Granularity Adjustmentin Dynamic Multiple Factor Models:Systematic vs Unsystematic Risks

P., GAGLIARDINI (1), and C., GOURIEROUX (2)

Preliminary and Incomplete Version: March 1, 2010

1Swiss Finance Institute and University of Lugano.2CREST, CEPREMAP and University of Toronto.

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Granularity Adjustment in Dynamic Multiple Factor Models:Systematic vs Unsystematic Risks

Abstract

The granularity principle [Gordy (2003)] allows for closed form expres-sions of the risk measures of a large portfolio at order 1/n, where n is theportfolio size. The granularity principle yields a decomposition of such riskmeasures that highlights the different effects of systematic and unsystematicrisks. This paper provides the granularity adjustment of the Value-at-Risk(VaR) for both static and dynamic risk factor models. The systematic factorcan be multidimensional. The technology is illustrated by several examples,such as the stochastic drift and volatility model, or the model for joint anal-ysis of default and loss given default.

Keywords: Value-at-Risk, Granularity, Large Portfolio, Credit Risk, Sys-tematic Risk, Loss Given Default, Basel 2 Regulation, Credibility Theory.

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1 Introduction

Risk measures such as the Value-at-Risk (VaR), the Expected Shortfall (alsocalled Tail VaR) and more generally the Distortion Risk Measures are thebasis of the new risk management policies and regulations in both Finance(Basel 2) and Insurance (Solvency 2). These measures are used to definethe minimum required capital needed to hedge risky investments; this is thePillar 1 in Basel 2. They are also used to monitor the risk by means ofinternal risk models (Pillar 2 in Basel 2).

These risk measures have in particular to be computed for large port-folios of individual contracts, which can be loans, mortgages, life insurancecontracts, or Credit Default Swaps (CDS), and for derivative assets writtenon such large portfolios, such as Mortgage Backed Securities, CollateralizedDebt Obligations, derivatives on a CDS index (such as iTraxx, or CDX),Insurance Linked Securities, or longevity bonds. The value of a portfolio riskmeasure is often difficult to derive even numerically due to

i) the large size of the support portfolio, which can include from aboutone hundred 3 to several thousands of individual contracts,

ii) the nonlinearity of individual risks, such as default, recovery, claimoccurrence, prepayment, surrender, lapse;

iii) the need to take into account the dependence between individual risks,that is induced by the systematic components of these risks.

The granularity principle has been introduced for static single factor mod-els during the discussion on the New Basel Capital Accord [BCBS (2001)],following the contributions by Gordy (2003) and Wilde (2001). The granular-ity principle allows for closed form expressions of the risk measures for largeportfolios at order 1/n, where n denotes the portfolio size. More precisely,any portfolio risk measure can be decomposed as the sum of an asymptoticrisk measure corresponding to an infinite portfolio size and (1/n) times an ad-justment term. The asymptotic portfolio risk measure, called Cross-SectionalAsymptotic (CSA) risk measure, captures the non diversifiable effect of sys-tematic risks on the portfolio. The adjustment term, called GranularityAdjustment (GA), summarizes the effect of the individual specific risks andtheir cross-effect with systematic risks, when the portfolio size is large, butfinite.

The static risk factor model is introduced in Section 2, whereas the gran-

3This corresponds to the number of names included in the iTraxx, or CDX indexes.

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ularity adjustment of the VaR is given in Section 3. Section 4 provides thegranularity adjustment for a variety of static single and multiple factor riskmodels. They include models with common stochastic drift and volatilityas well as a factor model for the joint analysis of stochastic probability ofdefault (PD) and Expected Loss Given Default (ELGD). The analysis isextended to dynamic risk factor models in Section 5. In the dynamic frame-work, two granularity adjustments are required. The first GA concerns theconditional VaR with current factor value assumed to be observed. The sec-ond GA takes into account the unobservability of the current factor value.This new decomposition relies on recent results on the granularity principleapplied to filtering problems [Gagliardini, Gourieroux (2009)]. The patternsof the granularity adjustment in the credit risk model with stochastic PDand ELGD are illustrated in Section 6. Section 7 concludes. The theoreticalderivations of the granularity adjustments are done in the Appendices.

2 Static Risk Factor Model

We first consider the static risk factor model to focus on the individual risksand their dependence structure. We omit the unnecessary time index.

2.1 Homogenous Portfolio

Let us assume that the individual risks (e.g. asset values, or default indica-tors) depend on some common factors and on individual specific effects:

yi = c(F, ui), i = 1, . . . , n, (2.1)

where yi denotes the individual risk, F the systematic factor and ui theidiosyncratic term. Both F and ui can be multidimensional, whereas yi isone-dimensional. Variables F and ui satisfy the following assumptions:

Distributional Assumptions: For any portfolio size n:A.1: F and (u1, . . . , un) are independent.

A.2: u1, . . . , un are independent and identically distributed.

The portfolio of individual risks is homogenous, since the joint distribu-tion of (y1, . . . , yn) is invariant by permutation of the n individuals, for anyn. This exchangeability property of the individual risks is equivalent to the

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fact that variables y1, . . . , yn are independent, identically distributed condi-tional on some factor F [de Finetti (1931), Hewitt, Savage (1955)]. Whenthe unobservable systematic factor F is integrated out, the individual riskscan become dependent.

2.2 Examples

The function c in equation (2.1) is given and generally nonlinear. We describebelow simple examples of static Risk Factor Model (RFM) (see Section 4 forfurther examples).

Example 2.1: Linear Single-Factor Model

We have:yi = F + ui,

where the specific error terms ui are Gaussian N(0, σ2) and the factor F isGaussian N(µ, η2). Since Corr (yi, yj) = η2/(η2 +σ2), for i 6= j, the commonfactor creates the (positive) dependence between individual risks, wheneverη2 6= 0. This model has been used rather early in the literature on individualrisks. For instance, it is the Buhlmann model considered in actuarial scienceand is the basis for credibility theory [Buhlmann (1967), Buhlmann, Straub(1970)].

Example 2.2: The Single Risk Factor Model for Default

The individual risk is the default indicator, that is yi = 1, if there is adefault of individual i, and yi = 0, otherwise. This risk variable is given by:

yi =

1, if F + ui < 0,0, otherwise,

where F ∼ N(µ, η2) and ui ∼ N(0, σ2). The quantity F + ui is often inter-preted as a log asset/liability ratio, when i is a company [see e.g. Merton(1974), Vasicek (1991)]. Thus, the company defaults when the asset valuebecomes smaller than the amount of debt.

The basic specifications in Examples 2.1 and 2.2 can be extended byintroducing additional individual heterogeneity, or multiple factors.

Example 2.3: Model with Stochastic Drift and Volatility

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The individual risks are such that :

yi = F1 + (exp F2)1/2ui,

where F1 (resp. F2) is a common stochastic drift factor (resp. stochasticvolatility factor). When yi is an asset return, we expect factors F1 and F2

to be dependent, since the (conditional) expected return generally containsa risk premium.

Example 2.4: Linear Single Risk Factor Model with Beta Hetero-geneity

This is a linear factor model, in which the individual risks may havedifferent sensitivities (called betas) to the systematic factor. The model is:

yi = βiF + vi,

where ui = (βi, vi)′ is bidimensional. In particular, the betas are assumed

unobservable and are included among the idiosyncratic risks. This type ofmodel is the basis of Arbitrage Pricing Theory (APT) [see e.g. Ross (1976),and Chamberlain, Rothschild (1983), in which similar assumptions are intro-duced on the beta coefficients].

3 Granularity Adjustment for Portfolio Risk

Measures

3.1 Portfolio Risk

Let us consider an homogenous portfolio including n individual risks. Thetotal portfolio risk is:

Wn =n∑

i=1

yi =n∑

i=1

c(F, ui). (3.1)

The total portfolio risk can correspond to a profit and loss (P&L), for instancewhen yi is an asset return and Wn/n the equally weighted portfolio return.In other cases, it corresponds to a loss and profit (L&P), for instance when yi

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is a default indicator and Wn/n the portfolio default frequency 4. As usual,we pass from a P&L to a L&P by a change of sign 5. The quantile of Wn ata given risk level is used to define a VaR (resp. the opposite of a VaR), ifWn is a L&P (resp. a P&L).

The distribution of Wn is generally unknown in closed form due to therisks dependence and the aggregation step. The density of Wn involves in-tegrals with a large dimension, which can reach dim(F ) + n dim(u) − 1.Therefore, the quantiles of the distribution of Wn, can also be difficult tocompute 6. To address this issue we consider a large portfolio perspective.

3.2 Asymptotic Portfolio Risk

The standard limit theorems such as the Law of Large Numbers (LLN) andthe Central Limit Theorem (CLT) cannot be applied directly to the sequencey1, . . . , yn due to the common factors. However, LLN and CLT can be appliedconditionally on the factor values, under Assumptions A.1 and A.2. This isthe condition of infinitely fine grained portfolio 7 in the terminology of Basel2 [BCBS (2001)].

Let us denote by

m(F ) = E[yi|F ] = E[c(F, ui)|F ], (3.2)

the conditional individual expected risk, and

σ2(F ) = V [yi|F ] = V [c(F, ui)|F ], (3.3)

4The results of the paper are easily extended to obligors with different exposures Ai, say.

In this case we have Wn =n∑

i=1

Aiyi =n∑

i=1

Aic(F, ui) =n∑

i=1

c∗(F, u∗i ), where the idiosyn-

cratic risks u∗i = (ui, Ai) contain the individual shocks ui and the individual exposures Ai

[see e.g. Emmer, Tasche (2005) in a particular case].5For instance, this means that asset returns will be replaced by opposite asset returns,

that are returns for investors with short positions.6The VaR can often be approximated by simulations, but these simulations can be

very time consuming, if the portfolio size is large and the risk level of the VaR is small,especially in dynamic factor models.

7Loosely speaking, “... the portfolio is infinitely fine grained, when the largest individ-ual exposure accounts for an infinitely small share of the total portfolio exposure” [Ebert,Lutkebohmert (2009)]. This condition is satisfied under Assumptions A.1 and A.2.

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the conditional individual volatility. By applying the CLT conditional on F ,we get:

Wn/n = m(F ) + σ(F )X√n

+ O(1/n), (3.4)

where X is a standard Gaussian variable independent of factor F . The termat order O(1/n) is zero mean, conditional on F , since Wn/n is a conditionallyunbiased estimator of m(F ).

Expansion (3.4) differs from the expansion associated with the standardCLT. Whereas the first term of the expansion is constant, equal to the un-conditional mean in the standard CLT, it is stochastic in expansion (3.4) andlinked with the second term of the expansion by means of factor F . More-over, each term in the expansion depends on the factor value, but also onthe distribution of idiosyncratic risk by means of functions m(.) and σ(.). Byconsidering expansion (3.4), the initial model with dim(F ) + n dim(u) − 1dimensions of uncertainty is replaced by a 3-dimensional model, with uncer-tainty summarized by m(F ), σ(F ) and X.

