CROSSCUTTING AREAS Conditional Value-at-Risk and Average ... · The notation Y Y0 means that Y4Š5...

OPERATIONS RESEARCHVol. 60, No. 4, July–August 2012, pp. 739–756ISSN 0030-364X (print) � ISSN 1526-5463 (online) http://dx.doi.org/10.1287/opre.1120.1072

© 2012 INFORMS

CROSSCUTTING AREAS

Conditional Value-at-Risk and Average Value-at-Risk:Estimation and Asymptotics

So Yeon ChunMcDonough School of Business, Georgetown University, Washington, DC 20057, [email protected]

Alexander ShapiroSchool of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, [email protected]

Stan UryasevDepartment of Industrial and Systems Engineering, University of Florida, Gainesville, Florida 32611, [email protected]

We discuss linear regression approaches to the estimation of law-invariant conditional risk measures. Two estimationprocedures are considered and compared; one is based on residual analysis of the standard least-squares method, and theother is in the spirit of the M-estimation approach used in robust statistics. In particular, value-at-risk and average value-at-risk measures are discussed in detail. Large sample statistical inference of the estimators is derived. Furthermore, finitesample properties of the proposed estimators are investigated and compared with theoretical derivations in an extensiveMonte Carlo study. Empirical results on the real data (different financial asset classes) are also provided to illustrate theperformance of the estimators.

Subject classifications : value-at-risk; average value-at-risk; linear regression; least-squares residuals; M-estimators;quantile regression; conditional risk measures; law-invariant risk measures; statistical inference.

Area of review : Financial Engineering.History : Received April 2011; revisions received August 2011, October 2011; accepted December 2011.

1. IntroductionIn the financial industry, sell-side analysts periodically pub-lish recommendations of underlying securities with tar-get prices (e.g., the Goldman Sachs Conviction Buy List).These recommendations reflect specific economic condi-tions and influence investors’ decisions and thus pricemovements. However, this type of analysis does not pro-vide risk measures associated with underlying companies.We see similar phenomena in buy-side analysis as well.Each analyst or team covers different sectors (e.g., the air-line industry versus semiconductor industry) and typicallymakes separate recommendations for the portfolio man-agers without associated risk measures. However, the riskmeasure of the companies that are covered is one of themost important factors for investment decision making. Inthis paper, we consider ways to estimate risk measures fora single asset at given market conditions. This informa-tion could be useful for investors and portfolio managersto compare prospective securities and pick the best ones.For example, when portfolio managers expect the crude oilprice to spike (due to inflation or geo-political conflicts),they could select securities less sensitive to oil price move-ments in the airline industry.

To formalize our discussion, let us introduce the follow-ing setting. Let 4ì1F5 be a measurable space equipped withprobability measure P . A measurable function Y 2 ì→� is

called a random variable. With random variable Y , we asso-ciate a number �4Y 5, which we refer to as a risk measure.We assume that “smaller is better,” i.e., between two pos-sible realizations of random data, we prefer the one with asmaller value of �4 · 5. The term “risk measure” is somewhatunfortunate since it can be confused with the term prob-ability measure. Moreover, in applications, one often triesto reach a compromise between minimizing the expecta-tion (i.e., minimizing on average) and controlling the associ-ated risk. Thus, some authors use the term “mean-risk mea-sure,” or “acceptability functional” (e.g., Pflug and Römisch2007). For historical reasons, we use here the “risk measure”terminology. Formally, risk measure is a function �2 Y →

� defined on an appropriate space Y of random variables.For example, in some applications, it is natural to use thespace Y = Lp4ì1F1 P5, with p ∈ 611�5, of random vari-ables having finite pth-order moments.

It was suggested in Artzner et al. (1999) that a “good”risk measure should have the following properties (axioms),and such risk measures were called coherent:

(A1) Monotonicity: If Y 1Y ′ ∈ Y and Y � Y ′, then�4Y 5¾ �4Y ′5.

(A2) Convexity:

�4tY + 41 − t5Y ′5¶ t�4Y 5+ 41 − t5�4Y ′5

for all Y 1Y ′ ∈Y and all t ∈ 60117.

739

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Chun, Shapiro, and Uryasev: Conditional Value-at-Risk and Average Value-at-Risk740 Operations Research 60(4), pp. 739–756, © 2012 INFORMS

(A3) T ranslation equivariance: If a ∈� and Y ∈Y, then�4Y + a5= �4Y 5+ a.

(A4) Positive homogeneity: If t ¾ 0 and Y ∈ Y, then�4tY 5= t�4Y 5.The notation Y � Y ′ means that Y 4�5 ¾ Y ′4�5 for a.e.� ∈ ì. We refer, e.g., to Detlefsen and Scandolo (2005),Weber (2006), and Föllmer and Schied (2011) for furtherdiscussion of mathematical properties of risk measures.

An important example of risk measures is the value-at-risk measure

V@R�4Y 5= inf8t2 FY 4t5¾ �91 (1)

where � ∈ 40115 and FY 4t5 = Pr4Y ¶ t5 is the cumulativedistribution function (cdf) of Y , i.e., V@R�4Y 5 = F −1

Y 4�5is the left-side �-quantile of the distribution of Y . Thisrisk measure satisfies axioms (A1), (A3), and (A4), but not(A2), and hence is not coherent. Another important exam-ple is the so-called average value-at-risk measure, whichcan be defined as

AV@R�4Y 5= inft∈�

8t + 41 −�5−1Ɛ6Y − t7+9 (2)

(cf., Rockafellar and Uryasev 2002), or equivalently

AV@R�4Y 5=1

1 −�

∫ 1

�V@R�4Y 5d�0 (3)

Note that AV@R�4Y 5 is finite iff Ɛ6Y 7+ < �. Therefore,it is natural to use the space Y = L14ì1F1 P5 of randomvariables having finite first order moment for the AV@R�

risk measure. The average value-at-risk measure is alsocalled the conditional value-at-risk or expected shortfallmeasure. (Because we discuss here “conditional” variantsof risk measures, we use the average value-at-risk ratherthan conditional value-at-risk terminology.)

The value-at-risk and average value-at-risk measures arewidely used to measure and manage risk in the financialindustry (see, e.g., Jorion 2003, Duffie and Singleton 2003,Gaglianone et al. 2011, for the financial background andvarious applications). Note that in the above two exam-ples, risk measures are functions of the distribution of Y .Such risk measures are called law invariant. Law-invariantrisk measures have been studied extensively in the finan-cial risk management literature (e.g., Acerbi 2002, Frey andMcNeil 2002, Scaillet 2004, Fermanian and Scaillet 2005,Chen and Tang 2005, Zhu and Fukushima 2009, Jacksonand Perraudin 2000, Berkowitz et al. 2002, Bluhm et al.2002, and references therein). Sometimes, we write a law-invariant risk measure as a function �4F 5 of cdf F .

Now let us consider a situation where there existsinformation composed of economic and market variablesX11 0 0 0 1Xk that can be considered as a set of predictorsfor a variable of interest Y . In that case, one can be inter-ested in estimation of a risk measure of Y conditionalon observed values of predictors X11 0 0 0 1Xk. For example,

suppose we want to measure (predict) the risk of a sin-gle asset given specific economic conditions represented bymarket index and interest rates. Then, for a random vectorX = 4X11 0 0 0 1Xk5

T of relevant predictors, the conditionalversion of a law-invariant risk measure �, denoted �4Y �X5or ��X4Y 5, is obtained by applying � to the conditional dis-tribution of Y given X. In particular, V@R�4Y � X5 is the�-quantile of the conditional distribution of Y given X, and

AV@R�4Y �X5=1

1 −�

∫ 1

�V@R�4Y �X5d�0 (4)

Recently, several researchers have paid attention to esti-mation of the conditional risk measures. For the conditionalvalue-at-risk, Chernozhukov and Umantsev (2001) used apolynomial type regression quantile model, and Engle andManganelli (2004) proposed the model that specifies theevolution of the quantile over time using a special type ofautoregressive processes. In both models, unknown param-eters were estimated by minimizing the regression quantileloss function. For conditional average value-at-risk, Scaillet(2005) and Cai and Wang (2008) utilized Nadaraya-Watson(NW) type nonparametric double kernel estimation, whilePeracchi and Tanase (2008) and Leorato et al. (2012) usedthe semiparametric method.

