integrating these two limiting results with respect to](119889119903 119901)119889119865119875(119901) yields the first two cases of (22) The inter-change of limit and integrals is justified by the boundedconvergence theorem

Example 10 (ICM reprise) For the 120596-weighted average ofconditionally comonotonic and conditionally independentBernoulli mixtures described in Example 9 we can easilyobtain the following limiting distribution (this is theweightedaverage of the obvious limit for the conditionally comono-tonic component and the convergence result in Vasicek

6 Journal of Probability and Statistics

[21] for the conditionally independent component of themixture)

lim119899rarrinfin

119865119871119899

(119909)

equiv 119865119871infin

(119909) = 120596 (1 minus 119864 [119875]) + (1 minus 120596) 119865119875 (119909) 0 le 119909 lt 1

1 119909 = 1

(27)

Then the mean and variance of the limiting distribution are

119864 [119871infin] = 119864 [119875] = lim119899rarrinfin

119864 [119878119899

119899]

Var [119871infin] = 120596 119864 [119875] minus (119864 [119875])2 + (1 minus 120596)Var [119875]

= lim119899rarrinfin

Var [119878119899

119899]

(28)

In general the 119896th moment of the limiting distribution is

119864 [119871119896infin] = (1 minus 120596) 119864 [119875

119896] + 120596119864 [119875] 119896 = 1 2 (29)

The limiting result (27) can be applied to the area offinancial credit risk management The tail risk measuresof a large-sample portfolio credit loss distribution are ofparticular interest for financial risk managers The two well-known risk measures value at risk (VaR) and conditional tailexpectation (CTE) can be approximated by using the limitingdistribution (27) Formally for a loss randomvariable119883withthe cumulative distribution function 119865(119909) VaR and CTE atconfidence level 120572 isin [0 1] are defined as follows

VaR120572 (119883) = 119865minus1(120572) = inf 119909 119865 (119909) ge 120572

CTE120572 (119883) = VaR120572 (119883) +Pr [119883 gt VaR120572 (119883)]

(1 minus 120572)

times119864 [119883 minus VaR120572 (119883) | 119883 gt VaR120572 (119883)]

(30)

Note that this definition reduces to

CTE120572 (119883) = 119864 [119883 | 119883 gt VaR120572 (119883)] (31)

if 119883 is a continuous random variable (see Hardy [22] orAcerbi and Tasche [23])

Then by the scaling property of these risk measures andthe limiting result (27) for a large 119899 the VaR of the sum of 119899Bernoulli mixtures at level 120572 can be approximated as follows

VaR120572 (119878119899)

= 119899VaR120572 (119871119899) asymp 119899VaR120572 (119871infin) = 119899119865minus1119871infin

(120572)

=

0 0 le 120572 le 120596 (1 minus 119864 [119875])

119899119865minus1119875 (120572lowast) 120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(32)

where 120572lowast = 120572 minus 120596(1 minus 119864[119875])(1 minus 120596)

Similarly the CTE at level 120572 can be approximated as

CTE120572 (119878119899)

= 119899CTE120572 (119871119899)

asymp 119899CTE120572 (119871infin)

=

(119899

1 minus 120572)119864 [119875] 0 le 120572 le 120596 (1 minus 119864 [119875])

119899

1 minus 120572(1 minus 120572lowast) (1 minus 120596)CTE120572lowast (119875) + 120596119864 [119875]

120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(33)

3 Beta Mixing Distribution

We now specialize the results of the preceding section tothe case where the mixing distribution 119865119875 is the Betadistribution with parameters 119886 gt 0 and 119887 gt 0 Specificallythe density 119891119875(119901) is given by

119891119875 (119901) =Γ (119886 + 119887)

Γ (119886) Γ (119887)119901119886minus1(1 minus 119901)

119887minus1

=1

119861 (119886 119887)119901119886minus1(1 minus 119901)

119887minus1 0 lt 119901 lt 1

(34)

where Γ(sdot) and 119861(sdot sdot) are the Gamma and Beta functionsrespectively

For a Beta Bernoulli mixture several computationallyconvenient expressions are available We first introduce anotation which is convenient for the cumulative distributionfunction

119865119875 (119901) = 119868119901 (119886 119887) =119861119901 (119886 119887)

119861 (119886 119887) (35)

where 119861119909(119886 119887) = int119909

0119904119886minus1(1 minus 119904)

119887minus1119889119904 is the incomplete

Beta function Equation (1) the pmf of the sum under theconditional independence assumption can be evaluated as

120583119868119899 (119896) = (

119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) (36)

The 119896th moments for 119875 and (1 minus 119875) in this case are

119864 [119875119896] =

119861 (119886 + 119896 119887)

119861 (119886 119887)

119864 [(1 minus 119875)119896] =

119861 (119886 119887 + 119896)

119861 (119886 119887)

(37)

The following lists several results for the ICMmodel withthe Beta mixing distribution (120573-ICMmodel)

Journal of Probability and Statistics 7

Example 11 (120573-ICM) The pmf of the binomial double mix-ture is given from (20) as

120583119899 (119896)

=

120596(119887

119886 + 119887) + (1 minus 120596)

119861 (119886 119887 + 119899)

119861 (119886 119887) 119896 = 0

(1 minus 120596)(119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) 119896 = 1 119899 minus 1

120596(119886

119886 + 119887) + (1 minus 120596)

119861 (119886 + 119899 119887)

119861 (119886 119887) 119896 = 119899

(38)

The limiting distribution of the empirical frequency (27) is

119865119871infin

(119909) =

120596(119887

119886 + 119887) + (1 minus 120596) 119868119909 (119886 119887) 0 le 119909 lt 1

1 119909 = 1

(39)

Since the inverse cumulative Beta distribution function doesnot admit a closed-form expression a numerical method isrequired to approximate the VaR of a large-sample creditportfolio Given the VaR at level 120572 the CTE can be writtenin a concise form Specifically

VaR120572 (119878119899) asymp

0 0 le 120572 le120596119887

119886 + 119887

119899119868minus1120572lowast (119886 119887) 120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(40)

where 120572lowast = 120572 minus 120596(119887(119886 + 119887))(1 minus 120596) and

CTE120572 (119878119899)

asymp

(119899

1 minus 120572)(

119886

119886 + 119887) 0 le 120572 le

120596119887

119886 + 119887

(119899

1 minus 120572)(

119886

119886 + 119887) [1 minus (1 minus 119908) 119868119901lowast (119886 + 1 119887)]

120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(41)

where 119901lowast = VaR120572lowast(119875) = 119868minus1120572lowast (119886 119887)

As can be seen from (27) the limiting distribution ofthe empirical frequency under the 120573-ICM model has pointmasses at both end points 0 and 1 (when 120596 = 0) This mayrestrict the use of a certain parameter estimation methodin the case that all observations are strictly inside thedistributional range which is often the case In the followingexample we consider a Beta distribution for the measures](sdot 119901) 0 lt 119901 lt 1 as well as a Beta mixing distribution for119865119875 We call the model a double Beta model with acronym 120573-120573 The limiting empirical frequency for the 120573-120573 model doesnot have point masses at the boundary points 0 and 1

Example 12 (120573-120573) In this example the choice for ](sdot 119901) is theBeta distribution with the two shape parameters 120579119901 and 120579(1minus119901)

] (119889119903 119901)

= 119891] (119903 119901) 119889119903

=1

119861 (120579119901 120579 (1 minus 119901))119903120579119901minus1 (1 minus 119903)

120579(1minus119901)minus1119889119903 0 lt 119903 lt 1

(42)

where parameter 120579 satisfies 0 lt 120579 lt infin Then using therecursive property of the Beta function (119861(120572+1 120573) = [120572(120572+120573)]119861(120572 120573))

119886119896 (119901) = int1

0

119903119896] (119889119903 119901) =

119896minus1

prod119895=0

120579119901 + 119895

120579 + 119895 (43)

with the usual convention that a product over an empty set ofindices equals 1 In particular

1198862 (119901) =120579

1 + 1205791199012+

1

1 + 120579119901 (44)

and thus the limiting correlation for this model is byTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) =

1

1 + 120579 (45)

(Note that this is identical to the result for an ICM modelwith 120596 = 1(1 + 120579) cf (19) in Example 9)

The pmf of the binomial double mixture is given byTheorem 4 as

120583119899 (119896) = int1

0

119891119899 (119896 119901) 119889119865119875 (119901)

= (119899

119896)int1

0

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) 119889119865119875 (119901)

= (119899

119896)

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894

times int1

0

119896+119894minus1

prod119895=0

120579119901 + 119895

120579 + 119895119889119865119875 (119901) 119896 = 0 119899

(46)

By (22) the limiting distribution of the empirical fre-quency is absolutely continuous and its pdf is

119891119871infin

(119909) = int1

0

119891] (119909 119901) 119889119865119875 (119901) 0 lt 119909 lt 1 (47)

Since (47) does not admit a closed-form expression anumerical method is required to calculate the approximateVaR and CTE of a large-sample credit portfolio

8 Journal of Probability and Statistics

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 00120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 10

Figure 1The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-ICMmodel

4 Numerical Examples

As an illustrative example we set 119886 = 15 and 119887 = 135

for the Beta mixing distribution for both the 120573-ICM and120573-120573 models (These prescribed parameter values are chosenfrom the authorsrsquo previous experience in portfolio credit riskmodeling) This gives 119864[119875] = 119886(119886 + 119887) = 01 and Var[119875] =119886119887[(119886 + 119887)

2(119886 + 119887 + 1)] = 0005625

41 120573-ICMModel In Figure 1 we plot the cumulative distri-bution functions of the 120573-ICM binomial double mixture 11987820for several different values of120596 (the limiting correlation)Theplot shows that both the left and right tails get thicker as thelevel of limiting correlation 120596 increases to 1

In Figures 2 and 3 we display the approximate riskmeasures VaR and CTE for a scale of confidence levels120572 isin 08 085 090 095 099 and by the level of the limitingcorrelation120596The results show that both VaR120572 and CTE120572 areincreasing in the level of limiting correlation for 120572 ge 090This is not the case for VaR120572 when the confidence level is lessthan 1 minus 119864[119875] = 119887(119886 + 119887) because

120572lowast=

1

1 minus 120596120572 minus 120596(

119887

119886 + 119887)

=1

1 minus 120596120572 minus

119887

119886 + 119887 +

119887

119886 + 119887

(48)

is decreasing in 120596 when 120572 lt 119887(119886 + 119887)More specifically Table 1 compares the approximatedVaR

and CTE at the level 120572 = 095

42 120573-120573Model Recall that the limiting correlation of the 120573-120573 model is 1(1 + 120579) For the purpose of comparison withthe 120573-ICM model we reparametrize the 120573-120573 mixture modelas follows 120579 = (1 minus 120596)120596 such that the limiting correlationbecomes 120596

Figure 4 displays the cumulative distribution functions ofthe 120573-120573 binomial double mixture 11987820 for several different

Valu

e at r

isk

0

200

400

600

800

1000

00 02 04 06 08 10120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 2The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