3.3 Granularity principle

The granularity principle has been introduced for static single risk factormodels by Gordy in 2000 for application in Basel 2 [Gordy (2003)]. We extendbelow this principle to multiple factor models. The granularity principlerequires several steps, which are presented below for a loss and profit variable.

i) A standardized risk measure

Instead of the VaR of the portfolio risk, which explodes with the portfoliosize, it is preferable to consider the VaR per individual risk (asset) included inthis portfolio. Since by Assumptions A.1-A.2 the individuals are exchange-able and a quantile function is homothetic, the VaR per individual risk issimply a quantile of Wn/n. Intuitively, it is defined to hedge jointly the id-iosyncratic risk and a fair part of the systematic risk. The VaR at risk levelα∗ = 1− α is denoted by V aRn(α) and is defined by the condition:

P [Wn/n < V aRn(α)] = α, (3.5)

where α is a positive number close to 1, typically α = 95%, 99%, whichcorrespond to probabilities of large losses equal to α∗ = 5%, 1%, respectively.

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ii) The CSA risk measure

Vasicek [Vasicek (1991)] proposed to first consider the limiting case of aportfolio with infinite size. Since

limn→∞

Wn/n = m(F ), a.s., (3.6)

the infinite size portfolio is not riskfree. Indeed, the systematic risk isundiversifiable. We deduce that the CSA risk measure:

V aR∞(α) = limn→∞

V aRn(α), (3.7)

is the α-quantile associated with the systematic component of the portfoliorisk:

P [m(F ) < V aR∞(α)] = α. (3.8)

The CSA risk measure is suggested in the Internal Ratings Based (IRB)approach of Basel 2 for minimum capital requirements. As seen below, thisapproach neglects the effect of unsystematic risks.

iii) Granularity Adjustment for the risk measure

The main result in granularity theory applied to risk measures providesthe next term in the asymptotic expansion of V aRn(α) with respect to n, forlarge n. It is given below for multiple factor model.

Proposition 1: In a static RFM, we have:

V aRn(α) = V aR∞(α) +1

nGA(α) + o(1/n),

where:

GA(α) = −1

2

d log g∞

dm[V aR∞(α)]E[σ2(F )|m(F ) = V aR∞(α)]

+d

dmE[σ2(F )|m(F ) = V aR∞(α)]

= −1

2E

[σ2(F )|m(F ) = V aR∞(α)

] d log g∞

dm[V aR∞(α)]

+d

dmlog E

[σ2(F )|m(F ) = V aR∞(α)

],

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and g∞ [resp. V aR∞(.)] denotes the probability density function [resp. thequantile function] of the random variable m(F ).

Proof: See Appendix 1.

This asymptotic expansion of the VaR in Proposition 1 is important forat least four reasons.

(*) The computation of quantitites V aR∞(α) and GA(α) does not requirethe evaluation of large dimensional integrals. Indeed, V aR∞(α) and GA(α)involve the distribution of transformations m(F ) and σ2(F ) of the systematicfactor only, which are independent of the porfolio size n.

(**) The second term in the expansion is of order 1/n, and not 1/√

n asmight have been expected from the Central Limit Theorem. This implies

that the approximation V aR∞(α) +1

nGA(α) is likely rather accurate, even

for rather small values of n such as n = 100.

(***) The expansion is valid for single as well as multiple factor models.

(****) The expansion is easily extended to the other Distortion Risk Mea-sures, which are weighted averages of VaR:

DRMn(H) =

∫V aRn(u)dH(u), say,

where H denotes the distortion measure [Wang (2000)]. The granularityadjustment for the DRM is simply:

1

n

∫GA(u)dH(u).

In particular, the Expected Shortfall corresponds to the distortion measurewith cumulative distribution function H(u; α) = (u − α)+/(1 − α), and thegranularity adjustment is

1

n(1− α)

∫ 1

α

GA(u)du,

that is an average of the granularity adjustments for VaR above level α.

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The GA in Proposition 1 depends on the tail magnitude of the systematic

risk component m(F ) by means ofd log g∞

dm[V aR∞(α)], which is expected to

be negative. The GA depends also ond

dmlog E

[σ2(F )|m(F ) = V aR∞(α)

],

which is a measure of the reaction of the individual volatility to shocks onthe individual drift. When yi,t is the opposite of an asset return, this reactionfunction is expected to be nonlinear and increasing for positive values of m,according to the leverage effect interpretation [Black (1976)].

When the tail effect is larger than the leverage effect, the GA is positive,which implies an increase of the required capital compared to the CSA riskmeasure. In the special case of independent stochastic drift and volatility, the

GA reduces to GA(α) = −1

2E

[σ2(F )|m(F ) = V aR∞(α)

] d log g∞dm

[V aR∞(α)],

which is generally positive. The adjustment involves both the tail of thesystematic risk component and the expected conditional variability of theindividual risks.

4 Examples

The aim of this section is to provide the closed form expressions of the GAfor the main models encountered in applications to Finance and Insurance.We first consider single factor models, in which the GA formula is greatlysimplified, then models with multiple factor.

4.1 Single Risk Factor Model

In a single factor model, the factor can generally be identified with the ex-pected individual risk:

m(F ) = F. (4.1)

Then, V aR∞(α) is the α-quantile of factor F and g∞ is its density function.The granularity adjustment of the VaR becomes:

GA(α) = −1

2

d log g∞

dF[V aR∞(α)]σ2[V aR∞(α)] +

dσ2

dF[V aR∞(α)]

. (4.2)

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This formula has been initially derived by Wilde (2001) [see also Martin,Wilde (2002), Gordy (2003, 2004)], based on the local analysis of theVaR [Gourieroux, Laurent, Scaillet (2000)] and Expected Shortfall[Tasche (2000)]. 8

Example 4.1: Linear Single Risk Factor Model [Gordy (2004)]

In the linear model yi = F + ui, with F ∼ N(µ, η2) and ui ∼ N(0, σ2),

we have m(F ) = F , σ2(F ) = σ2, g∞(F ) =1

ηϕ

(F − µ

η

), and V aR∞(α) =

µ + ηΦ−1(α), where ϕ (resp. Φ) is the density function (resp. cumulativedistribution function) of the standard normal distribution. We deduce:

GA(α) = −σ2

2

d log g∞dF

[V aR∞(α)] =σ2

2ηΦ−1(α).

In this simple Gaussian framework, the quantile V aRn(α) is known inclosed form. It is easily checked that the above GA corresponds with thefirst-order term in a Taylor expansion of V aRn(α) w.r.t. 1/n (see Section5.3). As expected the GA is positive for large α (α > 0.5); the GA increaseswhen the idiosyncratic risk increases, that is, when σ2 increases. Moreover,the GA is a decreasing function of η, which means that the adjustment issmaller, when systematic risk increases.

Example 4.2: Static RFM for Default

Let us assume that the individual risks follow independent Bernoulli dis-tributions conditional on factor F :

yi ∼ B(1, F ),

where B(1, p) denotes a Bernoulli distribution with probability p. This is thewell-known model with stochastic probability of default (PD), often called

8When the factor F is not identified with the conditional mean m(F ), but the functionm(.) is increasing, the formula (4.2) becomes:

GA(α) = −12

1g∞(F )

d

dF

σ2(F )g∞(F )

dm

dF(F )

F=m−1(V aR∞(α))

, (4.3)

where g∞ is the density of F (see Appendix 2 for the derivation).

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reduced form or stochastic intensity model in the Credit Risk literature. Inthis case Wn/n is the default frequency in the portfolio. It also correspondsto the portfolio loss given default, if the loans have a unitary nominal anda zero recovery rate. In this model we have m(F ) = F , σ2(F ) = F (1 − F ),and we deduce:

GA(α) = −1

2

d log g∞

dF[V aR∞(α)]V aR∞(α)[1− V aR∞(α)] + 1− 2V aR∞(α)

.

(4.4)

This formula appears for instance in Rau-Bredow (2005).Different specifications have been considered in the literature for stochas-

tic intensity F . 9 Let us assume that there exists an increasing transforma-tion A, say, from (−∞, +∞) to (0,1) such that:

A−1(F ) ∼ N(µ, η2). (4.5)

We get a probit (resp. logit) normal specification 10, when A is the cumulativedistribution function of the standard normal distribution (resp. the logistic

distribution). Let us denote a(y) =dA(y)

dythe associated derivative. We

have:

V aR∞(α) = A[µ + ηΦ−1(α)],

(4.6)

d log g∞dF

[V aR∞(α)] = − 1

a[µ + ηΦ−1(α)]

(Φ−1(α)

η+

d log a

dy[µ + ηΦ−1(α)]

).

i) Let us first consider the structural Merton (1974) - Vasicek (1991)model [see Example 2.2]. It is known that this model can be written in

9Gordy (2004) and Gordy, Lutkebohmert (2007) derive the GA in the CreditRisk+model [Credit Suisse Financial Products (1997)], which has been the basis for the granu-larity adjustment proposed in the New Basel Capital Accord [see BCBS (2001, Chapter 8)and Wilde (2001)]. This model has some limitations. First, it assumes that the stochasticPD F follows a gamma distribution, that admits values of PD larger than 1. Second, itassumes a constant expected loss given default. We present a multi-factor model withstochastic default probability and expected loss given default in Example 4.6 and Section6.

10For credit risk, a probit specification is proposed in KMV/Moody’s, whereas a logitspecification is used in Credit Portfolio View by Mc Kinsey.

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terms of two structural parameters, that are the unconditional probability ofdefault PD and the asset correlation ρ, such as:

yi = 1l−Φ−1(PD) +√

ρF ∗ +√

1− ρu∗i < 0,

where F ∗ and u∗i , for i = 1, · · · , n, are independent standard Gaussian vari-ables. The structural factor F ∗ is distinguished from the reduced form factorF which is the stochastic probability of default. We see that they are relatedby:

Φ−1(F ) =Φ−1(PD)√

1− ρ−

√ρ√

1− ρF ∗.

We deduce that:

µ =Φ−1(PD)√

1− ρ, η =

√ρ

1− ρ. (4.7)

Thus, from (4.4)-(4.7) we deduce the CSA VaR [Vasicek (1991)]:

V aR∞(α) = Φ

(Φ−1(PD) +

√ρΦ−1(α)√

1− ρ

), (4.8)

and the granularity adjustment:

GA(α) = −1

2

Φ−1(PD)√1− ρ

− 1− 2ρ√ρ(1− ρ)

Φ−1(α)

φ

(Φ−1(PD) +

√ρΦ−1(α)√

1− ρ

) V aR∞(α)[1− V aR∞(α)]

+1− 2V aR∞(α)

. (4.9)

The CSA VaR and the GA depend on both the unconditional probability ofdefault PD and the asset correlation parameter ρ.