In this paper, we discuss procedures for estimation ofconditional risk measures. Especially, we will pay atten-tion to estimation of conditional value-at-risk and aver-age value-at-risk measures. We assume the following linearmodel (linear regression):

Y = �0 +ÂTX+ �1 (5)

where �0 and Â= 4�11 0 0 0 1�k5T are (unknown) parameters

of the model and the error (noise) random variable � isassumed to be independent of random vector X. Meaningof the model (5) is that there is a true (population) value�∗

01Â∗ of the respective parameters for which (5) holds. We

will sometimes write this explicitly and sometimes suppressthis in the notation.

Let �4 · 5 be a law-invariant risk measure satisfyingaxiom (A3) (translation equivariance), and ��X4 · 5 be itsconditional analogue. Note that because of the indepen-dence of � and X, it follows that ��X4�5 = �4�5. Togetherwith axiom (A3), this implies

��X4Y 5= ��X4�0 +ÂTX+ �5= �0 +ÂTX+��X4�5

= �0 +ÂTX+�4�50 (6)

Because �0 + �4�5 = �4� + �05, we can set �4�5 = 0 byadding a constant to the error term. In that case, for thetrue values of the parameters, we have ��X4Y 5= �∗

0 +Â∗TX.Hence, the question is how to estimate these (true) values�∗

01 Â∗ of the respective parameters.

This paper is organized as follows. In §2, we describetwo different estimation procedures for the conditional

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Chun, Shapiro, and Uryasev: Conditional Value-at-Risk and Average Value-at-RiskOperations Research 60(4), pp. 739–756, © 2012 INFORMS 741

risk measures; one is based on residuals of the least-squares estimation procedure and the other is based on theM-estimation approach. Asymptotic properties of both esti-mators are provided in §3. In §4, we investigate the finitesample and asymptotic properties of the considered esti-mators. We present Monte Carlo simulation results underdifferent distribution assumptions of the error term. Later,we illustrate the performance of different methods on thereal data (different financial asset classes) in §5. Finally,§6 gives some conclusion remarks and suggestions forfuture research directions.

2. Basic Estimation ProceduresSuppose that we have N observations (data points) 4Yi1Xi5,i = 11 0 0 0 1N , which satisfy the linear regression model(5), i.e.,

Yi = �0 +ÂTXi + �i1 i = 11 0 0 0 1N 0 (7)

We assume that: (i) Xi, i = 11 0 0 0 1N , are iid (independentidentically distributed) random vectors and write X for ran-dom vector having the same distribution as Xi, (ii) theerrors �11 0 0 0 1 �N are iid with finite second-order momentsand independent of Xi. We denote by �2 = Var6�i7 thecommon variance of the error terms.

There are two basic approaches to estimation of the truevalues of �0 and Â. One approach is to apply the standardleast-squares (LS) estimation procedure and then to makean adjustment of the estimate of the intercept parameter �0.That is, let �0 and Â be the least-squares estimators of therespective parameters of the linear model (7) and

ei 2= Yi − �0 − ÂTXi1 i = 11 0 0 0 1N 1 (8)

be the corresponding residuals. By the standard theory ofthe LS method, we have that �0 and Â are unbiased esti-mators of the respective parameters of the linear model (5)provided Ɛ6�7 = 0. Therefore, we need to make the cor-rection �0 + �4�5 of the intercept estimator. If we knewthe true values �11 0 0 0 1 �N of the error term, we could esti-mate �4�5 by replacing the cdf F� of � by its empiricalestimate F�1N associated with �11 0 0 0 1 �N , i.e., to estimate�4F�5 by �4F�1N 5. Because true values of the error termare unknown, it is a natural idea to replace �11 0 0 0 1 �N bythe residual values e11 0 0 0 1 eN . Hence, we use the estima-tor �0 + �4Fe1N 5, where Fe1N is the empirical cdf of theresidual values, i.e., Fe1N is the cdf of the probability dis-tribution assigning mass 1/N to each point ei, i = 11 0 0 0 1N(see §3.1 for further discussion). We refer to this estimationapproach as the least-squares residuals (LSR) method.

An alternative approach is based on the following idea.Suppose that we can construct a function h4y1 �5 ofy ∈ � and � ∈ �, convex in �, such that the mini-mizer of ƐF 6h4Y 1 �57 will be equal to �4F 5, i.e., �4F 5 =

arg min� ƐF 6h4Y 1 �57. Because �4Y + a5 = �4Y 5 + a forany a ∈ �, it follows that the function h4y1 �5 should be

of the form h4y1 �5 = �4y − �5 for some convex function�2 �→�. We refer to �4 · 5 as the error function. There-fore, we need to construct an error function such that

�4F 5= arg min�

ƐF 6�4Y − �570 (9)

This is equivalent to solving the equation

ƐF 6�4Y − �57= 01 (10)

where �4t5 2= �′4t5. Note that the error function �4 · 5could be nondifferentiable, in which case the correspond-ing derivative function �4 · 5 is discontinuous. That is, thefunction �4 · 5 is monotonically nondecreasing.

The corresponding estimators �0 and Â are taken as solu-tions of the optimization problem

Min�01Â

N∑

i=1

�4Yi −�0 −ÂTXi50 (11)

In the statistics literature, such estimators are calledM-estimators (the terminology that we will follow), andfor an appropriate choice of the error function this is theapproach of robust regression (Huber 1981). For the V@R�

risk measure, the error function is readily available (recallthat 6t7+ = max801 t9):

�4t5 2= �6t7+ + 41 −�56−t7+0 (12)

The corresponding robust regression approach is known asthe quantile regression method (cf. Koenker 2005).

For coherent risk measures, the situation is more deli-cate. Let us make the following observations. Suppose therepresentation (9) holds. Let F1 and F2 be two cdf suchthat �4F15 = �4F25 = �. Then it follows by (9) (by (10))that �4tF1 + 41 − t5F25 = � for any t ∈ 60117. This is astrong necessary condition for existence of a representationof the form (9). It certainly doesn’t hold for the AV@R�,� ∈ 40115, risk measure (see Gneiting 2011, proof of theo-rem 11, p. 760).

This shows that for general coherent risk measures, pos-sibility of constructing the corresponding M-estimators isexceptional, and such estimators certainly do not exist forthe AV@R� risk measure. Nevertheless, it is possible toconstruct the following approximations (this construction isessentially due to Rockafellar et al. 2008).

Proposition 1. Let �j 2 � → �, j = 11 0 0 0 1 r , be convexfunctions, �j ∈� be such that

∑rj=1 �j = 1 and

E4Y 5 2= infÒ∈�r

{

Ɛ

[ r∑

j=1

�j4Y − �j5

]

2r∑

j=1

�j�j = 0}

0 (13)

Moreover, let Sj4Y 5 be a minimizer of Ɛ6�j4Y − �57 over� ∈ �. Then S4Y 5 2=

∑rj=1 �jSj4Y 5 is a minimizer of

E4Y − �5 over � ∈�.

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Proof. Consider the problem

Min�1Ò

Ɛ

[ r∑

j=1

�j4Y − �− �j5

]

1 s.t.r∑

j=1

�j�j = 00 (14)

By making change of variables �j = � + �j , j = 11 0 0 0 1 r ,we can write this problem in the form

Min�1Ç

Ɛ

[ r∑

j=1

�j4Y −�j5

]

1 s.t.r∑

j=1

�j�j = �0 (15)

Because Sj4Y 5 is a minimizer of Ɛ6�j4Y −�j57, it followsthat �j = Sj4Y 5, i = 11 0 0 0 1 r , � = S4Y 5 is an optimal solu-tion of problem (15). This completes the proof. �

In particular, we can consider functions �j4 · 5 of theform (12), i.e.,

�j4t5 2= �j 6t7+ + 41 −�j56−t7+1 (16)

for some �j ∈ 40115, j = 11 0 0 0 1 r . Then Sj4Y 5 =

V@R�j4Y 5, and hence the risk measure

∑rj=1 �jV@R�j

4Y 5is a minimizer of E4Y − �5. We can view

∑rj=1 �j ·

V@R�j4Y 5 as a discretization of the integral 41/41 −�55 ·

∫ 1�

V@R�4Y 5d� if we set ã 2= 41 −�5/r and take

�j 2= 41 −�5−1ã1 �j 2= �+ 4j − 0055ã1

j = 11 0 0 0 1 r0 (17)

For this choice of �j , �j , and by formula (3) we have that

AV@R�4Y 5=1

1 −�

∫ 1

�V@R�4Y 5d�

≈

r∑

j=1

�jV@R�j4Y 5= S4Y 50 (18)

Consider now the problem

Min�01Â

E4Y −�0 −ÂTX50 (19)

By the definition (13) of E4 · 5, we can write this problemin the following equivalent form:

MinÒ1�01Â

Ɛ

[ r∑

j=1

�j4Y −�0 −ÂTX−�j5

]

s.t.r∑

j=1

�j�j =00 (20)

The so-called sample average approximation (SAA) of thisproblem is

MinÒ1�01Â

1N

N∑

i=1

r∑

j=1

�j4Yi −�0 −ÂTXi − �j51

s0t0r∑

j=1

�j�j = 00

(21)

The above problem (21) can be formulated as a linear pro-gramming problem. Following Rockafellar et al. (2008), weconsider the following estimators.

