Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors Dobrislav Dobrev, Travis D. Nesmith, and Dong Hwan Oh 2016-065 Please cite this paper as: Dobrev,Dobrislav,TravisD.Nesmith,andDongHwanOh(2016). “AccurateEvaluationof ExpectedShortfallforLinearPortfolioswithEllipticallyDistributedRiskFactors,”Finance and Economics Discussion Series 2016-065. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2016.065r1. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors DobrislavDobrev1,†,‡*,TravisD.Nesmith2,† andDongHwanOh3,†,‡* 1 FederalReserveBoardofGovernors;dobrislav.p.dobrev@frb.gov 2 FederalReserveBoardofGovernors;travis.d.nesmith@frb.gov 3 FederalReserveBoardofGovernors;donghwan.oh@frb.gov * Correspondence:dobrislav.p.dobrev@frb.gov&donghwan.oh@frb.gov;Tel.:+1-202-[452-2953&973-7334] † Currentaddress:20thSt.andConstitutionAve.N.W.,Washington,D.C.20551 TheauthorsthankFederalReserveBoardResearchLibraryLeadLibrarianHelenKeil-Loschfor comprehensivebibliographicassistanceaswellasJustinSkillmanforrelatedresearchassistance. ‡ Theseauthorscontributedequallytothiswork. Abstract: Weprovideanaccurateclosed-formexpressionfortheexpectedshortfalloflinearportfolios with elliptically distributed risk factors. Our results aim to correct inaccuracies that originate in [1] and are present also in at least thirty other papers referencing it, including the recent survey [2] on estimation methods for expected shortfall. In particular, we show that the correction we provide in the popular multivariate Student t setting eliminates understatement of expected shortfall by a factor varying from at least 4 to more than 100 across different tail quantiles and degrees of freedom. As such,theresultingeconomicimpactinfinancialriskmanagementapplicationscouldbesignificant. We further correct such errors encountered also in closely related results in [3,4] for mixtures of elliptical distributions. More generally, our findings point to the extra scrutiny required when deploying new methodsforexpectedshortfallestimationinpractice. Keywords: Expected shortfall; elliptical distributions; multivariate Student t distribution; accurate closed-formexpression MSC:91G10 JEL:C46,G11 1. Introduction Importantadvantagesofexpectedshortfall(ES)overvalueatrisk(VaR)asacoherentriskmeasure [see5]havedrawntheattentionoffinancialriskmanagers,regulatorsandacademicsalike. Forinstance, a key element of a recent proposal by the Basel Committee on Banking Supervision [6] is moving the quantitative risk metrics system in regard to trading book capital requirement policies from 99% VaR to 97.5% ES. The surge in interest in ES estimation methods also has been reflected in the recent and extensivesurveybyNadarajahetal.[2],whichemphasizesmanynewdevelopmentsandcoversover140 referencesonthesubject.Inthiscontext,usingellipticallydistributedriskfactorsemergesasanappealing choiceinmultivariatesettings, becauseellipticaldistributionscanmodelheavy-tailed, andthusriskier, financial return distributions flexibly while remaining analytically tractable.1 These benefits, however, 1 Imposinganyparticularparametricdistributionalassumptionscanbeavoidedatthecostofsacrificinganalyticaltractability basedonnon-parametricapproachestoESestimationsuchasScaillet[7]and[8],amongmanyothers. SeeNadarajahetal.[2] forfurtherdetails.

2of16 require restricting all risk factors to have equally heavy tails. One popular elliptical example is given bythemultivariateStudenttdistribution, whichallowsforsettingthetailindex, andconsequentlythe thicknessofthetails,directlyasafunctionofthenumberofdegreesoffreedom. Further restricting attention to linear portfolios, the main purpose of this paper is to correct the inaccuracies in Kamdem [1] for the analytical expressions and numerical results for ES in both the elliptical and the multivariate Student t cases. In particular, the ES expressions derived in [1] for both thegeneralcaseofanyellipticaldistributionandthespecialcaseofamultivariateStudenttdistribution aretoolargebyafactoroftwo. Furthermore,wefindthatthepowerintheESformulaformultivariate Studenttshouldbe−(ν−1),ratherthan−(ν+1)asderivedin[1]. Thislatererrormorethanoffsetsthe 2 2 missingscalingfactorof1/2,implyingthatfixingbotherrorswillincreaseestimatesofESforportfolios of risk factors that are distributed according to a multivariate Student t distribution. Both the linear and nonlinear errors are propagated in the survey paper Nadarajah et al. [2] and are confounded by additionalnumericalerrorsin[1]inthetabulatedvaluesforESinthemultivariateStudenttcase. More specifically,wefindthattheinaccurateanalyticalexpressionderivedin[1]doesnotmatchthereported numerical values in [1] for ES in the multivariate Student t case and neither one of them matches the correctexpressionandvalueswederive. IncludingtherecentsurveyNadarajahetal.[2],theinaccurateresultsin[1]havebeenreferencedin atleast30subsequentEnglishpublicationswithoutcorrection[2–4,9–36].2 Inparticular,whileLu’sPh.D. thesis[36]containsastand-alonederivationofthecorrectanalyticalexpressionforESinthemultivariate Studenttcase,itstillreportstheinaccurateESexpressionin[1]forthegeneralellipticalcase;italsodoes not correct any of the numerical inaccuracies in [1] for the multivariate Student t case. Similarly, [18] provides a stand-alone derivation of the correct ES expression only for the multivariate Student t case, addressingneitherthegeneralellipticalcasenortheneedtocorrectnumericalresultsin[1]. Moreover,in additiontobeingincomplete,neitheroftheseimplicitcorrectionsofpartoftheresultsin[1]hasdrawn attentiontotheerrorsoriginatingin[1]thathavespilledoverintoalloftheaboveotherreferences. As such,theinaccuraciesin[1]thatweaimtocorrecthaveyettobeexplicitlyrecognizedandacknowledged morebroadly.Asevidencedalsobytheirpropagationtotherecentsurvey[2]bothinthegeneralelliptical caseandthespecialmultivariateStudenttcase,itislikelyfortheinaccurateresultstobefurtherutilized andpropagatedifleftunchecked. Intermsofmagnitudeoftheresultingtailriskmeasurementerrors, applyingourcorrectioninthe popularmultivariateStudenttsetting,particularlythederivationofthecorrectpower−(ν−1)inaddition 2 toapplyingthenecessaryscalingfactorof1/2fromthegeneralellipticalcase,eliminatesunderstatement of ES by a factor varying from at least 4 to more than 100 across different tail quantiles and degrees of freedom. Forthe97.5%quantilespecifiedby[6]andtherangeof3–8degreesoffreedomcommonlyused inriskmanagementapplications, thecorrectionsproduceESestimatesthatarearoundsixtimeslarger. Clearly, the resulting economic impact in financial risk management applications could be significant. As another contribution of our paper, we eliminate such economically significant inaccuracies that have propagated also in the closely related results in [3,4] for ES in the case of mixtures of elliptical distributions. AnotherviableapproachtoobtainingtheaccurateESexpressionforlinearportfoliosinelliptically distributed risk factors we derive would be to express the returns of such portfolios as corresponding univariate elliptically distributed random variables and making appropriate substitutions in the respective ES formula for the univariate case. We show the equivalence of this alternative approach by 2 TenmorepapersnotwritteninEnglishalsocite[1];ourabilitytoreadthesepapersislimited,buttheydonotseemtocorrect theinaccuraciesin[1]either.