In Figure 1 we display the CSA and GA VaR as a function of the confi-dence level for a portfolio of n = 100 companies and different combinationsof PD and ρ, that are PD = 0.005, 0.05, and ρ = 0.12, 0.24. The two val-ues of PD represent yearly probabilities of default for investment grade andspeculative grade companies, respectively. The two values of ρ correspondto the smallest and largest asset correlations, respectively, considered in theBasel 2 regulation [see BCBS(2001)].

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[Insert Figure 1: The CreditVaR as a function of the confidence level inthe Merton-Vasicek model.]

For instance, for probability of default PD = 0.005 and asset correlationρ = 0.12, the GA CreditVaR of the portfolio default frequency at α = 95%is about 0.02. As expected, the CreditVaR is increasing w.r.t. the confi-dence level α, the probability of default PD and the asset correlation ρ. Thegranularity adjustment is positive, that is, it implies a larger minimum re-quired capital. Moreover, the GA is larger for smaller values of the defaultcorrelation.

ii) In the logit normal specification, the transformation A−1(F ) = log[F/(1−F )] corresponds to the log of an odd ratio, and the formula for the GA sim-plifies considerably:

V aR∞(α) =1

1 + exp[−µ− ηΦ−1(α)],

GA(α) =1

2ηΦ−1(α).

In particular, the GA doesn’t depend on parameter µ.

Example 4.3: Static RFM for Count Data

This model is the basic specification for homogenous portfolios of motorinsurance contracts. The conditional distribution of the individual risks isyi ∼ P(F ) given F , where P(λ) denotes a Poisson distribution with param-eter λ. Thus, we get a Poisson model with stochastic claim intensity. Wehave m(F ) = σ2(F ) = F , and we get:

GA(α) = −1

2

d log g∞

dF[V aR∞(α)]V aR∞(α) + 1

.

Usually, the distribution of F is assumed to be log-normal with parame-ters µ, η2, that is,

log(F ) ∼ N(µ, η2). (4.10)

Then:

d log g∞(F )

dF= − 1

F

(log F − µ

η2+ 1

), V aR∞(α) = exp[µ + ηΦ−1(α)],

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and finally:

GA(α) =1

2ηΦ−1(α). (4.11)

The granularity adjustment is independent of parameter µ, which is a scaleparameter for the stochastic intensity. Since in a Poisson process with inten-sity λ the number of claims during a period of length T follows a distributionP(λT ), this means that the granularity adjustment is the same whether theVaR concerns the number of claims per month, or the number of claimsper year. Moreover, the granularity adjustment in the linear Gaussian RFM(Example 4.1), the RFM for default with logit normal specification (Example4.2), or the RFM for count data with log-normal intensity (Example 4.3), isthe same. Indeed, in these models the transformation A that links the factorF to a normal variable [see (4.5) and (4.10)] is such that function a[A−1(F )]is proportional to σ2(F ). In such a case, the GA in (4.2) simplifies to (4.11).

Example 4.4: Linear Static RFM with Beta Heterogeneity

Let us consider the model of Example 2.4. We have:

yi = βiF + vi,

where F ∼ N(µ, η2), vi ∼ N(0, σ2), and βi ∼ N(1, γ2), with all these vari-ables independent. Due to a problem of factor identification, the meanof βi can always be fixed to 1. This will also facilitate the comparisonwith the model with constant beta of Example 4.1. We get m(F ) = F ,σ2(F ) = σ2 + γ2F 2, and:

GA(α) =σ2

2ηΦ−1(α) + γ2

[V aR∞(α)2 Φ−1(α)

2η− V aR∞(α)

], (4.12)

where V aR∞(α) = µ + ηΦ−1(α). Thus, the CSA risk measure V aR∞(α) iscomputed in the homogenous model with factor sensitivity β = 1. The granu-larity adjustment accounts for beta heterogeneity in the portfolio through thevariance γ2 of the heterogeneity distribution. More precisely, the first termin the RHS of (4.12) is the GA already derived in Example 4.1, whereas thesecond term is specific of the beta heterogeneity.

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4.2 Multiple Risk Factor Model

The single risk factor model is too restrictive to capture the complexity ofsystematic risks. Multiple factors are needed for a joint analysis of stochasticdrift and volatility, of default and loss given default, of country and industrialsector specific effects, to monitor the risk of loans with guarantees, when theguarantors themself can default [Ebert, Lutkebohmert (2009)], or to distin-guish between trend and cycle effects.

Example 4.5: Stochastic Drift and Volatility Model

i) Let us assume that yi ∼ N(F1, exp F2), conditional on the bivariate factorF = (F1, F2)

′, and that

F ∼ N

[(µ1

µ2

),

(η2

1 ρη1η2

ρη1η2 η22

)].

This type of stochastic volatility model is standard for modelling the dynamicof asset returns, or equivalently of opposite asset returns.

We can introduce the regression equation:

F2 = µ2 +ρη2

η1

(F1 − µ1) + v,

where v is independent of F1, with Gaussian distribution N [0, η22(1 − ρ2)].

We have m(F ) = F1, σ2(F ) = exp(F2), and

E[σ2(F )|m(F )] = E[exp F2|F1]

= E[expµ2 +ρη2

η1

(F1 − µ1) + v|F1]

= exp[µ2 +ρη2

η1

(F1 − µ1)]E[exp v]

= exp[µ2 +ρη2

η1

(F1 − µ1) +η2

2(1− ρ2)

2].

In particular:

d

dmlog E

[σ2(F )|m(F )

]=

d

dF1

log E [exp F2|F1] =ρη2

η1

.

16

When yi is the opposite of an asset return, a positive value ρ > 0, i.e. anegative correlation between return and volatility, can represent a leverageeffect. From Proposition 1, we deduce that:

GA(α) =1

2η1

(Φ−1(α)− ρη2) exp[µ2 + η22/2] exp[ρη2Φ

−1(α) +ρ2η2

2

2].

The GA of the linear single-factor RFM (see Example 4.1) is obtained wheneither factor F2 is constant (η2 = 0), or factors F1 and F2 are independent(ρ = 0), by noting that E[exp F2] = exp[µ2 + η2

2/2].

ii) The model above has been written with two factors only, specific of thedrift and volatility, respectively. The model encountered in the applicationscan involve several Gaussian factors F ∼ N(µ, Ω), say, where F has dimen-sion d ≥ 2. The individual risks are such that :

yi ∼ N [β′F, exp(γ′F )],

where β and γ are d-dimensional vector of parameters. The granularityadjustment is directly deduced by applying the results of part i) with F1 =β′F, F2 = γ′F, µ2 = γ′µ, η2

1 = β′Ωβ, η22 = γ′Ωγ, ρη1η2 = β′Ωγ.

Example 4.6: Stochastic Probability of Default and Loss GivenDefault

Let us consider a portfolio invested in zero-coupon corporate bonds witha same time-to-maturity and identical exposure at default. The individualloss can be written as:

yi = ZiLGDi,

where Zi is the default indicator and LGDi is the loss given default (i.e. oneminus the recovery rate). Let F = (F1, F2)

′ denote the vector of factors. Weassume that, conditional on factor F , the default indicator Zi and the lossgiven default LGDi are independent 11, such that Zi ∼ B(1, F1) and LGDi

admits a beta distribution with parameters a(F ) > 0 and b(F ) > 0 thatdepend on factor F . Then:

E [LGDi|F ] =a(F )

a(F ) + b(F ), V [LGDi|F ] =

E [LGDi|F ] (1− E [LGDi|F])

a(F ) + b(F ) + 1.

11But can become dependent, when the unobervable factor is integrated out.

17

An equivalent parameterization of the beta distribution is in terms of theconditional mean µ(F ) = a(F )/[a(F )+ b(F )] and the conditional concentra-tion parameter γ(F ) = 1/[a(F ) + b(F ) + 1], that are stochastic parameterson (0, 1). The conditional concentration parameter γ(F ) measures the vari-ability of the conditional distribution of LGDi given F taking into accountthat the variance of a random variable on [0, 1] with mean µ, say, is upperbounded by µ (1− µ). 12 When the conditional concentration parameterγ(F ) approaches 0, the beta distribution degenerates to a point mass; whenthe conditional concentration parameter γ(F ) approaches 1 the beta distri-bution converges to a Bernoulli distribution.

In the sequel, we identify factor F2 with the conditional mean of lossgiven default, while the concentration parameter γ(F ) = γ, with 0 < γ < 1,is assumed constant, independent of the factor. This gives:

E [LGDi|F ] = F2, V [LGDi|F ] = γF2(1− F2).

We get a two-factor model, where the stochastic factors are the (conditional)Probability of Default (PD) and the (conditional) Expected Loss Given De-fault (ELGD) 13.

The conditional mean and volatility of the individual risks are:

m(F ) = E[LGDi|F ]E[Zi|F ] = F1F2, (4.13)

and:

σ2(F ) = E[LGD2i |F ]E[Zi|F ]− E[LGDi|F ]2E[Zi|F ]2

= γF2(1− F2)F1 + F1(1− F1)F22 , (4.14)

respectively. Thus, the conditional mean is the product of the two factors F1

and F2, while the conditional volatility is a fourth-order polynomial in thefactor values. The CSA and GA VaR can be derived by applying Proposition1 and using equations (4.13) and (4.14). The result is given and discussed inSection 6, where the model is extended to a dynamic framework.

12This shows that the mean and the variance cannot be fixed independently for thedistribution of a random variable on [0, 1], and explains why we focus on the conditionalconcentration parameter and not on the conditional variance of LGDi given F .

13In the standard credit risk models, the LGD is often assumed constant. In such acase the LGD coincides with both its conditional and unconditional expectations. In ourframework the LGD is stochastic as well as its conditional expectation.

18

4.3 Aggregation of Risk Measures

The aggregation of risk measures, in particular of VaR, is a difficult problem.In this section we consider this question in a large portfolio perspective fora single factor model.

Let us index by k = 1, · · · , K the different subpopulations, with respec-tive weights πk, k = 1, · · · , K. Within subpopulation k, the conditional meanand variance given factor F are denoted mk(F ) and σ2

k(F ), respectively. Atpopulation level, the conditional mean is:

m(F ) =K∑

k=1

πkmk(F ),

by the iterated expectation theorem, and the conditional variance is givenby:

σ2(F ) =K∑

k=1

σ2k(F ) +

K∑

k=1

πk [mk(F )− m(F )]2 ,

by the variance decomposition equation. We denote by Q∞, q∞ =dQ∞dF

,

G∞ and g∞ the quantile, quantile density, cdf and pdf of the distribution offactor F , respectively.

i) Aggregation of CSA risk measures

We assume:

A.3: The conditional means mk(F ) are increasing functions of F , for anyk = 1, · · · , K.