Mixed quantile estimator for AV@R�4Y � x5We refer to �0 + ÂTx as the mixed quantile estimator ofAV@R�4Y � x5, where 4Ò1 �01 Â5 is an optimal solution ofproblem (21).

This idea can be extended to a larger class of law-invariant risk measures. For example, consider a riskmeasure

�4Y 5 2= cƐ6Y 7+ 41 − c5AV@R�4Y 5 (22)

for some constants c ∈ 60117 and � ∈ 40115. Recall that theminimizer of Ɛ64Y − t527 is t∗ = Ɛ6Y 7. Therefore, by takingfunction �04t5 2= t2, functions �j4t5 of the form (16), �j ,and �j given in (17), we can construct the correspondingerror function:

E4Y 5 2= infÒ∈�r+1

{

Ɛ

[

�04Y − �05+

r∑

j=1

�j4Y − �j5

]

2

c�0 +

r∑

j=1

41 − c5�j�j = 0}

0 (23)

As another example, consider risk measures of the form

�4Y 5 2=∫ 1

0AV@R�4Y 5d�4�51 (24)

where � is a probability measure on the interval 60115. Bya result due to Kusuoka (2001), this measure forms a classof the comonotone law-invariant coherent risk measures.By (3), we can write such risk measure as

�4Y 5=

∫ 1

0

∫ 1

�41 −�5−1V@R�4Y 5d� d�4�5

=

∫ 1

0w4�5V@R�4Y 5d�1 (25)

where w4�5 2=∫ �

0 41 −�5−1 d�4�5. Such risk measures arealso called spectral risk measures (Acerbi 2002). By mak-ing a discretization of the above integral (25), we can pro-ceed as above.

It could be remarked here that while the LSR approachis general, the approach based on mixing M-estimatorsis somewhat restrictive. Constructing an appropriate errorfunction for a particular risk measure could be veryinvolved.

3. Large Sample Statistical InferenceIn the previous section, we formulated two approaches,the LSR estimators and mixed M-estimators, to estima-tion of the true (population) values of parameters �∗

01 Â∗

of the linear model (5), such that �4�5= 0. For the V@R�

risk measure, the corresponding M-estimators �0 and Â are

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taken as solutions of the optimization problem (11), withthe error function (12), and referred to as the quantileregression estimators. For the AV@R� risk measure andmore generally comonotone risk measures of the form (25),we constructed the corresponding mixed quantile estimatorsÒ1 �01 Â. In this section, we discuss statistical properties ofthese estimators. In particular, we address the question ofwhich of these two estimation procedures is more efficientby computing corresponding asymptotic variances.

3.1. Statistical Inference of Least-SquaresResidual Estimators

The linear model (7) can be written as

Y=�6�03Â7+ Å1 (26)

where Y = 4Y11 0 0 0 1 YN 5T is N × 1 vector of responses,

� is N × 4k + 15 data matrix of predictor variables withrows 411XT

i 5, i = 11 0 0 0 1N (i.e., first column of � is col-umn of ones), Â= 4�11 0 0 0 1�k5

T vector of parameters, andÅ = 4�11 0 0 0 1 �N 5

T is N × 1 vector of errors. By 6�03Â7,we denote 4k + 15 × 1 vector 4�01Â

T5T. We assume thatthe conditions (i) and (ii), specified at the beginning of §2,hold. It is also possible to view data points Xi as determin-istic. In that case, we assume that � has full column rankk+ 1.

Let �0 and Â be the least-squares estimators of therespective parameters of the linear model (7). Recall thatthese estimators are given by 6�03 Â7= 4�T�5−1�TY, vec-tor of residuals e 2=Y−�6�01 Â7 is given by

e= 4IN −H5Y= 4IN −H5Å1

where IN is the N × N identity matrix, and H =

�4�T�5−1�T is the so-called hat matrix. Note thattrace4H5= k+ 1 and we have that

�i − ei = 613XTi 74�

T�5−1�TÅ1 i = 11 0 0 0 1N 0 (27)

If we knew errors �11 0 0 0 1 �N , we could estimate �4�5by the corresponding sample estimate based on the empir-ical cdf

F�1N 4 · 5=N−1N∑

i=1

6�i1�54 · 51 (28)

where A4 · 5 denotes the indicator function of set A. How-ever, the true values of the errors are unknown. Therefore,in the LSR approach we replace them by the residuals com-puted by the least-squares method and hence estimate �4�5by employing the respective empirical cdf Fe1N 4 · 5 insteadof F�1N 4 · 5.

The first natural question is whether the LSR estimatorsare consistent, i.e., converge w.p.1 to their true values asthe sample size N tends to infinity. It is well known thatunder the specified assumptions, the LS estimators �0 and

Â are consistent, with �0 being consistent under the condi-tion Ɛ6�7= 0. The question of consistency of empirical esti-mates of law-invariant coherent risk measures was studiedin Wozabal and Wozabal (2009). It was shown that undermild regularity conditions, such estimators are consistent.In particular, the consistency holds for the comonotone riskmeasures of the form (25), i.e., �4F�1N 5 converges w.p.1to �4F�5 as N → �. It is also possible to show that thedifference �4F�1N 5−�4Fe1N 5 tends w.p.1 to zero and hence�4Fe1N 5 converges w.p.1 to �4F�5 as well. A rigorous proofof this could be very technical and will be beyond the scopeof this paper.

We have that the LS estimator 6�03 Â7 asymptotically hasnormal distribution with the asymptotic covariance matrixN−1�2ì−1, where Ì 2= Ɛ6X7, è 2= Ɛ6XXT7 and

ì 2=

[

1 ÌT

Ì è

]

0

Consequently, for a given x, the estimate �0 + xTÂ asymp-totically has normal distribution with the asymptotic vari-ance N−1�2613xT7ì−1613xT7T.

We also have that random vectors 4�01 Â5 and e areuncorrelated. Therefore, if errors �i have normal distribu-tion, then vectors 4�01 Â5 and e jointly have a multivari-ate normal distribution, and these vectors are independent.Consequently, �0 + xTÂ and �4Fe1N 5 are independent. Fornonnormal distribution, this independence holds asymptot-ically, and thus asymptotically �0 + xTÂ and �4Fe1N 5 areuncorrelated.

Asymptotics of empirical estimators of law invariantcoherent risk measures were studied in Pflug and Wozabal(2010) and Shapiro et al. (2009, §6.5). Derivation of theasymptotic variance of �4F�1N 5, for a general law-invariantrisk measure, could be very complex. Let us consider twoimportant cases of the V@R� and AV@R� risk measures.We give (below) a summary of basic results; for a moretechnical discussion we refer to the appendix.

In case of � 2= V@R�, the LSR estimate of V@R�4�5becomes

V@R�4e5 2= F −1e1N 4�5= e4�N��51 (29)

where e415 ¶1 · · · 1¶ e4N 5 are order statistics (i.e., num-bers e11 0 0 0 1 eN arranged in the increasing order) and �a�

denotes the smallest integer ¾ a. Suppose that the cdf F�4 · 5has nonzero density f�4 · 5= F ′

�4 · 5 at F −1� 4�5 and let

�2 2=�41 −�5

6f�4F−1� 4�5572

0 (30)

LSR estimator of V@R�4Y � x5Consider the LSR estimator �0 + xTÂ + V@R�4e5of V@R�4Y � x5. Suppose that the set of population�-quantiles is a singleton. Then the LSR estimator is a

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consistent estimator of V@R�4Y � x5, and the asymptoticvariance of this estimator can be approximated by

N−14�2+�2613xT7ì−1613xT7T50 (31)

Detailed derivation of the above asymptotics is discussedin Appendix A.