3of16 specializingtheexpressionswederivetotheunivariateStudenttcase. WefurthernotethatLandsman andValdez[37, Theorems1and2], precedingtheresultsin[1], haveformallyadvocatedthisapproach for representing ES for linear combinations of jointly elliptical variables. However, neither [1], nor the recentsurvey[2]onestimationmethodsforexpectedshortfalloranyoftheotherreferencesmentioned abovehavemadetheconnectionwemaketotheresultsin[37]asanalternativewaytocorrecttheerrors in[1].3 The paper proceeds as follows. Section 2 derives the correct ES expression in the general elliptical case. Section 3 deals with the additional correction that needs to be made in the multivariate Student t case. Section 4 validates the corrected ES expressions via a mapping to the univariate Student t case. Section 5 conducts an assessment of the resulting economic impact. Section 6 extends the correction to mixtures of elliptical distributions, and more particularly multivariate Student t mixtures. Section 7 concludes. 2. AccurateESintheGeneralEllipticalCase Following the notations in [1], we consider a linear portfolio with a weight row vector δ = (δ ,δ ,...,δ ) in n elliptically distributed risky returns X = (X ,...,X ) with mean µ, scale matrix 1 2 n 1 n Σ = AA(cid:48),andprobabilitydensityfunction(pdf)ofXtakingtheform (cid:16) (cid:17) f (x) = |Σ|−1/2g (x−µ)Σ−1(x−µ)(cid:48) X forsomenon-negativedensitygeneratorfunctiong,whereforanymatrix|·|representsthedeterminant. The expected shortfall associated with the continuous portfolio returns ∆Π ≡ δX(cid:48) = δ X +...+ 1 1 δ X isthengivenby4 n n −ES = E(∆Π|∆Π ≤ −VaR ) α α 1 = E(∆Π·1{∆Π ≤ −VaR }) α α 1 (cid:90) = δx (cid:48) f (x) dx, α {δx(cid:48)≤−VaRα } whereVaR isdefinedbyPr{∆Π < −VaR } = α.Thenotationfollowstheusualconventionofrecording α α portfolio losses as negative numbers, but stating VaR and ES as positive quantities of money. After the samechangeofvariablesasin[1,section2],wearriveat 1 (cid:90) (cid:16) (cid:17) −ES = (|δA|z +δµ)g (cid:107)z(cid:107)2 dz α 1 α {|δA|z1 ≤−δµ−VaRα } 1 (cid:90) (cid:16) (cid:17) = |δA|z g (cid:107)z(cid:107)2 dz+δµ, 1 α {|δA|z1 ≤−δµ−VaRα } 3 We thank an anonymous referee for suggesting to reconcile our results with the ones that can be obtained following the alternativeapproachbyLandsmanandValdez[37,Theorems1and2],whichprecedestheinaccurateresultsin[1]. 4 See[38]forvariousdefinitionsofESforcontinuousordiscontinuousportfolioreturns.

4of16 where the norm (cid:107)·(cid:107) is defined as the Euclidean norm.5 By writing (cid:107)z(cid:107)2 = z2+(cid:107)z(cid:48)(cid:107)2 and introducing 1 sphericalcoordinatesz(cid:48) =rξ,ξ ∈ S n−2 ,theintegralontherighthandsideabovecanbeexpressedas −ES = δµ+ |S n−2 | (cid:90) ∞ rn−2 (cid:20)(cid:90) −qα |δA|z g (cid:16) z2+r2 (cid:17) dz (cid:21) dr, α α 0 −∞ 1 1 1 where|S n−2 | = Γ 2π (n n − − 2 1 1 ) isthesurfacemeasureoftheunit-spherein Rn−1 andq α = δµ+ |δ V A a | Rα. Wecanthen 2 change the variable z to −z and the variable r to u as given by u = z2+r2, which then implies that 1 1 1 (cid:113) du = 2rdrsothat dr = du = √du andr = u−z2. Substitutingthechangeofvariablesleadstothe 2r 2 u−z2 1 1 followingequivalentexpressionfor−ES , α −ES = δµ− |S n−2 | (cid:90) ∞(cid:90) ∞ rn−2|δA|z g (cid:16) z2+r2 (cid:17) drdz α α qα 0 1 1 1 = δµ− |S n−2 ||δA| (cid:90) ∞(cid:90) ∞ z (cid:16) u−z2 (cid:17)n− 2 3 g(u)dudz . 2α qα z2 1 1 1 1 Changingtheorderofthetwointegralsfurtheryields √ −ES = δµ− |S n−2 ||δA| (cid:90) ∞(cid:90) u z (cid:16) u−z2 (cid:17)n− 2 3 g(u) dz du α 2α q2 α qα 1 1 1 √ = δµ− |S n−2 ||δA| (cid:90) ∞ g(u) (cid:90) u z (cid:16) u−z2 (cid:17)n− 2 3 dz du. 2α q2 α qα 1 1 1 Becausetheinnerintegralsimplifiesto 1 (cid:0) u−q2 (cid:1)n− 2 1 ,andbydefinition|δA| = |δΣδ(cid:48)|1/2 ,weobtainthe n−1 α followingfinalresult, ES = −δµ+|δA| |S n−2 | (cid:90) ∞ 1 (cid:16) u−q2 (cid:17)n− 2 1 g(u) du α 2α q2 α n−1 α = −δµ+ (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 2αΓ (cid:18) n 2 − π 1 n− 2 (cid:19) 1 (n−1) (cid:90) q2 α ∞(cid:16) u−q2 α (cid:17)n− 2 1 g(u) du 2 (cid:124) (cid:123)(cid:122) (cid:125) =2Γ(n+1) 2 = −δµ+ (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 π n− 2 1 (cid:90) ∞(cid:16) u−q2 (cid:17)n− 2 1 g(u) du. 2αΓ(n+ 2 1) q2 α α Thus,wehaveprovedthefollowingtheoremforESinthegeneralellipticalcase: 5 Thefirstchangeofvariablesin[1],y = (x−µ)A−1,transformsthedistributionintoasphericaldistributionwiththesame generatingfunction[39,Corollary2.1andDefinition2.2onpage17].Thesphericaldistributionisinvarianttorotationslikethe secondchangeofvariables,y=zRwhereRistherotationin[1].