00 02 04 06 08 10

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 3The approximatedCTEs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

Table 1The approximated VaR and CTE of 1198781000 at confidence level120572 = 095 under the 120573-ICMmodel

120596 VaR120572 CTE12057200 24751 3031401 25467 4477202 26483 5913503 28073 7333804 31100 8722305 100000 100000

values of 120596 (the limiting correlation) The plot shows that 11987820has a heavier left tail under the 120573-120573model than under the 120573-ICM model for each limiting correlation level Note that the120573-120573model does not admit the two extreme cases 120596 = 0 and120596 = 1

Figures 5 and 6 show the approximate risk measuresVaR and CTE for a scale of confidence levels 120572 isin

08 085 090 095 099 and by the level of the limiting

Journal of Probability and Statistics 9

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 01120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 09

Figure 4The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-120573model

Valu

e at r

isk

02 04 06 08

0

200

400

600

800

1000

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 5The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-120573model

correlation 120596 The approximate risk measures are smallerthan those of the 120573-ICMmodel for each confidence level andlimiting correlation compare for example Figures 2 and 3This indicates that the right tail of the 120573-120573 model is thinnerthan that of the 120573-ICMmodel for the same value of 120596

5 Models Fit to Real Data

In addition to the previous examples with prescribed param-eter values we illustrate the two binomial double mixturemodels with real data Bloomberg mortgage delinquencyrate index US residential mortgage loans are segmentedinto three buckets Prime Alt-A and Subprime based upon

02 04 06 08

0

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 6The approximatedCTEs of 1198781000 by confidence level120572 andthe limiting correlation 120596 under the 120573-120573model

Table 2Themethod ofmoments estimates of the shape parameters119886 and 119887 of Beta mixing distribution and the limiting correlation 120596

Model Parameter119886 119887 120596

120573-ICM 4038 14563 0009120573-I 3381 12191

the loan type Several delinquency rates for each bucketare calculated as the percentage of loans that are reportedto be delinquent beyond a certain number of days (3060 or 90 days) or are classified as real estate owned(REO) or foreclosure In this section we only use theAlt-A 60+ days (60+90+REO+foreclosure) delinquency rate(Ticker BBMDA60P) for the purpose of illustration Thesame approach can be taken for other delinquency indexessuch as the Prime (Tickers BBMDP60P BBMDPDLQ) orSubprime (Tickers BBMDS60P BBMDSDLQ) segmentsFigure 7 is taken from the Bloomberg interactive chart andit plots the monthly history of the delinquency rate duringMay 2007ndashMarch 2012

51 120573-ICMModel Note that none of the observations has thevalue 0 or 1 In this case the maximum likelihood estimateof the weight parameter 120596 is 0 because for 0 lt 119901 lt 1log119891119871

infin

(119901) = log(1 minus 120596) + log119891119875(119901) where 119891119875(119901) is thedensity of the Beta distribution (34) To avoid this issue weestimate the three parameters in the 120573-ICMmodel by meansof the method of moments (MMEs) which uses the first threemoments (29) and (37) of the limiting distribution of theempirical frequency

For comparison we also estimate the two shape parame-ters of the Beta distributionmdashthe limiting distribution of theclassical Beta-binomial (120573-I) modelmdashby matching the firstand second sample moments with the theoretical ones using(28) and (37) Table 2 gives the parameter estimates

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

Journal of Probability and Statistics 5

corresponds to the following choice for ](sdot 119901) which is adouble mixture of point masses

] (sdot 119901) = 120596 [1199011205751 + (1 minus 119901) 1205750] + (1 minus 120596) 120575119901 (18)

Note that int101199030](119889119903 119901) = 1 follows from the usual under-

standing that 00 = 1Then the limiting correlation is (cf (4))

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) = 120596 (19)

and the unconditional pmf of the sum 120583119899(119896) = Pr[119878119899 = 119896]is

120583119899 (119896)

=

120596 (1 minus 119864 [119875]) + (1 minus 120596) 120583119868119899 (0)

= 120596 (1 minus 119864 [119875])

+ (1 minus 120596) 119864 [(1 minus 119875)119899] 119896 = 0

(1 minus 120596) 120583119868119899 (119896) 119896 = 1 119899 minus 1

120596119864 [119875] + (1 minus 120596) 120583119868119899 (119896)

= 120596119864 [119875] + (1 minus 120596) 119864 [119875119899] 119896 = 119899

(20)

with 120583119868119899(119896) given in (1) The mean and variance of the sum 119878119899can be easily obtained as follows

119864 [119878119899] = 119899119864 [119875]

Var [119878119899] = 120596 1198992119864 [119875] minus 119899

2(119864 [119875])

2 + (1 minus 120596)

times 119899119864 [119875] minus 119899119864 [1198752] + 1198992119864 [1198752] minus 1198992(119864 [119875])

2

(21)

Note that both the mean and variance are the weightedaverages of those for the independent and comonotone cases(In general the 119896th moment is the weighted average of the119896th moment of the independent and comonotone cases)

23 Large-Sample Distribution of Empirical Frequency Theproportion of observed defaults to the total number of entitiesin a credit portfolio is of interest in practice For examplethe historical default rate for a homogenous credit portfoliois used to estimate the probability of default for a genericcounterparty in the portfolio

For a fixed 119899 let 119871119899 = 119878119899119899 denote the empiricalfrequency which can be considered as the percentage grossloss of a portfolio of 119899 loans in equal dollar amounts

The probability distribution of 119871119899 is

119865119871119899

(119909) = Pr [119871119899 le 119909] = Pr [119878119899 le 119899119909] 119909 ge 0 (22)

For the general binomial double mixture model basedon a family of probability measures ](sdot 119901) and mixingdistribution 119865119875 we have the following result

119865119871infin

(119909)

equiv lim119899rarrinfin

119865119871119899

(119909)

=

int1

0

] (0 119901) 119889119865119875 (119901) 119909 = 0

int1

0

[] ([0 119909) 119901) +1

2] (119909 119901)] 119889119865119875 (119901) 0 lt 119909 lt 1

1 119909 = 1

(23)

Here is the proof of (22) The case 119909 = 1 is trivial For0 le 119909 lt 1 first note that by (14)

Pr [119878119899 le 119899119909 | 119875 = 119901] =[119899119909]

sum119896=0

(119899

119896)119892119899 (119896 119901)

= int1

0

[119899119909]

sum119896=0

(119899

119896) 119903119896(1 minus 119903)

119899minus119896] (119889119903 119901)

(24)

where for a nonnegative number 119910 [119910] represents thegreatest integer less than or equal to 119910 The integrand is thecdf at 119909 of 1198781015840119899119899 where 1198781015840119899 sim Bin(119899 119903) so by the LLN as119899 rarr infin 1198781015840119899119899 rarr 119903 as Therefore for 0 lt 119909 lt 1

lim119899rarrinfin

Pr[1198781015840119899

119899le 119909] = 1[0119909) (119903) +

1

21119909 (119903) (25)

where 1119860 denotes the indicator function of a set 119860 (Thesecond term with the factor ldquo12rdquo comes from an applicationof the CLT to Pr[1198781015840119899 minus 119899119903 le 0] when 119909 = 119903) For 119909 = 0

Pr [119878119899 = 0 | 119875 = 119901]

= int1

0

(1 minus 119903)119899] (119889119903 119901) 997888rarr ] (0 119901) as 119899 997888rarr infin

(26)

Integrating these two limiting results with respect to](119889119903 119901)119889119865119875(119901) yields the first two cases of (22) The inter-change of limit and integrals is justified by the boundedconvergence theorem

Example 10 (ICM reprise) For the 120596-weighted average ofconditionally comonotonic and conditionally independentBernoulli mixtures described in Example 9 we can easilyobtain the following limiting distribution (this is theweightedaverage of the obvious limit for the conditionally comono-tonic component and the convergence result in Vasicek

6 Journal of Probability and Statistics

[21] for the conditionally independent component of themixture)

lim119899rarrinfin

119865119871119899

(119909)

equiv 119865119871infin

(119909) = 120596 (1 minus 119864 [119875]) + (1 minus 120596) 119865119875 (119909) 0 le 119909 lt 1

1 119909 = 1

(27)

Then the mean and variance of the limiting distribution are

119864 [119871infin] = 119864 [119875] = lim119899rarrinfin

119864 [119878119899

119899]

Var [119871infin] = 120596 119864 [119875] minus (119864 [119875])2 + (1 minus 120596)Var [119875]

= lim119899rarrinfin

Var [119878119899

119899]

(28)

In general the 119896th moment of the limiting distribution is

119864 [119871119896infin] = (1 minus 120596) 119864 [119875

119896] + 120596119864 [119875] 119896 = 1 2 (29)

The limiting result (27) can be applied to the area offinancial credit risk management The tail risk measuresof a large-sample portfolio credit loss distribution are ofparticular interest for financial risk managers The two well-known risk measures value at risk (VaR) and conditional tailexpectation (CTE) can be approximated by using the limitingdistribution (27) Formally for a loss randomvariable119883withthe cumulative distribution function 119865(119909) VaR and CTE atconfidence level 120572 isin [0 1] are defined as follows

VaR120572 (119883) = 119865minus1(120572) = inf 119909 119865 (119909) ge 120572

CTE120572 (119883) = VaR120572 (119883) +Pr [119883 gt VaR120572 (119883)]

(1 minus 120572)

times119864 [119883 minus VaR120572 (119883) | 119883 gt VaR120572 (119883)]

(30)

Note that this definition reduces to

CTE120572 (119883) = 119864 [119883 | 119883 gt VaR120572 (119883)] (31)

if 119883 is a continuous random variable (see Hardy [22] orAcerbi and Tasche [23])

Then by the scaling property of these risk measures andthe limiting result (27) for a large 119899 the VaR of the sum of 119899Bernoulli mixtures at level 120572 can be approximated as follows

VaR120572 (119878119899)

= 119899VaR120572 (119871119899) asymp 119899VaR120572 (119871infin) = 119899119865minus1119871infin

(120572)

=

0 0 le 120572 le 120596 (1 minus 119864 [119875])

119899119865minus1119875 (120572lowast) 120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(32)

where 120572lowast = 120572 minus 120596(1 minus 119864[119875])(1 minus 120596)

Similarly the CTE at level 120572 can be approximated as

CTE120572 (119878119899)

= 119899CTE120572 (119871119899)

asymp 119899CTE120572 (119871infin)

=

(119899

1 minus 120572)119864 [119875] 0 le 120572 le 120596 (1 minus 119864 [119875])

119899

1 minus 120572(1 minus 120572lowast) (1 minus 120596)CTE120572lowast (119875) + 120596119864 [119875]

120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(33)

3 Beta Mixing Distribution

We now specialize the results of the preceding section tothe case where the mixing distribution 119865119875 is the Betadistribution with parameters 119886 gt 0 and 119887 gt 0 Specificallythe density 119891119875(119901) is given by

119891119875 (119901) =Γ (119886 + 119887)

Γ (119886) Γ (119887)119901119886minus1(1 minus 119901)