For the � 2= AV@R� risk measure, the LSR estimate ofAV@R�4�5 is given by

AV@R�4e5

= inft∈�

{

t +1

41 −�5N

N∑

i=1

6ei − t7+

}

= V@R�4e5+1

41 −�5N

N∑

i=1

[

ei − V@R�4e5]

+

= e4�N��5 +1

41 −�5N

N∑

i=�N��+1

4e4i5 − e4�N��550 (32)

LSR estimator of AV@R�4Y � x5Consider the LSR estimator �0 + xTÂ + AV@R�4e5 ofAV@R�4Y � x5. This estimator is consistent, and its asymp-totic variance is given by

N−1(

�2+�2613xT7ì−1613xT7T

)

1 (33)

where �2 2= 41 −�5−2Var46�− V@R�4�57+5,

ì 2=

[

1 ÌT

Ì è

]

1

Ì 2= Ɛ6X7 and è 2= Ɛ6XXT7.

The above asymptotics are discussed in Appendix B.

Remark 1. It should be remembered that the aboveapproximate variances are asymptotic results. Suppose forthe moment that N < 41 − �5−1. Then �N�� = N , andhence V@R�4�5 = max8�11 0 0 0 1 �N 9. Consequently 6�i −V@R�4�57+ = 0 for all i = 11 0 0 0 1N , and hence

AV@R�4�5= V@R�4�5= max8�11 0 0 0 1 �N 90

In that case, the above asymptotics are inappropriate. Forthese asymptotics to be reasonable, N should be signifi-cantly bigger than 41 −�5−1.

The LSR approach can be easily applied to a consider-ably larger class of law-invariant risk measures. For exam-ple, let us consider the entropic risk measure �4Y 5 2=�−1 logƐ6e�Y 7, where � > 0 is a positive constant. Thisrisk measure satisfies axioms (A1)–(A3), but it is not posi-tively homogeneous (see Giesecke and Weber 2008 for thegeneral discussion of utility-based shortfall risk includingentropic risk measure). The empirical estimate of �4�5 is

�4F�1N 5= �−1 log(

N−1N∑

i=1

e��i)

0 (34)

Of course, as discussed above, the errors �i should bereplaced by the respective residuals ei in the constructionof the corresponding LSR estimators. By using lineariza-tions e�� = 1 +��+ o4��5 and log41 + x5= x+ o4x5, weobtain that N 1/26�4F�1N 5− �4�57 converges in distributionto normal with zero mean and variance �2 (by the Deltatheorem).

3.2. Statistical Inference of Quantile andMixed Quantile Estimators

As discussed in §2, the quantile regression is a particularcase of the M-estimation method with the error function�4 · 5 of the form (12). By the law of large numbers (LLN),we have that N−1 times the objective function in (11)converges (pointwise) w.p.1 to the function ë4�01Â5 2=Ɛ6�4Y −�0 −ÂTX57. We also have

ë4�01Â5= Ɛ[

�4�∗

0 +Â∗TX+ �−�0 −ÂTX5]

= Ɛ[

�4�− 4�0 −�∗

05− 4Â−Â∗5TX5]

0 (35)

Under mild regularity conditions, derivatives ofë4�01Â5 can be taken inside the integral (expectation) andhence

ï�0ë4�01Â5=Ɛ

[

ï�0�4�−4�0 −�∗

05−4Â−Â∗5TX5]

=−Ɛ[

�′4�−4�0 −�∗

05−4Â−Â∗5TX5]

1 (36)

ïÂë4�01Â5=Ɛ[

ïÂ�4�−4�0 −�∗

05−4Â−Â∗5TX5]

=−Ɛ[

�′4�−4�0 −�∗

05−4Â−Â∗5TX5X]

0 (37)

Because � and X are independent, we obtain that deriva-tives of ë4�01Â5 are zeros at 4�∗

01Â∗5 if the following

condition holds:

Ɛ6�′4�57= 00 (38)

Because function ë4 · 1 · 5 is convex, it follows that if con-dition (38) holds, then ë4 · 1 · 5 attains its minimum at4�∗

01Â∗5. If the minimizer 4�∗

01Â∗5 is unique, then the

estimator 4�01 Â5 converges w.p.1 to the population value4�∗

01Â∗5 as N → �, i.e., 4�01 Â5 is a consistent estima-

tor of 4�∗01Â

∗5 (cf. Huber 1981). That is, (38) is the basiccondition for consistency of 4�01 Â5.

For the error function (12) of the quantile regression, wehave

�′4t5=

{

�− 1 if t < 01

� if t > 00(39)

(Note that here the error function �4t5 is not differentiableat t = 0, and its derivative �′4t5 is discontinuous at t = 0.Nevertheless, all arguments can go through provided thatthe error term has a continuous distribution.) Consequently,

Ɛ6�′4�57=4�−15F�405+�41−F�4055=�−F�4051 (40)

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and hence condition (38) holds iff F�405 = �, or equiva-lently F −1

� 4�5= 0 provided this quantile is unique. In thatcase, the estimator 4�01 Â5 is consistent if the populationvalue �∗

0 is normalized such that V@R�4�5 = 0. That is,for this error function, �0 + ÂTx is a consistent estimatorof the conditional Value-at-Risk V@R�4Y � x5 of Y givenX= x.

It is also possible to derive asymptotics of the estimator4�01 Â5. That is, suppose that the cdf F�4 · 5 has nonzerodensity f�4 · 5= F ′

�4 · 5 at F −1� 4�5 and consider �2 defined in

(30). Then N 1/26�0 −�∗03 Â−Â∗7 converges in distribution

to normal with zero mean vector and covariance matrix (cf.,Koenker 2005)

�2613xT7ì−1613xT7T1 (41)

i.e., N−1 times the matrix given in (41) is the asymptoticcovariance matrix of 6�03 Â7.

Remark 2. Note that by LLN, we have that N−1∑Ni=1 Xi

and N−1∑Ni=1 XiX

Ti converge w.p.1 as N → � to the vec-

tor Ì and matrix è, respectively, and that è−ÌÌT is thecovariance matrix of X. In case of deterministic Xi, wesimply define vector Ì and matrix è as the respective lim-its of N−1∑N

i=1 Xi and N−1∑Ni=1 XiX

Ti , assuming that such

limits exist. It follows then that N−1�T�→ì as N → �.

The mixed quantile estimator �0 + ÂTx can be justifiedby the following arguments. We have that an optimal solu-tion 4Ò1 �01 Â5 of problem (21) converges w.p.1 as N →

� to the optimal solution 4Ò?1�?01Â

?5 of problem (20),

Figure 1. Normal Q–Q plot for different error distributions.

–4 –2 0 2 4–25

–20

–15

–10

–5

0

5

10

15

20

Standard normal quantiles

Qua

ntile

s of

inpu

t sam

ple

(a) N (0, 1) vs. t (3)

–4 –2 0 2 4–10

–5

0

5

10

15


Qua

ntile

s of

inpu

t sam

ple

(b) N (0, 1) vs. CN (1, 9)

–4 –2 0 2 4–10

0

10

20

30

40

50

60

70


Qua

ntile

s of

inpu

t sam

ple

(c) N (0, 1) vs. LN (0, 1)

provided (20) has unique optimal solution. Because of thelinear model (5), we can write problem (20) as

MinÒ1�01Â

Ɛ

[ r∑

j=1

��j4�+�∗

0 −�0 + 4Â∗−Â5TX− �j5

]

1

s0t0r∑

j=1

�j�j = 01

(42)

where �∗0 and Â∗ are population values of the parameters.