Under Assumption A.3, the systematic risk components mk(F ), k = 1, · · · , Kare co-monotonic. Therefore, we have:

P[m(F ) < V aR∞(α)

]= α ⇔ P

[F < m−1

(V aR∞(α)

)]= α,

and we deduce that the VaR at population level is:

V aR∞(α) = m[Q∞(α)] =K∑

k=1

πkmk[Q∞(α)] =K∑

k=1

πkV aRk,∞(α), (4.15)

19

by definition of the disaggregated VaR. Thus, we have the perfect aggregationformula for the CSA risk measure, that is, the aggregate VaR is obtained bysumming the disaggregated VaR with the subpopulation weights.

ii) Aggregation of Granularity Adjustments

We have the following Proposition:

Proposition 2: The aggregate granularity adjustment is given by:Proof: See Appendix 3.[TO BE CONTINUED]

5 Dynamic Risk Factor Model (DRFM)

The static risk factor model implicitly assumes that the past observationsare not informative to predict the future risk. In dynamic factor models, theVaR becomes a function of the available information. This conditional VaRhas to account for the unobservability of the current and lagged factor values.We show in this section that factor unobservability implies an additional GAfor the VaR. Despite this further layer of complexity in dynamic models,the granularity principle becomes even more useful compared to the staticframework. Indeed, the conditional cdf of the portfolio value at date t involvesan integral that can reach dimension (t + 1)dim(F ) + n dim(u) − 1, whichnow depends on t, due to the integration w.r.t. the factor path.

5.1 The Model

Dynamic features can easily be introduced in the following way:

i) We still assume a static relationship between the individual risks andthe systematic factors. This relationship is given by the static measurementequations:

yit = c(Ft, uit), (5.1)

where the idiosyncratic risks (ui,t) are independent, identically distributedacross individuals and dates, and independent of the factor process (Ft).

20

ii) Then, we allow for factor dynamic. The factor process (Ft) is Markovwith transition pdf g(ft|ft−1), say. Thus, all the dynamics of individual riskspass by means of the factor dynamic.

Let us now consider the future portfolio risk per individual asset defined

by Wn,t+1/n =1

n

n∑i=1

yi,t+1. The (conditional) VaR is defined by the equation:

P [Wn,t+1/n < V aRn,t(α)|In,t] = α, (5.2)

where the available information In,t includes all current and past individualrisks yi,t, yi,t−1, . . ., for i = 1, . . . , n, but not the current and past factorvalues. The conditional quantile V aRn,t(α) depends on the date t throughthe information In,t.

5.2 Granularity Adjustment

i) Asymptotic expansion of the portfolio risk

Let us first perform an asymptotic expansion of the portfolio risk. By thecross-sectional CLT applied conditional on the factor path, we have:

Wn,t+1/n = m(Ft+1) + σ(Ft+1)Xt+1√

n+ O(1/n),

where the variable Xt+1 is standard normal, independent of the factor pro-cess, and O(1/n) denotes a term of order 1/n, which is zero-mean conditionalon Ft+1, Ft, . . .. The functions m(.) and σ2(.) are defined analogously as in(3.2)-(3.3), and depend on Ft+1 only by the static measurement equations(5.1).

In order to compute the conditional cdf of Wn,t+1/n, it is useful to rein-troduce the current factor value in the conditioning set through the law of

21

iterated expectation. We have:

P [Wn,t+1/n < y|In,t] = E[P (Wn,t+1/n < y|Ft, In,t)|In,t]

= E[P (Wn,t+1/n < y|Ft)|In,t]

= E[P (m(Ft+1) + σ(Ft+1)Xt+1√

n+ O(1/n) < y|Ft)|In,t]

= E[a(y,Xt+1√

n+ O(1/n); Ft)|In,t], (5.3)

where function a is defined by:

a(y, ε; ft) = P [m(Ft+1) + σ(Ft+1)ε < y|Ft = ft]. (5.4)

ii) Cross-sectional approximation of the factor

Function a depends on the unobserved factor value Ft = ft, and we havefirst to explain how this value can be approximated from observed individ-ual variables. For this purpose, let us denote by h(yi,t|ft) the conditionaldensity of yi,t given Ft = ft, deduced from model (5.1), and define the cross-sectional maximum likelihood approximation of ft given by:

fn,t = arg maxft

n∑i=1

log h(yi,t|ft). (5.5)

Hence, the factor value ft is treated as an unknown parameter in the cross-sectional conditional model at date t and is approximated by the maximumlikelihood principle. Approximation fnt is a function of the current individualobservations and hence of the available information In,t.

iii) Granularity Adjustment for factor prediction

It might seem natural to replace the unobserved factor value ft by itscross-sectional approximation fn,t in the expression of function a, and thento use the GA of the static model in Proposition 1, for the distribution ofFt+1 given Ft = fnt. However, replacing ft by fn,t implies an approximation

22

error. It has been proved that this error is of order 1/n, that is, the sameorder expected for the GA. More precisely, we have the following result whichis given in the single factor framework for expository purpose [Gagliardini,Gourieroux (2009), Corollary 5.3]:

Proposition 3: Let us consider a dynamic single factor model. For a largehomogenous portfolio, the conditional distribution of Ft given In,t is approx-imately normal at order 1/n:

N

(fn,t +

1

nµn,t,

1

nJ−1

n,t

),

where:

µn,t = J−1n,t

∂ log g

∂ft

(fn,t|fn,t−1) +1

2J−2

n,t Kn,t,

Jn,t = − 1

n

n∑i=1

∂2 log h

∂f 2t

(yi,t|fn,t),

Kn,t =1

n

n∑i=1

∂3 log h

∂f 3t

(yi,t|fn,t).

Proposition 3 gives an approximation of the filtering distribution of factorFt given the information In,t. This approximation involves four summarystatistics, which are the cross-sectional maximum likelihood approximationsfn,t and fn,t−1, the Fisher information Jn,t for approximating the factor inthe cross-section at date t, and the statistic Kn,t in the bias adjustment.

iii) Expansion of the cdf of portfolio risk

From equation (5.3) and Proposition 3, the conditional cdf of the portfoliorisk can be written as:

P [Wn,t+1/n < y|In,t]

= E

[a

(y,

Xt+1√n

+ O(1/n); fn,t +1

nµn,t +

1√n

J−1/2n,t X∗

t

)|In,t

]+ o(1/n),

23

where X∗t is a standard Gaussian variable independent of Xt+1 and O(1/n),

of the factor path and of the available information 14.Then, we can expand the expression above with respect to n, up to order

1/n. By noting that fn,t, µn,t, Jn,t are functions of the available informationand that E[Xt+1] = E[X∗

t ] = E[O(1/n)] = 0, E[Xt+1X∗t ] = 0, E[X2

t+1] =E[(X∗

t )2] = 1, we get:

P [Wn,t+1/n < y|In,t] = a(y, 0; fn,t) +1

n

∂a(y, 0; fnt)

∂ft

µnt

+1

2n

[J−1

n,t

∂2a(y, 0; fn,t)

∂f 2t

+∂2a(y, 0, fnt)

∂ε2

]+ o(1/n).

The CSA conditional cdf of the portfolio risk is a(y, 0; fn,t), where:

a(y, 0; ft) = P [m(Ft+1) < y|Ft = ft]. (5.6)

It corresponds to the conditional cdf of m(Ft+1) given Ft = ft, where theunobservable factor value ft is replaced by its cross-sectional approximationfn,t. The GA for the cdf is the sum of two components corresponding to:

(*) The granularity adjustment for the conditional cdf with known Ft

equal to fn,t, that is,

1

2n

∂2a(y, 0; fn,t)

∂ε2. (5.7)

The second-order derivative of function a(y, ε; ft) w.r.t. ε at ε = 0 can becomputed by using Lemma a.1 in Appendix 1, which yields:

∂2a(y, 0; ft)

∂ε2=

d

dy

g∞(y; ft)E[σ2(Ft+1)|m(Ft+1) = y, Ft = ft]

,

where g∞(y; ft) denotes the pdf of m(Ft+1) conditional on Ft = ft.

(**) The granularity adjustment for filtering, that is,

∂a(y, 0; fn,t)

∂ft

µnt +1

2J−1

n,t

∂2a(y, 0; fnt)

∂f 2t

. (5.8)

14The independence between X∗t and Xt+1 is due to the fact that X∗

t simply representsthe numerical approximation of the filtering distribution of Ft given In,t and is not relatedto the stochastic features of the observations at t+1 [see Gagliardini, Gourieroux (2009)].

24

It involves the first- and second-order derivatives of the CSA cdf w.r.t. theconditioning factor value.

Due to the independence between variables Xt+1 and X∗t , there is no cross

GA.

iv) Granularity Adjustment for the VaR

The CSA cdf is used to define the CSA risk measure V aR∞(α; fn,t)through the condition:

P[m(Ft+1) < V aR∞(α; fn,t)|Ft = fn,t

]= α.

The CSA VaR depends on the current information through the cross-sectionalapproximation of the factor value fn,t only. The GA for the (conditional)VaR is directly deduced from the GA of the (conditional) cdf by applyingthe Bahadur’s expansion [Bahadur (1966); see Lemma a.3 in Appendix 1].We get the next Proposition:

Proposition 4: In a dynamic RFM the (conditional) VaR is such that:

V aRn,t(α) = V aR∞(α; fn,t) +1

n[GArisk(α) + GAfilt(α)] + o(1/n),

where:

GArisk(α) = −1

2

∂ log g∞(y; fnt)

∂yE[σ2(Ft+1)|m(Ft+1) = y, Ft = fn,t]

+∂

∂yE[σ2(Ft+1)|m(Ft+1) = y, Ft = fn,t]

y=V aR∞(α;fn,t)

,

and:

GAfilt(α) = − 1

g∞[V aR∞(α; fn,t); fn,t]

∂a[V aR∞(α, fnt), 0; fnt]

∂ft

µnt

+1

2J−1

n,t

∂2a[V aR∞(α; fnt), 0; fnt]

∂f 2t

,

and where g∞(.; ft) [resp. a(., 0; ft) and V aR∞(.; ft)] denotes the pdf [resp.the cdf and quantile] of m(Ft+1) conditional on Ft = ft.

25

Thus, the GA for the conditional VaR is the sum of two components. Thefirst one GArisk(α) is the analogue of the GA in the static factor model (seeProposition 1). However, the distribution of m(Ft+1) and σ2(Ft+1) is nowconditional on Ft = ft, and the unobservable factor value ft is replaced byits cross-sectional approximation fn,t. The second component GAfilt(α) isdue to the filtering of the unobservable factor value, and involves first- andsecond-order derivatives of the CSA cdf w.r.t. the conditioning factor value.