5of16 π n− 2 1 (cid:90) ∞(cid:16) u−q2 (cid:17)n− 2 1 g(u) du π n− 2 1 (cid:90) ∞(cid:16) u−q2 (cid:17)n− 2 1 g(u) du αΓ(n+ 2 1) q2 α α 2αΓ(n+ 2 1) q2 α α (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) Kamdem[1] Theorem1 Figure1.ComparisonoftheformulaforESαfrom[1,Theorem4.1]withtheaccurateonefromTheorem1 forthezero-meanunit-variancecase. Theorem1. TheexpectedshortfallES atquantileαofalinearportfolioδXinellipticallydistributedriskfactors α (cid:16) (cid:17) Xwithpdfdefinedby f (x) = |Σ|−1/2g (x−µ)Σ−1(x−µ)(cid:48) isgivenby X ES α = −δµ+ (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 2α π Γ( n n − 2 + 1 2 1) (cid:90) q2 α ∞(cid:16) u−q2 α (cid:17)n− 2 1 g(u) du, (1) whereq = δµ+VaRα.6 α |δΣδ(cid:48)|1/2 Comparing the above result to the corresponding expressions in [1, equation (4.1) in Theorem 4.1] aswellasin[2, equation(15)insection3.20], weconcludethatoursecondtermontherighthandside of equation (1) is smaller by a factor of 2. In particular, this corrects a two-fold overstatement of ES in the zero-mean case of typical interest in many short-term financial risk management applications. To highlight the difference, the formulas are compared for the zero-mean unit-variance case in Figure 1, wherethefactorhasbeenincreasedinsizeandcoloredred. 3. AccurateESintheMultivariateStudenttCase The formula in equation (1) can be specialized to derive ES for any specific distribution in the α familyofmultivariateellipticaldistributions,byreplacing g(u)bytheappropriategeneratingfunction. A special case commonly used in risk management applications is given by the multivariate Student t distribution,whichhasthefollowingpdf, Γ(ν+n) (cid:32) (x−µ)Σ−1(x−µ)(cid:48) (cid:33)−(n+ 2 ν) f X (x) = (cid:113) 2 1+ , Γ(ν) |Σ|(νπ)n ν 2 6 FromKamdem[1,Theorem2.1]qαistheuniquesolutionofatranscendentalequation;thespecificequationwilldependonthe typeofellipticaldistribution.

6of16 where Γ(a) = (cid:82)∞ e−tta−1dt is the Gamma function.7 Substituting g(u) = Γ(√ ν+ 2 n) (cid:0) 1+ u(cid:1)−(n+ 2 ν) in 0 Γ(ν) (νπ)n ν 2 equation(1)abovefurtherspecializestheobtainedgeneralexpressionforEStothemultivariateStudentt case,sothat ES α = −δµ+ (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 2α π Γ( n n − 2 + 1 2 1) Γ(ν Γ ) ( (cid:113) ν+ 2 ( n ν ) π)n (cid:90) q2 α ∞(cid:16) u−q2 α (cid:17)n− 2 1 (cid:16) 1+ u ν (cid:17)−(n+ 2 ν) du 2 = −δµ+ (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 2α (cid:113) 1 (ν)n π Γ( Γ ν 2 ( )Γ ν+ ( 2 n n+ 2 ) 1) (cid:90) q2 α ∞(cid:16) u−q2 α (cid:17)n− 2 1 (cid:16) 1+ u ν (cid:17)−(n+ 2 ν) du. By[1,Lemma2.1],whereB(a,b) = Γ(a)Γ(b) istheEulerBetafunction,wehavethefollowingequality Γ(a+b) (cid:90) q2 α ∞(cid:16) u−q2 α (cid:17)n− 2 1 (cid:16) 1+ u ν (cid:17)−(n+ 2 ν) du = ν n+ 2 ν (cid:16) q2 α +ν (cid:17)−(ν− 2 1) B (cid:18) ν− 2 1 , n+ 2 1 (cid:19) . Substitutingtheequalityintothepriorequationyieldsinturn: ES α = −δµ+ (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 2α (cid:113) 1 (ν)n π Γ( Γ 2 ν ( )Γ ν+ ( 2 n n+ 2 ) 1) ν n+ 2 ν (cid:16) q2 α +ν (cid:17)−(ν− 2 1) B (cid:18) ν− 2 1 , n+ 2 1 (cid:19) = −δµ+ (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 2α (cid:113) 1 (ν)n π Γ( Γ 2 ν ( )Γ ν+ ( 2 n n+ 2 ) 1) ν n+ 2 ν (cid:16) q2 α +ν (cid:17)−(ν− 2 1) Γ(ν− Γ 2 ( 1) n Γ + 2 ( ν n ) + 2 1) = −δµ+ (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 ν √ ν 2 Γ(ν− 2 1) (cid:16) q2+ν (cid:17)−(ν− 2 1) (2) 2α π Γ(ν) α 2 Withthis,weobtainthefollowingresultforESinthemultivariateStudenttcase: Theorem2. TheexpectedshortfallES atquantileαofalinearportfolioδXinriskfactorsXhavingmultivariate α Studenttdistributionwithpdf f (x) = Γ √ (ν+ 2 n) (cid:16) 1+ (x−µ)Σ−1(x−µ)(cid:48)(cid:17)−(n+ 2 ν) isgivenby X Γ(ν) |Σ|(νπ)n ν 2 ES α = −δµ+es α,ν · (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 , (3) with es = ν √ ν 2 Γ(ν− 2 1) (cid:16) q2 +ν (cid:17)−(ν− 2 1) (4) α,ν 2α π Γ(ν) α,ν 2 and δµ+VaR q = α , (5) α,ν |δΣδ(cid:48)|1/2 7 Following[1],weareusingwhat[40,pp. 1]calls“themostcommonandnaturalform”ofthepdfofamultivariateStudentt distribution.

7of16 √ ν ν 2 Γ(ν− 2 1) (cid:16) q2 +ν (cid:17)−(ν+ 2 1) ν √ ν 2 Γ(ν− 2 1) (cid:16) q2 +ν (cid:17)−(ν− 2 1) α π Γ(ν) α,ν 2α π Γ(ν) α,ν 2 2 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) Kamdem[1] Theorem2 Figure2.ComparisonoftheformulaforESαfrom[1,Theorem4.2]withtheaccurateonefromTheorem2 forthezero-meanunit-variancecase. whereq isuniquelydeterminedbysolvingatranscendentalequationgivenby[1,Theorem2.2]and[2,section α,ν 3.20].8 A close inspection of our expression for es in equation (4) above in comparison to the α,ν corresponding equations in [1, Theorem 4.2] and [2, section 3.20] reveals a difference of 1 in the power (cid:0) (cid:1)−(ν−1) ofthelastterm q2 +ν 2 inadditiontotheextrascalingfactorby2inheritedfromthecorrection α,ν made above in the general elliptical case. In particular, the correct power of that term is −(ν−1) rather 2 than −(ν+1). This nonlinear error is larger than the missing scaling factor. Combined, these two 2 (q2 +ν) correctionseliminateanunderstatementofESbyafactorof α,ν > 1inthezero-meanmultivariate 2 Student t setting with ν ≥ 2 degrees of freedom of particular interest in many applications. Again to highlight the difference, the formulas are compared for the zero-mean unit-variance case in Figure 2, wherethedifferentpowertermiscoloredredinadditiontothescalingfactorbeingincreasedinsizeand coloredred. Before conducting a numerical assessment of this combined effect of correcting the two separate inaccuraciesintheESexpressionsfoundinthegeneralellipticalcaseandthemultivariateStudenttcase overlooked in both [1] and [2], we first provide an alternative way to reconcile our results through the univariateStudenttcase. 4. ComparisontotheUnivariateStudenttCase TheESexpressioninTheorem2holdsforanylinearportfolio. Inparticular,itshouldholdifonlya singleassetisheld,forexampleifδ = [1,0,··· ,0]. Consequently,theformulaforexpectedshortfallfor themultivariateStudenttshouldreproducetheformulaforaunivariateStudentt. The results in [41, section 2.2.2] show that the expected shortfall for a zero-mean unit-variance univariateStudenttrandomvariableisgivenby9 1 Γ(ν+1) (cid:18) q2(cid:19)−(ν+ 2 1)(cid:18) ν+q2(cid:19) ES = √ √ 2 1+ α α α α π ν Γ(ν) ν ν−1 2 8 Accuratenumericalvaluesofqα,νfordifferentαandνofinterestarereproducedinTableA1intheappendixastabulatedalso by[1,section2.2]. 9 Wecanrestrictattentiontothezero-meanunit-variancecasewithoutlossofgeneralitygiventheaffinepropertiesofESfor Studenttoranyotherellipticaldistributionasafunctionofitsmeanandstandarddeviation.