Similar to the proof of Proposition 1, by making change ofvariables �j = �0 + �j , j = 11 0 0 0 1 r , we can write problem(42) in the following equivalent form:

MinÇ1�01Â

Ɛ

[ r∑

j=1

��j4�+�∗

0 −�j + 4Â∗−Â5TX5

]

1

s0t0r∑

j=1

�j�j = �00 (43)

It follows that if

r∑

j=1

�jV@R�j4�5= 01 (44)

then 4�?01Â

?5 = 4�∗01Â

∗5. That is, �0 + ÂTx is a consistentestimator of

∑rj=1 �jV@R�j

4Y � x5. Consequently, for �j

and �j given in (17), we can use �0 + ÂTx as an approxi-mation of AV@R�4Y � x5.

Asymptotics of the mixed quantile estimators aremore complicated. These asymptotics are discussed inAppendix C.

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Figure 2. Conditional VaR and AVaR: True vs. estimated (errors ∼ CN41195, �= 0095, N = 11000).

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0–2

–1

0

1

2

3

4

5

6

7

xk xk

Con

ditio

nal V

aRMAE(QVaR) = 0.4771, MAE(RVaR) = 0.2145

TrueQVaRRVaR

(a) Conditional VaR

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

7

8

Con

ditio

nal A

VaR

MAE(QAVaR) = 0.6336, MAE(RAVaR) = 0.2466

(b) Conditional AVaR

4. Simulation StudyTo illustrate the performance of the considered estima-tors, we perform the Monte Carlo simulations where errors(innovations) in linear model (7) are generated from the fol-lowing different distributions: (1) standard normal (denotedas N40115); (2) Student’s t distribution with 3 degrees offreedom (denoted as t435); (3) skewed contaminated nor-mal, where standard normal is contaminated with 20%N41195 errors (denoted as CN41195); (4) log-normal withparameter 0 and 1 (denoted as LN40115). Note that errordistributions (2)–(4) are heavy-tailed in contrast to the nor-mal errors as shown in Figure 1. In fact, financial innova-tions often follow heavy-tailed distributions. We consider� = 009, 0.95, 0.99, sample size N = 500, 1,000, 2,000,and R = 500 replications for each sample size. Condi-tional value-at-risk (VaR) and average value-at-risk (AVaR)are estimated and compared with true (theoretical) valuesat given 500 test points xk (k = 1121 0 0 0 1500), which areequally spaced between −2 and 2 for each replication.Estimators obtained from different methods are computed:quantile based estimator (referred to as QVaR) and LSRestimator (referred to as RVaR) for the conditional VaR,mixed quantile estimator (referred to as QAVaR), and LSRestimator (referred to as RAVaR) for the conditional AVaR(as described in §2).

Figure 2 displays an example of estimation results wherethe solid line is true (theoretical) VaR (AVaR), dash-circleline is QVaR (QAVaR), and dash-cross line is RVaR(RAVaR) given test points xk. In this example, errors fol-low CN41195, � = 0095 and N = 11000. In Figure 2(a),RVaR estimates are closer to true VaR values as mean abso-lute error (MAE) confirms (MAE(QVaR) = 004771 versus

MAE(RVaR) = 002145). Performance of both estimators areworse for AVaR, yet RAVaR estimates are still closer to trueAVaR values than QAVaR (MAE(QAVaR) = 006336 versusMAE(RAVaR) = 002466), as shown in Figure 2(b).

To compare estimators under different error distributions,MAE (averaged over all test points) and variance of MAE(in parenthesis) across 500 replications are obtained asshown in Table 1. Regardless of the error distributions,RVaR (RAVaR) works better than QVaR (QAVaR); MAEand the variance of MAE are smaller. As we can expect,both estimators perform better for the conditional VaRthan AVaR.

Figure 3 presents box-plots for both estimators (QAVaRand RAVaR) given x = 10006 across 500 replications. Find-ings are similar to the one from Table 1; there is someevidence to suggest that RAVaR has smaller MAE thanQAVaR. Also, RAVaR is more stable than QAVaR (MAEof QAVaR is more spread). Note that both estimatorswork better for normal distributions than other heavy-tailed

Table 1. MAE for different error distributions �= 0095,N = 11000 (averaged over all test points).

Error QVaR RVaR QAVaR RAVaR

N40115 000762 000575 000990 00067440000375 40000205 40000585 40000265

t435 001758 001290 004255 00323240001885 40000955 40008085 40006235

CN41195 003006 001955 003844 00231140005635 40002255 40008825 40003165

LN40115 003905 002670 008957 00643240009595 40004305 40038965 40024815

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Figure 3. MAE for conditional AVaR given x = 10006 under different error distributions (�= 0095, N = 11000).

QAVaR-N (0, 1) RAVaR-N(0, 1) QAVaR-t (3) RAVaR-t (3) QAVaR-CN (1, 9) RAVaR-CN(1, 9) QAVaR-LN(0, 1) RAVaR-LN(0, 1)

0

0.5

1.0

1.5

2.0

2.5

3.0M

AE

(m

ean

abso

lute

err

or)

Estimator-error distribution

distributions. We could observe the similar pattern for con-ditional VaR.

Table 2 illustrates sample size effect on MAE of estima-tors. As expected, both estimators perform better as samplesize increases. MAE of RVaR (RAVaR) is still smaller thanthat of QVaR (QAVaR) across all sample sizes.

Next, we obtain asymptotic variances (derived in §3)and compare that with empirical (finite sample) variances

Table 2. MAE for different sample size N with � =

0095 (averaged over all test points).

Error Estimator N = 500 N = 11000 N = 21000

N40115 QVaR 001129 000762 000569RVaR 000849 000575 000418QAVaR 001390 000990 000737RAVaR 000992 000674 000498

t435 QVaR 002420 001758 001277RVaR 001785 001290 000942QAVaR 005385 004255 003207RAVaR 004517 003232 002085

CN41195 QVaR 004322 003006 002180RVaR 002928 001955 001447QAVaR 005471 003844 002658RAVaR 003373 002311 001636

LN40115 QVaR 005814 003905 002959RVaR 004095 002670 001975QAVaR 101986 008957 007275RAVaR 009503 006432 004754

of both estimators. Figure 4 reports asymptotic and finitesample efficiencies of both estimators for the conditionalVaR where R= 500, and error follows N40115 (results aresimilar for other error distributions). In Figure 4(a), wesee that asymptotic variance of RVaR (dash-dot line) issmaller than that of QVaR (solid line) except at xk near 0.In fact, asymptotic variance is affected by how far xk isaway from 0 (which is the mean of explanatory variablein the simulation); when xk is further from the mean, thedifference between asymptotic variances of both estimatorsis bigger. Figure 4(b) provides empirical variance of bothestimators across 500 replications. Empirical variance ofRVaR is (equal or) smaller than that of QVaR at all xk. Fig-ures 4(c) and 4(d) compare asymptotic variances to empir-ical variances of both estimators. It is clear that asymptoticvariances are to provide a good approximation to the empir-ical ones for both estimators.

Figure 5 illustrates asymptotic and empirical variancesof both estimators for AVaR. Insights obtained from theresults are similar to the VaR case. However, Figure 4(c)indicates that empirical variances of QAVaR are largerthan asymptotic variances, especially when xk is far fromthe mean. For this case, asymptotic efficiency of QAVaRmay not very informative on its behavior in finite sample.Results are similar for other error distributions except t435.When the error follows t435, asymptotic (empirical) vari-ances of QAVaR are smaller than that of RAVaR exceptwhen xk is close to the boundary (as shown in Figure 6).

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Figure 4. Conditional VaR: Asymptotic and empirical variance (error ∼ N40115, �= 0095, N = 11000, R= 500).

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.00

0.005

0.010

0.015

0.020

0.025

0.030

0.035

xk

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0xk

xk xk

Var

ianc

e

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Var

ianc

e

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Var

ianc

e

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Var

ianc

e

QVaRRVaR

(a) Asymptotic variance

QVaRRVaR

(b) Empirical variance

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0

AsymptoticEmpirical

(c) QVaR: Asymptotic vs. empirical

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0

Asymptotic

Empirical

(d) RVaR: Asymptotic vs. empirical

To further investigate the finite sample efficiencies androbustness of both estimators compared to the asymptoticones, we provide empirical coverage probabilities (CP) ofa two-sided 95% (nominal) confidence interval (CI) inTable 3 (difference between CP and 0.95 is given in paren-theses). For each replication, the empirical confidence inter-val is calculated from the sample version of asymptoticvariance (when applied to the values of an observed sam-ple of a given size). Then, for given xk, the proportion ofthe 500 replications where the obtained confidence inter-val contains the true (theoretical) value is calculated, andthese proportions are averaged across all test points. ForN40115 and CN41195 error distributions, the resulting CPof RVaR (RAVaR) is very close to 0.95, while empirical CIfor QVaR (QAVaR) under-covers (resulting CP is smallerthan 0.95). For t435 and LN40115 error distributions, CPof RVaR (RAVaR) drops yet maintains somewhat adequate

CP, which is a lot better than CP of QVaR (QAVaR). CIof QAVaR under-covers seriously (resulting CP is about0.7), and this indicates QAVaR procedure might be veryunstable and needs rather wider CI than other estimators toovercome its sensitivity. Note that RVaR (RAVaR) is moreconservative than QVaR (QAVaR) regardless of the errordistributions.

