Munich Personal RePEc Archive
Conditional Value-at-Risk and Average
Value-at-Risk: Estimation and
Asymptotics
Chun, So Yeon and Shapiro, Alexander and Uryasev, Stan
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 Asymptotis
SoYeonChun*
ShoolofIndustrialandSystemsEngineering,GeorgiaInstituteofTehnology,shunisye.gateh.edu Alexander Shapiro
y
ShoolofIndustrialandSystemsEngineering,GeorgiaInstituteofTehnology,ashapiroisye.gateh.edu StanUryasev
DepartmentofIndustrialandSystemsEngineering,UniversityofFlorida,uryasevu.edu
Wedisusslinearregressionapproahestoestimationoflawinvariantonditionalriskmeasures.Two
esti-mation proeduresare onsidered and ompared; oneis based onresidual analysis of the standard least
squares method and the otheris in the spiritof the M-estimationapproah used inrobust statistis.In
partiular,Value-at-RiskandAverageValue-at-Riskmeasuresaredisussedindetails.Largesample
statis-tialinfereneoftheestimatorsisderived.Furthermore,nitesamplepropertiesoftheproposedestimators
are investigated andompared withtheoretial derivations inanextensive MonteCarlo study.Empirial
resultsonthereal-data(dierent nanialasset lasses) arealsoprovidedtoillustratetheperformane of
theestimators.
Keywords: Value-at-Risk,AverageValue-at-Risk,linearregression,leastsquaresresiduals,M-estimators,
quantileregression,onditionalriskmeasures,lawinvariantriskmeasures,statistial inferene
1. Introdution
Innanialindustry,sell-sideanalystsperiodiallypublishreommendationsof underlying
seuri-tieswithtargetpries(e.g.,GoldmanSah'sConvitionBuyList).Thosereommendationsreet
speieonomionditionsandinueneinvestors'deisionsandthuspriemovements.However,
thistypeof analysisdoesnotprovideriskmeasuresassoiatedwithunderlyingompanies.Wesee
similar phenomenainthe buy-side analysisas well. Eah analystor team overs dierent setors
(e.g.,AirlinesVS Semi-ondutors)andtypiallymakesseparatereommendationsforthe
portfo-liomanagerswithoutassoiatedrisk measures.However, riskmeasures of theoveringompanies
ResearhofthisauthorwaspartlysupportedbytheNSFawardDMS-0914785. y
are one of themost important fators for investment deision making. In this paper, we onsider
ways to estimate risk measures for a single asset at given market onditions. These information
ouldbe usefulforinvestorsandportfoliomanagerstoompareprospetive seuritiesandtopik
thebest.Forexample,whenportfoliomanagersexpettherudeoilprietohike(duetoination
or geo-politialonits), they ouldselet seuritiesless sensitive to oilprie movements in the
airlineindustry.
In order to formalize our disussion, let us introdue the following setting. Let (;F) be a
measurablespaeequippedwithprobabilitymeasureP.AmeasurablefuntionY :!R isalled
arandomvariable.WithrandomvariableY,weassoiateanumber(Y)towhihwe referasrisk
measure.Weassumethat\smallerisbetter",i.e.,betweentwopossiblerealizationsofrandomdata,
weprefertheonewithsmallervalueof().Theterm\riskmeasure"issomewhatunfortunatesine
itanbe onfusedwith theprobabilitymeasure.Moreover, inappliationsoneoftentriestoreah
aompromisebetweenminimizingtheexpetation(i.e.,minimizingonaverage)andontrollingthe
assoiatedrisk.Thus,someauthorsusetheterm\mean-riskmeasure",or\aeptabilityfuntional"
(e.g.PugandRomish2007).Forhistorialreasons,weuseherethe\riskmeasure"terminology.
Formally, risk measure is a funtion :Y !R dened on an appropriate spae Y of random
variables.For example, insome appliations it is naturalto use the spae Y=L p
(;F;P), with
p2[1;1), of randomvariableshavingnitep-th ordermoments.
It was suggestedin Artzneret al. (1999) that a \good" risk measure shouldhave the following
properties(axioms),andsuh risk measureswerealledoherent.
(A1) Monotoniity: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) TranslationEquivariane:Ifa2R andY 2Y,then (Y +a)=(Y)+a.
The notation Y Y 0
means that Y(!) Y 0
(!) for a.e. ! 2 . We may refer, e.g., to
DetlefsenandSandolo (2005), Weber (2006), FollmerandShied (2011) fora further disussion
of mathematialpropertiesof 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 umulative distribution funtion (df) of Y, i.e.,
V R
(Y)=F 1 Y
() isthe leftside-quantileof the distributionof Y.This risk measuresatises
axioms(A1),(A3) and(A4),butnot(A2), andheneisnotoherent.Another importantexample
is theso-alledAverageValue-at-Risk measure, whih anbe dened as
AV R
(Y)=inf t2R
t+(1 ) 1
E[ Y t℄ +
(2)
(f.,RokafellarandUryasev 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 spae Y =
L 1
(;F;P)ofrandomvariableshavingniterstordermomentfortheAV R
riskmeasure.The
Average Value-at-Risk measure is also alled the Conditional Value-at-Risk or Expeted
Short-fall measure. (Sine we disuss here \onditional" variants of risk measures, we use the Average
Value-at-Riskratherthan ConditionalValue-at-Riskterminology.)
TheValue-at-RiskandAverageValue-at-Riskmeasuresarewidelyusedtomeasureandmanage
risk in the nanial industry (see, e.g., Jorion 2003, DuÆeandSingleton 2003, Gaglianoneet al.
2011,forthenanialbakgroundandvariousappliations).Notethatintheabovetwoexamples,
risk measures are funtions of the distribution of Y. Suh risk measures are alled law
invari-ant. Law invariant risk measures have been studiedextensively inthe nanial risk management
literature (e.g., Aerbi 2002, Frey andMNeil 2002, Saillet 2004, FermanianandSaillet 2005,
2002,Bluhmet al.2002,andreferenetherein).Sometimes,wewritea lawinvariantriskmeasure
asa funtion(F) of dfF.
Nowletusonsiderasituationwherethereexistsinformationomposedofeonomiandmarket
variables X 1
;:::;X k
whih anbe onsideredas a set of preditors fora variableof interestY. In
thatase,oneanbeinterestedinestimationofariskmeasureofY onditionalonobservedvalues
of preditors X 1
;:::;X k
. For example, suppose we want to measure (predit) the risk of a single
assetgivenspeieonomionditionsrepresentedbymarketindexandinterestrates.Then,for
arandomvetorX=(X 1
;:::;X k
) T
ofrelevantpreditors,theonditionalversionofalawinvariant
riskmeasure,denoted(YjX)or jX
(Y),isobtainedbyapplyingtotheonditionaldistribution
of Y given X. In partiular,V R
(YjX) is the -quantile of the onditional distribution of Y
givenX,and
AV R
(YjX)= 1 1
Z 1
V R
(YjX)d: (4)
Reently,severalresearhershavepaid attentiontoestimationoftheonditionalriskmeasures.