119887minus1

=1

119861 (119886 119887)119901119886minus1(1 minus 119901)

119887minus1 0 lt 119901 lt 1

(34)

where Γ(sdot) and 119861(sdot sdot) are the Gamma and Beta functionsrespectively

For a Beta Bernoulli mixture several computationallyconvenient expressions are available We first introduce anotation which is convenient for the cumulative distributionfunction

119865119875 (119901) = 119868119901 (119886 119887) =119861119901 (119886 119887)

119861 (119886 119887) (35)

where 119861119909(119886 119887) = int119909

0119904119886minus1(1 minus 119904)

119887minus1119889119904 is the incomplete

Beta function Equation (1) the pmf of the sum under theconditional independence assumption can be evaluated as

120583119868119899 (119896) = (

119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) (36)

The 119896th moments for 119875 and (1 minus 119875) in this case are

119864 [119875119896] =

119861 (119886 + 119896 119887)

119861 (119886 119887)

119864 [(1 minus 119875)119896] =

119861 (119886 119887 + 119896)

119861 (119886 119887)

(37)

The following lists several results for the ICMmodel withthe Beta mixing distribution (120573-ICMmodel)

Journal of Probability and Statistics 7

Example 11 (120573-ICM) The pmf of the binomial double mix-ture is given from (20) as

120583119899 (119896)

=

120596(119887

119886 + 119887) + (1 minus 120596)

119861 (119886 119887 + 119899)

119861 (119886 119887) 119896 = 0

(1 minus 120596)(119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) 119896 = 1 119899 minus 1

120596(119886

119886 + 119887) + (1 minus 120596)

119861 (119886 + 119899 119887)

119861 (119886 119887) 119896 = 119899

(38)

The limiting distribution of the empirical frequency (27) is

119865119871infin

(119909) =

120596(119887

119886 + 119887) + (1 minus 120596) 119868119909 (119886 119887) 0 le 119909 lt 1

1 119909 = 1

(39)

Since the inverse cumulative Beta distribution function doesnot admit a closed-form expression a numerical method isrequired to approximate the VaR of a large-sample creditportfolio Given the VaR at level 120572 the CTE can be writtenin a concise form Specifically

VaR120572 (119878119899) asymp

0 0 le 120572 le120596119887

119886 + 119887

119899119868minus1120572lowast (119886 119887) 120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(40)

where 120572lowast = 120572 minus 120596(119887(119886 + 119887))(1 minus 120596) and

CTE120572 (119878119899)

asymp

(119899

1 minus 120572)(

119886

119886 + 119887) 0 le 120572 le

120596119887

119886 + 119887

(119899

1 minus 120572)(

119886

119886 + 119887) [1 minus (1 minus 119908) 119868119901lowast (119886 + 1 119887)]

120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(41)

where 119901lowast = VaR120572lowast(119875) = 119868minus1120572lowast (119886 119887)

As can be seen from (27) the limiting distribution ofthe empirical frequency under the 120573-ICM model has pointmasses at both end points 0 and 1 (when 120596 = 0) This mayrestrict the use of a certain parameter estimation methodin the case that all observations are strictly inside thedistributional range which is often the case In the followingexample we consider a Beta distribution for the measures](sdot 119901) 0 lt 119901 lt 1 as well as a Beta mixing distribution for119865119875 We call the model a double Beta model with acronym 120573-120573 The limiting empirical frequency for the 120573-120573 model doesnot have point masses at the boundary points 0 and 1

Example 12 (120573-120573) In this example the choice for ](sdot 119901) is theBeta distribution with the two shape parameters 120579119901 and 120579(1minus119901)

] (119889119903 119901)

= 119891] (119903 119901) 119889119903

=1

119861 (120579119901 120579 (1 minus 119901))119903120579119901minus1 (1 minus 119903)

120579(1minus119901)minus1119889119903 0 lt 119903 lt 1

(42)

where parameter 120579 satisfies 0 lt 120579 lt infin Then using therecursive property of the Beta function (119861(120572+1 120573) = [120572(120572+120573)]119861(120572 120573))

119886119896 (119901) = int1

0

119903119896] (119889119903 119901) =

119896minus1

prod119895=0

120579119901 + 119895

120579 + 119895 (43)

with the usual convention that a product over an empty set ofindices equals 1 In particular

1198862 (119901) =120579

1 + 1205791199012+

1

1 + 120579119901 (44)

and thus the limiting correlation for this model is byTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) =

1

1 + 120579 (45)

(Note that this is identical to the result for an ICM modelwith 120596 = 1(1 + 120579) cf (19) in Example 9)

The pmf of the binomial double mixture is given byTheorem 4 as

120583119899 (119896) = int1

0

119891119899 (119896 119901) 119889119865119875 (119901)

= (119899

119896)int1

0

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) 119889119865119875 (119901)

= (119899

119896)

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894

times int1

0

119896+119894minus1

prod119895=0

120579119901 + 119895

120579 + 119895119889119865119875 (119901) 119896 = 0 119899

(46)

By (22) the limiting distribution of the empirical fre-quency is absolutely continuous and its pdf is

119891119871infin

(119909) = int1

0

119891] (119909 119901) 119889119865119875 (119901) 0 lt 119909 lt 1 (47)

Since (47) does not admit a closed-form expression anumerical method is required to calculate the approximateVaR and CTE of a large-sample credit portfolio

8 Journal of Probability and Statistics

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 00120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 10

Figure 1The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-ICMmodel

4 Numerical Examples

As an illustrative example we set 119886 = 15 and 119887 = 135

for the Beta mixing distribution for both the 120573-ICM and120573-120573 models (These prescribed parameter values are chosenfrom the authorsrsquo previous experience in portfolio credit riskmodeling) This gives 119864[119875] = 119886(119886 + 119887) = 01 and Var[119875] =119886119887[(119886 + 119887)

2(119886 + 119887 + 1)] = 0005625

41 120573-ICMModel In Figure 1 we plot the cumulative distri-bution functions of the 120573-ICM binomial double mixture 11987820for several different values of120596 (the limiting correlation)Theplot shows that both the left and right tails get thicker as thelevel of limiting correlation 120596 increases to 1

In Figures 2 and 3 we display the approximate riskmeasures VaR and CTE for a scale of confidence levels120572 isin 08 085 090 095 099 and by the level of the limitingcorrelation120596The results show that both VaR120572 and CTE120572 areincreasing in the level of limiting correlation for 120572 ge 090This is not the case for VaR120572 when the confidence level is lessthan 1 minus 119864[119875] = 119887(119886 + 119887) because

120572lowast=

1

1 minus 120596120572 minus 120596(

119887

119886 + 119887)

=1

1 minus 120596120572 minus

119887

119886 + 119887 +

119887

119886 + 119887

(48)

is decreasing in 120596 when 120572 lt 119887(119886 + 119887)More specifically Table 1 compares the approximatedVaR

and CTE at the level 120572 = 095

42 120573-120573Model Recall that the limiting correlation of the 120573-120573 model is 1(1 + 120579) For the purpose of comparison withthe 120573-ICM model we reparametrize the 120573-120573 mixture modelas follows 120579 = (1 minus 120596)120596 such that the limiting correlationbecomes 120596

Figure 4 displays the cumulative distribution functions ofthe 120573-120573 binomial double mixture 11987820 for several different

Valu

e at r

isk

0

200

400

600

800

1000

00 02 04 06 08 10120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 2The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

00 02 04 06 08 10

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 3The approximatedCTEs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

Table 1The approximated VaR and CTE of 1198781000 at confidence level120572 = 095 under the 120573-ICMmodel

120596 VaR120572 CTE12057200 24751 3031401 25467 4477202 26483 5913503 28073 7333804 31100 8722305 100000 100000

values of 120596 (the limiting correlation) The plot shows that 11987820has a heavier left tail under the 120573-120573model than under the 120573-ICM model for each limiting correlation level Note that the120573-120573model does not admit the two extreme cases 120596 = 0 and120596 = 1

Figures 5 and 6 show the approximate risk measuresVaR and CTE for a scale of confidence levels 120572 isin

08 085 090 095 099 and by the level of the limiting

Journal of Probability and Statistics 9

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 01120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 09

Figure 4The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-120573model

Valu

e at r

isk

02 04 06 08

0

200

400

600

800

1000

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 5The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-120573model

correlation 120596 The approximate risk measures are smallerthan those of the 120573-ICMmodel for each confidence level andlimiting correlation compare for example Figures 2 and 3This indicates that the right tail of the 120573-120573 model is thinnerthan that of the 120573-ICMmodel for the same value of 120596

5 Models Fit to Real Data

In addition to the previous examples with prescribed param-eter values we illustrate the two binomial double mixturemodels with real data Bloomberg mortgage delinquencyrate index US residential mortgage loans are segmentedinto three buckets Prime Alt-A and Subprime based upon

02 04 06 08

0

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 6The approximatedCTEs of 1198781000 by confidence level120572 andthe limiting correlation 120596 under the 120573-120573model

Table 2Themethod ofmoments estimates of the shape parameters119886 and 119887 of Beta mixing distribution and the limiting correlation 120596

Model Parameter119886 119887 120596

120573-ICM 4038 14563 0009120573-I 3381 12191

the loan type Several delinquency rates for each bucketare calculated as the percentage of loans that are reportedto be delinquent beyond a certain number of days (3060 or 90 days) or are classified as real estate owned(REO) or foreclosure In this section we only use theAlt-A 60+ days (60+90+REO+foreclosure) delinquency rate(Ticker BBMDA60P) for the purpose of illustration Thesame approach can be taken for other delinquency indexessuch as the Prime (Tickers BBMDP60P BBMDPDLQ) orSubprime (Tickers BBMDS60P BBMDSDLQ) segmentsFigure 7 is taken from the Bloomberg interactive chart andit plots the monthly history of the delinquency rate duringMay 2007ndashMarch 2012

51 120573-ICMModel Note that none of the observations has thevalue 0 or 1 In this case the maximum likelihood estimateof the weight parameter 120596 is 0 because for 0 lt 119901 lt 1log119891119871

infin

(119901) = log(1 minus 120596) + log119891119875(119901) where 119891119875(119901) is thedensity of the Beta distribution (34) To avoid this issue weestimate the three parameters in the 120573-ICMmodel by meansof the method of moments (MMEs) which uses the first threemoments (29) and (37) of the limiting distribution of theempirical frequency

For comparison we also estimate the two shape parame-ters of the Beta distributionmdashthe limiting distribution of theclassical Beta-binomial (120573-I) modelmdashby matching the firstand second sample moments with the theoretical ones using(28) and (37) Table 2 gives the parameter estimates

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

6 Journal of Probability and Statistics

[21] for the conditionally independent component of themixture)

lim119899rarrinfin

119865119871119899

(119909)

equiv 119865119871infin

(119909) = 120596 (1 minus 119864 [119875]) + (1 minus 120596) 119865119875 (119909) 0 le 119909 lt 1

1 119909 = 1

(27)

Then the mean and variance of the limiting distribution are

119864 [119871infin] = 119864 [119875] = lim119899rarrinfin

119864 [119878119899

119899]