We could draw similar conclusions for other sample sizesand � values. That is, RVaR (RAVaR) performs better andprovides more stable results than QVaR (QAVaR) underdifferent error distributions.

In addition, we estimate another law-invariant risk mea-sure given in (22) with c = 007 using different procedures(mixed quantile based and residual based methods). Thequantile based estimator is referred to as “QRM,” and theLSR estimator is referred to as “RRM” for this risk mea-sure. As before, we compare these estimators under different

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Figure 5. Conditional AVaR: Asymptotic and empirical variance (error ∼ N40115, �= 0095, N = 11000, R= 500)

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.00

0.005

0.010

0.015

0.020

0.025

0.030

0.035

xk

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0xk

Var

ianc

e

QAVaRRAVaR


0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Var

ianc

e

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0xk

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0xk

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Var

ianc

e

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Var

ianc

e

QAVaRRAVaR


Asymptotic

Empirical

(c) QAVaR: Asymptotic vs. empirical

Asymptotic

Empirical

(d) RAVaR: Asymptotic vs. empirical

error distributions. Table 4 presents MAE (averaged over alltest points) and variance of MAE (in parenthesis) across 500replications of estimates. Similar to the cases of value-at-risk and average value-at-risk measures, RRM works better

Table 3. Coverage probability with � = 00951 N =

11000 (averaged over all test points).

Error QVaR RVaR QAVaR RAVaR

N40115 009167 009551 008442 00955240003335 4−0000515 40010585 4−0000525

t435 009044 009269 007088 00908040004565 40002315 40024125 40004205

CN41195 009262 009428 008824 00954840002385 40000725 40006765 4−0000485

LN40115 009185 009276 006930 00918540003155 40002245 40025705 40003155

than QRM. That is, MAE and the variance of MAE com-puted for RRM are smaller. These results indicate that LSRestimators perform better than their mixed quantile counter-parts for different risk measures.

Table 4. MAE for different error distributions � =

00951 N = 11000 of the risk measure (22).

Error QRM RRM

N40115 000661 00040440000295 40000095

t435 002045 00124040002465 40001065

CN41195 002158 00143940002985 40000965

LN40115 009971 10144240018325 40009445

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Figure 6. Conditional AVaR: Asymptotic and empirical variance (error∼t435, �= 0095, N = 11000, R= 500).

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

xk

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0xk

Var

ianc

e

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Var

ianc

e

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0xk

–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0xk

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Var

ianc

e

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Var

ianc

e

QAVaRRAVaR


QAVaRRAVaR


AsymptoticEmpirical

AsymptoticEmpirical

(c) QAVaR: Asymptotic vs. empirical (d) RAVaR: Asymptotic vs. empirical

5. Illustrative Empirical ExamplesIn this section, we demonstrate considered methods to esti-mate conditional VaR and AVaR with real data; differ-ent financial asset classes. Let us first present an exampleof credit default swap (CDS), which is the most popu-lar credit derivative in the rapidly growing credit markets(see FitchRatings 2006 for a detailed survey of the creditderivatives market. A CDS contract provides insuranceagainst a default by a particular company, a pool of com-panies, or sovereign entity. The rate of payments made peryear by the buyer is known as the CDS spread (in basispoints). We focus on the risk of CDS trading (long or shortposition) rather than on the use of a CDS to hedge creditrisk. The CDS data set obtained from Bloomberg consists

of 1,006 daily observations from January 2007 to January2011. Let the dependent variable Y be daily percent change,4Y 4t + 15 − Y 4t55/Y 4t5 ∗ 100, of Bank of America Corp(NYSE:BAC) 5-year CDS spread, explanatory variables X1

be daily return of BAC stock price, and X2 be daily percentchange of generic 5-year investment grade CDX spread(CDX.IG). We use the term “percent change” rather thanreturn because the return of CDS contract is not same asthe return of CDS spread (e.g., see O’Kane and Turnbull2003 for an overview of CDS valuation models). Residu-als obtained from this data set are heavy-tailed distributed(similar to Figure 4(b)).

Figure 7 shows estimated conditional VaR (RVaR) ofBAC CDS spread percent change (result of QVaR is sim-ilar). Because one can take either short or long position,

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Figure 7. Estimated conditional VaR (RVaR) for BACCDS spread percent change for � =

00011 0 0 0 10099

0 01/0706/07

11/0704/08

09/08

01/1108/1004/1011/0901/0906/09

Date0.10.2

0.30.4

0.50.6

0.70.8

0.9

Con

ditio

nal V

aR (

perc

ent c

hang

e)

–40

–30

–20

–10

0

10

20

30

�

we present both tail risk with all values of �, which rangesfrom 0.01 to 0.99; �< 005 corresponds to the left tail (shortposition) and right tail (long position), otherwise. It is clearthat RVaR of certain dates are much higher (lower) thannormal level due to the different daily economic condi-tions reflected by BAC stock price and CDX spread. Thisindicates the specific (daily) economic conditions should

Figure 8. Risk prediction of BAC CDS: QVaR and RVaR 4�= 00055.

09/08 01/09 06/09 11/09 04/10 08/10 01/11–50

–40

–30

–20

–10

0

10

20

30

Date

Con

ditio

nal V

aR (

perc

ent c

hang

e) CDSQVaR

09/08 01/09 06/09 11/09 04/10 08/10 01/11–50

–40

–30

–20

–10

0

10

20

30

Date

Con

ditio

nal V

aR (

perc

ent c

hang

e) CDSRVaR

Table 5. Risk prediction performance of BAC CDS.

� Event (%) Mean

In-sampleQVaR(QAVaR) 0001 009950 40019655RVaR(RAVaR) 0001 009950 4−0086305QVaR(QAVaR) 0005 409751 40022875RVaR(RAVaR) 0005 409751 4−0002695

Out-of-sampleQVaR(QAVaR) 0001 008292 41045465RVaR(RAVaR) 0001 008292 41010525QVaR(QAVaR) 0005 306484 41037405RVaR(RAVaR) 0005 404776 4−0037225

be taken into account for the accurate estimation of risk,and therefore emphasize the importance of conditionalrisk measures. Note that given a specific date, estimatedRVaR curve along the different � values is asymmetricbecause the distribution of CDS spread percent change isnot symmetric.

To compare the prediction performance of both esti-mators, we forecast 603 one-day-ahead (tomorrow’s) VaR(AVaR) given the current (today’s) value of explanatory vari-ables using a rolling window of the previous 403 days. Fig-ure 8 presents forecasting results of QVaR and RVaR with�= 0005 on 603 out-of-sample. Both estimators show simi-lar behaviors, but RVaR seems a little more stable. Followingideas in McNeil and Frey (2000) and Leorato et al. (2012),

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Table 6. Risk prediction performance of IBM stock.

� Event (%) Mean

In-sampleQVaR(QAVaR) 0001 100180 4−0013055RVaR(RAVaR) 0001 009397 4−0034815QVaR(QAVaR) 0005 500117 40004685RVaR(RAVaR) 0005 409334 4−0002255

Out-of-sampleQVaR(QAVaR) 0001 203511 40061715RVaR(RAVaR) 0001 108809 40050235QVaR(QAVaR) 0005 607398 40047875RVaR(RAVaR) 0005 601129 40047785

“violation event” is said to occur whenever an observed CDSspread percent change falls below the predicted VaR (wecan find a few violation events from Figure 8). Also, theforecast error of AVaR is defined as the difference betweenthe observed CDS spread percent change and the predictedAVaR under the violation event. By definition, the violationevent probability should be close to �, and the forecast errorshould be close to zero. Table 5 presents the prediction per-formance (violation event probability for VaR, mean of fore-cast error for AVaR in parenthesis) of both estimators for� = 0001 and 0005. In-sample statistics show that both esti-mators fit the data well; the violation event probabilities arevery close to �, and forecast errors are very small. Out-of-sample performances of both estimators are very similar for� = 0001, even though the forecast errors increase a littlecompared to in-sample cases. For �= 0005, RVaR (RAVaR)

Figure 9. Airline equities: RVaR conditional on crude oil price 4�= 00055.