8of16 Collectingtermsandusingknownidentitiesallowsustoequivalentlyexpressthisas 1 Γ(ν+1) (cid:16) (cid:17)−(ν+1) (ν)(cid:18) ν+q2(cid:19) ES = √ 2 ν+q2 2 ν 2 α α α π Γ(ν) α ν−1 2 1 ν−1Γ(ν−1) (cid:16) (cid:17)−(ν+1) (ν)(cid:18) ν+q2(cid:19) = √ 2 2 ν+q2 2 ν 2 α α π Γ(ν) α ν−1 2 1 Γ(ν−1) (cid:16) (cid:17)−(ν−1) (ν) = √ 2 ν+q2 2 ν 2 α π 2Γ(ν) α 2 = ν √ 2 ν Γ(ν− 2 1) (cid:16) q2+ν (cid:17)−(ν− 2 1) (6) 2α π Γ(ν) α 2 This result matches exactly our equation (4) from Theorem 2 on page 6 with n = 1 for the zero-mean unit-variance case. Landsman and Valdez [37, equations (23) and (27)] obtain a seemingly different closed-formESexpressionforunivariateStudenttdistribution. Althoughitisnotthesameasequation (6), substituting with known equalities and rearranging terms makes it identical to equation (6) in line withtheaccuracyofourexpressionspecializedtotheunivariatecase. Conversely, combining this expression from the univariate case with the location-scale properties of the multivariate Student t distribution (as an elliptical distribution) with respect to δµ and |δΣδ(cid:48)|1/2 canalsobeusedtoreconcileourexpressionsinTheorem2withtheonesthatcanbeobtainedfollowing the approach by Landsman and Valdez [37, Theorems 1 and 2], which precedes the inaccurate results in [1] we correct. The consistency of our results in the univariate case also readily applies to general ellipticaldistributions. Bycontrast,theformulascontainedinboth[1]and[2]forellipticaldistributions andmultivariateStudenttdistributionsarenotconsistentwiththerespectiveunivariateformulas. 5. EconomicImpactofTheCorrection In order to assess the resulting economic impact, we study numerically the combined effect of the above corrections of the inaccuracies in the ES expressions found in [1] and [2] as well as additional numerical errors we uncover in the respective tabulated values in [1]. Without loss of generality, the numerical results restrict attention to the zero-mean unit-variance multivariate Student t setting. We tabulateinTable1onthenextpagetheaccurate(panelA)andinaccurate(panelB)valuesofes aswell α,ν astheirratio(panelD)acrossdifferenttailquantilesα = 0.01,0.025,0.05anddegreesoffreedomν = 2, 3,4,5,6,7,8,9,10,100,200,250. (cid:0) (cid:1) Itstandsoutthattheratio q2 +ν /2oftheaccurateversusinaccuratevaluesreportedinpanelD α,ν ofTable1onthefollowingpageisquitelargeandvariesfromatleastjustabove4(forα =0.05andν =3 or4)tomorethan100(forν ≥ 200). Thelaterdiscrepanciesoccurastheresultsshouldbeconvergingto theresultsforaGaussiandistribution;theresultsinpanelAareconvergingtotheGaussianonesunlike thevaluesinpanelB. Importantly, in the case of α = 0.025 the minimum ratio of the accurate to inaccurate values of ES is about 6. Given that moving the quantitative risk metrics system in regard to trading book capital requirement policies from 99% VaR to 97.5% ES (i.e. in our notation α = 0.025) is a key element of the recent proposal by [6], the numerical results imply that the correction would eliminate at least a six-fold and potentially much larger understatement of risk in the popular zero-mean multivariate Studenttsetting. Clearly,theresultingeconomicimpactinfinancialriskmanagementapplicationscould besignificant. Noting again the magnitude and potential economic impact of our correction, one could look for possiblereasonsastowhysuchlargeinaccuraciesintheotherwisequitepopularmultivariateStudentt