For the onditional Value-at-Risk, ChernozhukovandUmantsev (2001) used a polynomial type
regression quantile model and EngleandManganelli (2004) proposed the model whih speies
the evolution of the quantile over time using a speial type of autoregressive proesses. In both
models, unknown parameterswere estimatedbyminimizingthe regression quantile lossfuntion.
ForonditionalAverageValue-at-Risk,Saillet(2005)andCaiandWang(2008)utilized
Nadaraya-Watson(NW)typenonparametridoublekernelestimationwhilePerahi andTanase(2008) and
Leoratoet al. (2010) usedthe semiparametrimethod.
In this paper, we disuss proedures forestimationof onditionalrisk measures. Espeially,we
willpayattentiontoestimationofonditionalValue-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
isatrue(population)value 0
;
of therespetiveparametersforwhih(5)holds.Sometimes,we
willwritethisexpliitlyandsometimessuppressthisinthe notation.
Let () be a law invariant risk measuresatisfying axiom(A3) (TranslationEquivariane),and
jX
() be its onditionalanalogue. Note that beause of the independeneof "and X,it follows
that jX
(")=(").Together withaxiom(A3), thisimplies
jX
(Y)= jX
( 0
+ T
X+")= 0
+ T
X+ jX
(")= 0
+ T
X+("): (6)
Sine 0
+(")=("+ 0
), we an set (")=0 by adding a onstant to the error term. In that
ase, for the truevalues of the parameters,we have jX
(Y)= 0
+ T
X. Hene, the question is
howtoestimatethese(true) values 0
;
of therespetiveparameters.
This paperis organizedasfollows.InSetion2,wedesribetwodierent estimationproedures
fortheonditionalriskmeasures;oneisbasedonresidualsoftheleastsquaresestimationproedure
andtheotherisbasedontheM-estimationapproah.Asymptotipropertiesofbothestimatorsare
provided inSetion3. In Setion4,we investigate thenite sampleandasymptotipropertiesof
theonsideredestimators.We presentMonteCarlosimulationresultsunder dierent distribution
assumptions of the error term. Later, we illustrate the performane of dierent methods on the
real data (dierent nanial asset lasses) in Setion 5. Finally, Setion6 gives some onlusion
remarks andsuggestionsforfutureresearh diretions.
2. Basi Estimation Proedures
Suppose that we have N observations (datapoints) (Y i
;X i
), i=1;:::;N, whih 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 identiallydistributed)randomvetors,
andwriteX forrandomvetorhavingthesamedistributionasX i
,(ii)theerrors" 1
;::;" N
are iid
withniteseondordermomentsandindependentof X i
.We denoteby 2
=Var[" i
℄the ommon
There aretwobasiapproahestoestimationofthetruevaluesof 0
and.One approahisto
apply thestandardLeast Squares(LS) estimationproedureandthen to make an adjustment of
the estimateof the intereptparameter 0
.That is,let ~ 0
and ~
be the least squares estimators
of therespetiveparameters ofthe linearmodel(7) and
e i
:=Y i
~ 0
~ T
X i
; i=1;:::;N; (8)
be the orrespondingresiduals. By the standard theory of the LS method, we have that ~ 0
and
~
are unbiased estimatorsof the respetive parametersof the linearmodel (5) provided E["℄=0.
Therefore,weneedtomake theorretion ~ 0
+(")oftheintereptestimator.Ifweknewthetrue
values" 1
;:::;" N
oftheerrorterm,weouldestimate(")byreplaingthedfF "
of"byitsempirial
estimate ^ F
";N
assoiated with" 1
;:::;" N
,i.e.,toestimate(F "
) by( ^ F
";N
).Sinetruevalues ofthe
error term are unknown, it is a naturalidea to replae" 1
;:::;" N
by the residual values e 1
;:::;e N
.
Hene, we use the estimator ~ 0
+( ^ F
e;N
), where ^ F
e;N
is the empirial df of the residual values,
i.e., ^ F
e;N
is thedfof theprobabilitydistributionassigningmass1=N toeahpointe i
,i=1;:::;N
(seesetion3.1 for furtherdisussion).We referto thisestimationapproah asthe Least Squares
Residuals (LSR)method.
Analternativeapproahisbasedonthefollowingidea.Supposethatweanonstrutafuntion
h(y;) of y2R and 2R, onvex in , suh that the minimizer of E F
[h(Y;)℄ will be equal to
(F),i.e.,(F)=argmin
E F
[h(Y;)℄: Sine(Y +a)=(Y)+aforanya2R, itfollowsthat the
funtion h(y;) should be of the form h(y;)= (y ) forsome onvex funtion :R !R. We
referto () asthe error funtion.Therefore,we needtoonstrutan errorfuntion suh that
(F)=argmin
E F
[ (Y )℄: (9)
This isequivalenttosolving theequation
E F
[(Y )℄=0; (10)
where(t):= 0
(t).Note thattheerrorfuntion () ouldbe nondierentiable,inwhihasethe
orrespondingderivativefuntion()is disontinuous.Thatis,thefuntion()ismonotonially
The orrespondingestimators ^ 0 and ^
aretakenassolutionsof theoptimizationproblem
Min 0 ; N X i=1 Y i 0 T X i : (11)
In the statistis literature, suh estimators are alled M-estimators (the terminology whih we
will follow) and for an appropriate hoie of the error funtion, this is the approah of robust
regression(Huber1981).FortheV R
risk measure,theerrorfuntionis readilyavailable(reall
that [t℄ +
=maxf0;tg):
(t):=[t℄ +
+(1 )[ t℄ +
: (12)
The orresponding robust regression approah is known as the quantile regression method (f.
Koenker2005).
Foroherentriskmeasures,thesituationismoredeliate.Letusmakethefollowingobservations.
Suppose that therepresentation (9) holds.Let F 1
andF 2
be two dfsuhthat (F 1
)=(F 2
)=.
Thenitfollowsby(9) (by(10))that(tF 1
+(1 t)F 2
)=foranyt2[0;1℄. Thisis quite astrong
neessary ondition for existene of a representation of the form (9). It ertainly doesn't hold
for the AV R
, 2(0;1), risk measure. Indeed, onsider the following probabilitydistributions
F 1 :=Æ a + 1 2 (1 )(Æ b +Æ d ),F 2 :=Æ
+(1 )Æ (b+d)=2 and 1 2 (F 1 +F 2 )= 1 2 Æ a + 1 4 (1 )Æ b + 1 2 Æ + 1 4 (1 )Æ d + 1 2 (1 )Æ (b+d)=2 ; whereÆ x
denotesmeasureofmassoneatx,2( 1 2
;1)anda<b<<daresuhthat< 1 2
(b+d).It
isstraightforwardtoalulatethatAV R
(F 1
)=AV R (F 2 )= 1 2
(b+d).However,sine2( 1 2
;1),
AV R 1 2 (F 1 +F 2 ) = 1 4
b+2(1 ) 1 +2d > 1 4
(b+2+2d)> 1 4
(2b++2d)> 1 2
(b+d):
These arguments areduetoGneiting(2009).