05/07 07/07 09/07 12/07 02/08 05/08 07/08–150

–120

–90

–60

–30

0

30

60

90

120

150Crude oilDALAMRLUV

seems to perform better; event probabilities are closer to 0.05and forecast errors are smaller.

Next, we apply considered methods to one of theU.S. equities: International Business Machines Corpora-tion (NYSE). The data set contains 1722 daily observationfrom December 2005 to December 2010. Let the dependentvariable Y be the daily log return, 100 ∗ log4Y 4t + 15/Y 4t55, of IBM stock price, explanatory variables X1 bethe log return of S&P 500index, and X2 be the laggedlog return. Similar to the CDS example, we forecast 638one-day-ahead (tomorrow’s) VaR (AVaR) given the current(today’s) value of explanatory variables using a rolling win-dow of the previous 639 days. Residuals obtained fromthis data set are heavy-tailed distributed. Table 6 comparesthe risk prediction performance of IBM stock return. Bothestimators perform well for in-sample prediction. For out-of-sample prediction, both estimators behave similarly for�= 0005, but violation event probability is larger than 0.05.For �= 0001, RVaR (RAVaR) seems a bit better, but eventprobability exceeds 0.01. We provide the additional infor-mation of estimated regression coefficients and confidenceintervals (upper and lower) for the empirical examples inTable D.1, Appendix D.

Finally, we illustrate how crude oil price had impactedthe U.S. airlines’ risk, as we mentioned in §1. Crude oilprices had continued to rise since May 2007 and peaked atan all-time high in July 2008, right before the brink of theU.S. financial system collapse. We compare the movementof estimated VaR for three airline stocks given crude oilprice change: Delta Airlines, Inc. (NYSE:DAL), American

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Airlines, Inc. (NYSE:AMR), and Southwest Airlines Co.(NYSE:LUV). Figure 9 depicts RVaR movement with �=

0005 from May 2007 to July 2008 (QVaR shows similarpatterns). For easy comparison, we standardize all units rel-ative to the starting date. As we can see, crude oil price hadjumped 150% during this time span. On the other hand,RVaR of LUV increased about 15%, while that of AMRincreased 120% and that of DAL increased 90% (in mag-nitude). In fact, different airlines have different strategiesto hedge the risk on oil price fluctuations, and this in turnaffects the risk of airlines’ stock movement. For exam-ple, Southwest Airlines is well known for hedging crudeoil prices aggressively. On the other hand, Delta Airlinesdoes little hedge against crude oil price but operates a lotof international flights. American Airlines does not havestrong hedging against crude oil price, either, and oper-ates fewer international flights than Delta Airlines. Ourestimation results confirm the firm’s specific risk regardingcrude oil price fluctuations.

6. ConclusionsValue-at-risk and average value-at-risk are widely usedmeasures of financial risk. To accurately estimate riskmeasures, taking into account the specific economic con-ditions, we considered two estimation procedures for con-ditional risk measures; one is based on residual analysisof the standard least-squares method (LSR estimator) andthe other on mixed M-estimators (mixed quantile estima-tor). Large sample statistical inferences of both estima-tors are derived and compared. In addition, finite sampleproperties of both estimators are investigated and com-pared as well. Monte Carlo simulation results, under dif-ferent error distributions, indicate that the LSR estimatorsperform better than their (mixed) quantiles counterparts.In general, MAE and asymptotic/empirical variance of theLSR estimators are smaller than that of quantile basedestimators. We also observe that asymptotic variance ofestimators approximates the finite sample efficiencies wellfor reasonable sample sizes used in practice. However, wemight need more samples to guarantee an acceptable effi-ciency of the quantile based estimator for average value-at-risk compared to other estimators. Prediction performanceson the real data example suggest similar conclusions. Infact, residual based estimators can be calculated easily,and therefore the LSR method could be implemented effi-ciently in practice. Moreover, the LSR method can be eas-ily applied to the general class of law-invariant risk mea-sures. In this study, we assume a static model with indepen-dent error distributions. Extension of considered estimationprocedures incorporating different aspects of (dynamic)time series models could be an interesting topic for thefurther study.

Appendix A. Asymptotics for LSR Estimator ofV@R�4Y � x5

Suppose, for the sake of simplicity, that support of the distributionof Xi is bounded, i.e., Xi is bounded w.p.1. Because N−1�T�converges w.p.1 to ì and by (27) we have that

��i − ei�¶Op4N−15

N∑

j=1

�j 0

We can assume here that Ɛ6�i7 = 0, and hence∑N

j=1 �j =

Op4N1/25. It follows that

∣

∣�4�N��5 − e4�N��5

∣

∣=Op4N−1/250 (A1)

Suppose now that the set of population �-quantiles is a sin-gleton. Then F −1

� 4�5 converges w.p.1 to the population quantileF −1� 4�5= V@R�4�5, and hence by (A1) we have that e4�N��5 con-

verges in probability to F −1� 4�5. That is, V@R�4e5 is a consis-

tent estimator of V@R�4�5, and hence the estimator �0 + xTÂ+V@R�4e5 is a consistent estimator of V@R�4Y � x5.

Let us consider the asymptotic efficiency of the residual basedV@R� estimator. It is known that �0 + xTÂ is an unbiased esti-mator of the true expected value �0 + xTÂ and N 1/26�0 − �∗

0 +

xT4Â− Â∗57 converges in distribution to normal with zero meanand variance

�2613xT7ì−1613xT7T0 (A2)

Also, N 1/24�4�N��5 − V@R�4�55 converges in distribution to nor-mal with zero mean and variance

�2 2=�41 −�5

6f�4F−1� 4�5572

1 (A3)

provided that distribution of � has nonzero density f�4 · 5 at thequantile F −1

� 4�5.Let us also estimate the asymptotic variance of the right-hand

side of (27). We have that N times variance of the second termin the right-hand side of (27) can be approximated by

�2Ɛ{

613XTi 7ì

−1613XTi 7

T}= �24k+ 150

We also have that random vectors 4�01 Â5 and e are uncorre-lated. Therefore, if errors �i have normal distribution, then vec-tors 4�01 Â5 and e have jointly a multivariate normal distribution,and these vectors are independent. Consequently, �0 + xTÂ andV@R�4e5 are independent. For not necessarily normal distribu-

tion, this independence holds asymptotically, and thus asymptoti-cally �0 + xTÂ and V@R�4e5 are uncorrelated.

Now we can calculate the asymptotic covariance of the cor-responding terms 4�4�N��5 − V@R�4�55 and 4�4�N��5 − e4�N��55 as4−�24k+ 155/2. Thus, asymptotic variance of the residual basedV@R� estimator can be approximated as

N−14�2+�2613xT7ì−1613xT7T50 (A4)

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Appendix B. Asymptotics for LSR Estimator ofAV@R�4Y � x5

The estimator AV@R�4e5 can be compared with the correspondingrandom variable that is based on the errors instead of residuals

AV@R�4�5

2= inft∈�

{

t +1

41 −�5N

N∑

i=1

6�i − t7+

}

= V@R�4�5+1

41 −�5N

N∑

i=1

6�i − V@R�4�57+

= �4�N��5 +1

41 −�5N

N∑

i=�N��+1

6�4i5 − �4�N��570 (B1)

Note that AV@R�4�5 is not an estimator because errors �i areunobservable.

By (A1), we have that∣

∣

V@R�4�5− V@R�4e5∣

∣=Op4N−1/251 (B2)

and it is known that AV@R�4�5 converges w.p.1 to AV@R�4�5as N → �, provided that � has a finite first-order moment. Itfollows that AV@R�4e5 converges in probability to AV@R�4�5,and hence �0 + xTÂ + AV@R�4e5 is a consistent estimator ofAV@R�4Y � x5.

