FRM二级培训项目：市场风险测量与管理（阅读版）

「市场风险测量V

\与管理Z

FRMPartIIProgram■基础班

讲师：CrystalGao

e［史由+叶间晞的|hProfQuiomGsvn

TopicWeightingsinFRMPartII

SessionNO.Content%

Session1MarketRiskMeasurementandManagement20

Session2CreditRiskMeasurementandManagement20

Session3OperationalRiskandResiliency20

LiquidityandTreasuryRiskMeasurementand

Session415

Management

Session5RiskManagementandInvestmentManagement15

Session6CurrentIssuesinFinancialMarket10

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ModelingDependence:CorrelationsAnd

Copulas

⑥Framework•SomeCorrelationBasics

i•EmpiricalPropertiesofCorrelation

、MarketRiskMeasurement

\/•FinancialCorrelationModeling

andManagement/EmpiricalApproachestoRiskMetricsand

Hedges

TermStructureModelsofInterestRates

•TheScienceofTermStructureModels

rVaRandotherRiskMeasures•TheEvolutionofShortRatesandthe

•ParametricApproachesShapeoftheTermStructure

•Non-parametricApproaches•TheArtofTermStructureModels:

•Semi-parametricApproachesDrift

•Extremevalue•TheArtofTermStructureModels:

,BacktestingVaRVolatilityandDistribution

•VaRNappingVolatilitySmiles

,RiskMeasurementfortheTradingBook

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VaRandotherRiskMeasures

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Parametric

.

Approaches

VaRandotherRiskMeasures

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♦l.ProfitandLoss

>Profit/Loss

P/L=Pt+Dt-P1

>ArithmeticReturnData:

Pt+Dt—Pt-iPt+Dt

r=-----------------=----------1

tPP

t-it-i

jGeometricReturnData:

P+D

Rt=皿与t-t-)=ln(l+r)

vt

t-i

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♦l.ProfitandLoss

>Thedifferencebetweenthetworeturnsisnegligiblewhenbothreturnsare

small,butthedifferencegrowsasthereturnsgetbigger-whichistobe

expected,asthegeometricisalogfunctionofthearithmeticreturn.

>Sincewewouldexpectreturnstobelowovershortperiodsandhigher

overlongerperiods,thedifferencebetweenthetwotypesofreturnis

negligibleovershortperiodsbutpotentiallysubstantialoverlongerones.

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♦2.NormalVaR

>Approach1:NormalVaR

•Weassumethatarithmeticreturnsarenormallydistributedwithmean叩

andstandarddeviationo

VaR=-(n-zaa)VaR=-(|i-ZaO)P.i

-10

Profit(-t-Vloss(-)

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♦2.NormalVaR

圜>Example:

•Assumethattheprofit/lossdistributionforXYZisnormally

distributedwithanannualmeanof$16millionandastandard

deviationof$11million.CalculatetheVaRatthe95%and99%

confidencelevelsusingaparametricapproach.

VaR(5%)=-$16million+Sllmillionx1.65

=$2.15million

VaR(l%)=-$16million+Sllmillionx2.33

=$9.63million

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♦3.LognormalVaR

>LognormalVaR

•Assumethatgeometricreturnsarenormallydistributedwithmeanp

andstandarddeviationo.Thisassumptionimpliesthatthenatural

logarithmofPtisnormallydistributed,orthatPtitselfislognormally

distributed.NormallydistributedgeometricreturnsimplythattheVaRis

lognormallydistributed.07

VaR=1-

3

=64

3

zQW

O

VaR=(l-e^«)PJ

d3

t-iO6.

2

-08-06-04-02002040808

Loss(4>Vbrofit(-)

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♦3.LognormalVaR

圜,Example:

•Adiversifiedportfolioexhibitsanormallydistributedgeometric

returnwithmeanandstandarddeviationof11%and21%,

respectively.Calculatethe5%and1%lognormalVaRassumingthe

beginningperiodportfoliovalueis$100.

LognormalVaR(5%)-100x(1-e011-0-21x1-65)-$21.06

LognormalVaR(l%)=100x(1-e011-0-21x2-33)=$31.57

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4.Quantile-QuantilePlots

>Weareinterestedinasking:

•Ifdatalooksrightwhenweuseparametricapproach?

•Whatwedois

JPlotourdataonahistogramandestimatetherelevantsummary

statistics.

/Considerwhatkindofdistributionmightfitourdata.