Thisshowsthatforgeneraloherentriskmeasures,possibilityofonstrutingtheorresponding
M-estimatorsisratherexeptional,andsuh estimatorsertainlydo notexistfortheAV R
risk
measure. Nevertheless, it is possibleto onstrut the following approximations (this onstrution
Proposition 1. Let j
:R!R, j=1;:::;r, be onvex funtions, j
2R be suh 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 of E[ 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 makinghangeof variables j
=+ j
, j=1;:::;r,we anwritethisprobleminthe form
Min ; E " r X j=1 j (Y j ) # s:t: r X j=1 j j
=: (15)
SineS 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 ompletestheprof.
In partiular,we anonsiderfuntions 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 hene the risk measure P r j=1 j V R j
(Y) isa minimizerof E(Y ).We anview P r j=1 j V R j
(Y) asa disretization
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 thishoieof 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
By thedenition(13) of E(),weanwritethis 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-alledSampleAverage 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) an be formulated as a linear programming problem. Following
Rokafellaretal. (2008),we onsiderthe 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).
Thisideaanbeextendedtoalargerlassoflawinvariantrisk measures.Forexample,onsider
a riskmeasure
(Y):=E[ Y℄+(1 )AV R
(Y) (22)
for some onstants 2[0;1℄ and 2(0;1). Reall that the minimizerof E[(Y t) 2
℄ is t
=E[Y℄.
Therefore, by taking funtions 0
(t):=t 2
, funtions j
(t) of the form (16), j
, and j
given in
(17),we anonstruttheorrespondingerrorfuntion
E(Y):= inf 2R r+1 ( E " 0 (Y 0 )+ r X j=1 j (Y j ) # : 0 + r X j=1 (1 ) j j =0 ) : (23)
As anotherexample,onsiderrisk 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 lass ofthe omonotelawinvariantoherentrisk measures.By (3),we anwrite
suhrisk measureas
(Y)= Z 1 Z 1 (1 ) 1 V R
(Y)dd()= Z
1
w()V R
wherew():= R
0
(1 ) 1
d(). Suhriskmeasuresarealsoalledspetralrisk measures(Aerbi
2002).By makinga disretizationof theabove integral(25),we anproeedasabove.
It ould be remarked here that while the LSR approah is quite general, the approah based
on mixingM-estimatorsissomewhatrestritive.Construtingan appropriateerrorfuntion fora
partiularriskmeasure ouldbe quiteinvolved.
3. Large Sample Statistial Inferene
In the previous setion, we formulated two approahes, the LSR estimators and mixed M
-estimators, toestimationof thetrue (population)valuesof parameters 0
;
of the linearmodel
(5) suh that (")=0.For theV R
risk measure,the orrespondingM-estimators ^ 0
and ^ are
takenas solutionsof theoptimization problem(11),with the error funtion(12), andreferredto
asthequantileregressionestimators.FortheAV R
riskmeasureandmoregenerallyomonotone
riskmeasuresoftheform(25),weonstrutedtheorrespondingmixedquantileestimators; 0
; .
In thissetion, we disuss statistialpropertiesof these estimators.In partiular,we address the
questionofwhihofthesetwoestimationproeduresismoreeÆientbyomputingorresponding
asymptotivarianes.
3.1. Statistial Inferene ofLeast Squares ResidualEstimators
The linearmodel(7) anbewrittenas
Y =X[ 0
;℄+ ; (26)
where Y =(Y 1
;:::;Y N
) T
is N 1 vetor of responses, X is N (k+1) data matrix of preditor
variableswithrows (1;X T i
),i=1;:::;N,(i.e.,rstolumnofX isolumnof ones),=( 1
;:::; k
) T
vetorof parametersand=(" 1
;::;" N
) T
isN1 vetorof errors.By [ 0
;℄, wedenote(k+1)1
vetor ( 0
; T
) T
.We assume that theonditions (i)and (ii),speied atthe beginningof setion
2,hold.ItisalsopossibletoviewdatapointsX i
asdeterministi.Inthatase, weassume thatX
hasfullolumn rankk+1.
Let ~ and
~
Reallthattheseestimatorsaregivenby[ ~ 0 ; ~ ℄=(X
T X)
1 X
T
Y,vetorofresidualse:=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-alledhat matrix. Note
that trae(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 ouldestimate(") bythe orrespondingsampleestimatebased
on theempirialdf
^ F
";N
()=N 1 N X i=1 I [" i ;1) (); (28) where I A
() denotes the indiator funtion of set A. However, the true values of the errors are
unknown.Therefore,intheLSRapproahwe replaethembytheresidualsomputedbytheleast
squaresmethod andhene estimate(")byemployingtherespetiveempirialdf ^ F e;N () instead of ^ F ";N ().
The rstnaturalquestion is whetherthe LSRestimatorsare onsistent,i.e., onvergew.p.1 to
their truevalues asthe samplesizeN tendsto innity.It iswellknownthat, under thespeied
assumptions, theLS estimators ~ 0
and ~
are onsistent, with ~ 0
beingonsistent under the
on-dition E["℄=0. The question of onsisteny of empirial estimates of law invariant oherent risk
measures was studiedin Wozabal andWozabal (2009). It was shown that, under mild regularity
onditions,suh estimatorsareonsistent.Inpartiular,theonsistenyholdsfortheomonotone
riskmeasures oftheform (25),i.e.,( ^ F
";N
) onvergesw.p.1 to(F "
) asN!1.It isalsopossible
toshowthatthedierene( ^ F ";N ) ( ^ F e;N
)tendsw.p.1tozeroandhene( ^ F
e;N
)onvergesw.p.1
to(F "
) aswell. A rigorousproofof thisouldbe quite tehnial andwill be beyondthesope of
thispaper.
We have thatthe LS estimator h ~ 0 ; ~ i
asymptotiallyhasnormaldistributionwith the
asymp-toti ovariane 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 ~
asymptotially has normal distribution with the
We also have that random vetors ( ~ 0 ; ~
) and e are unorrelated. Therefore, if errors " i
have
normaldistribution,thenvetors( ~ 0
; ~
)andejointlyhaveamultivariatenormaldistributionand
thesevetorsareindependent.Consequently, ~ 0 +x T ~ and( ^ F e;N
)areindependent.Fornonnormal
distribution,thisindependeneholdsasymptotiallyandthusasymptotially ~ 0 +x T ~ and( ^ F e;N ) areunorrelated.