Lets discuss asymptotic properties of the residual basedAV@R� estimator. As it was pointed out in Appendix A, randomvectors 4�01 Â5 and e are uncorrelated, and hence asymptotically�0 +xTÂ and AV@R�4e5 are independent and hence uncorrelated.Assuming that �-quantile of F�4 · 5 is unique, we have by Deltatheorem

AV@R�4e5= V@R�4�5+ 41 −�5−1N−1

·

N∑

i=1

6ei −V@R�4�57+ + op4N−1/251 (B3)

and

AV@R�4�5= V@R�4�5+ 41 −�5−1N−1

·

N∑

i=1

6�i −V@R�4�57+ + op4N−1/250 (B4)

Equation (B4) leads to the following asymptotic result(cf. Trindade et al. 2007; Shapiro et al. 2009, §6.5.1)

N 1/2[

AV@R�4�5−AV@R�4�5] D→ N401 �251 (B5)

where �2 = 41 −�5−2Var46�− V@R�4�57+5. Moreover, if distri-bution of � has nonzero density f�4 · 5 at V@R�4�5, then

Ɛ6 AV@R�4�57−AV@R�4�5

= −1 −�

2Nf�4V@R�4�55+ o4N−150 (B6)

From Equations (B3) and (B4), the asymptotic vari-ance of 4 AV@R�4�5 − AV@R�4e55 can be bounded by41 −�5−1N−2�24k+ 15, and we can approximate the asymptoticcovariance of the corresponding terms 4 AV@R�4�5−AV@R�4�55

and 4 AV@R�4�5− AV@R�4e55 as 4−41−�5−1N−2�24k+155/2.Thus, asymptotic variance of the residual based AV@R� estimatorcan be approximated as

N−1(

�2+�2613xT7ì−1613xT7T

)

0 (B7)

Appendix C. Asymptotics for theMixed Quantile Estimator

It is possible to derive asymptotics of the mixed quantile estima-tor. For the sake of simplicity, let us start with a sample estimateof S4X5, with �j and �j , j = 11 0 0 0 1 r , given in (17). That is, letX11 0 0 0 1XN be an iid sample (data) of the random variable X, andX415 ¶ · · · ¶ X4N 5 be the corresponding order statistics. Then thecorresponding sample estimate is obtained by replacing the truedistribution F of X by its empirical estimate F . Consequently,41 −�5−1S4X5 is estimated by

41 −�5−1r∑

j=1

�j F−14�j5=

1r

r∑

j=1

X4�N�j �50 (C1)

This can be compared with the following estimator of AV@R�4X5based on a sample version of (2):

X4�N��5 +1

41 −�5N

N∑

i=�N��+1

6X4i5 −X4�N��57

=

(

1−N −�N��

41−�5N

)

X4�N��5+1

41−�5N

N∑

i=�N��+1

X4i50 (C2)

Assuming that N� is an integer and taking r 2= 41 − �5N , weobtain that the right-hand sides of (C1) and (C2) are the same.

Asymptotic variance of the mixed quantile estimator can becalculated as follows. Consider problem (43). The optimal solu-tion of that problem is Â? = Â∗,

�?j = �∗

0 +V@R�j4�5= �∗

0 + F −1� 4�j51 j = 11 0 0 0 1 r1

and �?0 =

∑rj=1 �j�

?j = �∗

0. Assume that � has continu-ous distribution with cdf F�4 · 5 and density function f�4 · 5.Then conditional on X, the asymptotic covariance matrixof the corresponding sample estimator 4Â1 Ç5 of 4Â?1Ç?5 isN−1H−1èH−1, where H is the Hessian matrix of second-order partial derivatives of Ɛ6

∑rj=1 ��j

4� + �∗0 − �j +

4Â∗ −Â5TX57 at the point 4Â?1Ç?5, and è is the covariance matrixof the random vector

Z 2=r∑

j=1

ï��j4�+�∗

0 −�j + 4Â∗−Â5TX51

where the gradients are taken with respect to 4Â1Ç5 at 4Â1Ç5 =

4Â?1Ç?5 (e.g., Shapiro 1989). We have

r∑

j=1

ïÂ��j4�+�∗

0 −�j + 4Â∗−Â5TX5

= −

( r∑

j=1

�′

�j4�+�∗

0 −�j + 4Â∗−Â5TX5

)

X1

ï�j��j

4�+�∗

0 −�j + 4Â∗−Â5TX5

= −�′

�j4�+�∗

0 −�j + 4Â∗−Â5TX51

with �′�j4 · 5 is given in (39).

Note that Ɛ6�′�j4�−F −1

� 4�j57= 0, j = 11 0 0 0 1 r , (see (40)), andhence Ɛ6Z7= 0. Then è= Ɛ6ZZT7, and we can compute

è=

[

�Ɛ6XXT7 ëëT ã

]

1

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where

�= Ɛ

{[ r∑

j=1

�′

�j4�− F −1

� 4�j55]2}

1 ë = 6ë11 0 0 0 1ër 7

with

ëj = Ɛ

[( r∑

i=1

�′

�i4�− F −1

� 4�i55

)

�′

�j4�− F −1

� 4�j55X]

1

j = 11 0 0 0 1 r1

and ãij = Ɛ6�′�i4�− F −1

� 4�i55�′�j4�− F −1

� 4�j557, i1 j = 11 0 0 0 1 r .The Hessian matrix H can be computed as

H=

[

�Ɛ6XXT7 FFT D

]

1

where � =∑r

j=1 �j with

�j =¡Ɛ6�′

�j4�+�∗

0 −�?j + t57

¡t

∣

∣

∣

∣

t=0

=¡6�j41−F�4F

−1� 4�j5−t55+4�j −15F�4F

−1� 4�j5−t57

¡t

∣

∣

∣

∣

t=0

= �jf�4F−1� 4�j55− 41 −�j5f�4F

−1� 4�j55= f�4F

−1� 4�j551

j = 11 0 0 0 1 r1

F = 6F11 0 0 0 1Fr 7 with Fj = �jƐ6X7, j = 11 0 0 0 1 r , and D =

diag4�11 0 0 0 1 �r5.Because Â0 = ËTÇ, we have that Â0 + ÂTx = 6xT3ËT76Â3 Ç7,

and hence the asymptotic variance of Â0 + Â0Tx is given by

N−16xT3ËT7H−1èH−16x3Ë7.

Appendix D. Estimated Regression Coefficientsfor the Empirical Examples

Table D.1. Estimated coefficients, lower(LCI) andupper(UCI) confidence intervals for theempirical examples.

Variables Coefficients LCI UCI

BAC CDS spread example

Y (percent change of 003956 000587 007325BAC CDS)InterceptX1 (return of BAC −002164 −002849 −001479

stock price)X2 (percent change of 004555 003574 005536

generic five-year CDX.IG)

IBM stock example

Y (log return of IBM stock) 00036 −000479 0012InterceptX1 (daily log return of −001733 −002539 −000927

S&P 500 index)X2 (lagged log return) 000009 000001 000017

Acknowledgments

Research of So Yeon Chun was partly supported by the NationalScience Foundation [Award DMS-0914785]. Research of Alexan-der Shapiro was partly supported by the National Science Foun-dation [Award DMS-0914785] and the Office of Naval Research[Award N000140811104].

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So Yeon Chun is an assistant professor at McDonoughSchool of Business, Georgetown University. Her research inter-ests include risk management and revenue management (demandforecasting and pricing) utilizing methodologies in stochastic opti-mization and applied statistics. This paper is part of her Ph.D.dissertation under the supervision of Alexander Shapiro in theSchool of Industrial and Systems Engineering at Georgia Instituteof Technology.

Alexander Shapiro is a professor in the School of Indus-trial and Systems Engineering at Georgia Institute of Technology.His areas of interest include stochastic programming, simulation-based optimization, and multivariate statistical analysis.

Stan Uryasev is a professor, director of the Risk Managementand Financial Engineering Lab, and director of the Ph.D. pro-gram with concentration in quantitative finance at the Universityof Florida. His research is focused on efficient computer model-ing and optimization techniques and their applications in financeand military projects.

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CROSSCUTTING AREAS Conditional Value-at-Risk and Average ... · The notation Y Y0 means that Y4Š5...

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