9of16 Table1.NumericalComparisonoftheAccurateversusInaccurateExpressionforExpectedShortfallin theMultivariateStudenttCase.Thetablereportstheaccurateanalytical(panelA),inaccurateanalytical (panelB),andinaccuratenumericallytabulated(panelC)valuesofesα,ν aswellastherespectiveratios ofaccurateversusinaccurateanalytical(panelD)andaccurateanalyticalversusinaccuratenumerically tabulated(panelE)valuesacrossdifferenttailquantilesα=0.01,0.025,0.05(differentrows)anddegrees of freedom ν = 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, 200, 250 (different columns) of the multivariate Student t distributiongoverningtheindividualriskyreturnsinalinearportfolio.Theaccurateanalyticalexpression inpanelAreflectsthederivationsinthispaper(Theorems1and2).TheinaccurateexpressioninpanelBis originallyduetoKamdem[1,Theorem4.2]andisreproducedinNadarajahetal.[2,section3.20],while theinaccuratevaluesinpanelCaretheonesnumericallytabulatedin[1,section4.1]. ν 2 3 4 5 6 7 8 9 10 100 200 250 PanelA:Theaccurateanalyticalesα,ν = 2α ν√ ν 2 π Γ Γ ( ( ν− ν 2 1 ) ) (cid:0) q2 α,ν +ν (cid:1)−(ν− 2 1) derivedinsections2and3above 2 est 14.071 7.004 5.221 4.452 4.033 3.770 3.591 3.462 3.363 2.722 2.717 2.665 0.010,ν est 8.832 5.040 3.994 3.522 3.256 3.087 2.970 2.884 2.819 2.379 2.358 2.354 0.025,ν est 6.164 3.874 3.203 2.890 2.711 2.595 2.514 2.515 2.891 2.093 2.078 2.075 0.050,ν PanelB:Theinaccurateanalyticalesα,ν = α ν√2 ν π Γ Γ ( ( ν− ν 2 1 ) ) (cid:0) q2 α,ν +ν (cid:1)−(ν+ 2 1) in[1,theorem4.2]and[2,section3.20] 2 est 0.557 0.593 0.579 0.546 0.508 0.472 0.438 0.408 0.381 0.052 0.026 0.021 0.010,ν est 0.861 0.768 0.682 0.607 0.543 0.490 0.446 0.409 0.377 0.046 0.023 0.019 0.025,ν est 1.171 0.908 0.750 0.638 0.555 0.490 0.439 0.409 0.453 0.041 0.021 0.016 0.050,ν PanelC:Theinaccuratenumericallytabulatedesα,νin[1,section4.1] est 5.572 5.931 5.788 5.456 5.080 4.716 4.382 4.082 3.814 0.516 0.264 0.209 0.010,ν est 8.611 7.678 6.822 6.068 5.433 4.903 4.460 4.086 3.768 0.458 0.231 0.185 0.025,ν est 11.712 9.075 7.497 6.380 5.546 4.901 4.388 3.971 3.626 0.407 0.205 0.164 0.050,ν PanelD:Ratio (cid:0) q2 α,ν +ν (cid:1) /2oftheaccurateanalyticalesα,ν(panelA)totheinaccurateanalyticalesα,ν(panelB) est 25.253 11.808 9.020 8.161 7.938 7.994 8.195 8.480 8.819 52.795 102.741 127.750 0.010,ν est 10.256 6.564 5.854 5.804 5.994 6.296 6.659 7.059 7.482 51.968 101.944 126.939 0.025,ν est 5.263 4.269 4.272 4.530 4.888 5.295 5.729 6.143 6.378 51.378 101.365 126.363 0.050,ν PanelE:Ratiooftheaccurateanalyticalesα,ν(panelA)totheinaccuratenumericallytabulatedesα,ν(panelC) est 2.525 1.181 0.902 0.816 0.794 0.799 0.819 0.848 0.882 5.279 10.275 12.774 0.010,ν est 1.026 0.656 0.585 0.580 0.599 0.630 0.666 0.706 0.748 5.197 10.195 12.696 0.025,ν est 0.526 0.427 0.427 0.453 0.489 0.529 0.573 0.633 0.797 5.138 10.134 12.635 0.050,ν settingcouldhavegoneunnoticed. Oneimportantobservationtomakeinthisregardisthat,asrecently noted by [42] among others, the literature on backtesting ES is fairly new, thereby leaving a potential loopholeforanysucherrorsinEScomputationstogounnoticedforawhile,becausefinancialindustry applicationsmaynotyetperformroutineandpowerfulenoughESbacktesting. Anotherpossibilitytokeepinmindisthatinsteadofusingtheinaccurateexpressionsin[1]and[2] one could alternatively take directly the values of es tabulated by [1, section 4.1] and reproduced α,ν in panel C of Table 1. These numerical values happen to be offset by yet another separate mistake by a factor of 10, thereby mechanically, but still inaccurately, shifting the discrepancy in the opposite more conservative direction across much of the range of tabulated different tail quantiles and degrees offreedom. Toillustratethis, panelEreportstheratiooftheaccurateanalyticalvalues(panelA)tothe inaccurate numerically tabulated values (panel C). In particular, the magnitude of ES underestimation whenusingthewronglytabulatedvaluesinpanelCstillremainsverylargefor ν muchlargerthan10.

10of16 However,formostothervaluesofνthereisatleastsomepartialcancellationeffectwithallothererrorsas aresultofthisthirdinaccuracyin[1];theresultfor97.5%ESspecifiedby[6]andtherangeof3–8degrees of freedom commonly used in risk management applications would be to bring most discrepancies down to the order of 0.6, thereby shifting the direction of the numerical errors from underestimation tooverestimationofES. Lastbutnotleast,aspointedoutabove,theaccurateESexpressionsweprovidecanalsobeobtained followingthealternativeapproachbyLandsmanandValdez[37,Theorems1and2]preceding[1]. Thus, althoughtheinaccurateresultsin[1]havespreadacross[2–4,9–36],itcannotberuledoutthatatleastin somecasestheerrorsin[1]couldhavebeenavoidedbyfollowing[37]instead. Allinall,partialoffsettingofdifferenterrors,challengeswithESbacktesting,aswellasthepotential use of viable alternative approaches such as [37] could have played a role for ES discrepancies of even such large magnitude and potential economic impact as the ones we report in panel D of Table 1 to eludedetectionforsometime. Ourfindingsprovideawordofcautionaboutthescrutinyrequiredwhen deployinganynewmethodsforESestimationinpractice,asmaybehappeningasaresultoftheproposed newguidelinesissuedbytheBaselCommitteeonBankingSupervision[6]. 6. AccurateESforMixturesofEllipticalDistributionsandMultivariateStudenttMixtures Asanotherusefulapplicationwefurthereliminatesimilareconomicallysignificantinaccuraciesthat have propagated also in closely related results for ES in the case of mixtures of elliptical distributions studied by [3,4].10 In particular, using a similar derivation to [1], both [3] and [4] provide inaccurate closed-form ES expressions for the general case of mixtures of elliptical distributions as well as the special case of multivariate Student t mixtures with identical variance-covariance matrix Σ = Σ . The i expressionsforESin[3,4]haveinheritedtheomissionofthescalingfactor1/2correctedabove.Therefore, forthesakeofcompletenessweprovideaccurateESexpressionsfirstforthegeneralcaseofmultivariate elliptical distribution mixtures (Theorem 3) and then also for the special case of multivariate Student t mixtures(Theorem4). Theorem 3. The expected shortfall ES α,{βi } i m =1 ,{gi } i m =1 at quantile α of a linear portf (cid:16) olio δX in risk factors X (cid:17) following a mixture of elliptical distributions with pdf f (x) = ∑m β |Σ |−1/2g (x−µ )Σ−1(x−µ )(cid:48) X i=1 i i i i i i where∑m β =1isgivenby i=1 i ES α,{βi } i m =1 ,{gi } i m =1 = − i ∑ = m 1 β i (δµ i )+ 2αΓ π (cid:16) n− n 2 1 + 2 1 (cid:17) i ∑ = m 1 β i (cid:12) (cid:12)δΣ i δ (cid:48)(cid:12) (cid:12) 1/2 (cid:90) ∞(cid:16) (cid:17)n−1 × u−q2 2 g (u)du (7) α,i i q2 α,i where q = (δµ +VaR )/|δΣ δ(cid:48)|1/2 andVaR isdefinedbyPr{δY < −VaR } = α withY followingan α,i i α,i i α,i α,i (cid:16) (cid:17) ellipticaldistributionwithpdf f (y) = |Σ |−1/2g (y−µ )Σ−1(y−µ )(cid:48) . Y i i i i i Asabove,wefurtherspecializethisresultforthespecialcaseofmultivariateStudenttmixtureswith µ = µandΣ = Σfori =1,2,...,m. i i 10 Distributionmixturesareknowntoprovideaflexibleandtractablewayformodellingarichersetofheavytaileddistributions encounteredinmanyriskmanagementapplications.

