Optimization of Conditional V alue-at-Risk

1 (CVaR)ratherthanminimizingValue-at-Risk( VaR), ,alsocalledMeanExcessLoss,MeanShortfall, orTailVaR, ,brokerage rms,mutualfunds, , .edu,URL: . (VaR)hasaroleintheapproach,buttheemphasi sisonConditionalValue-at-Risk(CVaR),whic hisknownalsoasMeanExcessLoss,MeanShortfa ll, nitionwithrespecttoaspeci edprobabilitylevel ,the -VaRofaportfolioisthelowestamount suchthat,withprobability ,thelosswillnotexceed ,whereasthe -CVaRistheconditionalexpectationoflosses abovethatamount .Threevaluesof , nitionsensurethatthe -VaRisnevermorethanthe -CVaR, ,alongwithre-latedresources, ,approachestocalculatingVaRrelyonlineara pproximationoftheportfoliorisksandassume ajointnormal(orlog-normal)dis-tributiono ftheunderlyingmarketparameters,see,forin stance,Du eandPan(1997),Jorion(1996),Pritsker(1997 ),RiskMetrics(1996),Simons(1996),Beder(1 995),Stambaugh(1996).

2 Also,historicalorMonteCarlosimulation-ba sedtoolsareusedwhentheportfoliocontainsn on-linearinstrumentssuchasoptions(Bucaya ndRosen(1999),Jorion(1996),MauserandRose n(1999),Pritsker(1997),RiskMetrics(1996) ,Beder(1995),Stambaugh(1996)).Discussion sofoptimizationproblemsinvolvingVaRcanbe foundinpapersbyLitterman(1997a,1997b),Ka stetal.(1998),LucasandKlaassen(1998).Alt houghVaRisaverypopularmeasureofrisk,itha sundesirablemathematicalcharac-teristics suchasalackofsubadditivityandconvexity,s eeArtzneretal.(1997,1999).VaRiscoherento nlywhenitisbasedonthestandarddeviationof normaldistributions(foranormaldistributi onVaRisproportionaltothestandarddeviatio n).Forexample, ,VaRisdi (1999),McKayandKeefer(1996)showedthatVaR canbeill-behavedasafunctionofportfoliopo sitionsandcanexhibitmultiplelocalextrema , ,CVaRisknowntohavebetterpropertiesthanVa R,seeArtzneretal.

3 (1997),Embrechts(1999).Recently,P ug(2000)provedthatCVaRisacoherentriskmea surehavingthefollowingproperties:transit ion-equivariant,positivelyhomogeneous,co nvex, , (2000).AlthoughCVaRhasnotbecomeastandard inthe nanceindustry,CVaRisgainingintheinsuranc eindustry,seeEmbrechtsetal.(1997).Bucaya ndRosen(1999) (1999).SimilarmeasuresasCVaRhavebeenearl ierintroducedinthestochasticprogrammingl iterature,althoughnotin (1995) ,asalreadyobservedfromthede ordsaconvenientwayofevaluating linearandnonlinearderivatives(options,fu tures); market,credit,andoperationalrisks; circumstancesinanycorporationthatisexpos edto ,brokerage rms,mutualfunds, , ,seeforinstance,BirgeandLouveaux(1997),E rmolievandWets(1988),KallandWallace(1995 ),KanandKibzun(1996),P ug(1996),Prekopa(1995).