Var [119871infin] = 120596 119864 [119875] minus (119864 [119875])2 + (1 minus 120596)Var [119875]

= lim119899rarrinfin

Var [119878119899

119899]

(28)

In general the 119896th moment of the limiting distribution is

119864 [119871119896infin] = (1 minus 120596) 119864 [119875

119896] + 120596119864 [119875] 119896 = 1 2 (29)

The limiting result (27) can be applied to the area offinancial credit risk management The tail risk measuresof a large-sample portfolio credit loss distribution are ofparticular interest for financial risk managers The two well-known risk measures value at risk (VaR) and conditional tailexpectation (CTE) can be approximated by using the limitingdistribution (27) Formally for a loss randomvariable119883withthe cumulative distribution function 119865(119909) VaR and CTE atconfidence level 120572 isin [0 1] are defined as follows

VaR120572 (119883) = 119865minus1(120572) = inf 119909 119865 (119909) ge 120572

CTE120572 (119883) = VaR120572 (119883) +Pr [119883 gt VaR120572 (119883)]

(1 minus 120572)

times119864 [119883 minus VaR120572 (119883) | 119883 gt VaR120572 (119883)]

(30)

Note that this definition reduces to

CTE120572 (119883) = 119864 [119883 | 119883 gt VaR120572 (119883)] (31)

if 119883 is a continuous random variable (see Hardy [22] orAcerbi and Tasche [23])

Then by the scaling property of these risk measures andthe limiting result (27) for a large 119899 the VaR of the sum of 119899Bernoulli mixtures at level 120572 can be approximated as follows

VaR120572 (119878119899)

= 119899VaR120572 (119871119899) asymp 119899VaR120572 (119871infin) = 119899119865minus1119871infin

(120572)

=

0 0 le 120572 le 120596 (1 minus 119864 [119875])

119899119865minus1119875 (120572lowast) 120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(32)

where 120572lowast = 120572 minus 120596(1 minus 119864[119875])(1 minus 120596)

Similarly the CTE at level 120572 can be approximated as

CTE120572 (119878119899)

= 119899CTE120572 (119871119899)

asymp 119899CTE120572 (119871infin)

=

(119899

1 minus 120572)119864 [119875] 0 le 120572 le 120596 (1 minus 119864 [119875])

119899

1 minus 120572(1 minus 120572lowast) (1 minus 120596)CTE120572lowast (119875) + 120596119864 [119875]

120596 (1 minus 119864 [119875]) lt 120572 lt 1 minus 120596119864 [119875]

119899 1 minus 120596119864 [119875] le 120572 le 1

(33)

3 Beta Mixing Distribution

We now specialize the results of the preceding section tothe case where the mixing distribution 119865119875 is the Betadistribution with parameters 119886 gt 0 and 119887 gt 0 Specificallythe density 119891119875(119901) is given by

119891119875 (119901) =Γ (119886 + 119887)

Γ (119886) Γ (119887)119901119886minus1(1 minus 119901)

119887minus1

=1

119861 (119886 119887)119901119886minus1(1 minus 119901)

119887minus1 0 lt 119901 lt 1

(34)

where Γ(sdot) and 119861(sdot sdot) are the Gamma and Beta functionsrespectively

For a Beta Bernoulli mixture several computationallyconvenient expressions are available We first introduce anotation which is convenient for the cumulative distributionfunction

119865119875 (119901) = 119868119901 (119886 119887) =119861119901 (119886 119887)

119861 (119886 119887) (35)

where 119861119909(119886 119887) = int119909

0119904119886minus1(1 minus 119904)

119887minus1119889119904 is the incomplete

Beta function Equation (1) the pmf of the sum under theconditional independence assumption can be evaluated as

120583119868119899 (119896) = (

119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) (36)

The 119896th moments for 119875 and (1 minus 119875) in this case are

119864 [119875119896] =

119861 (119886 + 119896 119887)

119861 (119886 119887)

119864 [(1 minus 119875)119896] =

119861 (119886 119887 + 119896)

119861 (119886 119887)

(37)

The following lists several results for the ICMmodel withthe Beta mixing distribution (120573-ICMmodel)

Journal of Probability and Statistics 7

Example 11 (120573-ICM) The pmf of the binomial double mix-ture is given from (20) as

120583119899 (119896)

=

120596(119887

119886 + 119887) + (1 minus 120596)

119861 (119886 119887 + 119899)

119861 (119886 119887) 119896 = 0

(1 minus 120596)(119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) 119896 = 1 119899 minus 1

120596(119886

119886 + 119887) + (1 minus 120596)

119861 (119886 + 119899 119887)

119861 (119886 119887) 119896 = 119899

(38)

The limiting distribution of the empirical frequency (27) is

119865119871infin

(119909) =

120596(119887

119886 + 119887) + (1 minus 120596) 119868119909 (119886 119887) 0 le 119909 lt 1

1 119909 = 1

(39)

Since the inverse cumulative Beta distribution function doesnot admit a closed-form expression a numerical method isrequired to approximate the VaR of a large-sample creditportfolio Given the VaR at level 120572 the CTE can be writtenin a concise form Specifically

VaR120572 (119878119899) asymp

0 0 le 120572 le120596119887

119886 + 119887

119899119868minus1120572lowast (119886 119887) 120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(40)

where 120572lowast = 120572 minus 120596(119887(119886 + 119887))(1 minus 120596) and

CTE120572 (119878119899)

asymp

(119899

1 minus 120572)(

119886

119886 + 119887) 0 le 120572 le

120596119887

119886 + 119887

(119899

1 minus 120572)(

119886

119886 + 119887) [1 minus (1 minus 119908) 119868119901lowast (119886 + 1 119887)]

120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(41)

where 119901lowast = VaR120572lowast(119875) = 119868minus1120572lowast (119886 119887)

As can be seen from (27) the limiting distribution ofthe empirical frequency under the 120573-ICM model has pointmasses at both end points 0 and 1 (when 120596 = 0) This mayrestrict the use of a certain parameter estimation methodin the case that all observations are strictly inside thedistributional range which is often the case In the followingexample we consider a Beta distribution for the measures](sdot 119901) 0 lt 119901 lt 1 as well as a Beta mixing distribution for119865119875 We call the model a double Beta model with acronym 120573-120573 The limiting empirical frequency for the 120573-120573 model doesnot have point masses at the boundary points 0 and 1

Example 12 (120573-120573) In this example the choice for ](sdot 119901) is theBeta distribution with the two shape parameters 120579119901 and 120579(1minus119901)

] (119889119903 119901)

= 119891] (119903 119901) 119889119903

=1

119861 (120579119901 120579 (1 minus 119901))119903120579119901minus1 (1 minus 119903)

120579(1minus119901)minus1119889119903 0 lt 119903 lt 1

(42)

where parameter 120579 satisfies 0 lt 120579 lt infin Then using therecursive property of the Beta function (119861(120572+1 120573) = [120572(120572+120573)]119861(120572 120573))

119886119896 (119901) = int1

0

119903119896] (119889119903 119901) =

119896minus1

prod119895=0

120579119901 + 119895

120579 + 119895 (43)

with the usual convention that a product over an empty set ofindices equals 1 In particular

1198862 (119901) =120579

1 + 1205791199012+

1

1 + 120579119901 (44)

and thus the limiting correlation for this model is byTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) =

1

1 + 120579 (45)

(Note that this is identical to the result for an ICM modelwith 120596 = 1(1 + 120579) cf (19) in Example 9)

The pmf of the binomial double mixture is given byTheorem 4 as

120583119899 (119896) = int1

0

119891119899 (119896 119901) 119889119865119875 (119901)

= (119899

119896)int1

0

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) 119889119865119875 (119901)

= (119899

119896)

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894

times int1

0

119896+119894minus1

prod119895=0

120579119901 + 119895

120579 + 119895119889119865119875 (119901) 119896 = 0 119899

(46)

By (22) the limiting distribution of the empirical fre-quency is absolutely continuous and its pdf is

119891119871infin

(119909) = int1

0

119891] (119909 119901) 119889119865119875 (119901) 0 lt 119909 lt 1 (47)

Since (47) does not admit a closed-form expression anumerical method is required to calculate the approximateVaR and CTE of a large-sample credit portfolio

8 Journal of Probability and Statistics

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 00120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 10

Figure 1The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-ICMmodel

4 Numerical Examples

As an illustrative example we set 119886 = 15 and 119887 = 135

for the Beta mixing distribution for both the 120573-ICM and120573-120573 models (These prescribed parameter values are chosenfrom the authorsrsquo previous experience in portfolio credit riskmodeling) This gives 119864[119875] = 119886(119886 + 119887) = 01 and Var[119875] =119886119887[(119886 + 119887)

2(119886 + 119887 + 1)] = 0005625

41 120573-ICMModel In Figure 1 we plot the cumulative distri-bution functions of the 120573-ICM binomial double mixture 11987820for several different values of120596 (the limiting correlation)Theplot shows that both the left and right tails get thicker as thelevel of limiting correlation 120596 increases to 1

In Figures 2 and 3 we display the approximate riskmeasures VaR and CTE for a scale of confidence levels120572 isin 08 085 090 095 099 and by the level of the limitingcorrelation120596The results show that both VaR120572 and CTE120572 areincreasing in the level of limiting correlation for 120572 ge 090This is not the case for VaR120572 when the confidence level is lessthan 1 minus 119864[119875] = 119887(119886 + 119887) because

120572lowast=

1

1 minus 120596120572 minus 120596(

119887

119886 + 119887)

=1

1 minus 120596120572 minus

119887

119886 + 119887 +

119887

119886 + 119887

(48)

is decreasing in 120596 when 120572 lt 119887(119886 + 119887)More specifically Table 1 compares the approximatedVaR

and CTE at the level 120572 = 095

42 120573-120573Model Recall that the limiting correlation of the 120573-120573 model is 1(1 + 120579) For the purpose of comparison withthe 120573-ICM model we reparametrize the 120573-120573 mixture modelas follows 120579 = (1 minus 120596)120596 such that the limiting correlationbecomes 120596

Figure 4 displays the cumulative distribution functions ofthe 120573-120573 binomial double mixture 11987820 for several different

Valu

e at r

isk

0

200

400

600

800

1000

00 02 04 06 08 10120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 2The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

00 02 04 06 08 10

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 3The approximatedCTEs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

Table 1The approximated VaR and CTE of 1198781000 at confidence level120572 = 095 under the 120573-ICMmodel

120596 VaR120572 CTE12057200 24751 3031401 25467 4477202 26483 5913503 28073 7333804 31100 8722305 100000 100000

values of 120596 (the limiting correlation) The plot shows that 11987820has a heavier left tail under the 120573-120573model than under the 120573-ICM model for each limiting correlation level Note that the120573-120573model does not admit the two extreme cases 120596 = 0 and120596 = 1

Figures 5 and 6 show the approximate risk measuresVaR and CTE for a scale of confidence levels 120572 isin

08 085 090 095 099 and by the level of the limiting

Journal of Probability and Statistics 9

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 01120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 09

