MPRA
Munich Personal RePEc Archive
Conditional Value-at-Risk and Average
Value-at-Risk: Estimation and
Asymptotics
So Yeon Chun and Alexander Shapiro and Stan Uryasev
School of Industrial and Systems Engineering, Georgia Institute of
Technology, epartment of Industrial and Systems Engineering,
University of Florida
3 April 2011
Online at
https://mpra.ub.uni-muenchen.de/33115/
Value-at-Risk: Estimation and Asymptotics
SoYeonChun*
SchoolofIndustrialandSystemsEngineering,GeorgiaInstituteofTechnology,[email protected] Alexander Shapiro
y
SchoolofIndustrialandSystemsEngineering,GeorgiaInstituteofTechnology,[email protected] StanUryasev
DepartmentofIndustrialandSystemsEngineering,UniversityofFlorida,uryasev@u .edu
Wediscusslinearregressionapproachestoestimationoflawinvariantconditionalriskmeasures.Two esti-mation proceduresare considered and compared; oneis based onresidual analysis of the standard least squares method and the otheris in the spiritof the M-estimationapproach used inrobust statistics.In particular,Value-at-RiskandAverageValue-at-Riskmeasuresarediscussedindetails.Largesample statis-ticalinferenceoftheestimatorsisderived.Furthermore, nitesamplepropertiesoftheproposedestimators are investigated andcompared withtheoretical derivations inanextensive MonteCarlo study. Empirical resultsonthereal-data(di erent nancialasset classes) arealsoprovidedtoillustratetheperformance of theestimators.
Keywords: Value-at-Risk,AverageValue-at-Risk,linearregression,leastsquaresresiduals,M-estimators, quantileregression,conditionalriskmeasures,lawinvariantriskmeasures,statistical inference
1. Introduction
In nancialindustry,sell-sideanalystsperiodicallypublishrecommendationsof underlying securi-tieswithtargetprices(e.g.,GoldmanSach'sConvictionBuyList).Thoserecommendationsre ect speci ceconomicconditionsandin uenceinvestors'decisionsandthuspricemovements.However, thistypeof analysisdoesnotprovideriskmeasuresassociatedwithunderlyingcompanies.Wesee similar phenomenainthe buy-side analysisas well. Each analystor team covers di erent sectors (e.g.,AirlinesVS Semi-conductors)andtypicallymakesseparaterecommendationsforthe portfo-liomanagerswithoutassociatedrisk measures.However, riskmeasures of thecoveringcompanies
ResearchofthisauthorwaspartlysupportedbytheNSFawardDMS-0914785. y
are one of themost important factors for investment decision making. In this paper, we consider ways to estimate risk measures for a single asset at given market conditions. These information couldbe usefulforinvestorsandportfoliomanagerstocompareprospective securitiesandtopick thebest.Forexample,whenportfoliomanagersexpectthecrudeoilpricetohike(duetoin ation or geo-politicalcon icts), they couldselect securitiesless sensitive to oilprice movements in the airlineindustry.
In order to formalize our discussion, let us introduce the following setting. Let (;F) be a measurablespaceequippedwithprobabilitymeasureP.AmeasurablefunctionY :!R iscalled arandomvariable.WithrandomvariableY,weassociateanumber(Y)towhichwe referasrisk measure.Weassumethat\smallerisbetter",i.e.,betweentwopossiblerealizationsofrandomdata, weprefertheonewithsmallervalueof().Theterm\riskmeasure"issomewhatunfortunatesince itcanbe confusedwith theprobabilitymeasure.Moreover, inapplicationsoneoftentriestoreach acompromisebetweenminimizingtheexpectation(i.e.,minimizingonaverage)andcontrollingthe associatedrisk.Thus,someauthorsusetheterm\mean-riskmeasure",or\acceptabilityfunctional" (e.g.P ugandRomisch2007).Forhistoricalreasons,weuseherethe\riskmeasure"terminology. Formally, risk measure is a function :Y !R de ned on an appropriate space Y of random variables.For example, in someapplications it is naturalto use the space Y=L
p
(;F;P), with p2[1;1), of randomvariableshaving nitep-th ordermoments.
It was suggestedin Artzneret al. (1999) that a \good" risk measure shouldhave the following properties(axioms),andsuch risk measureswerecalledcoherent.
(A1) Monotonicity:If Y;Y 0
2Y andY Y 0
,then (Y)(Y 0 ). (A2) Convexity: (tY +(1 t)Y 0 )t(Y)+(1 t)(Y 0 ) for allY;Y 0 2Y andallt2[0;1].
(A3) TranslationEquivariance:Ifa2R andY 2Y,then (Y +a)=(Y)+a. (A4) Positive Homogeneity:If t0andY 2Y,then (tY)=t(Y).
The notation Y Y 0
means that Y(!) Y 0
(!) for a.e. ! 2 . We may refer, e.g., to DetlefsenandScandolo (2005), Weber (2006), FollmerandSchied (2011) fora further discussion of mathematicalpropertiesof risk measures.
An importantexampleof risk measuresis theValue-at-Riskmeasure
V @R (Y)=infft:F Y (t) g; (1) where 2(0;1) and F Y
(t)=Pr(Y t) is the cumulative distribution function (cdf) of Y, i.e., V @R
(Y)=F 1 Y
( ) isthe leftside -quantileof the distributionof Y.This risk measuresatis es axioms(A1),(A3) and(A4),butnot(A2), andhenceisnotcoherent.Another importantexample is theso-calledAverageValue-at-Risk measure, which canbe de ned as
AV @R (Y)=inf t2R t+(1 ) 1 E[ Y t] + (2)
(cf.,RockafellarandUryasev 2002),orequivalently
AV @R (Y)= 1 1 Z 1 V @R (Y)d : (3) Note that AV @R (Y) is nite i E[ Y] +
<1. Therefore, it is natural to use the space Y = L
1
(;F;P)ofrandomvariableshaving nite rstordermomentfortheAV @R
riskmeasure.The Average Value-at-Risk measure is also called the Conditional Value-at-Risk or Expected Short-fall measure. (Since we discuss here \conditional" variants of risk measures, we use the Average Value-at-Riskratherthan ConditionalValue-at-Riskterminology.)
TheValue-at-RiskandAverageValue-at-Riskmeasures arewidelyusedtomeasureandmanage risk in the nancial industry (see, e.g., Jorion 2003, DuÆeandSingleton 2003, Gaglianoneet al. 2011,forthe nancialbackgroundandvariousapplications).Notethatintheabovetwoexamples, risk measures are functions of the distribution of Y. Such risk measures are called law invari-ant. Law invariant risk measures have been studiedextensively inthe nancial risk management literature (e.g., Acerbi 2002, Frey andMcNeil 2002, Scaillet 2004, FermanianandScaillet 2005, ChenandTang 2005, Zhu andFukushima 2009, JacksonandPerraudin 2000, Berkowitzet al.
2002,Bluhmet al.2002,andreferencetherein).Sometimes,wewritea lawinvariantriskmeasure asa function(F) of cdfF.