Asymptotis of empirial estimators of law invariant oherent risk measures were studied in
PugandWozabal (2010) and Shapiroetal. (2009, setion 6.5). Derivation of the asymptoti
varianeof( ^ F
";N
),foragenerallawinvariantriskmeasure,ouldbequiteinvolved.Letusonsider
two important ases of the V R
and AV R
risk measures. We give below a summary of basi
results,fora moretehnial disussionwe refertotheAppendix.
In aseof :=V R
,theLSRestimateof V R (")beomes [ V R (e):= ^ F 1 e;N
()=e (dNe)
; (29)
wheree (1)
:::e (N)
areorderstatistis(i.e.,numberse 1
;:::;e N
arrangedintheinreasingorder),
anddaedenotesthesmallestintegera.SupposethatthedfF "
()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 onsistent estimator of
V R
(Yjx), andtheasymptotivarianeof thisestimatoranbeapproximatedby
N 1 ! 2 + 2 [1;x T ℄ 1 [1;x T ℄ T : (31)
Detailedderivationofabove asymptotisis disussedinAppendix A.
Forthe:=AV R
risk measure,the LSRestimateofAV R
(")is givenby \
AV R
LSR estimator of AV R
(Yjx )
ConsidertheLSRestimator ~ 0
+x T
~ +
\ AV R
(e)ofAV R
(Yjx).Thisestimatorisonsistent
anditsasymptotivarianeis 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 aboveasymptotisare disussedin AppendixB.
Remark 1. Itshouldberememberedthattheaboveapproximatevarianesareasymptotiresults.
SupposeforthemomentthatN<(1 ) 1
.ThendNe=N andhene [ V R
(")=maxf" 1
;:::;" N
g.
Consequently
" i
[ V R
(")
+
=0 foralli=1;:::;N,andhene
\ AV R
(")=
[ V R
(")=maxf" 1
;:::;" N
g:
Inthatasetheaboveasymptotisareinappropriate.Inorderfortheseasymptotistobe
reason-able,N shouldbesigniantlybiggerthan(1 ) 1
.
LSRapproahanbeeasilyappliedtoaonsiderablylargerlassoflawinvariantriskmeasures.
Forexample,letusonsidertheentropiriskmeasure(Y):= 1
logE[ e Y
℄,where>0isa
pos-itive onstant.This riskmeasuresatises axioms(A1){(A3),butitis notpositivelyhomogeneous
(see Gieseke andWeber 2008, for the general disussion of utility-based shortfallrisk inluding
entropi riskmeasure).The empirialestimateof (")is
( ^ F
";N )=
1 log
N
1 P
N i=1
e "
i
: (34)
Ofourse,asitwasdisussedabove,theerrors" i
shouldbereplaedbytherespetiveresidualse i
in
theonstrutionof theorrespondingLSRestimators.Byusinglinearizationse "
=1+"+o(")
andlog(1+x)=x+o(x),we obtainthat N 1=2
[( ^ F
";N
) (")℄onverges indistributiontonormal
withzeromeanandvariane 2
3.2. Statistial Inferene ofQuantile and Mixed Quantile Estimators
As it was disussed in setion2, the quantile regression is a partiular aseof the M-estimation
method with the error funtion () of the form (12). By the Law of Large Numbers (LLN), we
have that N 1
times the objetive funtion in (11) onverges (pointwise) w.p.1 to the funtion
( 0
;):=E (Y 0 T X)
.We alsohave
( 0
;) =E 0 + T
X+" 0
T
X
=E[ (" ( 0 0 ) ( ) T X)℄: (35)
Under mildregularityonditions,derivativesof ( 0
;)anbetakeninsidetheintegral
(expe-tation)andhene
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)
Sine" andX are independent,we obtainthat derivativesof ( 0
;) arezeros at( 0
;
) ifthe
followingonditionholds
E[ 0
(")℄=0: (38)
Sine funtion (;) is onvex, it follows that if ondition (38) holds, then (;) attains its
minimumat( 0
;
).Iftheminimizer( 0
;
)isunique,thentheestimator( ^ 0
; ^
)onvergesw.p.1
to the population value ( 0
;
) as N !1, i.e., ( ^ 0
; ^
) is a onsistent estimator of ( 0
;
) (f.
Huber1981). Thatis,(38) isthebasi onditionforonsistenyof ( ^ 0 ; ^ ).
Fortheerrorfuntion (12) ofthe quantile regression,we have
0 (t)=
1ift<0; ift>0:
(39)
(Note that here the error funtion (t) is not dierentiable at t=0 and its derivative 0
(t) is
disontinuous at t=0. Nevertheless, all arguments an go through provided that the error term
hasa ontinuousdistribution.)Consequently,
E[ 0
andheneondition (38) holds i F "
(0)=, or equivalentlyF 1 "
()=0 provided thisquantile is
unique. In that ase, the estimator ( ^ 0
; ^
) is onsistent if the population value 0
is normalized
suh that V R
(")=0. That is,for this errorfuntion, ^ 0 + ^ T
x is a onsistent estimator of the
onditionalValue-at-RiskV R
(Yjx)of Y givenX=x.
It is also possible to derive asymptotis of the estimator ( ^ 0
; ^
). That is, suppose that the
df F "
() has nonzero density f "
() =F 0 "
() at F 1 "
() and onsider ! 2
dened in (30). Then
N 1=2 h ^ 0 0 ; ^ i
onverges in distribution to normal with zero mean vetor and ovariane
matrix(f., Koenker2005)
! 2 [1;x T ℄ 1 [1;x T ℄ T ; (41) i.e.,N 1
timesthe matrixgivenin(41) is theasymptotiovarianematrixof 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 onverge w.p.1
as N!1 tothe vetor andmatrix respetivelyandthat T
is theovarianematrix
of X. In ase of deterministi X i
, we simply dene vetor and matrix as the respetive
limitsof N 1 P N i=1 X i and N 1 P N i=1 X i X T i
, assumingthat suhlimitsexist. It follows then that
N 1
X T
X! asN!1.
Themixedquantileestimator 0 + T
xanbejustiedbythefollowingarguments.Wehavethat
an optimal solution (; 0
;
) of problem (21) onverges w.p.1 as N !1 to theoptimal solution
( ? ; ? 0 ; ?
)ofproblem(20),provided(20)hasuniqueoptimalsolution.Beauseofthelinearmodel
(5),we anwriteproblem(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,
bymakinghangeof variables j
= 0
+ j
,j=1;:::;r,we anwrite problem(42) in thefollowing
−4
−2
0
2
4
−25
−20
−15
−10
−5
0
5
10
15
20
Standard Normal Quantiles
Quantiles of Input Sample
(a)N(0,1)vs.t(3)
−4
−2
0
2
4
−10
−5
0
5
10
15
Standard Normal Quantiles
Quantiles of Input Sample
(b)N(0,1)vs.CN(1,9)
−4
−2
0
2
4
−10
0
10
20
30
40
50
60
70
Standard Normal Quantiles
Quantiles of Input Sample
()N(0,1)vs.LN(0,1) Figure1 NormalQ-Qplotfor dierenterrordistributions
It follows thatif
r X
j=1
j V R
j
(")=0; (44)
then ( ? 0
; ?