5.3 Linear RFM with AR(1) factor

As an illustration, let us consider the model given by:

yi,t = Ft + ui,t, i = 1, . . . , n,

where:Ft = µ + ρ(Ft−1 − µ) + vt,

and ui,t, vt are independent, with uit ∼ IIN(0, σ2), vt ∼ IIN(0, η2). In thisGaussian framework the (conditional) VaR can be computed explicitly, whichallows for a comparison with the granularity approximation in order to assessthe accuracy of the two GA components and their (relative) magnitude.

i) The static case

Let us first set ρ = 0, which corresponds to the static case (see Example4.1). The distribution of:

Wn,t+1/n = yn,t+1 = Ft+1 + un,t+1 = µ + vt+1 + un,t+1,

is Gaussian with mean µ and variance η2 +σ2

n, where yn,t+1 denotes the

cross-sectional average of the individual risks at date t + 1 and similarly forun,t+1. We deduce the static VaR given by:

V aRn(α) = µ +

√η2 +

σ2

nΦ−1(α).

By expanding the square root term at order 1/n, we get:

V aRn(α) = µ + ηΦ−1(α) +1

n

σ2

2ηΦ−1(α) + o(1/n),

26

where the first term is the VaR associated with factor Ft and the second termis the GA derived in Example 4.1.

ii) The dynamic case

Let us now consider the general case. The individual observations can besummarized by their cross-sectional averages 15 and we have:

yn,t+1 = Ft+1 + un,t+1,

Ft+1 = µ + ρ(Ft − µ) + vt+1.(5.9)

This implies:

yn,t+1 = µ +1

1− ρLvt+1 + un,t+1

= µ +1

1− ρL(vt+1 + un,t+1 − ρun,t),

where L denotes the lag operator. The process Zt+1 = vt+1 + un,t+1− ρun,t isa Gaussian MA(1) process that can be written as Zt+1 = εt+1 − θnεt, wherethe variables εt are IIN(0, γ2

n), say, and |θn| < 1. The new parameters θn

and γn are deduced from the expressions of the variance and autocovarianceat lag 1 of process (Zt). They satisfy:

η2 +σ2

n(1 + ρ2) = γ2

n(1 + θ2n),

ρσ2

n= θnγ

2n.

(5.10)

Hence:

θn =bn −

√b2n − 4ρ2

2ρ, γ2

n =ρσ2

nθn

, (5.11)

where bn = 1 + ρ2 +nη2

σ2and we have selected the root θn such that |θn| < 1.

Therefore, variable yn,t+1 follows a Gaussian ARMA(1,1) process, and we can

15By writing the likelihood of the model, it is seen that the cross-sectional averages aresufficient statistics.

27

write:

yn,t+1 = µ +1− θnL

1− ρLεt+1

⇐⇒ yn,t+1 = µ +

(1− 1− ρL

1− θnL

)(yn,t+1 − µ) + εt+1

⇐⇒ yn,t+1 = µ +ρ− θn

1− θnL(yn,t − µ) + εt+1

⇐⇒ yn,t+1 = µ + (ρ− θn)∞∑

j=0

θjn(yn,t−j − µ) + εt+1.

We deduce that the conditional distribution of Wn,t+1/n = yn,t+1 given In,t

is Gaussian with mean µ + (ρ− θn)∞∑

j=0

θjn(yn,t−j − µ) and variance γ2

n.

Proposition 5: In the linear RFM with AR(1) Gaussian factor, the condi-tional VaR is given by:

V aRn,t(α) = µ + (ρ− θn)∞∑

j=0

θjn(yn,t−j − µ) + γnΦ−1(α),

where θn and γn are given in (5.11).

Thus, the conditional VaR depends on the information In,t through a weightedsum of current and lagged cross-sectional individual risks averages. Theweights decay geometrically with the lag as powers of parameter θn. More-over, the information In,t impacts the VaR uniformly in the risk level α.

Let us now derive the expansion of V aRn,t(α) at order 1/n for large n.From (5.11), the expansions of parameters θn and γn are:

θn =ρσ2

η2n+ o(1/n), γn = η +

σ2

2nη(1 + ρ2) + o(1/n).

As n → ∞, the MA parameter θn converges to zero and the variance ofthe shocks γ2

n converges to η2. Hence, the ARMA(1,1) process of the cross-sectional averages yn,t approaches the AR(1) factor process (Ft) as expected.

28

By plugging the expansions for θn and γn into the expression of V aRn,t(α)in Proposition 5, we get:

V aRn,t(α) = µ + ρ(yn,t − µ) + ηΦ−1(α)

+1

n

σ2

2η(1 + ρ2)Φ−1(α)− ρσ2

η2[(yn,t − µ)− ρ(yn,t−1 − µ)]

+ o(1/n).

The first row on the RHS provides the CSA VaR:

V aR∞(α; fn,t) = µ + ρ(yn,t − µ) + ηΦ−1(α). (5.12)

which depends on the information through the cross-sectional maximum like-lihood approximation of the factor fn,t = yn,t. The CSA VaR is the quantileof the normal distribution with mean µ + ρ(yn,t − µ) and variance η2, thatis the conditional distribution of Ft+1 given Ft = yn,t. The GA involves theinformation In,t through the current and lagged cross-sectional averages yn,t

and yn,t−1. The other lagged values yn,t−j for j ≥ 2 are irrelevant at ordero(1/n).

Let us now identify the risk and filtering GA components. We have:

a(y, 0; ft) = P (Ft+1 < y|Ft = ft) = Φ

(y − µ− ρ(ft − µ)

η

).

We deduce:

g∞(y; ft) =∂a(y, 0; ft)

∂y=

1

ηϕ

[y − µ− ρ(ft − µ)

η

],

and:

∂a(y, 0; ft)

∂ft

= −ρ

ηϕ

[y − µ− ρ(ft − µ)

η

],

∂2a(y, 0; ft)

∂f 2t

=ρ2

η2

[y − µ− ρ(ft − µ)

η

]ϕ

(y − µ− ρ(ft − µ)

η

).

Moreover, the statistics involved in the approximate filtering distribution of

Ft given In,t (see Proposition 3) are µn,t = −σ2

η2[(yn,t − µ)− ρ(yn,t−1 − µ)] ,

Jn,t = 1/σ2 and Kn,t = 0. From Proposition 3 and equation (5.12), we get:

29

GArisk(α) =σ2

2ηΦ−1(α),

GAfilt(α) =σ2ρ2

2ηΦ−1(α)− ρσ2

η2[(yn,t − µ)− ρ(yn,t−1 − µ)] .

The GA for risk is the same as in the static model for ρ = 0 [see Sectioni) and Example 4.1], since in this Gaussian framework the current factor ft

impacts the conditional distribution of m(Ft+1) = Ft+1 given Ft = ft throughthe mean only, and σ2(Ft+1) = σ2 is constant. The GA for filtering dependson both the risk level α and the information through (yn,t−µ)−ρ(yn,t−1−µ).By the latter effect, GAfilt(α) can take any sign. Moreover, this term inducesa stabilization effect on the dynamics of the GA VaR compared to the CSAVaR. To see this, let us assume ρ > 0 and suppose there is a large upwardaggregate shock on the individual risks at date t, such that yn,t−µ is positiveand (much) larger than ρ(yn,t−1−µ). The CSA VaR in (5.12) reacts linearlyto the shock and features a sharp increase. Since (yn,t−µ)−ρ(yn,t−1−µ) > 0,the GA term for filtering is negative and reduces the reaction of the VaR.

In Figure 2 we display the patterns of the true, CSA and GA VaR curvesas a function of the risk level for a specific choice of parameters.

[Insert Figure 2: The VaR as a function of the risk level in the linearRFM with AR(1) factor.]

The mean and the autoregressive coefficient of the factor are µ = 0 andρ = 0.5, respectively. The idiosyncratic and systematic variance parametersσ2 and η2 are selected in order to imply an unconditional standard devia-tion of the individual risks

√η2/(1− ρ2) + σ2 = 0.15, and an unconditional

correlation between individual risksη2/(1− ρ2)

σ2 + η2/(1− ρ2)= 0.10. The portfolio

size is n = 100. The available information In,t is such that yn,t−j = µ = 0for all lags j ≥ 2, and we consider four different cases concerning the currentand the most recent lagged cross-sectional averages, yn,t and yn,t−1, respec-tively. Let us first assume yn,t = yn,t−1 = 0 (upper-left Panel), that is, bothcross-sectional averages are equal to the unconditional mean. As expected,all VaR curves are increasing w.r.t. the confidence level. The true VaR isabout 0.10 at confidence level 99%. The CSA VaR underestimates the true

30

VaR (that is, underestimates the risk) by about 0.01 uniformly in the risklevel. The GA for risk corrects most of this bias and dominates the GAfor filtering. The situation is different when yn,t = −0.30 and yn,t−1 = 0(upper-right Panel), that is, when we have a downward aggregate shock inrisk of two standard deviations at date t. The CSA VaR underestimates thetrue VaR by about 0.02. The GA for risk corrects only a rather small partof this bias, while including the GA for filtering allows for a quite accurateapproximation. The GA for filtering is about five times larger than the GAfor risk. When yn,t = 0.30 and yn,t−1 = 0 (lower-left Panel), there is a largeupward aggregate shock in risk at date t, and the CSA VaR overestimatesthe true VaR (that is, overestimates the risk). The GA correction for riskfurther increases the VaR and the bias, while including the GA correctionyields a good approximation of the true VaR. The results are similar in thecase yn,t = 0.30 and yn,t−1 = 0.30 (lower-right Panel), that is, in case of apersistent downward aggregate shock in risk. Finally, by comparing the fourpanels in Figure 2, it is seen that the CSA risk measure is more sensitive tothe current information than the true VaR and the GA VaR. To summarize,Figure 2 shows that the CSA VaR can either underestimate or overestimatethe risk, the GA for filtering can dominate the GA for risk, and the completeGA can yield a good approximation of the true VaR even for portfolio sizesof some hundreds of contracts (at least in the specific linear RFM consideredin our illustration). Moreover, the GA component reduces the undue timeinstability of the CSA VaR.

6 Stochastic Probability of Default and Ex-

pected Loss Given Default

6.1 Two-factor dynamic model

Let us consider a two-factor dynamic model with stochastic probability ofdefault and expected loss given default. We extend Example 4.6 in Section4.2 to a dynamic framework. The future value of the zero-coupon corporatebond maturying at t + 1 is:

yi,t+1 = LGDi,t+1Zi,t+1,

where Zi,t+1 is the default indicator and LGDi,t+1 is the loss given default.Conditional on the path of the bivariate factor Ft = (F1,t, F2,t)

′, the default

31

indicator Zi,t+1 and the loss given default LGDi,t+1 are independent, suchthat Zi,t+1 ∼ B(1, F1,t+1) and LGDi,t+1 admits a beta distribution with con-ditional mean and volatility given by:

E [LGDi,t+1|Ft+1] = F2,t+1, V [LGDi,t+1|Ft+1] = γF2,t+1(1− F2,t+1),

where the concentration parameter γ ∈ (0, 1) is constant. The dynamic fac-tors F1,t and F2,t correspond to the (conditional) Probability of Default (PD)and the (conditional) Expected Loss Given Default (ELGD), respectively.