Nowletusconsiderasituationwherethereexistsinformationcomposedofeconomicandmarket variables X
1 ;:::;X
k
which canbe consideredas a set of predictors fora variableof interestY. In thatcase,onecanbeinterestedinestimationofariskmeasureofY conditionalonobservedvalues of predictors X
1 ;:::;X
k
. For example, suppose we want to measure (predict) the risk of a single assetgivenspeci ceconomicconditionsrepresentedbymarketindexandinterestrates.Then,for arandomvectorX=(X 1 ;:::;X k ) T
ofrelevantpredictors,theconditionalversionofalawinvariant riskmeasure,denoted(YjX)or
jX
(Y),isobtainedbyapplyingtotheconditionaldistribution of Y given X. In particular,V @R
(YjX) is the -quantile of the conditional distribution of Y givenX,and AV @R (YjX)= 1 1 Z 1 V @R (YjX)d : (4)
Recently,severalresearchershavepaid attentiontoestimationoftheconditionalriskmeasures. For the conditional Value-at-Risk, ChernozhukovandUmantsev (2001) used a polynomial type regression quantile model and EngleandManganelli (2004) proposed the model which speci es the evolution of the quantile over time using a special type of autoregressive processes. In both models, unknown parameterswere estimatedbyminimizingthe regression quantile lossfunction. ForconditionalAverageValue-at-Risk,Scaillet(2005)andCaiandWang(2008)utilized Nadaraya-Watson(NW)typenonparametricdoublekernelestimationwhilePeracchi andTanase(2008) and Leoratoet al. (2010) usedthe semiparametricmethod.
In this paper, we discuss procedures forestimationof conditionalrisk measures. Especially,we willpayattentiontoestimationofconditionalValue-at-RiskandAverageValue-at-Riskmeasures. We assume thefollowinglinearmodel(linear regression)
Y = 0 + T X+"; (5) where 0 and =( 1 ;:::; k ) T
are(unknown)parametersofthemodelandtheerror(noise)random variable"isassumedtobeindependentofrandomvectorX.Meaningofthemodel(5)isthatthere
isatrue(population)value 0
;
of therespectiveparametersforwhich(5)holds.Sometimes,we willwritethisexplicitlyandsometimessuppressthisinthe notation.
Let () be a law invariant risk measuresatisfying axiom(A3) (TranslationEquivariance),and
jX
() be its conditionalanalogue. Note that because of the independenceof "and X,it follows that
jX
(")=(").Together withaxiom(A3), thisimplies
jX (Y)= jX ( 0 + T X+")= 0 + T X+ jX (")= 0 + T X+("): (6) Since 0 +(")=("+ 0
), we can set (")=0 by adding a constant to the error term. In that case, for the truevalues of the parameters,we have
jX (Y)= 0 + T
X. Hence, the question is howtoestimatethese(true) values
0
;
of therespectiveparameters.
This paperis organizedasfollows.InSection2,wedescribetwodi erent estimationprocedures fortheconditionalriskmeasures;oneisbasedonresidualsoftheleastsquaresestimationprocedure andtheotherisbasedontheM-estimationapproach.Asymptoticpropertiesofbothestimatorsare provided inSection3. In Section4,we investigate the nite sampleandasymptoticpropertiesof theconsideredestimators.We presentMonteCarlosimulationresultsunder di erent distribution assumptions of the error term. Later, we illustrate the performance of di erent methods on the real data (di erent nancial asset classes) in Section 5. Finally, Section6 gives some conclusion remarks andsuggestionsforfutureresearch directions.
2. Basic Estimation Procedures
Suppose that we have N observations (datapoints) (Y i
;X i
), i=1;:::;N, which satisfy the linear regressionmodel(5),i.e.,
Y i = 0 + T X i +" i ; i=1;:::;N: (7)
We assume that:(i)X i
,i=1;:::;N, areiid(independent identicallydistributed)randomvectors, andwriteX forrandomvectorhavingthesamedistributionasX
i
,(ii)theerrors" 1
;::;" N
are iid with nitesecondordermomentsandindependentof X
i .We denoteby 2 =Var[" i ]the common varianceof theerror terms.
There aretwobasicapproachestoestimationofthetruevaluesof 0
and .One approachisto apply thestandardLeast Squares(LS) estimationprocedureandthen to make an adjustment of the estimateof the interceptparameter
0
.That is,let ~ 0
and ~
be the least squares estimators of therespectiveparameters ofthe linearmodel(7) and
e i :=Y i ~ 0 ~ T X i ; i=1;:::;N; (8)
be the correspondingresiduals. By the standard theory of the LS method, we have that ~ 0
and ~
are unbiased estimatorsof the respective parametersof the linearmodel (5) provided E["]=0. Therefore,weneedtomake thecorrection
~ 0
+(")oftheinterceptestimator.Ifweknewthetrue values"
1 ;:::;"
N
oftheerrorterm,wecouldestimate(")byreplacingthecdfF " of"byitsempirical estimate ^ F ";N associated with" 1 ;:::;" N ,i.e.,toestimate(F " ) by( ^ F ";N
).Sincetruevalues ofthe error term are unknown, it is a naturalidea to replace"
1 ;:::;"
N
by the residual values e 1
;:::;e N
. Hence, we use the estimator
~ 0 +( ^ F e;N ), where ^ F e;N
is the empirical cdf of the residual values, i.e.,
^ F
e;N
is thecdfof theprobabilitydistributionassigningmass1=N toeachpointe i
,i=1;:::;N (seesection3.1 for furtherdiscussion).We referto thisestimationapproach asthe Least Squares Residuals (LSR)method.
Analternativeapproachisbasedonthefollowingidea.Supposethatwecanconstructafunction h(y;) of y2R and 2R, convex in , such that the minimizer of E
F
[h(Y;)] will be equal to (F),i.e.,(F)=argmin
E
F
[h(Y;)]: Since(Y +a)=(Y)+aforanya2R, itfollowsthat the function h(y;) should be of the form h(y;)= (y ) forsome convex function :R !R. We referto () asthe error function.Therefore,we needtoconstructan errorfunction such that
(F)=argmin
E F
[ (Y )]: (9)
This isequivalenttosolving theequation
E F
[ (Y )]=0; (10)
where (t):= 0
(t).Note thattheerrorfunction () couldbe nondi erentiable,inwhichcasethe correspondingderivativefunction ()is discontinuous.Thatis,thefunction ()ismonotonically nondecreasing.
The correspondingestimators ^ 0 and ^
aretakenassolutionsof theoptimizationproblem
Min 0 ; N X i=1 Y i 0 T X i : (11)
In the statistics literature, such estimators are called M-estimators (the terminology which we will follow) and for an appropriate choice of the error function, this is the approach of robust regression(Huber1981).FortheV @R
risk measure,theerrorfunctionis readilyavailable(recall that [t] + =maxf0;tg): (t):= [t] + +(1 )[ t] + : (12)
The corresponding robust regression approach is known as the quantile regression method (cf. Koenker2005).
Forcoherentriskmeasures,thesituationismoredelicate.Letusmakethefollowingobservations. Suppose that therepresentation (9) holds.Let F
1 andF
2
be two cdfsuchthat (F 1
)=(F 2
)=. Thenitfollowsby(9) (by(10))that(tF
1
+(1 t)F 2
)=foranyt2[0;1]. Thisis quite astrong necessary condition for existence of a representation of the form (9). It certainly doesn't hold for the AV @R
, 2(0;1), risk measure. Indeed, consider the following probabilitydistributions F 1 := Æ a + 1 2 (1 )(Æ b +Æ d ),F 2 := Æ c +(1 )Æ (b+d)=2 and 1 2 (F 1 +F 2 )= 1 2 Æ a + 1 4 (1 )Æ b + 1 2 Æ c + 1 4 (1 )Æ d + 1 2 (1 )Æ (b+d)=2 ; whereÆ x
denotesmeasureofmassoneatx, 2( 1 2
;1)anda<b<c<daresuchthatc< 1 2
(b+d).It isstraightforwardtocalculatethatAV @R
(F 1 )=AV @R (F 2 )= 1 2 (b+d).However,since 2( 1 2 ;1), AV @R 1 2 (F 1 +F 2 ) = 1 4 b+2c (1 ) 1 +2d > 1 4 (b+2c+2d)> 1 4 (2b+c+2d)> 1 2 (b+d):
These arguments areduetoGneiting(2009).