Figure 4The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-120573model

Valu

e at r

isk

02 04 06 08

0

200

400

600

800

1000

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 5The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-120573model

correlation 120596 The approximate risk measures are smallerthan those of the 120573-ICMmodel for each confidence level andlimiting correlation compare for example Figures 2 and 3This indicates that the right tail of the 120573-120573 model is thinnerthan that of the 120573-ICMmodel for the same value of 120596

5 Models Fit to Real Data

In addition to the previous examples with prescribed param-eter values we illustrate the two binomial double mixturemodels with real data Bloomberg mortgage delinquencyrate index US residential mortgage loans are segmentedinto three buckets Prime Alt-A and Subprime based upon

02 04 06 08

0

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 6The approximatedCTEs of 1198781000 by confidence level120572 andthe limiting correlation 120596 under the 120573-120573model

Table 2Themethod ofmoments estimates of the shape parameters119886 and 119887 of Beta mixing distribution and the limiting correlation 120596

Model Parameter119886 119887 120596

120573-ICM 4038 14563 0009120573-I 3381 12191

the loan type Several delinquency rates for each bucketare calculated as the percentage of loans that are reportedto be delinquent beyond a certain number of days (3060 or 90 days) or are classified as real estate owned(REO) or foreclosure In this section we only use theAlt-A 60+ days (60+90+REO+foreclosure) delinquency rate(Ticker BBMDA60P) for the purpose of illustration Thesame approach can be taken for other delinquency indexessuch as the Prime (Tickers BBMDP60P BBMDPDLQ) orSubprime (Tickers BBMDS60P BBMDSDLQ) segmentsFigure 7 is taken from the Bloomberg interactive chart andit plots the monthly history of the delinquency rate duringMay 2007ndashMarch 2012

51 120573-ICMModel Note that none of the observations has thevalue 0 or 1 In this case the maximum likelihood estimateof the weight parameter 120596 is 0 because for 0 lt 119901 lt 1log119891119871

infin

(119901) = log(1 minus 120596) + log119891119875(119901) where 119891119875(119901) is thedensity of the Beta distribution (34) To avoid this issue weestimate the three parameters in the 120573-ICMmodel by meansof the method of moments (MMEs) which uses the first threemoments (29) and (37) of the limiting distribution of theempirical frequency

For comparison we also estimate the two shape parame-ters of the Beta distributionmdashthe limiting distribution of theclassical Beta-binomial (120573-I) modelmdashby matching the firstand second sample moments with the theoretical ones using(28) and (37) Table 2 gives the parameter estimates

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

Journal of Probability and Statistics 7

Example 11 (120573-ICM) The pmf of the binomial double mix-ture is given from (20) as

120583119899 (119896)

=

120596(119887

119886 + 119887) + (1 minus 120596)

119861 (119886 119887 + 119899)

119861 (119886 119887) 119896 = 0

(1 minus 120596)(119899

119896)119861 (119896 + 119886 119899 minus 119896 + 119887)

119861 (119886 119887) 119896 = 1 119899 minus 1

120596(119886

119886 + 119887) + (1 minus 120596)

119861 (119886 + 119899 119887)

119861 (119886 119887) 119896 = 119899

(38)

The limiting distribution of the empirical frequency (27) is

119865119871infin

(119909) =

120596(119887

119886 + 119887) + (1 minus 120596) 119868119909 (119886 119887) 0 le 119909 lt 1

1 119909 = 1

(39)

Since the inverse cumulative Beta distribution function doesnot admit a closed-form expression a numerical method isrequired to approximate the VaR of a large-sample creditportfolio Given the VaR at level 120572 the CTE can be writtenin a concise form Specifically

VaR120572 (119878119899) asymp

0 0 le 120572 le120596119887

119886 + 119887

119899119868minus1120572lowast (119886 119887) 120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(40)

where 120572lowast = 120572 minus 120596(119887(119886 + 119887))(1 minus 120596) and

CTE120572 (119878119899)

asymp

(119899

1 minus 120572)(

119886

119886 + 119887) 0 le 120572 le

120596119887

119886 + 119887

(119899

1 minus 120572)(

119886

119886 + 119887) [1 minus (1 minus 119908) 119868119901lowast (119886 + 1 119887)]

120596119887

119886 + 119887lt 120572 lt 1 minus

120596119886

119886 + 119887

119899 1 minus120596119886

119886 + 119887le 120572 le 1

(41)

where 119901lowast = VaR120572lowast(119875) = 119868minus1120572lowast (119886 119887)

As can be seen from (27) the limiting distribution ofthe empirical frequency under the 120573-ICM model has pointmasses at both end points 0 and 1 (when 120596 = 0) This mayrestrict the use of a certain parameter estimation methodin the case that all observations are strictly inside thedistributional range which is often the case In the followingexample we consider a Beta distribution for the measures](sdot 119901) 0 lt 119901 lt 1 as well as a Beta mixing distribution for119865119875 We call the model a double Beta model with acronym 120573-120573 The limiting empirical frequency for the 120573-120573 model doesnot have point masses at the boundary points 0 and 1

Example 12 (120573-120573) In this example the choice for ](sdot 119901) is theBeta distribution with the two shape parameters 120579119901 and 120579(1minus119901)

] (119889119903 119901)

= 119891] (119903 119901) 119889119903

=1

119861 (120579119901 120579 (1 minus 119901))119903120579119901minus1 (1 minus 119903)

120579(1minus119901)minus1119889119903 0 lt 119903 lt 1

(42)

where parameter 120579 satisfies 0 lt 120579 lt infin Then using therecursive property of the Beta function (119861(120572+1 120573) = [120572(120572+120573)]119861(120572 120573))

119886119896 (119901) = int1

0

119903119896] (119889119903 119901) =

119896minus1

prod119895=0

120579119901 + 119895

120579 + 119895 (43)

with the usual convention that a product over an empty set ofindices equals 1 In particular

1198862 (119901) =120579

1 + 1205791199012+

1

1 + 120579119901 (44)

and thus the limiting correlation for this model is byTheorem 1

120588119894119895 = 2 minus lim119901rarr1

11988610158402 (119901) =

1

1 + 120579 (45)

(Note that this is identical to the result for an ICM modelwith 120596 = 1(1 + 120579) cf (19) in Example 9)

The pmf of the binomial double mixture is given byTheorem 4 as

120583119899 (119896) = int1

0

119891119899 (119896 119901) 119889119865119875 (119901)

= (119899

119896)int1

0

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) 119889119865119875 (119901)

= (119899

119896)

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894

times int1

0

119896+119894minus1

prod119895=0

120579119901 + 119895

120579 + 119895119889119865119875 (119901) 119896 = 0 119899

(46)

By (22) the limiting distribution of the empirical fre-quency is absolutely continuous and its pdf is

119891119871infin

(119909) = int1

0

119891] (119909 119901) 119889119865119875 (119901) 0 lt 119909 lt 1 (47)

Since (47) does not admit a closed-form expression anumerical method is required to calculate the approximateVaR and CTE of a large-sample credit portfolio

8 Journal of Probability and Statistics

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 00120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 10

Figure 1The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-ICMmodel

4 Numerical Examples

As an illustrative example we set 119886 = 15 and 119887 = 135

for the Beta mixing distribution for both the 120573-ICM and120573-120573 models (These prescribed parameter values are chosenfrom the authorsrsquo previous experience in portfolio credit riskmodeling) This gives 119864[119875] = 119886(119886 + 119887) = 01 and Var[119875] =119886119887[(119886 + 119887)

2(119886 + 119887 + 1)] = 0005625

41 120573-ICMModel In Figure 1 we plot the cumulative distri-bution functions of the 120573-ICM binomial double mixture 11987820for several different values of120596 (the limiting correlation)Theplot shows that both the left and right tails get thicker as thelevel of limiting correlation 120596 increases to 1

In Figures 2 and 3 we display the approximate riskmeasures VaR and CTE for a scale of confidence levels120572 isin 08 085 090 095 099 and by the level of the limitingcorrelation120596The results show that both VaR120572 and CTE120572 areincreasing in the level of limiting correlation for 120572 ge 090This is not the case for VaR120572 when the confidence level is lessthan 1 minus 119864[119875] = 119887(119886 + 119887) because

120572lowast=

1

1 minus 120596120572 minus 120596(

119887

119886 + 119887)

=1

1 minus 120596120572 minus

119887

119886 + 119887 +

119887

119886 + 119887

(48)

is decreasing in 120596 when 120572 lt 119887(119886 + 119887)More specifically Table 1 compares the approximatedVaR

and CTE at the level 120572 = 095

42 120573-120573Model Recall that the limiting correlation of the 120573-120573 model is 1(1 + 120579) For the purpose of comparison withthe 120573-ICM model we reparametrize the 120573-120573 mixture modelas follows 120579 = (1 minus 120596)120596 such that the limiting correlationbecomes 120596

Figure 4 displays the cumulative distribution functions ofthe 120573-120573 binomial double mixture 11987820 for several different

Valu

e at r

isk

0

200

400

600

800

1000

00 02 04 06 08 10120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 2The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

00 02 04 06 08 10

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 3The approximatedCTEs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

Table 1The approximated VaR and CTE of 1198781000 at confidence level120572 = 095 under the 120573-ICMmodel

120596 VaR120572 CTE12057200 24751 3031401 25467 4477202 26483 5913503 28073 7333804 31100 8722305 100000 100000

values of 120596 (the limiting correlation) The plot shows that 11987820has a heavier left tail under the 120573-120573model than under the 120573-ICM model for each limiting correlation level Note that the120573-120573model does not admit the two extreme cases 120596 = 0 and120596 = 1

Figures 5 and 6 show the approximate risk measuresVaR and CTE for a scale of confidence levels 120572 isin

08 085 090 095 099 and by the level of the limiting

Journal of Probability and Statistics 9

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 01120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 09

Figure 4The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-120573model

Valu

e at r

isk

02 04 06 08

0

200

400

600

800

1000

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 5The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-120573model

correlation 120596 The approximate risk measures are smallerthan those of the 120573-ICMmodel for each confidence level andlimiting correlation compare for example Figures 2 and 3This indicates that the right tail of the 120573-120573 model is thinnerthan that of the 120573-ICMmodel for the same value of 120596

5 Models Fit to Real Data

In addition to the previous examples with prescribed param-eter values we illustrate the two binomial double mixturemodels with real data Bloomberg mortgage delinquencyrate index US residential mortgage loans are segmentedinto three buckets Prime Alt-A and Subprime based upon

02 04 06 08

0

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 6The approximatedCTEs of 1198781000 by confidence level120572 andthe limiting correlation 120596 under the 120573-120573model

Table 2Themethod ofmoments estimates of the shape parameters119886 and 119887 of Beta mixing distribution and the limiting correlation 120596