11of16 Theorem 4. The expected shortfall ES α,{βi } i m =1 ,{νi } i m =1 at quantile α of a linear portfolio δX in risk factors X following a mixture of multivariate Student t distributions with pdf f (x) = X ∑m β Γ (cid:16) √ νi+ 2 n(cid:17) (cid:16) 1+ (x−µ)Σ−1(x−µ)(cid:48)(cid:17)− (n+ 2 νi ) where∑m β =1isgivenby i=1 i Γ(ν 2 i) |Σ|(νiπ)n νi i=1 i ES α,{βi } i m =1 ,{νi } i m =1 = −δµ+est α,{βi } i m =1 ,{νi } i m =1 · (cid:12) (cid:12)δΣδ (cid:48)(cid:12) (cid:12) 1/2 (8) with est = ∑ m β ν √i ν 2 i Γ(νi − 2 1) (cid:16) q2 +ν (cid:17)− (cid:16)νi 2 −1(cid:17) (9) α,{βi } i m =1 ,{νi } i m =1 i=1 i 2α π Γ(ν 2 i) α,{βi } i m =1 ,{νi } i m =1 i and q α,{βi } i m =1 ,{νi } i m =1 = δµ+Va |δ R Σ α, δ { (cid:48) β | i 1 } / i m = 2 1 ,{νi } i m =1 , (10) (cid:16) (cid:17) wh (cid:110) ere q α,{βi } i m =1 ,{νi } i m =1 = (cid:111) δµ+VaR α,{βi } i m =1 ,{νi } i m =1 /|δΣδ(cid:48)|1/2 ,VaR α,{βi } i m =1 ,{νi } i m =1 is defined by Pr δX < −VaR α,{βi } i m =1 ,{νi } i m =1 = α and q α,{βi } i m =1 ,{νi } i m =1 is uniquely determined by solving a transcendental equationgivenby[3,Corollary3.8].11 The difference between equation (7) and the corresponding equations in [3, Theorem 5.1] and [4, Theorem4.1]isthescalingfactoroftwointhedenominator.Likewise,thedifferencebetweenequation(9) andthecorrespondingequationin[3,Theorem5.4]and[4,Theorem4.2]isonlythescalingfactoroftwo inthedenominator. Importantly,thereisnoadditionalerrorinthepowerofthelastterm,unliketheone encountered in [1] and corrected in section 3 on page 5. Nonetheless, the numerically tabulated values forESin[3,4]differfromtheaccurateonesbyasignificantlylargerscalingfactorthantwo. To illustrate these additional numerical errors, we tabulate the correct and incorrect analytically obtained values of ES for a mixture of multivariate Student t distributions respectively in panel A and panelBofTable2;theratiooftheaccuratevaluesinpanelAtotheinaccurateonesinpanelBisreported inpanelDandisexactly1/2asexpected. However,thenumericallytabulatedvaluesinboth[3]and[4], which are reproduced in panel C of Table 2, are completely different from those in either panel B or panelAofTable2. Itisunclearhow[3,4]havegeneratedthesevalues,becausetheycannotbeobtained from the formula in [3, Theorem 5.4] and [4, Theorem 4.2] and do not match the accurate formula we provideinTheorem4either.12 PanelEofTable2reportstheratiosoftheaccuratevaluesinpanelAto theinaccuratenumericallytabulatedonesinpanelC.WithmostratiosinpanelEsignificantlylargerthan one,thecorrectionweprovidecaneliminateverylargeandeconomicallysignificantunderestimationof ESalsofortheconsideredmixturesofdistributionsinTheorems3and4above. 11 Accuratenumericalvaluesofq α,{βi}2 i=1,{νi}2 i=1 fordifferentα,β1,β2 (=1−β1 ),ν1,andν2ofinterestarereproducedinTableA2 intheappendixastabulatedalsoin[3,tables2and3]and[4,tables1and2]. 12 Inaddition,thevaluesinTable4of[3]areexactlythesameastheincorrectonesin[1]reflectingtheinaccuratetabulationofES inthecaseofamultivariateStudenttdistribution.

12of16 Table2. NumericalComparisonoftheAccurateversusInaccurateExpressionforExpectedShortfall in the Case of a Multivariate Student t Mixture. The table reports the accurate analytical (panel A), inaccurate analytical (panel B), and inaccurate numerically tabulated (panel C) values of est as α,β,ν1,ν2 well as the respective ratios of accurate versus inaccurate analytical (panel D) and accurate analytical versusinaccuratenumericallytabulated(panelE)valuesacrossdifferenttailquantilesα=0.01and0.001 (left and right panel), different mixture weights β = 0.25,0.30,0.35,0.40,0.45,0.50 (different rows), and differentpairsofdegreesoffreedom(ν ,ν )=(2,3),(3,4),(4,6),(7,15)(differentcolumns)foramixture 1 2 of multivariate Student t distributions governing the individual risky returns in a linear portfolio. The accurateexpressioninpanelAreflectsthederivationsinthispaper(Theorems3and4). Theinaccurate expressioninpanelBisoriginallyduetoKamdem[3,Theorem5.4]andKamdem[4,Theorem4.2],while theinaccuratevaluesinpanelCaretheonesnumericallytabulatedinKamdem[3, tables5and6]and Kamdem[4,tables3and4]. α=0.01 α=0.001 (ν1,ν2 ) (2,3) (3,4) (4,6) (7,15) (2,3) (3,4) (4,6) (7,15) PanelA:Theaccurateanalyticalest derivedinsection6 α,β,ν1,ν2 est 8.994 5.709 4.366 3.290 24.981 11.474 7.510 4.790 α,0.25,ν1,ν2 est 9.372 5.803 4.430 3.327 26.634 11.795 7.699 4.882 α,0.30,ν1,ν2 est 9.745 5.896 4.492 3.362 28.220 12.105 7.879 4.969 α,0.35,ν1,ν2 est 10.111 5.988 4.554 3.398 29.743 12.406 8.052 5.051 α,0.40,ν1,ν2 est 10.471 6.078 4.614 3.432 31.210 12.697 8.218 5.128 α,0.45,ν1,ν2 est 10.825 6.168 4.674 3.466 32.625 12.979 8.377 5.201 α,0.50,ν1,ν2 PanelB:Theinaccurateanalyticalest derivedin[3,4] α,β,ν1,ν2 est 17.987 11.418 8.732 6.580 49.961 22.949 15.020 9.580 α,0.25,ν1,ν2 est 18.745 11.606 8.859 6.653 53.269 23.590 15.398 9.764 α,0.30,ν1,ν2 est 19.489 11.792 8.984 6.725 56.440 24.211 15.759 9.937 α,0.35,ν1,ν2 est 20.222 11.976 9.108 6.795 59.486 24.811 16.104 10.101 α,0.40,ν1,ν2 est 20.942 12.157 9.229 6.864 62.420 25.393 16.435 10.256 α,0.45,ν1,ν2 est 21.650 12.336 9.348 6.932 65.251 25.957 16.753 10.402 α,0.50,ν1,ν2 PanelC:Theinaccuratetabulatedest in[3,tables5-6]and[4,tables3-4] α,β,ν1,ν2 est 6.366 1.294 0.243 0.003 20.896 3.033 0.577 0.007 α,0.25,ν1,ν2 est 7.019 1.410 0.279 0.003 23.164 3.323 0.666 0.007 α,0.30,ν1,ν2 est 7.647 1.523 0.314 0.004 25.271 3.588 0.716 0.008 α,0.35,ν1,ν2 est 8.251 1.631 0.348 0.004 27.239 3.837 0.776 0.009 α,0.40,ν1,ν2 est 8.834 1.737 0.380 0.005 29.089 4.071 0.831 0.009 α,0.45,ν1,ν2 est 9.396 1.839 0.410 0.005 30.835 4.290 0.881 0.010 α,0.50,ν1,ν2 PanelD:Ratio1/2oftheaccuratevaluesofest inPanelAtothoseinPanelB α,β,ν1,ν2 est 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 α,0.25,ν1,ν2 est 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 α,0.30,ν1,ν2 est 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 α,0.35,ν1,ν2 est 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 α,0.40,ν1,ν2 est 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 α,0.45,ν1,ν2 est 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 α,0.50,ν1,ν2 PanelE:Ratiooftheaccuratevaluesofest inPanelAtothoseinPanelC α,β,ν1,ν2 est 1.413 4.412 17.967 1096.633 1.195 3.783 13.016 684.271 α,0.25,ν1,ν2 est 1.335 4.116 15.877 1108.833 1.150 3.550 11.560 697.414 α,0.30,ν1,ν2 est 1.274 3.871 14.306 840.575 1.117 3.374 11.005 621.088 α,0.35,ν1,ν2 est 1.225 3.671 13.085 849.375 1.092 3.233 10.376 561.167 α,0.40,ν1,ν2 est 1.185 3.499 12.143 686.400 1.073 3.119 9.889 569.756 α,0.45,ν1,ν2 est 1.152 3.354 11.400 693.180 1.058 3.025 9.508 520.110 α,0.50,ν1,ν2