Thisshowsthatforgeneralcoherentriskmeasures,possibilityofconstructingthecorresponding M-estimatorsisratherexceptional,andsuch estimatorscertainlydo notexistfortheAV @R
risk measure. Nevertheless, it is possibleto construct the following approximations (this construction is essentiallyduetoRockafellaretal. (2008)).
Proposition 1. Let j :R!R, j=1;:::;r, be convex functions, j 2R be such that P r j=1 j =1 and E(Y):= inf 2R r ( E " r X j=1 j (Y j ) # : r X j=1 j j =0 ) : (13) Moreover, let S j (Y) be a minimizer ofE[ j
(Y )] over2R. Then S(Y):= P r j=1 j S j (Y) is a minimizerof E(Y ) over2R.
Proof Considertheproblem
Min ; E " r X j=1 j (Y j ) # s:t: r X j=1 j j =0: (14)
By makingchangeof variables j
=+ j
, j=1;:::;r,we canwritethisprobleminthe form
Min ; E " r X j=1 j (Y j ) # s:t: r X j=1 j j =: (15) SinceS j (Y) is a minimizerof E[ j (Y j )], it follows that j =S j (Y), i=1;::;r,=S(Y), is an optimalsolutionof problem(15).This completestheprof.
In particular,we canconsiderfunctions j
() oftheform (12), i.e.,
j (t):= j [t] + +(1 j )[ t] + ; (16) for some j 2 (0;1), j = 1;:::;r. Then S j (Y) = V @R j
(Y) and hence the risk measure P r j=1 j V @R j
(Y) isa minimizerof E(Y ).We canview P r j=1 j V @R j (Y) asa discretization of theintegral 1 1 R 1 V @R
(Y)d ifwe set :=(1 )=rand take
j :=(1 ) 1 ; j := +(j 0:5);j=1;:::;r: (17) For thischoiceof j , j
,andby formula(3),we have that
AV @R (Y)= 1 1 Z 1 V @R (Y)d r X j=1 j V @R j (Y)=S(Y): (18)
Considernowthe problem
Min ; E(Y 0 T X): (19)
By thede nition(13) of E(),wecanwritethis probleminthefollowingequivalent form Min ; 0 ; E h P r j=1 j (Y 0 T X i ) i s:t: P r j=1 j j =0: (20)
The so-calledSampleAverage Approximation(SAA)of thisproblemis
Min ; 0 ; 1 N N X i=1 r X j=1 j (Y i 0 T X i i ) s:t: r X j=1 j j =0: (21)
The above problem (21) can be formulated as a linear programming problem. Following Rockafellaretal. (2008),we considerthe followingestimators.
Mixed quantile estimator for AV @R (Yjx) We refer to 0 + T
x as the mixed quantile estimator of AV @R (Yjx), where ( ; 0 ; ) is an optimalsolutionof problem(21).
Thisideacanbeextendedtoalargerclassoflawinvariantrisk measures.Forexample,consider a riskmeasure
(Y):=cE[ Y]+(1 c)AV @R
(Y) (22)
for some constants c2[0;1] and 2(0;1). Recall that the minimizerof E[(Y t) 2
] is t
=E[Y]. Therefore, by taking functions
0 (t):=t 2 , functions j (t) of the form (16), j , and j given in (17),we canconstructthecorrespondingerrorfunction
E(Y):= inf 2R r+1 ( E " 0 (Y 0 )+ r X j=1 j (Y j ) # :c 0 + r X j=1 (1 c) j j =0 ) : (23)
As anotherexample,considerrisk measuresof theform
(Y):= Z 1 0 AV @R (Y)d( ); (24)
where is a probability measure on the interval [0;1). By a result due to Kusuoka (2001), this measures forma class ofthe comonotelawinvariantcoherentrisk measures.By (3),we canwrite suchrisk measureas
(Y)= Z 1 Z 1 (1 ) 1 V @R (Y)d d( )= Z 1 w( )V @R (Y)d ; (25)
wherew( ):= R 0 (1 ) 1
d( ). Suchriskmeasuresarealsocalledspectralrisk measures(Acerbi 2002).By makinga discretizationof theabove integral(25),we canproceedasabove.
It could be remarked here that while the LSR approach is quite general, the approach based on mixingM-estimatorsissomewhatrestrictive.Constructingan appropriateerrorfunction fora particularriskmeasure couldbe quiteinvolved.
3. Large Sample Statistical Inference
In the previous section, we formulated two approaches, the LSR estimators and mixed M -estimators, toestimationof thetrue (population)valuesof parameters
0
;
of the linearmodel (5) such that (")=0.For theV @R
risk measure,the correspondingM-estimators ^ 0 and ^ are takenas solutionsof theoptimization problem(11), withthe error function(12), andreferredto asthequantileregressionestimators.FortheAV @R
riskmeasureandmoregenerallycomonotone riskmeasuresoftheform(25),weconstructedthecorrespondingmixedquantileestimators ;
0 ; . In thissection, we discuss statisticalpropertiesof these estimators.In particular,we address the questionofwhichofthesetwoestimationproceduresismoreeÆcientbycomputingcorresponding asymptoticvariances.
3.1. Statistical Inference ofLeast Squares ResidualEstimators The linearmodel(7) canbewrittenas
Y =X[ 0 ; ]+ ; (26) where Y =(Y 1 ;:::;Y N ) T
is N 1 vector of responses, X is N (k+1) data matrix of predictor variableswithrows (1;X
T i
),i=1;:::;N,(i.e., rstcolumnofX iscolumnof ones), =( 1
;:::; k
) T vectorof parametersand=("
1 ;::;" N ) T isN1 vectorof errors.By [ 0 ; ], wedenote(k+1)1 vector ( 0 ; T ) T
.We assume that theconditions (i)and (ii),speci ed atthe beginningof section 2,hold.ItisalsopossibletoviewdatapointsX
i
asdeterministic.Inthatcase, weassume thatX hasfullcolumn rankk+1.
Let ~ and
~
Recallthattheseestimatorsaregivenby[ ~ 0 ; ~ ]=(X T X) 1 X T Y,vectorofresidualse:=Y X[ ~ 0 ; ~ ] is givenby e=(I N H)Y =(I N H); where I N
is the NN identitymatrix, and H=X(X T
X) 1
X T
is the so-calledhat matrix. Note that trace(H)=k+1 andwe have that
" i e i =[1;X T i ](X T X) 1 X T ; i=1;:::;N: (27) If we knewerrors" 1 ;::;" N
, we couldestimate(") bythe correspondingsampleestimatebased on theempiricalcdf ^ F ";N ()=N 1 N X i=1 I [" i ;1) (); (28) where I A
() denotes the indicator function of set A. However, the true values of the errors are unknown.Therefore,intheLSRapproachwe replacethembytheresidualscomputedbytheleast squaresmethod andhence estimate(")byemployingtherespectiveempiricalcdf
^ F e;N () instead of ^ F ";N ().