)=( 0
;
). Thatis, 0
+ T
x is a onsistent estimatorof P
r j=1
j
V R
j
(Yjx).
Con-sequentlyfor j
and j
givenin (17), we anuse 0
+ T
xas anapproximationof AV R
(Yjx).
Asymptotisofthemixedquantileestimatorsaremoreinvolved.Theseasymptotisaredisussed
inAppendix C.
4. Simulation Study
Toillustratetheperformaneoftheonsideredestimators,weperformtheMonteCarlosimulations
whereerrors(innovations)inlinearmodel(7)are generatedfrom followingdierent distributions;
(1) StandardNormal (denotedas N(0;1)),(2)Student's tdistributionwith 3 degreesof freedom
(denotedast(3)),(3)SkewedContaminatedNormalwherestandardnormalis ontaminatedwith
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-tailedinontrasttothenormalerrorsas
showninFigure4.Infat,nanialinnovationsoftenfollowheavy-taileddistributions.Weonsider
=0:9;0:95;0:99, samplesize N =500;1000;2000 andR=500 repliationsfor eah sample size.
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
−1
0
1
2
3
4
5
6
7
x
k
conditional VaR
MAE(QVaR)=0.4771, MAE(RVaR)=0.2145
True
QVaR
RVaR
(a)ConditionalVaR
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
1
2
3
4
5
6
7
8
x
k
conditional AVaR
MAE(QAVaR)=0.6336, MAE(RAVaR)=0.2466
True
QAVaR
RAVaR
(b)ConditionalAVaR Figure2 ConditionalVaRandAVaR:Truevs.Estimated(ErrorsCN(1;9),=0:95,N=1000)
withtrue(theoretial)valuesatgiven500testpointsx k
(k=1;2;:::;500),whihareequallyspaed
between -2and2 foreah repliation.Estimatorsobtainedfrom dierent methodsareomputed;
quantile based estimator (referred to as \QVaR") and LSR estimator (referred to as \RVaR")
for the onditional VaR, mixed quantile estimator (referred to as \QAVaR") and LSRestimator
(referredtoas\RAVaR") fortheonditionalAVaR (as desribedinSetion2).
Figure 4 displays an example of estimation results where solid line is true (theoretial)
VaR (AVaR), dash-irle line is QVaR (QAVaR), and dash-ross 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 loser to true VaR values as Mean Absolute Error (MAE)
onrms (MAE(QVaR)=0.4771 vs. MAE(RVaR)=0.2145). Performane of both estimators are
worse for AVaR, yet RAVaR estimates are still loser to true AVaR values than QAVaR
(MAE(QAVaR)=0.6336vs.MAE(RAVaR)=0.2466)asshowninFigure4-(b).
To ompareestimatorsunder dierent errordistributions,MAE (averaged over alltest points)
and variane of MAE (in parenthesis) aross 500 repliations are obtainedas shown in Table 1.
Regardless of the error distributions,RVaR (RAVaR) works better than QVaR (QAVaR); MAE
Table1 MAEfordierenterrordistributions=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)
0
0.5
1
1.5
2
2.5
3
MAE (Mean Absolute Error)
Estimator − Error Distribution
Figure3 MAEforonditionalAVaRgivenx=1:006underdierenterrordistributions(=0:95,N=1000)
onditionalVaR thanAVaR.
Figure 3 presents box-plots for both estimators (QAVaR and RAVaR) given x=1:006 aross
500repliations.Findingsaresimilar totheonefrom Table1; therearesomeevidene tosuggest
that RAVaR hassmaller MAEthan QAVaR. Also, RAVaR is more stable thanQAVaR (MAE of
QAVaRismorespread).Notethatbothestimatorsworkbetterfornormaldistributionsthanother
Table2 MAEfordierentsamplesizeN 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
Table2illustratessamplesizeeetonMAEofestimators.Asexpeted,bothestimatorsperform
betterassamplesizeinreases.MAEofRVaR(RAVaR)isstillsmallerthanthatofQVaR(QAVaR)
aross allsamplesizes.
Next, we obtain asymptotivarianes (derived in Setion3) andompare that with empirial
(nitesample)varianesofbothestimators.Figure4reportsasymptotiandnitesample
eÆien-ies of bothestimators forthe onditional VaR where R=500, anderror follows N(0;1) (results
aresimilarforothererrordistributions).InFigure4-(a),weseethatasymptotivarianeofRVaR
(dash-dot line)is smallerthan that of QVaR (solid line) exept atx k
near 0.In fat,asymptoti
variane is aeted by how far x k
is away from 0 (whih is the mean of explanatory variable in
thesimulation);whenx k
is furtherfromthemean,thedierenebetweenasymptotivarianesof
bothestimators is bigger. Figure4-(b) provides empirial variane of both estimatorsaross 500
repliations.Empirial variane of RVaR is (equal or) smaller than that of QVaR at allx k
.
Fig-ure4-()andFigure4-(d)ompareasymptotivarianestoempirialvarianesofbothestimators.
Itislearthatasymptotivarianesaretoprovideagoodapproximationtotheempirialonesfor
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x
k
Variance
QVaR
RVaR
(a)Asymptotivariane
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x
k
Variance
QVaR
RVaR
(b)Empirialvariane
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x
k
Variance
Asymptotic
Empirical
() QVaR:Asymptotivs.Empirial
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x
k
Variance
Asymptotic
Empirical
(d)RVaR:Asymptotivs.Empirial
Figure4 ConditionalVaR:asymptotiandempirialvariane(ErrorN(0;1),=0:95,N=1000,R=500)
Figure 4 illustrates asymptoti and empirial varianes of both estimators for AVaR. Insights
obtainedfromtheresultsaresimilartotheVaRase.However,Figure4-()indiatesthatempirial
varianesofQAVaRarelargerthanasymptotivarianes,espeiallywhenx k
isfarfromthemean.
For this ase, asymptotieÆieny of QAVaR may not very informative on its behavior innite
sample. Results are similarfor othererror distributions exept t(3). Whenthe error follows t(3),
asymptoti (empirial) varianes of QAVaR are smaller than that of RAVaR exept when x k
is
losetothe boundary(asshowninFigure4).
TofurtherinvestigatethenitesampleeÆieniesandrobustnessofbothestimatorsomparedto
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x
k
Variance
QAVaR
RAVaR
(a)Asymptotivariane
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x
k
Variance
QAVaR
RAVaR
(b)Empirialvariane
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x
k
Variance
Asymptotic
Empirical
()QAVaR:Asymptotivs.Empirial
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x
k
Variance
Asymptotic
Empirical
(d)RAVaR:Asymptotivs.Empirial
Figure5 ConditionalAVaR:asymptotiandempirialvariane(ErrorN(0;1),=0:95,N=1000,R=500)
ondene interval (CI) in Table 3 (dierene between CP and 0.95 is given in parentheses).