4 Nitefamilyofscenariosisselectedasanappro ximation, nancearea,see,forinstance,Zenios(1996),Z iembaandMulvey(1998).2 DESCRIPTIONOFTHEAPPROACHLetf(x;y)bethelo ssassociatedwiththedecisionvectorx,tobec hosenfromacertainsubsetXofIRn,andtherand omvectoryinIRm.(Weuseboldfacetypeforvect orstodistinguishthemfromscalars.)Thevect orxcanbeinterpretedasrepresentingaportfo lio,withXasthesetof3availableportfolios( subjecttovariousconstraints), , ,thatcana ,ine ect, ,thelossf(x;y) ,whichwedenotebyp(y).However,asitwillbes hownlater,ananalyticalexpressionp(y) (code)whichgeneratesrandomsamplesfromp(y ).Atwostepprocedurecanbeusedtoderiveanal yticalexpressionforp(y)orconstructaMonte Carlosimulationcodefordrawingsamplesfrom p(y)(see,forinstance,RiskMetrics(1996)): (1)modelingofriskfactorsinIRm1,(withm1<m ),(2)basedonthecharacteristicsofinstrume nti,i=;:::;n,thedistributionp(y)canbeder ivedorcodetransformingrandomsamplesofris kfactorstotherandomsamplesfromdensityp(y ) (x;y)notexceedingathreshold isgiventhenby (x; )=Zf(x;y) p(y)dy:(1)Asafunctionof for xedx, (x; ) , (x; )isnondecreasingwithrespectto andcontinuousfromtheright, ,orinotherwords,that (x; )iseverywherecontinuouswithrespectto.

5 Thisassumption,likethepreviousoneaboutde nsityiny, ,eveninthede nitionofCVaR, ,therequiredcontinuityfollowsfrompropert iesoflossf(x;y)andthedensityp(y);seeUrya sev(1995).The -VaRand -CVaRvaluesforthelossrandomvariableassoc iatedwithxandanyspeci edprobabilitylevel in(0;1)willbedenotedby (x)and (x).Inoursettingtheyaregivenby (x)=minf 2IR: (x; ) g(2)and (x)=(1 ) 1Zf(x;y) (x)f(x;y)p(y)dy:(3)4 Inthe rstformula, (x)comesoutastheleftendpointofthenonempt yintervalconsistingofthevalues suchthatactually (x; )= .(Thisfollowsfrom (x; )beingcontinuousandnondecreasingwithresp ectto .Theintervalmightcontainmorethanasinglep ointif has\ atspots.")Inthesecondformula,theprobabil itythatf(x;y) (x)isthereforeequalto1.

6 Thus, (x)comesoutastheconditionalexpectationof thelossassociatedwithxrelativetothatloss being (x) (x)and (x)intermsofthefunctionF onX IRthatwenowde nebyF (x; )= +(1 ) 1Zy2 IRm[f(x;y) ]+p(y)dy;(4)where[t]+=twhent>0but[t]+=0w hent ,undertheassumptionsmadeabove, ,whichisakeypropertyinoptimizationthatin particulareliminatesthepossibilityofaloc alminimumbeingdi erentfromaglobalminimum,seeRockafellar(1 970),Shor(1985), ,F (x; )isconvexandcontinuouslydi -CVaRofthelossassociatedwithanyx2 Xcanbedeterminedfromtheformula (x)=min 2 IRF (x; ):(5)Inthisformulathesetconsistingofthev aluesof forwhichtheminimumisattained,namelyA (x)=argmin 2 IRF (x; );(6)isanonempty,closed,boundedinterval( perhapsreducingtoasinglepoint),andthe -VaRofthelossisgivenby (x)=leftendpointofA (x).

7 (7)Inparticular,onealwayshas (x)2argmin 2 IRF (x; )and (x)=F (x; (x)):(8) (1 )F (x; )asminimizeF (x; ).Thiswouldavoiddividingtheintegralby1 andmightbebetternumericallywhen1 -CVaRcan5becalculatedwithout rsthavingtocalculatethe -VaRonwhichitsde nitiondepends, -VaRmaybeobtainedinsteadasabyproduct,but theextrae ortthatthismightentail(indeterminingthei ntervalA (x)andextractingitsleftendpoint,ifitcont ainsmorethanonepoint)canbeomittedif -VaRisn' ,theintegralinthede nition(4)ofF (x; ) ,thiscanbedonebysamplingtheprobabilitydi stributionofyaccordingtoitsdensityp(y).I fthesamplinggeneratesacollectionofvector sy1;y2;:::;yq,thenthecorrespondingapprox imationtoF (x; )is~F (x; )= +1q(1 )qXk=1[f(x;yk) ]+:(9)Theexpression~F (x; )isconvexandpiecewiselinearwithrespectto .