The rstnaturalquestion is whetherthe LSRestimatorsare consistent,i.e., convergew.p.1 to their truevalues asthe samplesizeN tendsto in nity.It iswellknownthat, under thespeci ed assumptions, theLS estimators
~ 0
and ~
are consistent, with ~ 0
beingconsistent under the con-dition E["]=0. The question of consistency of empirical estimates of law invariant coherent risk measures was studiedin Wozabal andWozabal (2009). It was shown that, under mild regularity conditions,such estimatorsareconsistent.Inparticular,theconsistencyholdsforthecomonotone riskmeasures oftheform (25),i.e.,(
^ F ";N ) convergesw.p.1 to(F " ) asN!1.It isalsopossible toshowthatthedi erence(
^ F ";N ) ( ^ F e;N
)tendsw.p.1tozeroandhence( ^ F e;N )convergesw.p.1 to(F "
) aswell. A rigorousproofof thiscouldbe quite technical andwill be beyondthescope of thispaper.
We have thatthe LS estimator h ~ 0 ; ~ i
asymptoticallyhasnormaldistributionwith the asymp-totic covariance matrix N
1 2 1 , where :=E[ X]; :=E XX T and := 1 T : Conse-quently, for a given x, the estimate
~ 0 +x T ~
asymptotically has normal distribution with the asymptoticvarianceN 1 2 [1;x T ] 1 [1;x T ] T .
We also have that random vectors ( ~ 0 ; ~
) and e are uncorrelated. Therefore, if errors " i
have normaldistribution,thenvectors(
~ 0
; ~
)andejointlyhaveamultivariatenormaldistributionand thesevectorsareindependent.Consequently,
~ 0 +x T ~ and( ^ F e;N
)areindependent.Fornonnormal distribution,thisindependenceholdsasymptoticallyandthusasymptotically
~ 0 +x T ~ and( ^ F e;N ) areuncorrelated.
Asymptotics of empirical estimators of law invariant coherent risk measures were studied in P ugandWozabal (2010) and Shapiroetal. (2009, section 6.5). Derivation of the asymptotic varianceof(
^ F
";N
),foragenerallawinvariantriskmeasure,couldbequiteinvolved.Letusconsider two important cases of the V @R
and AV @R
risk measures. We give below a summary of basic results,fora moretechnical discussionwe refertotheAppendix.
In caseof :=V @R ,theLSRestimateof V @R (")becomes [ V @R (e):= ^ F 1 e;N ( )=e (dN e) ; (29) wheree (1) :::e (N)
areorderstatistics(i.e.,numberse 1
;:::;e N
arrangedintheincreasingorder), anddaedenotesthesmallestintegera.SupposethatthecdfF
"
()hasnonzerodensityf " ()=F 0 " () atF 1 " ( ) andlet ! 2 := (1 ) [f " (F 1 " ( ))] 2 : (30) LSR estimator of V @R (Yjx) Consider the LSR estimator
~ 0 +x T ~ + [ V @R (e) of V @R
(Yjx ). Suppose that the set of population -quantiles is a singleton. Then the LSR estimator is a consistent estimator of V @R
(Yjx), andtheasymptoticvarianceof thisestimatorcanbeapproximatedby
N 1 ! 2 + 2 [1;x T ] 1 [1;x T ] T : (31)
Detailedderivationofabove asymptoticsis discussedinAppendix A. Forthe:=AV @R
risk measure,the LSRestimateofAV @R (")is givenby \ AV @R (e)=inf t2R n t+ 1 (1 )N P N i=1 [e i t] + o = [ V @R (e)+ 1 (1 )N P N i=1 h e i [ V @R (e) i + =e (dN e) + 1 P N i=dN e+1 e (i) e (dN e) : (32)
LSR estimator of AV @R
(Yjx ) ConsidertheLSRestimator
~ 0 +x T ~ + \ AV @R (e)ofAV @R
(Yjx).Thisestimatorisconsistent anditsasymptoticvarianceis givenby
N 1 2 + 2 [1;x T ] 1 [1;x T ] T ; (33) where 2 :=(1 ) 2 Var [" V @R (")] + ,:= 1 T ;:=E[X]and:=E XX T .
The aboveasymptoticsare discussedin AppendixB.
Remark 1. Itshouldberememberedthattheaboveapproximatevariancesareasymptoticresults. SupposeforthemomentthatN<(1 )
1
.ThendN e=N andhence [ V @R (")=maxf" 1 ;:::;" N g. Consequently " i [ V @R (") +
=0 foralli=1;:::;N,andhence
\ AV @R (")= [ V @R (")=maxf" 1 ;:::;" N g:
Inthatcasetheaboveasymptoticsareinappropriate.Inorderfortheseasymptoticstobe reason-able,N shouldbesigni cantlybiggerthan(1 )
1 .
LSRapproachcanbeeasilyappliedtoaconsiderablylargerclassoflawinvariantriskmeasures. Forexample,letusconsidertheentropicriskmeasure(Y):=
1 logE[ e
Y
],where >0isa pos-itive constant.This riskmeasuresatis es axioms(A1){(A3),butitis notpositivelyhomogeneous (see Giesecke andWeber 2008, for the general discussion of utility-based shortfallrisk including entropic riskmeasure).The empiricalestimateof (")is
( ^ F ";N )= 1 log N 1 P N i=1 e " i : (34)
Ofcourse,asitwasdiscussedabove,theerrors" i
shouldbereplacedbytherespectiveresidualse i
in theconstructionof thecorrespondingLSRestimators.Byusinglinearizationse
"
=1+ "+o( ") andlog(1+x)=x+o(x),we obtainthat N
1=2 [(
^ F
";N
) (")]converges indistributiontonormal withzeromeanandvariance
2
3.2. Statistical Inference ofQuantile and Mixed Quantile Estimators
As it was discussed in section2, the quantile regression is a particular caseof the M-estimation method with the error function () 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 ( 0 ; ):=E (Y 0 T X) .We alsohave ( 0 ; ) =E 0 + T X+" 0 T X =E[ (" ( 0 0 ) ( ) T X)]: (35)
Under mildregularityconditions,derivativesof ( 0
; )canbetakeninsidetheintegral (expec-tation)andhence
r 0 ( 0 ; ) =E[r 0 (" ( 0 0 ) ( ) T X)] = E[ 0 (" ( 0 0 ) ( ) T X)]; (36) r ( 0 ; ) =E[r (" ( 0 0 ) ( ) T X)] = E[ 0 (" ( 0 0 ) ( ) T X)X]: (37)
Since" andX are independent,we obtainthat derivativesof ( 0 ; ) arezeros at( 0 ; ) ifthe followingconditionholds
E[ 0
(")]=0: (38)
Since function (;) is convex, it follows that if condition (38) holds, then (;) attains its minimumat( 0 ; ).Iftheminimizer( 0 ;
)isunique,thentheestimator( ^ 0 ; ^ )convergesw.p.1 to the population value (
0 ; ) as N !1, i.e., ( ^ 0 ; ^ ) is a consistent estimator of ( 0 ; ) (cf. Huber1981). Thatis,(38) isthebasic conditionforconsistencyof (
^ 0 ; ^ ). Fortheerrorfunction (12) ofthe quantile regression,we have
0 (t)= 1ift<0; ift>0: (39)
(Note that here the error function (t) is not di erentiable at t=0 and its derivative 0
(t) is discontinuous at t=0. Nevertheless, all arguments can go through provided that the error term hasa continuousdistribution.)Consequently,
E[ 0
andhencecondition (38) holds i F "
(0)= , or equivalentlyF 1 "
( )=0 provided thisquantile is unique. In that case, the estimator (
^ 0
; ^
) is consistent if the population value 0
is normalized such that V @R
(")=0. That is,for this errorfunction, ^ 0 + ^ T
x is a consistent estimator of the conditionalValue-at-RiskV @R
(Yjx)of Y givenX=x. It is also possible to derive asymptotics of the estimator (
^ 0
; ^
). That is, suppose that the cdf F
"
() has nonzero density f " () =F 0 " () at F 1 " ( ) and consider ! 2 de ned in (30). Then N 1=2 h ^ 0 0 ; ^ i
converges in distribution to normal with zero mean vector and covariance matrix(cf., Koenker2005) ! 2 [1;x T ] 1 [1;x T ] T ; (41) i.e.,N 1
timesthe matrixgivenin(41) is theasymptoticcovariancematrixof h ^ 0 ; ^ i .