For eah repliation, the empirial ondene interval is alulated from the sample version of
asymptotivariane(whenappliedtothevaluesof an observedsampleof a givensize).Then,for
given x k
, the proportion of the 500 repliations where the obtained ondeneinterval ontains
thetrue(theoretial)valueisalulated,andtheseproportionsareaveragedaross alltestpoints.
For N(0;1) and CN(1;9) error distributions, the resulting CP of RVaR (RAVaR) is very lose
to 0.95 whileempirial CI for QVaR (QAVaR) under-overs (resulting CP is smaller than 0.95).
For t(3) and LN(0;1) error distributions,CP of RVaR (RAVaR) drops, yet maintains somewhat
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
x
k
Variance
QAVaR
RAVaR
(a)Asymptotivariane
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
x
k
Variance
QAVaR
RAVaR
(b)Empirialvariane
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
x
k
Variance
Asymptotic
Empirical
()QAVaR:Asymptotivs.Empirial
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
x
k
Variance
Asymptotic
Empirical
(d)RAVaR:Asymptotivs.Empirial
Figure6 ConditionalAVaR:asymptotiandempirialvariane(Errort(3),=0:95,N=1000,R=500)
(resultingCP is about0.7)andthisindiatesQAVaR proeduremay be veryunstable andneeds
rather wider CI than other estimators to overome its sensitivity. Note that RVaR (RAVaR) is
moreonservativethanQVaR (QAVaR) regardlessof theerrordistributions.
Weoulddrawsimilaronlusionsforothersamplesizesand values.Thatis,RVaR(RAVaR)
performsbetterandprovidesstableresultsthanQVaR(QAVaR)underdierenterrordistributions.
5. Illustrative Empirial Examples
In thissetion, we demonstrate onsideredmethods toestimateonditional VaR andAVaR with
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 redit derivative in the rapidly growing redit markets (see
FithRatings2006, fora detailedsurvey of the reditderivativesmarket). CDSontratprovides
insuraneagainsta defaultbya partiularompany,a poolofompanies,orsovereign entity.The
rate of payments made per year bythe buyer is known as the CDS spread (in basis points). We
fousontherisk ofCDStrading(longorshortposition)ratherthanontheuseofaCDStohedge
redit risk. The CDS dataset obtainedfrom Bloomberg onsists of 1006 daily observations from
January2007toJanuary2011.LetthedependentvariableY bedailyperent hange,(Y(t+1)
Y(t))=Y(t)100,ofBankofAmeriaCorp(NYSE:BAC)5-yearCDSspread,explanatoryvariables
X 1
bedailyreturnofBACstokprie,andX 2
bedailyperenthangeofgeneri5-yearinvestment
gradeCDX spread (CDX.IG). We use the term \perent hange"ratherthan return beausethe
return of CDSontrat 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 estimatedonditionalVaR (RVaR) of BAC CDS spread perenthange (result
ofQVaRissimilar).Sineoneantakeeithershortorlongposition,we presentbothtailriskwith
allvaluesof whihranges from0.01 to0.99;<0:5orrespondstothelefttail(shortposition)
and right tail (long position), otherwise. It is lear that RVaR of ertain dates are muh higher
(lower) than normal level due to the dierent daily eonomi onditions reetedby BAC stok
prie and CDX spread. This indiates the spei (daily) eonomi onditions should be taken
Figure7 EstimatedonditionalVaR(RVaR)forBACCDSspreadperenthangefor=0:01;:::;0:99
risk measures.Note that givena speidate,estimatedRVaR urve alongthedierent values
is asymmetrisinethedistributionof CDSspreadperenthange isnot symmetri.
To ompare the predition performane of both estimators, we foreast 603 one-day-ahead
(tomorrow's)VaR(AVaR)giventheurrent(today's)valueofexplanatoryvariablesusingarolling
window of the previous 403 days. Figure 8 presents foreasting results of QVaR and RVaR with
=0:05on603out-of-sample.Bothestimatorsshowsimilarbehaviors,butRVaRseemslittlemore
stable. Following ideas in MNeilandFrey(2000) and Leorato etal. (2010), \violation event" is
said to our whenever observed CDS spread perent hange falls below the predited VaR (we
an nda few violation events from Figure8). Also, the foreasterror of AVaR is dened as the
dierene between the observed CDS spread perent hange and the predited AVaR under the
violationevent.Bydenition,theviolationeventprobabilityshouldbelosetoandtheforeast
error should be lose to zero. Table 4 presents the predition performane (violationevent
prob-abilityfor VaR,mean andMAEof foreasterror forAVaR inparenthesis)of bothestimatorsfor
09/08
01/09
06/09
11/09
04/10
08/10
01/11
−50
−40
−30
−20
−10
0
10
20
30
Date
conditional VaR (percent change)
CDS
QVaR
09/08
01/09
06/09
11/09
04/10
08/10
01/11
−50
−40
−30
−20
−10
0
10
20
30
Date
conditional VaR (percent change)
CDS
RVaR
Figure8 RiskpreditionofBACCDS:QVaRandRVaR(=0.05)
Table4 RiskpreditionperformaneofBACCDS 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 lose to and foreast errors are very small. Out-of-sample
perfor-manes of both estimatorsare very similar for =0:01, even thoughthe foreast errorsinrease
a little ompared to in-sample ases. For =0:05, RVaR (RAVaR) seems perform better; event
probabilitiesare loserto 0.05andforeasterrorsaresmaller.
Next, we applyonsideredmethodstoone of theUS equities; InternationalBusinessMahines
Table5 RiskpreditionperformaneofIBMstok 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 log return,100*log(Y(t+1)/Y(t)),of IBM stok
prie, 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 foreast638 one-day-ahead (tomorrow's) VaR (AVaR) given
the urrent (today's) value of explanatory variables using a rolling window of the previous 639
days.Residualsobtainedfrom thisdatasetareheavy-taileddistributed.Table4 omparestherisk
predition performane of IBM stok return. Bothestimators perform well for in-sample
predi-tion. For out-of-sample predition, 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
probabilityexeeds0.01.