Both stochastic factors F1,t and F2,t admit values in the interval (0, 1).We assume that the transformed factors F ∗

t = (F ∗1,t, F

∗2,t)

′ defined by F ∗l,t =

log[Fl,t/(1 − Fl,t)], for l = 1, 2 (logistic transformation), follow a bivariateGaussian VAR(1) process:

F ∗t = c + ΦF ∗

t−1 + εt,

with εt ∼ IIN(0, Ω) and Ω =

(σ2

1 ρσ1σ2

ρσ1σ2 σ22

).

6.2 CSA VaR and Granularity Adjustment

The CSA VaR and the GA are derived from the results in Section 5.2. Letus first consider the cross-sectional factor approximations. It is proved inAppendix 4 that:

f1,n,t =Nt

n, (5.13)

where Nt =n∑

i=1

1lyi,t>0 is the number of defaults at date t, and:

f2,n,t = arg maxf2,t

f2,t

(1− γ

γ

) ∑i:yi,t>0

log

(yi,t

1− yi,t

)

−Nt log Γ

[(1− γ

γ

)f2,t

]−Nt log Γ

[(1− γ

γ

)(1− f2,t)

],

(5.14)

where the sum is over the companies that default at date t, and Γ(.) denotesthe Gamma function. Thus, the approximation f1,n,t of the conditional PD

32

is the cross-sectional default frequency at date t, while the approximationf2,n,t of the conditional ELGD is obtained by maximizing the cross-sectionallikelihood associated with the conditional beta distribution of the LGD atdate t. Proposition 3 on the approximate filtering distribution can be easilyextended to multiple factor. Since the cross-sectional log-likelihood can bewritten as the sum of a component involving f1,t only, and a componentinvolving f2,t only, the approximate filtering distribution is such that F1,t

and F2,t are independent conditional on information In,t at order 1/n, with

Gaussian distributions N

(fl,n,t +

1

nµl,n,t,

1

nJ−1

l,n,t

), for l = 1, 2, where:

µ1,n,t = −e′1Ω−1(f ∗n,t − c− Φf ∗n,t−1),

(5.15)

µ2,n,t = − J−12,n,t

f2,n,t(1− f2,n,t)

[e′2Ω

−1(f ∗n,t − c− Φf ∗n,t−1) + 1− 2f ∗2,n,t

]+

1

2J−2

2,n,tK2,n,t,

with f ∗n,t =(log[f1,n,t/(1− f1,n,t)], log[f2,n,t/(1− f2,n,t)]

)′and vectors e1 =

(1, 0)′, e2 = (0, 1)′, and:

J1,n,t =1

f1,n,t(1− f1,n,t),

(5.16)

J2,n,t = f1,n,t

(1− γ

γ

)2 Ψ′

[(1− γ

γ

)f2,n,t

]+ Ψ′

[(1− γ

γ

)(1− f2,n,t)

],

with Ψ(s) =d log Ψ(s)

dsand:

K2,n,t = −f1,n,t

(1− γ

γ

)3 Ψ′′

[(1− γ

γ

)f2,n,t

]−Ψ′′

[(1− γ

γ

)(1− f2,n,t)

].

(5.17)

Let us now derive the CSA VaR and the GA. From the results in Section5.2 iv), we get the next Proposition.

Proposition 6: In the model with stochastic conditional PD and ELGD:i) The CSA VaR at risk level α is the solution of the equation:

a(V aR∞(α; fn,t), 0; fn,t

)= α,

33

where:

a(w, 0; ft) = Φ

(log[w/(1− w)]− c1,t

σ1

)

+

∫ 1

w

Φ

log[w/(y − w)]− c2,t − ρσ2

σ1

(log[y/(y − 1)]− c1,t)

σ2

√1− ρ2

· 1

σ1

ϕ

(log[y/(y − 1)]− c1,t

σ1

)1

y(1− y)dy.

and cl,t = cl + Φl,1 log[f1,t/(1− f1,t)] + Φl,2 log[f2,t/(1− f2,t)], for l = 1, 2.

ii) The GA at risk level α is GA(α) = GArisk(α)+GAfilt(α), where GArisk(α)is computed from Proposition 4 with g∞(w; ft) = b(w, 0; ft) and:

E[σ2(Ft+1)|m(Ft+1) = w, Ft = ft] = w(γ − w) + (1− γ)wb(w, 1; ft)

b(w, 0; ft),

where:

b(w, k; ft) =

∫ 1

w

yk−1

σ1σ2

ϕ

(log[w/(y − w)]− c1,t

σ1

,log[y/(1− y)]− c2,t

σ2

; ρ

)

1

(1− y)w(1− w/y)dy,

and ϕ(., .; ρ) denotes the pdf of the standard bivariate Gaussian distributionwith correlation ρ; the component GAfilt(α) is given by:

GAfilt(α) = − 1

g∞[V aR∞(α; fn,t); fn,t]

2∑

l=1

∂a[V aR∞(α, fnt), 0; fl,nt]

∂fl,t

µl,nt

+1

2J−1

l,n,t

∂2a[V aR∞(α; fnt), 0; fl,nt]

∂f 2l,t

,

where µl,n,t and Jl,n,t are given in (5.15) and (5.16).

Proof: See Appendix 4.

The CSA and GA VaR in Proposition 6 are given in closed form, up to afew one-dimensional integrals. [TO BE CONTINUED]

34

7 Concluding Remarks

For large homogenous portfolios and a variety of both single-factor and multi-factor dynamic risk models, closed form expressions of the VaR and otherdistortion risk measures can be derived at order 1/n. Two granularity adjust-ments are required. The first GA concerns the conditional VaR with currentfactor value assumed to be observed. The second GA takes into account theunobservability of the current factor value and is specific to dynamic factormodels. This explains why this GA has not been taken into account in theearlier literature which focuses on static models.

These GA assume given the link function c and the distributions of boththe factor and idiosyncratic risks. In practice the link function and thedistribution depend on unknown parameters, which have to be estimated.This creates an additional error on the VaR, which has been consideredneither here, nor in the previous literature. This estimation error can belarger than the GA derived in this paper. However, such a separate analysisis compatible with the Basel 2 methodology. Indeed, the GA in this paperare useful to compute the reserves for Credit Risk, whereas the adjustmentfor estimation concerns the reserves for Estimation Risk.

The granularity adjustment principle appeared in Pillar 1 of the NewBasel Capital Accord in 2001 [BCBS (2001)], concerning the minimum cap-ital requirement. It has been suppressed from Pillar 1 in the most recentversion of the Accord in 2003 [BCBS (2003)], and assigned to Pillar 2 oninternal risk models. The recent financial crisis has shown that systematicrisks, which include in particular systemic risks 16, have to be distinguishedfrom unsystematic risk, and in the new organization these two risks will besupervised by two different regulators. This shows the importance of takinginto account this distinction in computing the reserves, that is, also at Pil-lar 1 level. For instance, one may fix different risk levels α1 and α2 in theCSA and GA VaR components, and smooth differently these componentsover the cycle in the definition of the required capital. The recent literatureon granularity shows that the technology is now in place and rather easy toimplement, at least for nonlinear static, and linear dynamic, factor models.

16A systemic risk is a systematic risk which can seriously damage the Financial System.

35

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Basel Committee on Banking Supervision (2001): ”The New Basel Cap-ital Accord”, Consultative Document of the Bank for International Settle-ments, April 2001, Part 2: Pillar 1, Section “Calculation of IRB GranularityAdjustment to Capital”.

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Black, F. (1976): “Studies of Stock Price Volatility Changes”, Proceed-ings of the 1976 Meeting of the American Statistical Association, Businessand Economics Statistics Section, 177-181.

Buhlmann, H. (1967): ”Experience Rating and Credibility”, ASTIN Bul-letin, 4, 199-207.

Buhlmann, H., and L., Straub (1970): ”Credibility for Loss Ratios”,ARCH, 1972-2.

Chamberlain, G., and M., Rothschild (1983): ”Arbitrage, Factor Struc-ture and Mean-Variance Analysis in Large Markets”, Econometrica, 51, 1281-1304.

Credit Suisse Financial Products (1997): ”CreditRisk+: A Credit RiskManagement Framework“, Credit Suisse Financial Products, London.

de Finetti, B. (1931): ”Funzione Caratteristica di un Fenomeno Aleato-rio”, Atti della R. Accademia dei Lincei, 6, Memorie. Classe di ScienzeFisiche, Matematiche e Naturali, 4, 251-299.

Ebert, S., and E., Lutkebohmert (2009): ”Treatment of Double DefaultEffects Within the Granularity Adjustment for Basel II“, University of Bonn,Working Paper.

36

Emmer, S., and D., Tasche (2005): ”Calculating Credit Risk CapitalCharges with the One-Factor Model“, Journal of Risk, 7, 85-103.

Gagliardini, P., and C., Gourieroux (2009): ”Approximate DerivativePricing in Large Class of Homogeneous Assets with Systematic Risk“, CREST,DP.

Gagliardini, P., and C., Gourieroux (2010): ”Granularity Theory”, CREST,DP.

Gordy, M. (2003): ”A Risk-Factor Model Foundation for Rating-BasedBank Capital Rules”, Journal of Financial Intermediation, 12, 199-232.

Gordy, M. (2004): ”Granularity Adjustment in Portfolio Credit Risk Mea-surement”, in Risk Measures for the 21.st Century, ed. G. Szego, Wiley,109-121.

Gordy, M., and E., Lutkebohmert (2007): ”Granularity Adjustment forBasel II”, Deutsche Bank, DP.

Gourieroux, C., Laurent, J.P., and O., Scaillet (2000): ”Sensitivity Anal-ysis of Value-at-Risk”, Journal of Empirical Finance, 7, 225-245.

Gourieroux, C., and A., Monfort (2009): ”Granularity in a QualitativeFactor Model“, Journal of Credit Risk, 5, 29-65.

Hewitt, E, and L., Savage (1955): ”Symmetric Measures on CartesianProducts”, Transaction of the American Mathematical Society, 80, 470-501.

Martin, R., and T., Wilde (2002): ”Unsystematic Credit Risk”, Risk, 15,123-128.

Merton, R. (1974): ”On the Pricing of Corporate Debt: The Risk Struc-ture of Interest Rates”, Journal of Finance, 29, 449-470.