Remark 2. Note that by LLN, we have that N 1 P N i=1 X i andN 1 P N i=1 X i X T i converge w.p.1 as N!1 tothe vector andmatrix respectivelyandthat
T
is thecovariancematrix of X. In case of deterministic X
i
, we simply de ne vector and matrix as the respective limitsof N 1 P N i=1 X i and N 1 P N i=1 X i X T i
, assumingthat suchlimitsexist. It follows then that N
1 X
T
X! asN!1.
Themixedquantileestimator 0 + T
xcanbejusti edbythefollowingarguments.Wehavethat an optimal solution ( ; 0 ;
) of problem (21) converges w.p.1 as N !1 to theoptimal solution ( ? ; ? 0 ; ?
)ofproblem(20),provided(20)hasuniqueoptimalsolution.Becauseofthelinearmodel (5),we canwriteproblem(20) as
Min ; 0 ; E h P r j=1 j ("+ 0 0 +( ) T X j ) i s:t: P r j=1 j j =0; (42) where 0 and
are population values of the parameters. Similarto the proof of Proposition1, bymakingchangeof variables
j =
0 +
j
,j=1;:::;r,we canwrite problem(42) in thefollowing equivalentform Min ; ; E h P r j=1 j ("+ 0 j +( ) T X) i s:t: P r j=1 j j = 0 : (43)
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(c)N(0,1)vs.LN(0,1) Figure1 NormalQ-Qplotfor di erenterrordistributions
It follows thatif r X j=1 j V @R j (")=0; (44) then ( ? 0 ; ? )=( 0 ; ). Thatis, 0 + T x is a consistent estimatorof P r j=1 j V @R j (Yjx). Con-sequentlyfor j and j
givenin (17), we canuse 0 + T xas anapproximationof AV @R (Yjx). Asymptoticsofthemixedquantileestimatorsaremoreinvolved.Theseasymptoticsarediscussed inAppendix C.
4. Simulation Study
Toillustratetheperformanceoftheconsideredestimators,weperformtheMonteCarlosimulations whereerrors(innovations)inlinearmodel(7)are generatedfrom followingdi erent distributions; (1) StandardNormal (denotedas N(0;1)),(2)Student's tdistributionwith 3 degreesof freedom (denotedast(3)),(3)SkewedContaminatedNormalwherestandardnormalis contaminatedwith 20% N(1;9) errors (denoted as CN(1;9)), (4) Log-Normal with parameter 0 and 1 (denotedas LN(0;1)).Notethaterrordistributions(2)-(4)areheavy-tailedincontrasttothenormalerrorsas showninFigure4.Infact, nancialinnovationsoftenfollowheavy-taileddistributions.Weconsider =0:9;0:95;0:99, samplesize N =500;1000;2000 andR=500 replicationsfor each sample size. ConditionalValue-at-Risk(VaR)andAverageValue-at-Risk(AVaR)areestimatedandcompared
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(b)ConditionalAVaR Figure2 ConditionalVaRandAVaR:Truevs.Estimated(ErrorsCN(1;9), =0:95,N=1000)withtrue(theoretical)valuesatgiven500testpointsx k
(k=1;2;:::;500),whichareequallyspaced between -2and2 foreach replication.Estimatorsobtainedfrom di erent methodsarecomputed; quantile based estimator (referred to as \QVaR") and LSR estimator (referred to as \RVaR") for the conditional VaR, mixed quantile estimator (referred to as \QAVaR") and LSRestimator (referredtoas\RAVaR") fortheconditionalAVaR (as describedinSection2).
Figure 4 displays an example of estimation results where solid line is true (theoretical) VaR (AVaR), dash-circle line is QVaR (QAVaR), and dash-cross line is RVaR (RAVaR) given test points x
k
. In this example, errors follow CN(1;9), = 0:95 and N = 1000. In Fig-ure 4-(a), RVaR estimates are closer to true VaR values as Mean Absolute Error (MAE) con rms (MAE(QVaR)=0.4771 vs. MAE(RVaR)=0.2145). Performance of both estimators are worse for AVaR, yet RAVaR estimates are still closer to true AVaR values than QAVaR (MAE(QAVaR)=0.6336vs.MAE(RAVaR)=0.2466)asshowninFigure4-(b).
To compareestimatorsunder di erent errordistributions,MAE (averaged over alltest points) and variance of MAE (in parenthesis) across 500 replications are obtainedas shown in Table 1. Regardless of the error distributions,RVaR (RAVaR) works better than QVaR (QAVaR); MAE and the varianceof MAE are smaller. As we can expect,both estimators perform better for the
Table1 MAEfordi erenterrordistributions =0:95;N=1000(averagedoveralltestpoints) Error QVaR RVaR QAVaR RAVaR
N(0;1) 0.0762 0.0575 0.0990 0.0674 (0.0037) (0.0020) (0.0058) (0.0026) t(3) 0.1758 0.1290 0.4255 0.3232 (0.0188) (0.0095) (0.0808) (0.0623) CN(1;9) 0.3006 0.1955 0.3844 0.2311 (0.0563) (0.0225) (0.0882) (0.0316) LN(0;1) 0.3905 0.2670 0.8957 0.6432 (0.0959) (0.0430) (0.3896) (0.2481)
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)
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Estimator − Error Distribution
Figure3 MAEforconditionalAVaRgivenx=1:006underdi erenterrordistributions( =0:95,N=1000)
conditionalVaR thanAVaR.
Figure 3 presents box-plots for both estimators (QAVaR and RAVaR) given x=1:006 across 500replications.Findingsaresimilar totheonefrom Table1; therearesomeevidence tosuggest that RAVaR hassmaller MAEthan QAVaR. Also, RAVaR is more stable thanQAVaR (MAE of QAVaRismorespread).Notethatbothestimatorsworkbetterfornormaldistributionsthanother heavy-tailed distributions.We couldobserve thesimilarpattern forconditionalVaR.
Table2 MAEfordi erentsamplesizeN with =0:95(averagedoveralltestpoints) Error Estimator N=500 N =1000 N =2000 N(0;1) QVaR 0.1129 0.0762 0.0569 RVaR 0.0849 0.0575 0.0418 QAVaR 0.1390 0.0990 0.0737 RAVaR 0.0992 0.0674 0.0498 t(3) QVaR 0.2420 0.1758 0.1277 RVaR 0.1785 0.1290 0.0942 QAVaR 0.5385 0.4255 0.3207 RAVaR 0.4517 0.3232 0.2085 CN(1;9) QVaR 0.4322 0.3006 0.2180 RVaR 0.2928 0.1955 0.1447 QAVaR 0.5471 0.3844 0.2658 RAVaR 0.3373 0.2311 0.1636 LN(0;1) QVaR 0.5814 0.3905 0.2959 RVaR 0.4095 0.2670 0.1975 QAVaR 1.1986 0.8957 0.7275 RAVaR 0.9503 0.6432 0.4754
Table2illustratessamplesizee ectonMAEofestimators.Asexpected,bothestimatorsperform betterassamplesizeincreases.MAEofRVaR(RAVaR)isstillsmallerthanthatofQVaR(QAVaR) across allsamplesizes.