13of16 7. Conclusion Elliptically distributed risk factors are popular in financial risk management applications, because they can model heavy tails while still offering a great deal of flexibility and analytical tractability. Our accurate closed-form expressions for the expected shortfall of linear portfolios with elliptically-distributed risk factors correct major inaccuracies in the results by Kamdem [1] for both the general elliptical case and the special multivariate Student t case. The inaccurate results in [1] have been referenced by at least thirty other papers [2–4,9–36], including the recent comprehensive survey of ES estimation methods by Nadarajah et al. [2]. We note that our accurate results can be reconciled alsofollowingthealternativeapproachbyLandsmanandValdez[37, Theorems1and2]precedingthe inaccuratederivationsby[1]andalloftheseotherstudiesreferringto[1]. Intermsofitsmagnitude,our correctioninthezero-meanmultivariateStudenttsettingeliminatespotentialunderstatementofESbya factor varying from at least 4 to more than 100 across different tail quantiles and degrees of freedom. As such, the economic impact from using our accurate ES expressions in financial risk management applications with elliptically-distributed risk factors can be significant. We also eliminate economically significant inaccuracies that have further propagated in the closely related results in [3,4] for ES of mixturesofellipticaldistributions. Another important application for the accurate closed-form results for ES with elliptically distributed, or mixtures of elliptically distributed, risk factors is gauging the statistical precision of non-parametric ES estimation methods relying on Monte Carlo simulations in the spirit of the analysis in [43]. In particular, the ability to study the performance of alternative non-parametric ES estimators incontrolledexperimentsformultivariateheavy-tailedsettingswithaccuratelyknownanalyticalresults canhelpprovidesomeusefulguidanceinthecontextoftheproposalbytheBaselCommitteeonBanking Supervision[6]tomovethequantitativeriskmetricssysteminregardtotradingbookcapitalrequirement policiesfrom99%VaRto97.5%ES.Moregenerally,ourfindingspointtotheextrascrutinyrequiredwhen deployingnewmethodsforESestimationinpractice,especiallyalsoinlightofthewidelyacknowledged separatechallengeswithbacktestingexpectedshortfall. AuthorContributions:TheresearchproblemwasidentifiedbyD.D.;theanalyticalsolutionandallnumericalresults were obtained independently by both D.D. and D.O.; the paper was written jointly by D.D. and D.O., and it was editedbyT.D.N. ConflictsofInterest:AllauthorscurrentlyworkattheFederalReserveBoardofGovernors.Theviewsexpressedin thispaperarethoseoftheauthorsandshouldnotbeinterpretedasreflectingtheviewsoftheFederalReserveBoard ofGovernorsoranyotherpersonassociatedwiththeFederalReserveSystem.

14of16 Abbreviations Thefollowingabbreviationsareusedinthismanuscript: ES:Expectedshortfall VaR:Value-at-risk Appendix. SupplementalTables TableA1.AccurateNumericalValuesofqα,νforDifferentαandνintheCaseofMultivariateStudentt. Thetablereproducesnumericalvaluesofqα,ν fordifferentαandνofinterestastabulatedby[1,section 2.2]forthepurposesofcomputingES. ν 2 3 4 5 6 7 8 9 10 100 200 250 q 0.010,ν 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.364 2.341 2.345 q0.025,ν 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 1.984 1.972 1.969 q0.050,ν 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.812 1.660 1.660 1.653 1.651 TableA2. AccurateNumericalValuesofq forDifferentα,β ,β (=1−β ),ν ,andν in α,{βi }2 i=1 ,{νi }2 i=1 1 2 1 1 2 theCaseofMultivariateStudenttMixture. Thetablereproducesnumericalvaluesofq for α,{βi }2 i=1 ,{νi }2 i=1 differentα,β ,β (=1−β ),ν ,andν ofinterestastabulatedby[3,tables2and3]and[4,tables1and2] 1 2 1 1 2 forthepurposesofcomputingES. α=0.01 α=0.001 (ν1,ν2 ) (2,3) (3,4) (4,6) (7,15) (2,3) (3,4) (4,6) (7,15) qα,0.25,ν1,ν2 5.103 3.940 3.291 2.700 13.558 8.014 5.775 4.051 qα,0.30,ν1,ν2 5.221 3.980 3.321 2.720 14.221 8.177 5.883 4.111 qα,0.35,ν1,ν2 5.341 4.019 3.351 2.740 14.874 8.338 5.990 4.169 qα,0.40,ν1,ν2 5.463 4.059 3.381 2.760 15.517 8.497 6.094 4.226 qα,0.45,ν1,ν2 5.585 4.099 3.412 2.780 16.148 8.654 6.196 4.282 qα,0.50,ν1,ν2 5.709 4.139 3.442 2.800 16.767 8.808 6.296 4.335 Bibliography 1. Kamdem,J.S.Value-at-riskandexpectedshortfallforlinearportfolioswithellipticallydistributedriskfactors. InternationalJournalofTheoreticalandAppliedFinance2005,8,537–551. 2. Nadarajah, S.; Zhang, B.; Chan, S. Estimation methods for expected shortfall. Quantitative Finance 2014, 14,271–291. 3. Kamdem,J.S. ∆-VaRand∆-TVaRforportfolioswithmixtureofellipticdistributionsriskfactorsandDCC. Insurance:MathematicsandEconomics2009,44,325–336. 4. Kamdem,J.S. VaRandESforlinearportfolioswithmixtureofellipticdistributionsriskfactors. Computing andVisualizationinScience2007,10,197–210. 5. Artzner,P.;Delbaen,F.;Eber,J.M.;Heath,D.Coherentmeasuresofrisk. MathematicalFinance1999,9,203–228. 6. Basel Committee on Banking Supervision. Consultative Document, Fundamental Review of the Trading Book:ArevisedMarketRiskframework. BIS,Basel,Switzerland2013. 7. Scaillet,O. NonparametricEstimationandSensitivityAnalysisofExpectedShortfall. MathematicalFinance 2004,14,115–129. 8. Scaillet,O. Nonparametricestimationofconditionalexpectedshortfall. Insuranceandriskmanagementjournal 2005,74,639–660.