Finally, we illustrate how rude oil prie had impated the US airlines' risk as we mentioned
in Setion 1. Crude oil pries had ontinuedto rise sine May 2007 and peaked alltime high in
July2008,rightbeforethebrinkoftheUSnanialsystemollapse.Weomparethemovementof
estimatedVaRforthreeairlinestoksgivenrudeoilpriehange;DeltaAirlines,In(NYSE:DAL),
Amerian Airlines,In(NYSE:AMR), andSouthwestAirlines Co(NYSE:LUV). Figure9 depits
RVaR movement with =0:05 from May2007 to July 2008 (QVaR shows similar patterns).For
easy omparison, we standardize all units relative to the starting date. As we an see, rude oil
priehadjumped150%duringthistimespan.On theotherhand,RVaR of LUVinreasedabout
15% whilethat of AMR inreased120% andthat of DALinreased 90% (in magnitude). Infat,
dierent airlines have dierent strategies to hedge the risk on oil prie utuations and this in
05/07
07/07
09/07
12/07
02/08
05/08
07/08
−150
−120
−90
−60
−30
0
30
60
90
120
150
Crude Oil
DAL
AMR
LUV
Figure9 Airlineequities:RVaRonditionalonrudeoilprie(=0.05)
hedging rudeoil priesaggressively.On the other hand, Delta Airlines does little hedge against
rudeoilprie,butoperates alot of internationalights. Amerian Airlinesdoesnothave strong
hedging againstrude oilprie either, andoperates less internationalights than Delta Airlines.
Our estimationresultsonrm therm speirisk regardingrudeoilprieutuations.
6. Conlusions
Value-at-Risk and Average Value-at-Risk are widely used measures of nanialrisk. In order to
auratelyestimateriskmeasures,takingintoaountthespeieonomionditions,we
onsid-ered two estimationproedures for onditionalrisk measures;oneis basedon residual analysisof
thestandardleastsquaresmethod(LSRestimator)andtheotherisbasedonmixedM-estimators
(mixedquantileestimator).Largesamplestatistialinferenesof bothestimatorsarederivedand
ompared. Inaddition, nitesamplepropertiesof bothestimatorsareinvestigatedand ompared
aswell.MonteCarlosimulationresults,under dierent error distributions,indiatethat the LSR
estimatorsperformbetterthantheir(mixed)quantilesounterparts.Ingeneral,MAEand
We alsoobservethat asymptotivarianeof estimatorsapproximatesthe nitesampleeÆienies
wellforreasonablesamplesizes usedin pratie.However, we mayneed moresamplesto
guaran-teean aeptableeÆienyof thequantilebasedestimatorforAverageValue-at-risk omparedto
other estimators. Predition performanes on the real data example suggest similar onlusions.
Infat,residualbasedestimatorsanbe alulatedeasilyandthereforetheLSRmethodouldbe
implemented eÆiently in pratie. Moreover, LSR method an be easily applied to the general
lass of law invariant risk measures. In this study, we assume a stati model with independent
error distributions.Extension of onsideredestimationproedures inorporatingdierent aspets
of (dynami)timeseriesmodelsouldbe an interestingtopiforthefurtherstudy.
Appendix A: Asymptotis for LSR Estimatorof V R
(Yjx)
Suppose, forthe sake of simpliity, that support of the distributionof X i
is bounded,i.e., X i
is
boundedw.p.1. SineN 1
X T
X onvergesw.p.1 toandby(27),wehavethat
j" i
e i
jO p (N 1 ) N X j=1 " j :
We an assumehere thatE[ " i
℄=0,andhene P N j=1 " j =O p (N 1=2
). Itfollowsthat
" (dNe) e (dNe) =O p (N 1=2 ): (45)
Supposenowthatthesetof population-quantilesisasingleton.Then ^ F
1 "
()onvergesw.p.1
tothe populationquantileF 1 "
()=V R
("), and heneby (45), we have that e (dNe)
onverges
in probabilityto F 1 "
(). That is, [ V R
(e) is a onsistent estimator of V R
("), and henethe
estimator ~ 0 +x T ~ + [ V R
(e)isa onsistentestimatorof V R
(Yjx).
Let us onsider the asymptoti eÆieny of the residual based V R
estimator. It is
known that ~ 0 +x T ~
is an unbiased estimator of the true expeted value 0 + x T and N 1=2 h ~ 0 0 +x T ( ~ ) i
onvergesin distributiontonormalwith zeromeanandvariane
2 [1;x T ℄ 1 [1;x T ℄ T : (46) Also, N 1=2 " (dNe) V R (")
onverges indistributiontonormalwith zeromeanandvariane
provided thatdistributionof "hasnonzerodensityf "
() atthe quantile F 1 "
().
Let us also estimate the asymptoti variane of the right hand side of (27). We have that N
timesvarianeof theseondterm intheright handsideof (27) anbe approximatedby
2 E [1;X T i ℄ 1 [1;X T i ℄ T = 2
(k+1):
Wealsohavethatrandomvetors( ~ 0
; ~
)andeareunorrelated.Therefore,iferrors" i
havenormal
distribution, then vetors ( ~ 0
; ~
) and e have jointlya multivariatenormal distributionandthese
vetorsareindependent.Consequently, ~ 0 +x T ~ and [ V R
(e)areindependent.Fornotneessarily
normaldistribution,thisindependeneholdsasymptotiallyandthusasymptotially ~ 0 +x T ~ and [ V R
(e)areunorrelated.
Now,weanalulatetheasymptotiovarianeoftheorrespondingterms " (dNe) V R (") and " (dNe) e (dNe) as 2 k +1 2
.Thus,asymptotivarianeoftheresidualbasedV R
estima-toran be approximatedas
N 1 ! 2 + 2 [1;x T ℄ 1 [1;x T ℄ T : (48)
Appendix B: Asymptotis for LSR Estimatorof AV R
(Yjx)
Theestimator \ AV R
(e)anbeomparedwiththeorrespondingrandomvariablewhihisbased
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 + = " (dNe) + 1 (1 )N P N i=dNe+1 " (i) " (dNe) : (49) Note that \ AV R
(")isnotan estimatorsine 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
(") onverges w.p.1 to AV R
(") as N!1, provided that " has a
nite rst order moment. It follows that \ AV R
(e) onverges in probability to AV R ("), and hene ~ +x T ~ + \
Lets disuss asymptoti properties of the residual based AV R
estimator. As it was pointed
out in Appendix A , random vetors ( ~ 0
; ~
) and e are unorrelated, and hene asymptotially
~ 0 +x T ~ and \ AV R
(e) are independent and hene unorrelated. 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)leadstothefollowingasymptotiresult(f.Trindadeet al.2007,Shapiroetal.2009,
setion6.5.1)
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 asymptotivariane of
\ AV R
(")
\ AV R
(e)
an be
boundedby(1 ) 1
N 2
2
k+1
andwe anapproximatetheasymptotiovarianeof the
or-respondingterms,
\ AV R
(") AV R (") and \ AV R
(") \ AV R (e) as (1 ) 1 N 2 2 k +1 2 .
Thus,asymptotivarianeof theresidualbasedAV R
estimatoranbe approximatedas
N 1 2 + 2 [1;x T ℄ 1 [1;x T ℄ T : (55)
Appendix C: Asymptotisfor the Mixed QuantileEstimator
It is possible to derive asymptotis of the mixed quantile estimator. For the sake of simpliity,
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
thetruedistributionF ofXbyitsempirialestimate ^
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 anbe omparedwith thefollowingestimatorof AV R
(X) basedon sampleversion of (2):
X (dNe) + 1 (1 )N P N i=dNe+1 X (i) X (dNe) = 1 N dNe (1 )N X (dNe) + 1 (1 )N P N i=dNe+1 X (i) : (57)
Assuming that N is an integerandtakingr:=(1 )N, we obtainthat the right handsides of
(56) and(57) are thesame.