Model Parameter119886 119887 120596

120573-ICM 4038 14563 0009120573-I 3381 12191

the loan type Several delinquency rates for each bucketare calculated as the percentage of loans that are reportedto be delinquent beyond a certain number of days (3060 or 90 days) or are classified as real estate owned(REO) or foreclosure In this section we only use theAlt-A 60+ days (60+90+REO+foreclosure) delinquency rate(Ticker BBMDA60P) for the purpose of illustration Thesame approach can be taken for other delinquency indexessuch as the Prime (Tickers BBMDP60P BBMDPDLQ) orSubprime (Tickers BBMDS60P BBMDSDLQ) segmentsFigure 7 is taken from the Bloomberg interactive chart andit plots the monthly history of the delinquency rate duringMay 2007ndashMarch 2012

51 120573-ICMModel Note that none of the observations has thevalue 0 or 1 In this case the maximum likelihood estimateof the weight parameter 120596 is 0 because for 0 lt 119901 lt 1log119891119871

infin

(119901) = log(1 minus 120596) + log119891119875(119901) where 119891119875(119901) is thedensity of the Beta distribution (34) To avoid this issue weestimate the three parameters in the 120573-ICMmodel by meansof the method of moments (MMEs) which uses the first threemoments (29) and (37) of the limiting distribution of theempirical frequency

For comparison we also estimate the two shape parame-ters of the Beta distributionmdashthe limiting distribution of theclassical Beta-binomial (120573-I) modelmdashby matching the firstand second sample moments with the theoretical ones using(28) and (37) Table 2 gives the parameter estimates

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

8 Journal of Probability and Statistics

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 00120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 10

Figure 1The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-ICMmodel

4 Numerical Examples

As an illustrative example we set 119886 = 15 and 119887 = 135

for the Beta mixing distribution for both the 120573-ICM and120573-120573 models (These prescribed parameter values are chosenfrom the authorsrsquo previous experience in portfolio credit riskmodeling) This gives 119864[119875] = 119886(119886 + 119887) = 01 and Var[119875] =119886119887[(119886 + 119887)

2(119886 + 119887 + 1)] = 0005625

41 120573-ICMModel In Figure 1 we plot the cumulative distri-bution functions of the 120573-ICM binomial double mixture 11987820for several different values of120596 (the limiting correlation)Theplot shows that both the left and right tails get thicker as thelevel of limiting correlation 120596 increases to 1

In Figures 2 and 3 we display the approximate riskmeasures VaR and CTE for a scale of confidence levels120572 isin 08 085 090 095 099 and by the level of the limitingcorrelation120596The results show that both VaR120572 and CTE120572 areincreasing in the level of limiting correlation for 120572 ge 090This is not the case for VaR120572 when the confidence level is lessthan 1 minus 119864[119875] = 119887(119886 + 119887) because

120572lowast=

1

1 minus 120596120572 minus 120596(

119887

119886 + 119887)

=1

1 minus 120596120572 minus

119887

119886 + 119887 +

119887

119886 + 119887

(48)

is decreasing in 120596 when 120572 lt 119887(119886 + 119887)More specifically Table 1 compares the approximatedVaR

and CTE at the level 120572 = 095

42 120573-120573Model Recall that the limiting correlation of the 120573-120573 model is 1(1 + 120579) For the purpose of comparison withthe 120573-ICM model we reparametrize the 120573-120573 mixture modelas follows 120579 = (1 minus 120596)120596 such that the limiting correlationbecomes 120596

Figure 4 displays the cumulative distribution functions ofthe 120573-120573 binomial double mixture 11987820 for several different

Valu

e at r

isk

0

200

400

600

800

1000

00 02 04 06 08 10120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 2The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

00 02 04 06 08 10

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 3The approximatedCTEs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-ICMmodel

Table 1The approximated VaR and CTE of 1198781000 at confidence level120572 = 095 under the 120573-ICMmodel

120596 VaR120572 CTE12057200 24751 3031401 25467 4477202 26483 5913503 28073 7333804 31100 8722305 100000 100000

values of 120596 (the limiting correlation) The plot shows that 11987820has a heavier left tail under the 120573-120573model than under the 120573-ICM model for each limiting correlation level Note that the120573-120573model does not admit the two extreme cases 120596 = 0 and120596 = 1

Figures 5 and 6 show the approximate risk measuresVaR and CTE for a scale of confidence levels 120572 isin

08 085 090 095 099 and by the level of the limiting

Journal of Probability and Statistics 9

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 01120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 09

Figure 4The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-120573model

Valu

e at r

isk

02 04 06 08

0

200

400

600

800

1000

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 5The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-120573model

correlation 120596 The approximate risk measures are smallerthan those of the 120573-ICMmodel for each confidence level andlimiting correlation compare for example Figures 2 and 3This indicates that the right tail of the 120573-120573 model is thinnerthan that of the 120573-ICMmodel for the same value of 120596

5 Models Fit to Real Data

In addition to the previous examples with prescribed param-eter values we illustrate the two binomial double mixturemodels with real data Bloomberg mortgage delinquencyrate index US residential mortgage loans are segmentedinto three buckets Prime Alt-A and Subprime based upon

02 04 06 08

0

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 6The approximatedCTEs of 1198781000 by confidence level120572 andthe limiting correlation 120596 under the 120573-120573model

Table 2Themethod ofmoments estimates of the shape parameters119886 and 119887 of Beta mixing distribution and the limiting correlation 120596

Model Parameter119886 119887 120596

120573-ICM 4038 14563 0009120573-I 3381 12191

the loan type Several delinquency rates for each bucketare calculated as the percentage of loans that are reportedto be delinquent beyond a certain number of days (3060 or 90 days) or are classified as real estate owned(REO) or foreclosure In this section we only use theAlt-A 60+ days (60+90+REO+foreclosure) delinquency rate(Ticker BBMDA60P) for the purpose of illustration Thesame approach can be taken for other delinquency indexessuch as the Prime (Tickers BBMDP60P BBMDPDLQ) orSubprime (Tickers BBMDS60P BBMDSDLQ) segmentsFigure 7 is taken from the Bloomberg interactive chart andit plots the monthly history of the delinquency rate duringMay 2007ndashMarch 2012

51 120573-ICMModel Note that none of the observations has thevalue 0 or 1 In this case the maximum likelihood estimateof the weight parameter 120596 is 0 because for 0 lt 119901 lt 1log119891119871

infin

(119901) = log(1 minus 120596) + log119891119875(119901) where 119891119875(119901) is thedensity of the Beta distribution (34) To avoid this issue weestimate the three parameters in the 120573-ICMmodel by meansof the method of moments (MMEs) which uses the first threemoments (29) and (37) of the limiting distribution of theempirical frequency

For comparison we also estimate the two shape parame-ters of the Beta distributionmdashthe limiting distribution of theclassical Beta-binomial (120573-I) modelmdashby matching the firstand second sample moments with the theoretical ones using(28) and (37) Table 2 gives the parameter estimates

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

Journal of Probability and Statistics 9

0 10 15 20

00

02

04

06

08

10

k

F(k)

5

120596 = 01120596 = 02120596 = 04

120596 = 06120596 = 08120596 = 09

Figure 4The cumulative distribution functions of 11987820 for a scale oflimiting correlations 120596 under the 120573-120573model

Valu

e at r

isk

02 04 06 08

0

200

400

600

800

1000

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 5The approximated VaRs of 1198781000 by confidence level 120572 andthe limiting correlation 120596 under the 120573-120573model

correlation 120596 The approximate risk measures are smallerthan those of the 120573-ICMmodel for each confidence level andlimiting correlation compare for example Figures 2 and 3This indicates that the right tail of the 120573-120573 model is thinnerthan that of the 120573-ICMmodel for the same value of 120596

5 Models Fit to Real Data

In addition to the previous examples with prescribed param-eter values we illustrate the two binomial double mixturemodels with real data Bloomberg mortgage delinquencyrate index US residential mortgage loans are segmentedinto three buckets Prime Alt-A and Subprime based upon

02 04 06 08

0

200

400

600

800

1000

Con

ditio

nal t

ail e

xpec

tatio

n

120596

120572 = 080120572 = 085120572 = 090

120572 = 095120572 = 099

Figure 6The approximatedCTEs of 1198781000 by confidence level120572 andthe limiting correlation 120596 under the 120573-120573model

Table 2Themethod ofmoments estimates of the shape parameters119886 and 119887 of Beta mixing distribution and the limiting correlation 120596

Model Parameter119886 119887 120596

120573-ICM 4038 14563 0009120573-I 3381 12191

the loan type Several delinquency rates for each bucketare calculated as the percentage of loans that are reportedto be delinquent beyond a certain number of days (3060 or 90 days) or are classified as real estate owned(REO) or foreclosure In this section we only use theAlt-A 60+ days (60+90+REO+foreclosure) delinquency rate(Ticker BBMDA60P) for the purpose of illustration Thesame approach can be taken for other delinquency indexessuch as the Prime (Tickers BBMDP60P BBMDPDLQ) orSubprime (Tickers BBMDS60P BBMDSDLQ) segmentsFigure 7 is taken from the Bloomberg interactive chart andit plots the monthly history of the delinquency rate duringMay 2007ndashMarch 2012

51 120573-ICMModel Note that none of the observations has thevalue 0 or 1 In this case the maximum likelihood estimateof the weight parameter 120596 is 0 because for 0 lt 119901 lt 1log119891119871

infin

(119901) = log(1 minus 120596) + log119891119875(119901) where 119891119875(119901) is thedensity of the Beta distribution (34) To avoid this issue weestimate the three parameters in the 120573-ICMmodel by meansof the method of moments (MMEs) which uses the first threemoments (29) and (37) of the limiting distribution of theempirical frequency

For comparison we also estimate the two shape parame-ters of the Beta distributionmdashthe limiting distribution of theclassical Beta-binomial (120573-I) modelmdashby matching the firstand second sample moments with the theoretical ones using(28) and (37) Table 2 gives the parameter estimates

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

10 Journal of Probability and Statistics

Sep 2008 May Sep 2009 May Sep 2010 May Sep 2011 May Sep 2012

40

30

20

10

40

30

20

10

[] a comparison Add BBMDA60PIN Open 000 High 000 Low 000 Close 3076

Overlay Indicators Annotations Settings 1D 1W 1M 3M 6M 1Y 3Y 5 Y YTD

Print

Figure 7 Bloomberg (60+ days) delinquency rate for Alt-A mortgage segment (May 2007ndashMarch 2012) Source Bloomberg Finance LPused with permission of Bloomberg LP Copyright 2012 All rights reserved

The table shows that the MMEs for the two shapeparameters of the Beta mixing distribution of the generalmodel (120596 = 0) are larger than those of the reduced model(120596 = 0) As a result the right tail of the fitted Beta distributionin the general model is slightly thinner than that of the fittedBeta distribution in the reduced model This is explained asfollows in the general model the tail thickness in the datais explained by the estimates of both the shape parametersand 120596 On the other hand the reduced model is less flexibleand the two shape parameter estimates attempt to capture theobserved tail

In order to identify the implications of the two fittedmodels (with and without 120596) for the risk measures we plotapproximate VaRs and CTEs of 119878119899 with 119899 = 10