15of16 9. Bormetti,G.;Cisana,E.;Montagna,G.;Nicrosini,O. Anon-Gaussianapproachtoriskmeasures. PhysicaA: StatisticalMechanicsanditsApplications2007,376,532–542. 10. Birbil,S.;Frenk,J.;Kaynar,B.;Noyan,N.Riskmeasuresandtheirapplicationsinassetmanagement.Technical report,EconometricInstitute,ErasmusUniversityRotterdam,2008. 11. Kamdem, J.S.; Genz, A. Approximation of multiple integrals over hyperboloids with application to a quadraticportfoliowithoptions. ComputationalStatistics&DataAnalysis2008,52,3389–3407. 12. Elliott,R.J.; Miao,H. VaRandexpectedshortfall: anon-normalregimeswitchingframework. Quantitative Finance2009,9,747–755. 13. Genz,A.;Bretz,F. Computationofmultivariatenormalandtprobabilities;Springer,2009. 14. Kamdem,J.S. VaRandESforlinearportfolioswithmixtureofgeneralizedLaplacedistributionsriskfactors. AnnalsofFinance2009,8,123–150. 15. Bormetti,G.;Cazzola,V.;Livan,G.;Montagna,G.;Nicrosini,O.AgeneralizedFouriertransformapproachto riskmeasures. JournalofStatisticalMechanics:TheoryandExperiment2010,2010. 16. Kamdem,J.S. SharpestimatesfortheCDFofquadraticformsofMPErandomvectors. JournalofMultivariate Analysis2010,101,1755–1771. 17. Yueh, M.L.; Wong, M.C. Analytical VaR and expected shortfall for quadratic portfolios. The Journal of Derivatives2010,17,33–44. 18. Jiang,C.F.;Yang,Y.K. Tailconditionalvarianceofportfolioandapplicationsinfinancialengineering. Systems EngineeringProcedia2011,2,213–221. 19. Kamdem,J.S. Businessesrisksaggregationwithcopula. JournalofQuantitativeEconomics2011,9,58–72. 20. Paindaveine,D.;Šiman,M.Ondirectionalmultiple-outputquantileregression. JournalofMultivariateAnalysis 2011,102,193–212. 21. Chen, Z.; Song, Z. Dynamic portfolio optimization under multi-factor model in stochastic markets. OR Spectrum2012,34,885–919. 22. Broda,S.A. Theexpectedshortfallofquadraticportfolioswithheavy-tailedriskfactors. MathematicalFinance 2012,22,710–728. 23. Pimenova, I. Semi-parametricestimationofellipticaldistributionincaseofhighdimensionality. Master’s thesis,Humboldt-UniversitätzuBerlin,2012. 24. Liu,Y.J.;Zhang,W.G. Fuzzyportfoliooptimizationmodelunderrealconstraints. Insurance:Mathematicsand Economics2013,53,704–711. 25. Makdissi,P.;Sylla,D.;Yazbeck,M. Decomposinghealthachievementandsocioeconomichealthinequalities inpresenceofmultiplecategoricalinformation. EconomicModelling2013,35,964–968. 26. Date, P.; Bustreo, R. Measuring the risk of a non-linear portfolio with fat-tailed risk factors through a probabilityconservingtransformation. IMAJournalofManagementMathematics2014. 27. Moussa, A.M.; Kamdem, J.S.; Shapiro, A.; Terraza, M. CAPM with fuzzy returns and hypothesis testing. Insurance:MathematicsandEconomics2014,55,40–57. 28. Moussa, A.M.; Kamdem, J.S.; Terraza, M. Fuzzy value-at-risk and expected shortfall for portfolios with heavy-tailedreturns. EconomicModelling2014,39,247–256. 29. Sun, C.; Zhang, Y.; Peng, S.; Zhang, W. The inequalities of public utility products in China: From the perspectiveoftheAtkinsonindex. RenewableandSustainableEnergyReviews2015,51,751–760. 30. Jiang,C.F.;Peng,H.Y.;Yang,Y.K. TailvarianceofportfolioundergeneralizedLaplacedistribution. Applied MathematicsandComputation2016,282,187–203. 31. Cisana,E. Non-Gaussianstochasticmodelsandtheirapplicationsineconophysics. PhDthesis,Universityof Pavia,2007. 32. Diallo, A.; Mbairadjim Moussa, A. Addressing agent specific extreme price risk in the presence of heterogeneousdatasources:Afoodsafetyperspective. Technicalreport,LAMETA,UniversitédeMontpellier I,2014. 33. Abad, P.; Benito, S. Variance reduction technique for calculating value at risk in fixed income portfolios. StatisticsandOperationsResearchTransactions2010,34,21–44.

16of16 34. Spangler,M.;Werner,R. Potentialfuturemarketrisk: Copingwithlongtermmodelrisk. Technicalreport, AvailableatSSRN:https://ssrn.com/abstract=1587937,2010. 35. Kamdem, J.S. Downside risk And Kappa index of non-Gaussian portfolio with LPM. Technical report, AvailableatHAL:https://hal.archives-ouvertes.fr/hal-00733043,2011. 36. Lu,N. Statisticalissuesincoherentriskmanagement. PhDthesis,NorthCarolinaStateUniversity,2004. 37. Landsman,Z.; Valdez,E. Tailconditionalexpectationsforellipticaldistributions. NorthAmericanActuarial Journal2003,7,55–71. 38. Acerbi, C.; Tasche, D. On the coherence of expected shortfall. Journal of Banking and Finance 2002, 26,1487–1503. 39. Gupta, A.K.; Varga, T.; Bodnar, T. Elliptically Contoured Models in Statistics and Portfolio Theory, 2nd ed.; Springer,2013. 40. Kotz,S.;Nadarajah,S. MultivariateT-DistributionsandTheirApplications,1sted.;CambridgeUniversityPress, 2004. 41. Broda,S.A.; Paolella,M.S.,Expectedshortfallfordistributionsinfinance. InStatisticalToolsforFinanceand Insurance;Cˇížek,P.;Härdle,K.W.;Weron,R.,Eds.;Springer:Berlin,Heidelberg,2011;pp.57–99. 42. Acerbi,C.;Szekely,B. Backtestingexpectedshortfall. Risk2014. 43. Yamai, Y.; Yoshiba, T. Comparativeanalysesofexpectedshortfallandvalue-at-risk: theirestimationerror, decomposition,andoptimization. MonetaryandEconomicStudies2002,20,87–121.