Rau-Bredow, H. (2005): ”Granularity Adjustment in a General FactorModel“, University of Wuerzburg, DP.

Ross, S. (1976): ”The Arbitrage Theory of Capital Asset Pricing”, Jour-nal of Economic Theory, 13, 341-360.

37

Tasche, D. (2000): ”Conditional Expectation as Quantile Derivative”,Bundesbank, DP.

Vasicek, O. (1991): ”Limiting Loan Loss Probability Distribution”, KMVCorporation, DP.

Vasicek, O. (2002): ”The Distribution of Loan Portfolio Value“, Risk, 15,160-162.

Wang, S. (2000): ”A Class of Distortion Operators for Pricing Financialand Insurance Risks”, Journal of Risk and Insurance, 67, 15-36.

Wilde, T. (2001): ”Probing Granularity”, Risk, 14, 103-106.

38

Figure 1: CreditVaR as a function of the confidence level in the Merton-Vasicek model.

0.95 0.96 0.97 0.98 0.99

0.02

0.04

0.06

0.08

α

VaR

n,t(α

)

PD = 0.005, ρ = 0.12

0.95 0.96 0.97 0.98 0.99

0.02

0.04

0.06

0.08

α

VaR

n,t(α

)

PD = 0.005, ρ = 0.24

0.95 0.96 0.97 0.98 0.990.1

0.15

0.2

0.25

0.3

0.35

α

VaR

n,t(α

)

PD = 0.05, ρ = 0.12

0.95 0.96 0.97 0.98 0.990.1

0.15

0.2

0.25

0.3

0.35

α

VaR

n,t(α

)

PD = 0.05, ρ = 0.24

In each Panel we display the CSA VaR (dashed line) and the GA VaR (solid line) of thedefault frequency in a portfolio of n = 100 companies, as a function of the confidencelevel α. The four panels correspond to different values of the unconditional probability ofdefault PD and asset correlation ρ, that are PD = 0.005 and ρ = 0.12 in the upper-leftpanel, PD = 0.005 and ρ = 0.24 in the upper-right panel, PD = 0.05 and ρ = 0.12 in thelower-left panel, and PD = 0.05 and ρ = 0.24 in the lower-right panel, respectively.

39

Figure 2: VaR as a function of the risk level in the linear RFM with AR(1)factor.

0.95 0.96 0.97 0.98 0.990.06

0.07

0.08

0.09

0.1

0.11

0.12

α

VaR

n,t(α

)

yn,t = 0, yn,t−1 = 0

0.95 0.96 0.97 0.98 0.99−0.1

−0.08

−0.06

−0.04

−0.02

0

αV

aR

n,t(α

)

yn,t = −0.30, yn,t−1 = 0

0.95 0.96 0.97 0.98 0.990.2

0.22

0.24

0.26

α

VaR

n,t(α

)

yn,t = 0.30, yn,t−1 = 0

0.95 0.96 0.97 0.98 0.990.2

0.22

0.24

0.26

α

VaR

n,t(α

)

yn,t = 0.30, yn,t−1 = 0.30

In each Panel we display the true VaR (solid line), the CSA VaR (dashed line), the GAVaR accounting for risk only (dashed-dotted line) and the GA VaR accounting for bothrisk and filtering (dotted line), as a function of the confidence level α. The four Panelscorrespond to different available informations In,t, that are yn,t = yn,t−1 = 0 in thefirst Panel, yn,t = −0.30, yn,t−1 = 0 in the second, yn,t = 0.30, yn,t−1 = 0 in the third,and yn,t = 0.30, yn,t−1 = 0.30 in the fourth Panel, respectively. The portfolio size isn = 100. The model parameters are such that the unconditional standard deviation ofthe individual risks is 0.15, the unconditional correlation between individual risks is 0.10,the factor mean is µ = 0, and the factor autoregressive coefficient is ρ = 0.5.

40

Appendix 1

Asymptotic Expansions

1. Expansion of the cumulative distribution function

Let us consider a pair (X, Y ) of real random variables, where X is acontinuous random variable with pdf f1 and cdf G1. The aim of this sectionis to derive an expansion of the function:

a(x, ε) = P [X + εY < x], (a.1)

in a neighbourhood of ε = 0.

The following Lemma has been first derived by Gourieroux, Laurent,Scaillet (2000) for the second-order expansion, and by Martin, Wilde (2002)for any order. We will extend the proof in Gourieroux, Laurent, Scaillet(2000), which avoids the use of characteristic functions and shows moreclearly the needed regularity conditions (RC) (see below).

Lemma a.1: Under regularity conditions (RC), we have:

a(x, ε) = G1(x) +J∑

j=1

(−1)j

j!εj dj−1

dxj−1[g1(x)E(Y j|X = x)]

+ o(εJ).

Proof: The proof requires two steps. First, we consider the case of a bivariatecontinuous random vector (X, Y ). Then, we extend the result when Y andX are in a deterministic relationship.

i) Bivariate continuous vector

Let us denote by g1,2(x, y) [resp. g1|2(x|y) and G1|2(x|y)] the joint pdf of(X, Y ) (resp. the conditional pdf and cdf of X given Y ). We have:

41

a(x, ε) = P [X + εY < x]

= EP [X < x− εY |Y ]

= E[G1|2(x− εY |Y )]

= E[G1|2(x|Y )] +J∑

j=1

(−1)jεj

j!E[Y j dj−1

dxj−1g1|2(x|Y )]

+ o(εJ)

= G1(x) +J∑

j=1

(−1)jεj

j!

dj−1

dxj−1(E[Y jg1|2(x|Y )])

+ o(εJ)

= G1(x) +J∑

j=1

(−1)jεj

j!

dj−1

dxj−1[g1(x)E[Y j|X = x]]

+ o(εJ).

ii) Variables in deterministic relationship

Let us now consider the case of a function a(x, ε) = P [X + εc(X) < x],where the direction of expansion Y = c(X) is in a deterministic relationshipwith variable X. Let us introduce a variable Z independent of X with agamma distribution γ(ν, ν), and study the function:

a(x, ε; ν) = P [X + εc(X)Z < x].

The joint distribution of [X,Y ∗ = c(X)Z] is continuous; thus, the results ofpart i) of the proof can be applied. We get:

a(x, ε; ν) = G1(x) +J∑

j=1

(−1)jεj

j!

dj−1

dxj−1(g1(x)E[c(X)jZj|X = x])

+ o(εJ)

= G1(x) +J∑

j=1

(−1)jεj

j!

dj−1

dxj−1[g1(x)c(x)jE(Zj)]

+ o(εJ)

= G1(x) +J∑

j=1

(−1)jεj

j!µj(ν)

dj−1

dxj−1[g1(x)c(x)j]

+ o(εJ),

42

where:

µj(ν) = E(Zj) = (1− 1

ν)(1− 2

ν) . . . (1− j − 1

ν), j = 1, . . . , J.

Since the moments µj(ν), j = 1, . . . , J tend uniformly to 1, and Y ∗ =Zc(X) tends to Y = c(X), when ν tends to infinity, we get :

a(x, ε) = P [X + εc(X) < x]

= limν→∞ a(x, ε; ν)

= G1(x) +J∑

j=1

(−1)jεj

j!

dj−1

dxj−1[g1(x)c(x)j]

+ o(εJ).

QED

2. Application to large portfolio risk

Let us consider the asymptotic expansion (3.2) :

Wn/n = m(F ) + σ(F )X/√

n + O(1/n),

where X is independent of F with distribution N(0, 1) and the term O(1/n)is zero-mean, conditional on F . We have:

an(x) = P [Wn/n < x]

= P [m(F ) + σ(F )X/√

n + O(1/n) < x].

Then, the expansion in Lemma a.1 can be applied at order 2, noting that:

E[O(1/n)|F ] = 0, E[σ(F )X|m(F )] = E[σ(F )|m(F )]E[X] = 0,

E[σ2(F )X2|m(F )] = E[X2]E[σ2(F )|m(F )] = E[σ2(F )|m(F )].

We deduce the following Lemma:

Lemma a.2: We have:

an(x) = P [Wn/n < x]

= P [m(F ) < x] +1

2n

d

dx

g∞(x)E[σ2(F )|m(F ) = x]

+ o(1/n),

43

where g∞ is the pdf of m(F ).

Note that this approximation at order 1/n is exact and not itself approx-imated by the cdf based on a bivariate distribution as in Vasicek (2002).

3. Expansion of the VaR

The expansion of the VaR per individual is deduced from the Bahadur’sexpansion [see Bahadur (1966), or Gagliardini, Gourieroux (2010), Section6.2].

Lemma a.3 (Bahadur’s expansion): Let us consider a sequence of cdf ’sFn tending to a limiting cdf F∞ at uniform rate 1/n:

Fn(x) = F∞(x) + O(1/n).

Let us denote by Qn and Q∞ the associated quantile functions and assumethat the limiting distribution is continuous with density f∞. Then:

Qn(α)−Q∞(α) = −Fn[Q∞(α)]− F∞[Q∞(α)]

f∞[Q∞(α)]+ o(1/n).

Lemma a.3 can be applied to the standardized portfolio risk. The limitingdistribution is the distribution of m(F ) with pdf g∞ and quantile functionV aR∞. By using the expansion of Lemma a.2, we get:

V aRn(α) = V aR∞(α)− 1

2n

1

g∞[V aR∞(α)]

[d

dx

g∞(x)E[σ2(F )|m(F ) = x]

]

x=V aR∞(α)

+o(1/n)

= V aR∞(α)− 1

2n

d log g∞

dx[V aR∞(α)]E[σ2(F )|m(F ) = V aR∞(α)]

+

[d

dxE[σ2(F )|m(F ) = x]

]

x=V aR∞(α)

+ o(1/n).

This is the result in Proposition 1.

44

Appendix 2: GA when m(F ) is monotone increasing

From Proposition 1 we have:

GA(α) = −1

2

(1

g∞(z)

d

dz

g∞(z)E

[σ2(F )|m(F ) = z

])

z=V aR∞(α)

= −1

2

(1

g∞(z)

d

dz

g∞(z)σ2[m−1(z)]

)

z=V aR∞(α)

= −1

2

dm

dF[m−1(z)]

g∞[m−1(z)]

d

dz

g∞[m−1(z)]σ2[m−1(z)]

dm

dF[m−1(z)]

z=V aR∞(α)

= −1

2

1

g∞[m−1(z)]

d

dm−1(z)

g∞[m−1(z)]σ2[m−1(z)]

dm

dF[m−1(z)]

z=V aR∞(α)

= −1

2

1

g∞(f)

d

df

g∞(f)σ2(f)

dm

dF(f)

f=m−1[V aR∞(α)]

.