Asymptoti variane of the mixed quantile estimator an be alulated 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 ontinuous distribution with df F "
() and density
funtion f "
(). Then onditional on X, the asymptoti ovariane matrix of the orresponding
sample estimator (
;) of ( ?
; ?
) is N 1
H 1
H 1
, where H is the Hessian matrix of seond
orderpartialderivativesof E h P r j=1 j ("+ 0 j +( ) T X) i
atthepoint ( ?
; ?
),andis
theovarianematrixof therandom vetor
Z:= r X j=1 r j "+ 0 j +( ) T X ;
where the gradients are taken with respet 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 an ompute = 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 anbeomputedasH= 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 ). Sine 0 = T
, we have that 0 + T
x=[x T
; T
℄[
;℄, and hene the asymptoti variane of
0 + 0 T
x isgivenbyN 1 [x T ; T ℄H 1 H 1
[x ;℄.
Referenes
Aerbi,C.2002. Spetralmeasuresofrisk:aoherentrepresentationofsubjetiveriskaversion. Journalof
Banking &Finane 261505{1518.
Artzner,P., F.Delbaen, J.-M.Eber,D. Heath.1999. Coherentmeasures of risk. Mathematial Finane 9
203{228.
Berkowitz,J,M Pritsker,M Gibson,H Zhou. 2002. How aurateare value-at-riskmodelsat ommerial
banks. Journal ofFinane 571093{1111.
Bluhm, Christian, Ludger Overbek, Christoph Wagner. 2002. An Introdution to Credit Risk Modeling
(Chapman&Hall/CrFinanial Mathematis Series). 1sted.Chapman andHall/CRC.
Cai, Z., X. Wang. 2008. Nonparametri estimation of onditional var and expeted shortfall. Journal of
Eonometris 147(1)120{130.
Chen, S. X., C. Y. Tang. 2005. Nonparametri inferene of value-at-riskfor dependent nanial returns.
Chernozhukov, V., L. Umantsev. 2001. Conditional value-at-risk: Aspets of modeling and estimation.
Empirial Eonomis 26(1)271{292.
Detlefsen,K.,G.Sandolo.2005. Conditionalanddynamionvexriskmeasures.FinaneandStohastis .
DuÆe,D.,K.J.Singleton.2003.CreditRisk:Priing,Measurement,andManagement.Prineton,Prineton
UniversityPress.
Engle,R. F., S Manganelli.2004. Caviar:Conditional autoregressivevalueat riskby regressionquantiles.
JournalofBusiness&EonomiStatistis 22367{381.
Fermanian, J.D.,O. Saillet. 2005. Sensitivity analysis of var and expeted shortfall for portfolios under
nettingagreements. Journalof BankingandFinane 29927{958.
FithRatings.2006. Globalredit derivativessurvey.
Follmer,H.,A.Shied.2011.Stohastinane:Anintrodutionindisretetime. WalterdeGruyter&Co.,
Berlin.
Frey,R.,A. J.MNeil.2002. Varandexpetedshortfallin portfoliosofdependentreditrisks:Coneptual
andpratialinsights. JournalofBanking &Finane 26(7)1317{1334.
Gaglianone,W.,L.Lima,O.Linton,SmithD. 2011. Varandexpetedshortfallin portfoliosof dependent
reditrisks:Coneptualandpratialinsights.JournalofBusiness&EonomiStatistis 29150{160.
Gieseke,K.,S. Weber.2008. Measuringthe riskoflargelosses. Journal of InvestmentManagement 6(4)
1{15.
Gneiting, T. 2009. Evaluating point foreasts. Institut fur Angewandte Mathematik, Universitpreprint,
preprint.
Huber,P.J.1981. RobustStatistis. Wiley,NewYork.
Jakson,Patriia, William Perraudin. 2000. Regulatory impliations of redit risk modelling. Journal of
Banking &Finane 24(1-2)1{14.
Jorion,P.2003. Finanial Risk Manager Handbook. 2nded. Wiley,NewYork.
Koenker,R.2005. QuantileRegression. CambridgeUniversityPress,Cambridge,UK.
Leorato,S.,F.Perahi,A.V.Tanase.2010. Asymptotiallyeientestimationoftheonditionalexpeted
MNeil,A.J.,R.Frey.2000.Estimationoftail-relatedriskmeasuresforheterosedastinanialtimeseries:
Anextremevalueapproah. Journalof Empirial Finane 7271{300.
O'Kane, D., S. Turnbull. 2003. Valuation of redit default swaps. Quantitative redit researh quarterly,
LehmanBrothers.
Perahi,F.,A.V.Tanase.2008.Onestimatingtheonditionalexpetedshortfall.AppliedStohastiModels
inBusinessandIndustry 24471493.
Pug,G., W. Romish. 2007. Modeling, Measuring and Managing Risk. World SientiPublishing Co.,
London.
Pug,G.,N.Wozabal.2010. Asymptotidistributionoflaw-invariantriskfuntionals. Finaneand
Stohas-tis 14397{418.
Rokafellar, R. T., S. Uryasev. 2002. Conditional value-at-risk for general lossdistributions. Journal of
Banking&Finane 26(7)1443{1471.
Rokafellar,R.T.,S. Uryasev,M.Zabarankin.2008. Risktuning withgeneralizedlinearregression.
Math-ematis ofOperationsResearh 33712{729.
Saillet, O. 2004. Nonparametri estimation and sensitivity analysis of expeted shortfall. Mathematial
Finane 14(1)115{129.
Saillet, O.2005. Nonparametriestimationofonditionalexpeted shortfall. InsuraneandRisk
Manage-mentJournal 72639{660.
Shapiro, A. 1989. Asymptoti properties of statistial estimators in stohasti programming. Annals of
Statistis 17841{858.
Shapiro,A.,D.Dentheva,A.Ruszzynski.2009.LeturesonStohastiProgramming:ModelingandTheory.
SIAM,Philadelphia.
Trindade,A., S.Uryasev,A. Shapiro,G.Zrazhevsky.2007. Finanial preditionwithonstrainedtailrisk.
JournalofBanking &Finane 313524{3538.
Weber,S. 2006. Distribution-invariantriskmeasures,information,anddynami onsisteny. Mathematial
Finane 16(2)419{441.
Wozabal, D., N. Wozabal. 2009. Asymptoti onsisteny of risk funtionals. Journal of Nonparametri
Zhu, S., M. Fukushima. 2009. Worst-ase onditional value-at-risk with appliation to robust portfolio