6 at variousconfidence levels (The chosen number of entities is purelynominal since we are using the large-sample results Accord-ing to the National Delinquency Survey issued by MortgageBankers Association the average number of conventionalsubprime mortgage loans during 2011 is over four million)

Figure 8 shows that the risk measures calculated underthe conditional independence (120573-I) model are larger thanthose of the more general model the 120573-ICM model (17)at confidence levels up to 0996 and 0945 for VaR andCTE respectively Note that the estimated 120596 is small to theextent that the effect is not well recognized in VaR until theconfidence level takes an extremely high value The condi-tional tail expectation however of the more general modelbecomes larger than that under the conditional independenceassumption at confidence levels which are typically used inthe calculation of economic capital for a credit portfolioThe nonzero 120596 allows further dependence between entities

and thus this result suggests that the use of the conditionalindependence model may result in the underestimation ofextreme tail risk

52 120573-120573 Model Here we illustrate the 120573-120573 model ofExample 12 Note that the large-sample distributions (of theempirical frequency) are continuous for both the 120573-120573 and 120573-I models and none of the observations are either 0 or 1Thusthe maximum likelihood estimation (MLE) method can beapplied by using the limiting distribution of the empiricalfrequency (47) However the number of observations 57data points may not be sufficient to yield statistically reliableMLE fitting results For the sake of statistical precision andmodel comparison we reduce the number of parameters inboth the 120573-120573 model and the 120573-I model which correspondsto the case of conditional independence Specifically we usethe following theoretical relationship between the first twomoments of the Beta mixing distribution and the two shapeparameters (119886 119887)

119887 = (1205902

1205833)119886 (119886 + 120583) (49)

where 120583 and 1205902 denote the mean and variance of the Betamixing distribution whichwe estimate with the samplemeanand variance 119909 = 02171 and 1199042 = 00103 respectively of thedelinquency rate 119909 Thus we use the following form for 119887 inthe Beta density

119887 = (00103

021713) 119886 (119886 + 02171) (50)

to obtain a single-parameter family of Beta distributions

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

Journal of Probability and Statistics 11

090 092 094 096 098 100

300

400

500

600

700

800

900

1000

VaR

(uni

t 10

00)

Confidence level

120573-1120573-ICM

(a)

300

400

500

600

700

800

900

1000

090 092 094 096 098 100

Con

ditio

nal t

ail e

xpec

tatio

n (u

nit

1000

)

Confidence level

120573-1120573-ICM

(b)

Figure 8 The approximated VaR120572 and CTE120572 of 119878119899 119899 = 106 the number of delinquencies in a month at confidence levels 09 le 120572 le 0999

based on 120573-ICM and 120573-I models

Table 3 The maximum likelihood estimates of the model parame-ters under the 120573-120573model and the 120573-I model

Model 119886 119887 120596 Log-likelihood120573-120573 3481 12900 00231 48248120573-I 3161 10703 45607

Table 3 gives the maximum likelihood estimates for theparameter 119886 of the Beta mixing distribution the limitingcorrelation parameter120596 = 1(1+120579) and the values of the log-likelihood functions under the 120573-120573model and the 120573-I modelrespectively

Figure 9 displays the differences in estimates of each of thetwo risk measures VaR and CTE under the 120573-120573 model andthe120573-Imodel For illustration the approximate riskmeasuresare calculated assuming that there are one million entities inthe mortgage portfolio

The result shows that both the VaRs and the CTEs underthe 120573-120573 model are larger than those under the 120573-I modelespecially at high confidence levels and the differencesincrease rapidly as the confidence level tends to 1

This result demonstrates the role of the limiting correla-tion parameter 120596 in explaining the tail risk resulting from aninterdependency among names within a credit portfolio

6 Conclusion

In this paper we study the conditional correlation andthe distribution of the sum of Bernoulli mixtures under ageneral dependence structure We show in particular under

090 092 094 096 098 100

minus10000

minus5000

0

5000

10000

15000

20000

Confidence level

Diff

eren

ce in

risk

mea

sure

s

CTE(120573-120573) minus CTE(120573-I)VaR(120573-120573) minus VaR(120573-I)

Figure 9 The differences between approximated risk measuresunder the 120573-120573model and the 120573-I model of 119878119899 119899 = 10

6 the numberof delinquencies in a month for the Alt-A residential mortgagesegment with (nominally) one million entities at confidence levels09 le 120572 le 0995

the typical conditional independence assumption that theconditional correlation between two Bernoulli mixtures con-verges to 0 given that the mixing random variable is largerthan a threshold tending to 1 We propose a method toconstruct a general dependence structure in the form of

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

12 Journal of Probability and Statistics

a doublemixturemodel inwhich the conditional iid assump-tion is replaced by the more general assumption of condi-tional exchangeability As a simple illustration we consider aweighted average of two cases conditional independence andcomonotonicity forwhich the limiting correlation is includedas a model parameter

The large-sample distribution of the empirical frequencyand its use in approximating the risk measures VaR andCTE are presented Several tractable results for two Bernoullidouble mixture models with Beta mixing distribution the120573-ICM model and the 120573-120573 model are given as illustrativenumerical examples and also applied to real data

Note that there is a strong demand for a credit riskmodel with an appropriate level of dependence in a stressedeconomic environment The most popular model the Beta-binomial mixture however cannot properly explain theempirically observed default correlation under stress On theother hand the model framework presented in this paper issimple but flexible enough to accommodate the required levelof limiting correlation and thus can be effectively applied toportfolio credit risk models

Future directions of research include the application ofdouble mixture models to pricing of credit derivatives suchas CDOs on a completely homogeneous pool (eg Bae et al[24])

Appendices

A Proof of Theorem 1

Let

119865119875 (119901) = 1 minus 119865119875 (119901) = int1

119901

119891119875 (119904) 119889119904 (A1)

be the survival probability function Then from (2)

Cov [119883119894 119883119895 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

1198862 (119904) 119891119875 (119904) 119889119904

minus (1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904)

2

Var [119883119894 | 119875 gt 119901] =1

119865119875 (119901)int1

119901

119904119891119875 (119904) 119889119904

minus (1

119865119875(119901)int1

119901

119904119891119875(119904)119889119904)

2

(A2)

Thus the conditional correlation given 119875 gt 119901 0 lt 119901 lt 1between119883119894 and119883119895 is

120588119894119895 (119901)

=(1119865119875 (119901)) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

(1119865119875 (119901)) int1

119901119904119891119875 (119904) 119889119904 minus ((1119865119875(119901)) int

1

119901119904119891119875(119904)119889119904)

2

=119865119875 (119901) int

1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

(A3)

Denote 119892(119901) = (1 minus 119901)119891119875(119901)119865119875(119901) and 119903(119901) =

int1

119901119904119891119875(119904)119889119904119865119875(119901) By assumption lim119901rarr1119892(119901) = 119871 lt infin

Note that by LrsquoHospitalrsquos rule

lim119901rarr1

119903 (119901) = lim119901rarr1

int1

119901119904119891119875 (119904) 119889119904

119865119875 (119901)= lim119901rarr1

minus119901119891119875 (119901)

minus119891119875 (119901)= 1

lim119901rarr1

1198862 (119901) minus 1199012

1 minus 119901= 2 minus lim

119901rarr111988610158402 (119901) equiv 2 minus 119886

10158402 (1minus)

(A4)

Finally to identify the limit of 120588119894119895(119901) as 119901 rarr 1 we applyLrsquoHospitalrsquos rule two more times

120588119894119895

= lim119901rarr1

120588119894119895 (119901)

= lim119901rarr1

119865119875 (119901) int1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875 (119904) 119889119904)

2

119865119875 (119901) int1

119901119904119891119875 (119904) 119889119904 minus (int

1

119901119904119891119875(119904)119889119904)

2

= lim119901rarr1

minusint1

1199011198862 (119904) 119891119875 (119904) 119889119904 minus 1198862 (119901) 119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

minus int1

119901119904119891119875 (119904) 119889119904 minus 119901119865119875 (119901) + 2119901 int

1

119901119904119891119875 (119904) 119889119904

= lim119901rarr1

((2 (1198862 (119901) minus 1199012) 119891119875 (119901) minus 119886

10158402 (119901) 119865119875 (119901)

+ 2int1

119901

119904119891119875 (119904) 119889119904)

times (2(119901 minus 1199012)119891119875(119901) minus 119865119875(119901) + 2int

1

119901

119904119891119875(119904)119889119904)

minus1

)

equiv lim119901rarr1

2 ((1198862 (119901) minus 1199012) (1 minus 119901)) 119892 (119901) minus 11988610158402 (119901) + 2119903 (119901)

2119901119892 (119901) minus 1 + 2119903 (119901)

=2 (2 minus 11988610158402 (1

minus)) 119871 minus 11988610158402 (1minus) + 2

2119871 minus 1 + 2

=2 (2119871 + 1) minus (2119871 + 1) 119886

10158402 (1minus)

2119871 + 1= 2 minus 119886

10158402 (1minus)

(A5)

Remark If it were possible that 119871 = infin then the previousproof is easily modified to cover that case In the third-to-lastline of the final derivation one simply divides the numerator

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

Journal of Probability and Statistics 13

and denominator by 119892(119901) before taking the limit which thenbecomes

lim119901rarr1 ((1198862 (119901) minus 1199012) (1 minus 119901)) minus 0

lim119901rarr1 (2119901) minus 0= 2 minus lim

119901rarr111988610158402 (119901)

(A6)

B Proof of Theorem 4

As remarked after the statement of the theorem the proof isbased on parts of the proof of de Finettirsquos theorem as givenin Feller [20] but for the case of a finite sequence As it iswell known that de Finettirsquos theorem does not hold in generalfor finite sequences (see eg Example 4 at the end of Section4 of Feller [20]) we will take care to elucidate exactly theingredients of the proof which are valid in our case

To begin we recall some identities for numericalsequences of finite length For any numerical sequence 119886 =(119886119895 0 le 119895 le 119899) denote the difference operator by Δ

(Δ119886)119895 = 119886119895+1 minus 119886119895 0 le 119895 le 119899 minus 1 (B1)

Then the 119898th-order difference operator denoted by Δ119898 isdefined recursively by

Δ119898+1

= Δ ∘ Δ119898 Δ

1equiv Δ (B2)

It is easily shown by induction using the elementary identity( 119898119894minus1 ) + (

119898119894 ) = (

119898+1119894 ) that for119898 le 119899 and 119895 le 119899 minus 119898

(Δ119898119886)119895=

119898

sum119894=0

(minus1)119898minus119894(119898

119894) 119886119895+119894 (B3)

(This is equation (17) in Chapter VII of Feller [20]) Next wecite the identity (19) from Chapter VII of Feller [20]

1198860 =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (B4)

Finally if for some probabilitymeasure ] 119886119895 = int1

0119903119895](119889119903)

for all 119895 then it is easily established by induction that

(Δ119898119886)119895= (minus1)

119898int1

0

119903119895(1 minus 119903)

119898] (119889119903) (B5)

(This is equation (33) in Chapter VII of Feller [20]) Putting(B3) and (B5) together yields