45

Appendix 4Stochastic probability of default and expected loss given default

In this Appendix we give a detailed derivation of the granularity adjust-ment in the model with stochastic probability of default and expected lossgiven default presented in Section 6.

i) Cross-sectional factor approximation

Let us consider date t and assume that we observe the individual lossesyi,t, i = 1, · · · , n, of the zero-coupon corporate bonds maturing at date t.Equivalently, we observe the default indicator Zi,t = 1lyi,t>0 for all individualcompanies, and the Loss Given Default LGDi,t = yi,t if Zi,t = 1. Thus, themodel is equivalent to a truncated model and the cross-sectional likelihoodconditional on the factor value is:n∏

i=1

h(yi,t|ft) =

[n∏

i=1

fZi,t

1,t (1− f1,t)1−Zi,t

] ∏i:Zi,t=1

Γ(at + bt)

Γ(at)Γ(bt)LGDat−1

i,t (1− LGDi,t)bt−1

= fNt1,t (1− f1,t)

n−Nt

(Γ[(1− γ)/γ]

Γ(at)Γ(bt)

)Nt

∏

i:yi,t>0

(yi,t)

at−1 ∏

i:yi,t>0

(1− yi,t)

bt−1

,

where Nt =n∑

i=1

1lyi,t>0 is the number of defaults at date t, and at =

(1− γ

γ

)f2,t

and bt =

(1− γ

γ

)(1− f2,t). We get the cross-sectional log-likelihood:

n∑i=1

log h(yi,t|ft) = Nt log f1,t + (n−Nt) log(1− f1,t) + Nt log Γ

[1− γ

γ

]

−Nt log Γ

[(1− γ

γ

)f2,t

]−Nt log Γ

[(1− γ

γ

)(1− f2,t)

]

+

[(1− γ

γ

)f2,t − 1

] ∑i:yi,t>0

log yi,t

+

[(1− γ

γ

)(1− f2,t)− 1

] ∑i:yi,t>0

log(1− yi,t)

= L1,n,t(f1,t) + L2,n,t(f2,t), say. (a.2)

46

which is decomposed as the sum of a function of f1,t and a function of f2,t.Therefore, the cross-sectional approximations of the two factors can be com-puted separately, and we get (5.13) and (5.14).

ii) Approximate filtering distribution of ft given In,t

From a multiple factor version of Corollary 5.3 in Gagliardini, Gourieroux(2009), and the log-likelihood decomposition in (a.2), it follows that theapproximate filtering distribution is such that F1,t and F2,t are independentconditional on information In,t at order 1/n, with Gaussian distributions

N

(fl,n,t +

1

nµl,n,t,

1

nJ−1

l,n,t

), for l = 1, 2, where:

µl,n,t = J−1l,n,t

∂ log g

∂fl,t

(fn,t|fn,t−1) +1

2J−2

l,n,tKl,n,t,

Jl,n,t = − 1

n

n∑i=1

∂2 log h

∂f 2l,t

(yi,t|fn,t) = − 1

n

∂2Ll,n,t

∂f 2l,t

(fl,n,t),

Kl,n,t =1

n

n∑i=1

∂3 log h

∂f 3l,t

(yi,t|fn,t) =1

n

∂3Ll,n,t

∂f 3l,t

(fl,n,t).

From (a.2), equations (5.16) and (5.17) follow, as well as K1,n,t = 21− 2f1,n,t

f 21,n,t(1− f1,n,t)2

.

Moreover:

∂ log g

∂fl,t

(ft|ft−1) = − 1

a[A−1(fl,t)]

e′l

[Ω−1(f ∗t − c− Φf ∗t−1)

]+

d log a

dy[A−1(fl,t)]

= − 1

fl,t(1− fl,t)

e′l

[Ω−1(f ∗t − c− Φf ∗t−1)

]+ 1− 2fl,t

,

for l = 1, 2, where A(y) = [1 + exp(−y)]−1 and a(y) = dA(y)/dy. Thenequations (5.15) follow.

iii) A useful lemma

The derivation of the CSA VaR and the GA below uses the next lemma.

47

Lemma a.4: Let X and Y be random variables on [0, 1]. Denote by f(x, y),f2(y), F2(y) and F1|2(x|y) the joint pdf of (X, Y ), the pdf of Y , the cdf of Yand the conditional cdf of X given Y , respectively. Let Z = XY . Then:

(i) P [Z ≤ z] =

∫ 1

z

F1|2(z/y|y)f2(y)dy + F2(z), for any z ∈ [0, 1];

(ii) The pdf of Z is g(z) =

∫ 1

z

1

yf(z/y, y)dy, z ∈ [0, 1];

(iii) E[Y |Z = z] =

∫ 1

z

f(z/y, y)dy

∫ 1

z

1

yf(z/y, y)dy

, for any z ∈ [0, 1].

Proof: (i) We have:

P [Z ≤ z] = EP [Z ≤ z|Y ] = EP [X ≤ z/Y |Y ].

Now:P [X ≤ z/Y |Y ] = 1lz≤Y F1|2(z/Y |Y ) + 1lz>Y .

Thus, we get:

P [Z ≤ z] =

∫ 1

z

F1|2(z/y|y)f2(y)dy + F2(z).

(ii) By differentiating the cdf found in (i) we get:

g(z) =d

dz

(∫ 1

z

F1|2(z/y|y)f2(y)dy + F2(z)

)

=

∫ 1

z

1

yf(z/y|y)f2(y)dy − [

F1|2(z/y|y)f2(y)]y=z

+ f2(z)

=

∫ 1

z

1

yf(z/y, y)dy.

(iii) Let us consider the change of variables from (x, y) to (z, y), where z = xy.

The Jacobian is

∣∣∣∣det

[∂(z, y)

∂(x, y)

]∣∣∣∣ = y. Thus, the joint density of Z and Y is

48

g(z, y) =1

yf(z/y, y), for 0 ≤ z ≤ y ≤ 1. We get:

E[Y |Z = z] =

∫ 1

z

yg(z, y)dy

∫ 1

z

g(z, y)dy

=

∫ 1

z

f(z/y, y)dy

∫ 1

z

1

yf(z/y, y)dy

.

QED

iv) CSA risk measure

Let us first compute function a(w, 0; f). From the dynamic version ofequation (4.13), we have:

m(Ft+1) = F1,t+1F2,t+1. (a.3)

Then:

a(w, 0; ft) = P [m(Ft+1) ≤ w|Ft = ft] = P [F1,t+1F2,t+1 ≤ w|Ft = ft].

We apply Lemma a.4 (i) with X = F2,t+1 and Y = F1,t+1, conditionally onFt = ft. The distribution of F ∗

2,t+1 conditional on F ∗1,t+1 and Ft = ft is:

N

(c2,t +

ρσ2

σ1

(F ∗1,t+1 − c1,t), σ

22(1− ρ2)

),

where c1,t = c1 + Φ11A−1(f1,t) + Φ12A

−1(f2,t) and c2,t = c2 + Φ21A−1(f1,t) +

Φ22A−1(f2,t). Thus, the conditional cdf of F2,t+1 given F1,t+1 = y and Ft = ft

is:

P [F2,t+1 ≤ x|F1,t+1 = y, Ft = ft] = Φ

A−1(x)− c2,t − ρσ2

σ1

(A−1(y)− c1,t)

σ2

√1− ρ2

,

for x, y ∈ (0, 1). The cdf and the pdf of F1,t+1 conditional on Ft = ft are:

F (y|ft) = Φ

(A−1(y)− c1,t

σ1

), f(y|ft) =

1

σ1

ϕ

(A−1(y)− c1,t

σ1

)1

a[A−1(y)],

49

respectively, where a(y) = dA(y)/dy. Thus, from Lemma a.4 we get:

a(w, 0; ft) = Φ

(A−1(w)− c1,t

σ1

)

−∫ 1

w

Φ

A−1(w/y)− c2,t − ρσ2

σ1

(A−1(y)− c1,t)

σ2

√1− ρ2

· 1

σ1

ϕ

(A−1(y)− c1,t

σ1

)1

a[A−1(y)]dy.

By using that a[A−1(y)] = y(1− y) and A−1(w/y) = log[w/(y − w)], Propo-sition 6 i) follows.

v) Granularity adjustment

Let us first compute the density g∞(.; ft). We use equation (a.3) andapply Lemma a.4 (ii) with X = F1,t+1 and Y = F2,t+1 conditionally onFt = ft. The joint density of (X,Y ) conditionally on Ft = ft is:

f(x, y) =1

σ1σ2

ϕ

(A−1(x)− c1,t

σ1

,A−1(y)− c2,t

σ2

; ρ

)1

a[A−1(x)]a[A−1(y)],

where ϕ(., .; ρ) denotes the pdf of a bivariate standard Gaussian distributionwith correlation parameter ρ. We get:

g∞(w; ft) =

∫ 1

w

1

y

1

σ1σ2

ϕ

(A−1 (w/y)− c1,t

σ1

,A−1(y)− c2,t

σ2

; ρ

)

1

a[A−1 (w/y)]a[A−1(y)]dy.

Let us now compute E[σ2(Ft+1)|m(Ft+1) = w, Ft = ft]. From the dy-namic version of equation (4.14), we have:

σ(Ft+1) = γF2,t+1(1− F2,t+1)F1,t+1 + F1,t+1(1− F1,t+1)F22,t+1,

By replacing F1,t+1F2,t+1 with m(Ft+1) in equation (4.14), function σ2(Ft+1)can be rewritten as:

σ2(Ft+1) = γ(F1,t+1F2,t+1)(1− F2,t+1) + (F1,t+1F2,t+1)(F2,t+1 − F1,t+1F2,t+1)

= m(Ft+1)[γ −m(Ft+1)] + (1− γ)m(Ft+1)F2,t+1.

50

Thus:

E[σ2(Ft+1)|m(Ft+1) = w,Ft = ft] = w(γ − w)

+(1− γ)wE[F2,t+1|F1,t+1F2,t+1 = w, Ft = ft].

(a.4)

From Lemma a.4 (iii) with X = F1,t+1 and Y = F2,t+1, conditionally onFt = ft we get:

E[F2,t+1|m(Ft+1) = w, Ft = ft] =b(w, 1; ft)

g∞(w; ft),

where:

b(w, 1; ft) =

∫ 1

w

1

σ1σ2

ϕ

(A−1 (w/y)− µ1,t

σ1

,A−1(y)− µ2,t

σ2

; ρ

)1

a[A−1 (w/y)]a[A−1(y)]dy.

Then, from equation (a.4) we get:

E[σ2(Ft+1)|m(Ft+1) = w, Ft = ft] = w(γ − w) + (1− γ)wb(w, 1; ft)

g∞(w; ft).

By using a[A−1 (w/y)]a[A−1(y)] = w(1 − y)(1 − w)(1 − w/y), Proposition 6ii) follows.

51