Next, we obtain asymptotic variances (derived in Section3) andcompare that with empirical ( nitesample)variancesofbothestimators.Figure4reportsasymptoticand nitesample eÆcien-cies of bothestimators forthe conditional VaR where R=500, anderror follows N(0;1) (results aresimilarforothererrordistributions).InFigure4-(a),weseethatasymptoticvarianceofRVaR (dash-dot line)is smallerthan that of QVaR (solid line) except atx
k
near 0.In fact,asymptotic variance is a ected by how far x
k
is away from 0 (which is the mean of explanatory variable in thesimulation);whenx
k
is furtherfromthemean,thedi erencebetweenasymptoticvariancesof bothestimators is bigger. Figure4-(b) provides empirical variance of both estimatorsacross 500 replications.Empirical variance of RVaR is (equal or) smaller than that of QVaR at allx
k . Fig-ure4-(c)andFigure4-(d)compareasymptoticvariancestoempiricalvariancesofbothestimators. Itisclearthatasymptoticvariancesaretoprovideagoodapproximationtotheempiricalonesfor bothestimators.
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Figure4 ConditionalVaR:asymptoticandempiricalvariance(ErrorN(0;1), =0:95,N=1000,R=500)
Figure 4 illustrates asymptotic and empirical variances of both estimators for AVaR. Insights obtainedfromtheresultsaresimilartotheVaRcase.However,Figure4-(c)indicatesthatempirical variancesofQAVaRarelargerthanasymptoticvariances,especiallywhenx
k
isfarfromthemean. For this case, asymptoticeÆciency of QAVaR may not very informative on its behavior in nite sample. Results are similarfor othererror distributions except t(3). Whenthe error follows t(3), asymptotic (empirical) variances of QAVaR are smaller than that of RAVaR except when x
k is closetothe boundary(asshowninFigure4).
Tofurtherinvestigatethe nitesampleeÆcienciesandrobustnessofbothestimatorscomparedto theasymptoticones,weprovideempiricalcoverageprobabilities(CP)ofatwo-sided95%(nominal)
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Figure5 ConditionalAVaR:asymptoticandempiricalvariance(ErrorN(0;1), =0:95,N=1000,R=500)
con dence interval (CI) in Table 3 (di erence between CP and 0.95 is given in parentheses). For each replication, the empirical con dence interval is calculated from the sample version of asymptoticvariance(whenappliedtothevaluesof an observedsampleof a givensize).Then,for given x
k
, the proportion of the 500 replications where the obtained con denceinterval contains thetrue(theoretical)valueiscalculated,andtheseproportionsareaveragedacross alltestpoints. For N(0;1) and CN(1;9) error distributions, the resulting CP of RVaR (RAVaR) is very close to 0.95 whileempirical CI for QVaR (QAVaR) under-covers (resulting CP is smaller than 0.95). For t(3) and LN(0;1) error distributions,CP of RVaR (RAVaR) drops, yet maintains somewhat adequateCPwhichisalotbetterthanCPofQVaR(QAVaR).CIofQAVaRunder-coversseriously
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Figure6 ConditionalAVaR:asymptoticandempiricalvariance(Errort(3), =0:95,N=1000,R=500)
(resultingCP is about0.7)andthisindicatesQAVaR proceduremay be veryunstable andneeds rather wider CI than other estimators to overcome its sensitivity. Note that RVaR (RAVaR) is moreconservativethanQVaR (QAVaR) regardlessof theerrordistributions.
Wecoulddrawsimilarconclusionsforothersamplesizesand values.Thatis,RVaR(RAVaR) performsbetterandprovidesstableresultsthanQVaR(QAVaR)underdi erenterrordistributions.
5. Illustrative Empirical Examples
In thissection, we demonstrate consideredmethods toestimateconditional VaR andAVaR with realdata; di erent nancialasset classes.Let us rstpresent an example of CreditDefaultSwap
Table3 Coverage probabilitywith =0:95;N=1000(averagedoveralltestpoints) Error QVaR RVaR QAVaR RAVaR
N(0;1) 0.9167 0.9551 0.8442 0.9552 (0.0333) (-0.0051) (0.1058) (-0.0052) t(3) 0.9044 0.9269 0.7088 0.9080 (0.0456) (0.0231) (0.2412) (0.0420) CN(1;9) 0.9262 0.9428 0.8824 0.9548 (0.0238) (0.0072) (0.0676) (-0.0048) LN(0;1) 0.9185 0.9276 0.6930 0.9185 (0.0315) (0.0224) (0.2570) (0.0315)
(CDS). CDS is the most popular credit derivative in the rapidly growing credit markets (see FitchRatings2006, fora detailedsurvey of the creditderivativesmarket). CDScontractprovides insuranceagainsta defaultbya particularcompany,a poolofcompanies,orsovereign entity.The rate of payments made per year bythe buyer is known as the CDS spread (in basis points). We focusontherisk ofCDStrading(longorshortposition)ratherthanontheuseofaCDStohedge credit risk. The CDS dataset obtainedfrom Bloomberg consists of 1006 daily observations from January2007toJanuary2011.LetthedependentvariableY bedailypercent change,(Y(t+1) Y(t))=Y(t)100,ofBankofAmericaCorp(NYSE:BAC)5-yearCDSspread,explanatoryvariables X
1
bedailyreturnofBACstockprice,andX 2
bedailypercentchangeofgeneric5-yearinvestment gradeCDX spread (CDX.IG). We use the term \percent change"ratherthan return becausethe return of CDScontract is not same as the return of CDSspread (e.g., see O'KaneandTurnbull 2003,for an overview of CDSvaluationmodels).Residuals obtainedfrom thisdatasetare heavy-taileddistributed(similartoFigure4-(b)).
Figure 7 shows estimatedconditionalVaR (RVaR) of BAC CDS spread percentchange (result ofQVaRissimilar).Sinceonecantake eithershortorlongposition,we presentbothtailriskwith allvaluesof whichranges from0.01 to0.99; <0:5correspondstothelefttail(shortposition) and right tail (long position), otherwise. It is clear that RVaR of certain dates are much higher (lower) than normal level due to the di erent daily economic conditions re ectedby BAC stock price and CDX spread. This indicates the speci c (daily) economic conditions should be taken accountfortheaccurateestimationofrisk,andthereforeemphasizetheimportanceof conditional
Figure7 EstimatedconditionalVaR(RVaR)forBACCDSspreadpercentchangefor =0:01;:::;0:99
risk measures.Note that givena speci cdate,estimatedRVaR curve alongthedi erent values is asymmetricsincethedistributionof CDSspreadpercentchange isnot symmetric.