119898

sum119894=0

(minus1)119894(119898

119894) 119886119896+119894 = int

1

0

119903119895(1 minus 119903)

119898] (119889119903) ge 0 (B6)

We now proceed with the proof of Theorem 4 startingwith the verification of (11) and (12) Temporarily overridingthe definition (12) denote

119892119899 (119896 119901) = Pr [(1198831 119883119896 119883119896+1 119883119899)

= (1 1 0 0) | 119875 = 119901] (B7)

Due to the exchangeability assumption the conditional jointprobability that the randomvector (1198831 119883119896 119883119896+1 119883119899)takes on any permutation of the vector (1 1 0 0) is119892119899(119896 119901) Therefore

119891119899 (119896 119901) = Pr [119878119899 = 119896 | 119875 = 119901]

= (119899

119896)119892119899 (119896 119901) 119896 = 0 119899

(B8)

and thus for (11) and (12) we must show that

119892119899 (119896 119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901) (B9)

As shown in (46) in Feller [20] (identifying our 119892119899(119896 119901) and119886119896(119901) with his 119901119896119899 and 119888119896 resp)

119892119899 (119896 119901) = (minus1)119899minus119896Δ119899minus119896119886119896 (119901) =

119899minus119896

sum119894=0

(119899 minus 119896

119894) (minus1)

119894119886119896+119894 (119901)

(B10)

where for the second equality we have used the result (B3)with119898 = 119899 minus 119896

For the converse part of the theorem we first show thatsum119899119896=0 119891119899(119896 119901) = 1 regardless of how the sequence (119886119896(119901) 0 le119896 le 119899) is chosen as long as 1198860(119901) = 1 By (B3) we recognizethat

119892119899 (119894 119901) = (minus1)119899minus119894Δ119899minus119894119886119894 (119901) 1 le 119894 le 119899 (B11)

Therefore by the identity (B4) with 119886119894 equiv 119886119894(119901)

1 = 1198860 (119901) =

119899

sum119894=0

(119899

119894) (minus1)

119899minus119894Δ119899minus119894119886119894 (119901)

=

119899

sum119894=0

(119899

119894) 119892119899 (119894 119901) =

119899

sum119894=0

119891119899 (119894 119901)

(B12)

Therefore to show that 119891119899 is a valid pmf we mustshow that each term 119891119899(119896 119901) or equivalently 119892119899(119896 119901) isnonnegative To achieve that we simply apply (B6) with119898 = 119899 minus 119896 to 119886119896 equiv 119886119896(119901) in case 119886119896(119901) = int

1

0119903119896](119889119903)

119896 = 0 1 as in the hypothesis of the converse part of thetheorem As a byproduct we obtain the conditional binomialrepresentation

119892119899 (119896 119901) = int1

0

119903119896(1 minus 119903)

119899minus119896] (119889119903 119901) (B13)

Finally with the conditional pmf as described at (13)(119883119894)119899119894=1 is clearly exchangeable The proof of the converse will

be complete as soon as it is shown that Pr[(1198831 119883119896) =(1 1) | 119875 = 119901] = 119886119896(119901) for all 119896 = 1 119899 For 119896 = 119899 itfollows by definition that 119892119899(119899 119901) = 119886119899(119901) For fixed 119896 lt 119899denote

X119895 = 119909 isin 0 1119899minus119896

119899minus119896

sum119894=1

119909119894 = 119895 0 le 119895 le 119899 minus 119896

(B14)

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

14 Journal of Probability and Statistics

that is each 119909 inX119895 has precisely 119895 components equal to 1 and119899 minus 119896 minus 119895 components equal to 0 Then

Pr [(1198831 119883119896) = (1 1) | 119875 = 119901]

=

119899minus119896

sum119895=0

sum119909isinX119895

Pr [(1198831 119883119899) = (1 1 119909) | 119875 = 119901]

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)119892119899 (119896 + 119895 119901)

=

119899minus119896

sum119895=0

(119899 minus 119896

119895)

times int1

0

119903119896+119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901) by (B13)

= int1

0

119903119896119899minus119896

sum119895=0

(119899 minus 119896

119895) 119903119895(1 minus 119903)

119899minus119896minus119895] (119889119903 119901)

= int1

0

119903119896] (119889119903 119901) = 119886119896 (119901)

(B15)

where the second line follows from the exchangeability of(119883119894)119899119894=1 the second-to-last equality follows from the binomial

expansion of 1 = [119903 + (1 minus 119903)]119899minus119896

Disclaimer

The views expressed in this paper are those of the authorsand might not represent the views of the authorsrsquo affiliatedinstitutions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Bloomberg for granting permis-sion to use the delinquency data in the examples of Section 5Taehan Bae is supported by the Discovery Grant Program ofthe Natural Science and Engineering Council of Canada

References

[1] C Donnelly and P Embrechts ldquoThe devil is in the tails actu-arial mathematics and the subprime mortgage crisisrdquo ASTINBulletin vol 40 no 1 pp 1ndash33 2010

[2] Basel Committee on Banking Supervision ldquoInternational con-vergence of capital measurement and capital standards arevised frameworkrdquo 2006 httpwwwbisorgpublbcbs128apdf

[3] D Li ldquoOn default correlation a copula function approachrdquoJournal of Fixed Income vol 9 pp 43ndash54 2000

[4] LAndersen and J Sidenius ldquoExtensions to theGaussian copularandom recovery and random factor loadingsrdquo Journal of CreditRisk vol 1 pp 29ndash70 2005

[5] X Burtschell J Gregory and J P Laurent ldquoBeyond the gaussiancopula stochastic and local correlationrdquo Journal of Credit Riskvol 3 pp 31ndash62 2007

[6] X Burtschell J Gregory and J-P Laurent ldquoA comparativeanalysis of CDO pricing models under the factor copulaframeworkrdquo Journal of Derivatives vol 16 no 4 pp 9ndash37 2009

[7] P J Schonbucher and D Schubert Copula Dependent DefaultRisk in Intensity Models 2001

[8] C Bluhm L Overbeck and C Wagner Introduction to CreditRisk Modeling Chapman amp HallCRC London UK 2003

[9] A Cousin and J-P Laurent ldquoComparison results for exchange-able credit risk portfoliosrdquo Insurance Mathematics amp Eco-nomics vol 42 no 3 pp 1118ndash1127 2008

[10] R Frey and A J McNeil ldquoVaR and expected shortfall inportfolios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[11] R Frey and A McNeil ldquoDependent defaults in models ofportfolio credit riskrdquo Journal of Risk vol 6 pp 59ndash92 2003

[12] KMV-Corporation ldquoModelling default riskrdquo 1997 httpwwwmoodysanalyticscom

[13] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Princeton University Press Princeton NJ USA2005

[14] A J McNeil and J P Wendin ldquoBayesian inference for gener-alized linear mixed models of portfolio credit riskrdquo Journal ofEmpirical Finance vol 14 no 2 pp 131ndash149 2007

[15] F Moraux ldquoSensitivity analysis of credit risk measures in thebeta binomial frameworkrdquoThe Journal of Fixed Income vol 19no 3 pp 66ndash76 2010

[16] RiskMetrics-Group ldquoCreditMetricsmdashTechnical Documentrdquo1997 httpwwwdefaultriskcom

[17] T Bae and I Iscoe ldquoCorrelations under stressrdquo InternationalReview of Applied Financial Issues and Economics vol 2 pp248ndash271 2010

[18] M Kalkbrener and N Packham ldquoCorrelation under stress innormal variance mixture modelsrdquo Mathematical Finance Toappear

[19] Sh Sharakhmetov and R Ibragimov ldquoA characterization ofjoint distribution of two-valued random variables and itsapplicationsrdquo Journal of Multivariate Analysis vol 83 no 2 pp389ndash408 2002

[20] W Feller An Introduction to Probability Theory and Its Appli-cations vol 2 John Wiley amp Sons New York NY USA 2ndedition 1970

[21] O Vasicek ldquoLimiting loan loss probability distributionrdquo 1991httpwwwmoodysanalyticscom

[22] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[23] C Acerbi and D Tasche ldquoExpected shortfall a natural coherentalternative to value at riskrdquo Economic Notes vol 31 no 2 pp379ndash388 2002

[24] T Bae I Iscoe and C Kim ldquoValuing retail credit tranches withstructural double mixture modelsrdquo Journal of Futures MarketsTo appear

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 15: Research Article Sum of Bernoulli Mixtures: Beyond ...downloads.hindawi.com/journals/jps/2014/838625.pdfSum of Bernoulli Mixtures: Beyond Conditional Independence TaehanBae 1 andIanIscoe

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Pascal & Bernoulli

Pascal & Bernoulli

Bernoulli theorm

Bernoulli theorm

Bernoulli Chapter 3

Bernoulli Chapter 3

Bernoulli Automatic Coarse Filters - HiFlux Filtrationhiflux-filtration.dk/media/23534/bernoulli-uk.pdf · Bernoulli Automatic Coarse Filters The Bernoulli filter is an automatic

Bernoulli Automatic Coarse Filters - HiFlux Filtrationhiflux-filtration.dk/media/23534/bernoulli-uk.pdf · Bernoulli Automatic Coarse Filters The Bernoulli filter is an automatic

Multi-label Active Learning with Conditional Bernoulli Mixtures · 2018. 8. 10. · mixtures (CBM) [11] was proposed to be a state-of-the-art multi-label learning method. It is a

Multi-label Active Learning with Conditional Bernoulli Mixtures · 2018. 8. 10. · mixtures (CBM) [11] was proposed to be a state-of-the-art multi-label learning method. It is a

Bernoulli BalanceMacroscopico

Bernoulli BalanceMacroscopico

Lift bernoulli principle

Lift bernoulli principle

Bernoulli Lab Report

Bernoulli Lab Report

Bernoulli solution

Bernoulli solution

SPECIAL CHARACTERIZATIONS OF STANDARD ...The Bernoulli and Poisson processes are described as genera-tors of such discrete models. A characterization of distributions by mixtures is

SPECIAL CHARACTERIZATIONS OF STANDARD ...The Bernoulli and Poisson processes are described as genera-tors of such discrete models. A characterization of distributions by mixtures is

Informe Bernoulli

Informe Bernoulli

Bernoulli 0704

Bernoulli 0704

Bernoulli WriteUp

Bernoulli WriteUp

The Bernoulli Brothers

The Bernoulli Brothers

Bernoulli Resolve

Bernoulli Resolve

Final Bernoulli

Final Bernoulli

Taking the “Convoluted” out of Bernoulli Convolutionsmdelcour/DelcourtEngberg.pdf · Conclusion Bernoulli convoluted Functional equation q = 2=3 Bernoulli convoluted A Bernoulli

Taking the “Convoluted” out of Bernoulli Convolutionsmdelcour/DelcourtEngberg.pdf · Conclusion Bernoulli convoluted Functional equation q = 2=3 Bernoulli convoluted A Bernoulli

Ch3 Bernoulli Equation

Ch3 Bernoulli Equation

03 Bernoulli

03 Bernoulli

Bernoulli - Oxfordstrat

Bernoulli - Oxfordstrat