8 Althoughitisnotdi erentiablewithrespectto ,itcanreadilybeminimized, -CVaRofthelossassociatedwithxoverallx2 XisequivalenttominimizingF (x; )overall(x; )2X IR,inthesensethatminx2X (x)=min(x; )2X IRF (x; );(10)wheremoreoverapair(x ; )achievesthesecondminimumifandonlyifx achievesthe rstminimumand 2A (x ).Inparticular,therefore,incircumstances wheretheintervalA (x )reducestoasinglepoint(asistypical),them inimizationofF(x; )over(x; )2X IRproducesapair(x ; ),notnecessarilyunique,suchthatx minimizesthe -CVaRand givesthecorresponding ,F (x; )isconvexwithrespectto(x; ),and (x)isconvexwithrespecttox,whenf(x;y)isco nvexwithrespecttox,inwhichcase,ifthecons traintsaresuchthatXisaconvexset, , ,itisnotnecessary,forthepurposeofdetermi ninganxthatyieldsminimum -CVaR,toworkdirectlywiththefunction (x),whichmaybehardtodobecauseofthenature ofitsde nitionintermsofthe -VaRvalue (x) ,6onecanoperateonthefarsimplerexpression F (x.)

9 Withitsconvexityinthevariable andeven,verycommonly,withrespectto(x; ).TheoptimizationapproachsupportedbyTheo rem2canbecombinedwithideasforapprox-imat ingtheintegralinthede nition(4)ofF (x; ) (x;y)withrespecttoxproducesconvexityofth eapproximatingexpression~F (x; )in(9), overX IRfallsintothecategoryofstochasticoptimi zation,ormorespeci callystochasticprogramming,becauseofpres enceofan\expectation"inthede nitionofF (x; ).Atleastforthecasesinvolvingconvexity,t hereisavastliteratureonsolvingsuchproble ms(BirgeandLouveaux(1997),ErmolievandWet s(1988),KallandWallace(1995),KanandKibzu n(1996),P ug(1996),Prekopa(1995)).Theorem2opensthe doortoapplyingthattotheminimizationof ,weconsidernowthecasewherethedecisionvec torxrepresentsaportfolioof nancialinstrumentsinthesensethatx=(x1;:: :;xn)withxjbeingthepositionininstrumentj andxj 0forj=1;:::;n;withXnj=1xj=1:(11)Denoting byyjthereturnoninstrumentj,wetaketherand omvectortobey=(y1;:::;yn).

10 Thedistributionofyconstitutesajointdistr ibutionofthevariousreturnsandisindepende ntofx;ithasdensityp(y).Thereturnonaportf olioxisthesumofthereturnsontheindividual instrumentsintheportfolio, ,beingthenegativeofthis,isgiventherefore byf(x;y)= [x1y1+ +xnyn]= xTy:(12)Aslongasp(y)iscontinuouswithresp ecttoy,thecumulativedistributionfunction sforthelossassociatedwithxwillitselfbeco ntinuous;seeKanandKibzun(1996),Uryasev(1 995).AlthoughVaRandCVaRusuallyisde nedinmonetaryvalues,herewede (thismaynotbetruefortheportfolioswithzer onetinvestment).Inthissection,wecomparet heminimumCVaRmethodologywiththeminimumva rianceapproach,therefore, -VaRand -CVaRisF (x; )= +(1 ) 1Zy2 IRn[ xTy ]+p(y)dy:(13)It'simportanttoobservethat, inthissetting,F (x; )isconvexasafunctionof(x; ),notjust.