To compare the prediction performance of both estimators, we forecast 603 one-day-ahead (tomorrow's)VaR(AVaR)giventhecurrent(today's)valueofexplanatoryvariablesusingarolling window of the previous 403 days. Figure 8 presents forecasting results of QVaR and RVaR with =0:05on603out-of-sample.Bothestimatorsshowsimilarbehaviors,butRVaRseemslittlemore stable. Following ideas in McNeilandFrey(2000) and Leorato etal. (2010), \violation event" is said to occur whenever observed CDS spread percent change falls below the predicted VaR (we can nda few violation events from Figure8). Also, the forecasterror of AVaR is de ned as the di erence between the observed CDS spread percent change and the predicted AVaR under the violationevent.Byde nition,theviolationeventprobabilityshouldbecloseto andtheforecast error should be close to zero. Table 4 presents the prediction performance(violation event prob-abilityfor VaR,mean andMAEof forecasterror forAVaR inparenthesis)of bothestimatorsfor =0:01 and 0:05. In-sample statistics show that both estimators tthe data well; the violation
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Figure8 RiskpredictionofBACCDS:QVaRandRVaR( =0.05)
Table4 RiskpredictionperformanceofBACCDS In-sample Event(%) Mean MAE QVaR(QAVaR) 0.01 0.9950 (0.1965) (1.3118)
RVaR(RAVaR) 0.01 0.9950 (-0.8630) (2.8183) QVaR(QAVaR) 0.05 4.9751 (0.2287) (2.5016) RVaR(RAVaR) 0.05 4.9751 (-0.0269) (2.8090) Out-of-sample Event(%) Mean MAE QVaR(QAVaR) 0.01 0.8292 (1.4546) (2.4421)
RVaR(RAVaR) 0.01 0.8292 (1.1052) (4.0615) QVaR(QAVaR) 0.05 3.6484 (1.3740) (3.1099) RVaR(RAVaR) 0.05 4.4776 (-0.3722) (3.3681)
event probabilitiesare very close to and forecast errors are very small. Out-of-sample perfor-mances of both estimatorsare very similar for =0:01, even thoughthe forecast errorsincrease a little compared to in-sample cases. For =0:05, RVaR (RAVaR) seems perform better; event probabilitiesare closerto 0.05andforecasterrorsaresmaller.
Next, we applyconsideredmethodstoone of theUS equities; InternationalBusinessMachines Corp (NYSE). The dataset contains 1722 daily observation from December 2005 to December
Table5 RiskpredictionperformanceofIBMstock In-sample Event(%) Mean MAE QVaR(QAVaR) 0.01 1.0180 (-0.1305) (0.5727)
RVaR(RAVaR) 0.01 0.9397 (-0.3481) (0.8926) QVaR(QAVaR) 0.05 5.0117 (0.0468) (1.0204) RVaR(RAVaR) 0.05 4.9334 (-0.0225) (1.1579) Out-of-sample Event(%) Mean MAE QVaR(QAVaR) 0.01 2.3511 (0.6171) (1.1028)
RVaR(RAVaR) 0.01 1.8809 (0.5023) (0.6827) QVaR(QAVaR) 0.05 6.7398 (0.4787) (1.3086) RVaR(RAVaR) 0.05 6.1129 (0.4778) (1.2387)
2010.Let the dependent variableY be the daily logreturn, 100*log(Y(t+1)/Y(t)),of IBM stock price, explanatory variables X
1
be the log return of S&P 500 index, and X 2
be the lagged log return.Similar toCDS example,we forecast638 one-day-ahead (tomorrow's) VaR (AVaR) given the current (today's) value of explanatory variables using a rolling window of the previous 639 days.Residualsobtainedfrom thisdatasetareheavy-taileddistributed.Table4 comparestherisk prediction performance of IBM stock return. Bothestimators perform well for in-sample predic-tion. For out-of-sample prediction, both estimators behave similarly for =0:05, but violation event probability is largerthan 0.05.For =0:01, RVaR (RAVaR) seems a bit better,but event probabilityexceeds0.01.
Finally, we illustrate how crude oil price had impacted the US airlines' risk as we mentioned in Section 1. Crude oil prices had continuedto rise since May 2007 and peaked alltime high in July2008,rightbeforethebrinkoftheUS nancialsystemcollapse.Wecomparethemovementof estimatedVaRforthreeairlinestocksgivencrudeoilpricechange;DeltaAirlines,Inc(NYSE:DAL), American Airlines,Inc(NYSE:AMR), andSouthwestAirlines Co(NYSE:LUV). Figure9 depicts RVaR movement with =0:05 from May2007 to July 2008 (QVaR shows similar patterns).For easy comparison, we standardize all units relative to the starting date. As we can see, crude oil pricehadjumped150%duringthistimespan.On theotherhand,RVaR of LUVincreasedabout 15% whilethat of AMR increased120% andthat of DALincreased 90% (in magnitude). Infact, di erent airlines have di erent strategies to hedge the risk on oil price uctuations and this in turna ectstheriskofairlines'stockmovement.Forexample,SouthwestAirlinesiswellknownfor
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Figure9 Airlineequities:RVaRconditionaloncrudeoilprice( =0.05)
hedging crudeoil pricesaggressively.On the other hand, Delta Airlines does little hedge against crudeoilprice,butoperates alot of international ights. American Airlinesdoesnothave strong hedging againstcrude oilprice either, andoperates less international ights than Delta Airlines. Our estimationresults con rmthe rm speci crisk regardingcrudeoilprice uctuations.
6. Conclusions
Value-at-Risk and Average Value-at-Risk are widely used measures of nancialrisk. In order to accuratelyestimateriskmeasures,takingintoaccountthespeci ceconomicconditions,we consid-ered two estimationprocedures for conditionalrisk measures;oneis based on residual analysisof thestandardleastsquaresmethod(LSRestimator)andtheotherisbasedonmixedM-estimators (mixedquantileestimator).Largesamplestatisticalinferencesof bothestimatorsarederivedand compared. Inaddition, nitesamplepropertiesof bothestimatorsareinvestigatedand compared aswell.MonteCarlosimulationresults,under di erent error distributions,indicatethat the LSR estimatorsperformbetterthantheir(mixed)quantilescounterparts.Ingeneral,MAEand asymp-totic/empiricalvarianceof theLSRestimatorsaresmallerthanthatof quantilebasedestimators.
We alsoobservethat asymptoticvarianceof estimatorsapproximatesthe nitesampleeÆciencies wellforreasonablesamplesizes usedin practice.However, we mayneed moresamplesto guaran-teean acceptableeÆciencyof thequantilebasedestimatorforAverageValue-at-risk comparedto other estimators. Prediction performances on the real data example suggest similar conclusions. Infact,residualbasedestimatorscanbe calculatedeasilyandthereforetheLSRmethodcouldbe implemented eÆciently in practice. Moreover, LSR method can be easily applied to the general class of law invariant risk measures. In this study, we assume a static model with independent error distributions.Extension of consideredestimationprocedures incorporatingdi erent aspects of (dynamic)timeseriesmodelscouldbe an interestingtopicforthefurtherstudy.
Appendix A: Asymptotics for LSR Estimatorof V @R
(Yjx)
Suppose, forthe sake of simplicity, that support of the distributionof X i is bounded,i.e., X i is boundedw.p.1. SinceN 1 X T
X convergesw.p.1 toandby(27),wehavethat
j" i e i jO p (N 1 ) N X j=1 " j :
We can assumehere thatE[ " i ]=0,andhence P N j=1 " j =O p (N 1=2 ). Itfollowsthat " (dN e) e (dN e) =O p (N 1=2 ): (45)
Supposenowthatthesetof population -quantilesisasingleton.Then ^ F
1 "
( )convergesw.p.1 tothe populationquantileF
1 "
( )=V @R
("), and henceby (45), we have that e (dN e) converges in probabilityto F 1 " ( ). That is, [ V @R
(e) is a consistent estimator of V @R
("), and hencethe estimator ~ 0 +x T ~ + [ V @R
(e)isa consistentestimatorof V @R
(Yjx).
Let us consider the asymptotic eÆciency of the residual based V @R estimator. It is known that ~ 0 +x T ~
is an unbiased estimator of the true expected value 0 + x T and N 1=2 h ~ 0 0 +x T ( ~ ) i
convergesin distributiontonormalwith zeromeanandvariance
2 [1;x T ] 1 [1;x T ] T : (46) Also, N 1=2 " (dN e) V @R (")
converges indistributiontonormalwith zeromeanandvariance
! 2 := (1 ) [f " (F 1 ( ))] 2 ; (47)
provided thatdistributionof "hasnonzerodensityf " () atthe quantile F 1 " ( ).
Let us also estimate the asymptotic variance of the right hand side of (27). We have that N timesvarianceof thesecondterm intheright handsideof (27) canbe approximatedby
2 E [1;X T i ] 1 [1;X T i ] T = 2 (k+1):
Wealsohavethatrandomvectors( ~ 0
; ~
)andeareuncorrelated.Therefore,iferrors" i
havenormal distribution, then vectors (
~ 0
; ~
) and e have jointlya multivariatenormal distributionandthese vectorsareindependent.Consequently,
~ 0 +x T ~ and [ V @R
(e)areindependent.Fornotnecessarily normaldistribution,thisindependenceholdsasymptoticallyandthusasymptotically
~ 0 +x T ~ and [ V @R
(e)areuncorrelated.
Now,wecancalculatetheasymptoticcovarianceofthecorrespondingterms " (dN e) V @R (") and " (dN e) e (dN e) as 2 k +1 2
.Thus,asymptoticvarianceoftheresidualbasedV @R
estima-torcan be approximatedas
N 1 ! 2 + 2 [1;x T ] 1 [1;x T ] T : (48)
Appendix B: Asymptotics for LSR Estimatorof AV @R (Yjx) Theestimator \ AV @R
(e)canbecomparedwiththecorrespondingrandomvariablewhichisbased on theerrorsinsteadofresiduals
\ AV @R ("):=inf t2R n t+ 1 (1 )N P N i=1 [" i t] + o = [ V @R (")+ 1 (1 )N P N i=1 h " i [ V @R (") i + = " (dN e) + 1 (1 )N P N i=dN e+1 " (i) " (dN e) : (49) Note that \ AV @R
(")isnotan estimatorsince errors" i are unobservable. By (45), we have that [ V @R (") [ V @R (e) =O p (N 1=2 ) (50)
and it is known that \ AV @R
(") converges w.p.1 to AV @R
(") as N!1, provided that " has a nite rst order moment. It follows that
\ AV @R
(e) converges in probability to AV @R ("), and hence ~ +x T ~ + \
Lets discuss asymptotic properties of the residual based AV @R
estimator. As it was pointed out in Appendix A , random vectors (
~ 0
; ~
) and e are uncorrelated, and hence asymptotically ~ 0 +x T ~ and \ AV @R
(e) are independent and hence uncorrelated. Assuming that -quantile of F
"
() isunique,wehave byDelta theorem
\ AV @R (e)=V @R (")+(1 ) 1 N 1 N X i=1 [e i V @R (")] + +o p (N 1=2 ) (51) and \ AV @R (")=V @R (")+(1 ) 1 N 1 N X i=1 [" i V @R (")] + +o p (N 1=2 ): (52)
Equation(52)leadstothefollowingasymptoticresult(cf.Trindadeet al.2007,Shapiroetal.2009, section6.5.1) N 1=2 \ AV @R (") AV @R (") D !N(0; 2 ); (53) where 2 =(1 ) 2 Var [" V @R (")] +
. Moreover, ifdistributionof " hasnonzerodensityf " () atV @R ("), then E \ AV @R (") AV @R (")= 1 2Nf " (V @R (")) +o(N 1 ): (54)
From the equation (51) and (52), the asymptoticvariance of
\ AV @R (") \ AV @R (e) can be boundedby(1 ) 1 N 2 2 k+1
andwe canapproximatetheasymptoticcovarianceof the cor-respondingterms, \ AV @R (") AV @R (") and \ AV @R (") \ AV @R (e) as (1 ) 1 N 2 2 k +1 2 . Thus,asymptoticvarianceof theresidualbasedAV @R
estimatorcanbe approximatedas
N 1 2 + 2 [1;x T ] 1 [1;x T ] T : (55)
Appendix C: Asymptoticsfor the Mixed QuantileEstimator
It is possible to derive asymptotics of the mixed quantile estimator. For the sake of simplicity, let us start with a sample estimate of S(X), with
j
and j
, j=1;:::;r, given in (17). That is, let X
1 ;:::;X
N
be an iid sample (data) of the random variable X, and X (1)
:::X (N)
be the corresponding order statistics. Then the corresponding sampleestimate is obtainedby replacing
thetruedistributionF ofXbyitsempiricalestimate ^ F.Consequently,(1 ) 1 S(X)isestimated by (1 ) 1 r X j=1 j ^ F 1 ( j )= 1 r r X j=1 X (dN j e) : (56)
This canbe comparedwith thefollowingestimatorof AV @R
(X) basedon sampleversion of (2): X (dN e) + 1 (1 )N P N i=dN e+1 X (i) X (dN e) = 1 N dN e (1 )N X (dN e) + 1 (1 )N P N i=dN e+1 X (i) : (57)
Assuming that N is an integerandtakingr:=(1 )N, we obtainthat the right handsides of (56) and(57) are thesame.
Asymptotic variance of the mixed quantile estimator can be calculated as follows. Consider problem(43).The optimalsolution ofthat problemis
? = , ? j = 0 +V @R j (")= 0 +F 1 " ( j );j=1;:::;r; and ? 0 = P r j=1 j ? j = 0
. Assume that " has continuous distribution with cdf F "
() and density function f
"
(). Then conditional on X, the asymptotic covariance matrix of the corresponding sample estimator ( ;) of ( ? ; ? ) is N 1 H 1 H 1
, where H is the Hessian matrix of second orderpartialderivativesof E
h P r j=1 j ("+ 0 j +( ) T X) i atthepoint ( ? ; ? ),andis thecovariancematrixof therandom vector
Z:= r X j=1 r j "+ 0 j +( ) T X ;
where the gradients are taken with respect to ( ;) at ( ;)=( ? ; ? ) (e.g., Shapiro 1989). We have r X j=1 r j "+ 0 j +( ) T X = r X j=1 0 j "+ 0 j +( ) T X ! X; r j j "+ 0 j +( ) T X = 0 j "+ 0 j +( ) T X ; with 0 j () is givenin(39). Note that E[ 0 (" F 1 " ( j
E
ZZ T
and we can compute = E XX T T , where =E h P r j=1 0 j (" F 1 " ( j )) i 2 , =[ 1 ;:::; r ]with j =E " r X i=1 0 i " F 1 " ( i ) ! 0 j " F 1 " ( j ) X # ;j=1;:::;r; and ij =E h 0 i (" F 1 " ( i )) 0 j (" F 1 " ( j )) i ,i;j=1;:::;r. The Hessianmatrix H canbecomputedasH=
E XX T F F T D ,where = P r j=1 j with j = @E h 0 j ("+ 0 ? j +t) i @t t=0 = @[ j (1 F " (F 1 " ( j ) t))+( j 1)F " (F 1 " ( j ) t)] @t t=0 = j f " (F 1 " ( j )) (1 j )f " (F 1 " ( j ))=f " (F 1 " ( j )); j=1;:::;r; F =[F 1 ;:::;F r ]with F j = j E[X],j=1;:::;r, andD=diag( 1 ;:::; r ). Since 0 = T , we have that 0 + T x=[x T ; T ][
;], and hence the asymptotic variance of 0 + 0 T x isgivenbyN 1 [x T ; T ]H 1 H 1 [x ;]. References
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