>Aplotofthequantilesoftheempiricaldistributionagainstthoseofsome

specifieddistribution.TheshapeoftheQQplottellsusalotabouthowthe

>Inparticular,iftheQQplotislinear,thenthespecifieddistributionfitsthe

data,andwehaveidentifiedthedistributiontowhichourdatabelong.

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4.Quantile-QuantilePlots

4

3

2

8

=

c

1

cn

b

e-

-2O

d-

E-

山

-1

-2

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♦4.Quantile-QuantilePlots

-10

Normalquantiles

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Non-parametric

Approaches

VaRandotherRiskMeasures

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♦l.HistoricalSimulation

>Allnon-parametricapproachesarebasedontheunderlyingassumptionthat

•Withnon-parametricmethods,therearenoproblemsdealingwith

va种甲nce-covarianciematrices,cursesofdimensionality；etc.~

Loss(+)/profit(-)

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♦l.HistoricalSimulation

>BootstrappedHistoricalSimulation

■Thebootstrapisveryintuitiveaodeasytoapply.

•Wecreatealargenumberofnewsamples,eachobservationofwhichis

obtainedbydrawingatrandomfromouroriginalsampleandreplacing

theobservationafterithasbeendrawn.

•Eachnew'resampled'samplegivesusanewVaRestimate,andwecan

takeour'best'estimatetobethemeanoftheseresample-based

estimates.Thesameapproachcanalsobeusedtoproduceresample-

basedESestimates-eachoneofwhichwouldbetheaverageofthe

lossesineachresampleexceedingtheresampleVaR—andour'best'ES

estimatewouldbethemeanoftheseestimates.

>Abootstrappedestimatewilloftenbemoreaccuratethana'raw'sample

estimate,andbootstrapsarealsousefulforgaugingtheprecisionofour

estimates.

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♦l.HistoricalSimulation

>DrawbacksofHS

•BasicHShasthepracticaldrawbackthatitonlyallowsustoestimate

VaRsatdiscreteconfidenceintervalsdeterminedbythesizeofourdata

set.

•Forinstance,theVaRatthe95.1%confidencelevelisaproblembecause

thereisnocorrespondinglossobservationtogowithit.

•Withnobservations,basicHSonlyallowsustoestimatetheVaRs

associatedwith,at-best,ndifferentconfidencelevels.

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♦l.HistoricalSimulation

>Non-parametricDensityEstimation

•Non-paQmetricdensityestimationoffersapotentialsolution.

•Drawinstraightlinesconnectingthemid-pointsatthetopofeach

histogrambar(Polygon).

•Treatingtheareaunderthelinesasapdfthenenablesustoestimate

VaRsatanyconfidencelevel.

(a)Originalhistogram(b)SurrogAfedensin*function

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♦2.ExpectedShortfall

>TheConditionalVaR(expectedshortfall)

•TheexpectedvalueofthelosswhenitexceedsVaR.

•Measurestheaverageofthelossconditionalonthefactthatitisgreater

thanVaR.

•CVaRindicatesthepotentiallossiftheportfoliois"hit"beyondVaR.

BecauseCVaRisanaverageofthetailloss,onecanshowthatitqualifies

asasubadditiveriskmeasure.

04

3

O.H

^

全o

z

wO.2

a

d

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♦2.ExpectedShortfall

圜,Example:

•Giventhefollowing30orderedpercentagereturnsofanasset:

-16,-14,-10z-7Z-7Z-5Z-4-—L-L0,0,0,L22Z4Z

6,7,8,9,11,12,12,14,18,21f23.

CalculatetheVaRandexpectedshortfallata90%confidencelevel:

•Solution:

VaR(90%)=7,ExpectedShortfall=13.3

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♦3.VaRvsES

>VaRcurveandEScurve:plotsofVaRorESagainsttheconfidencelevel.

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♦3.VaRvsES

>Thelongerthewindow,thesparsertheVaRcurve.

>TheVaRcurveisfairlyunsteady,asitdirectlyreflectstherandomnessof

individuallossobservations,buttheEScurveissmoother,becauseeach

ESisanaverageoftaillosses.

jAstheholdingperiodrises,thenumberofobservationsrapidlyfalls,

andwesoonfindthatwedon'thaveenoughdata.

>Evenifwehadaverylongrunofdata,theolderobservationsmight

haveverylittlerelevanceforcurrentmarketconditions.

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♦4.A/DofNon-parametricMethods

>Advantages

•Intuitiveandconceptuallysimple;

•Donotdependonparametricassumptions;

•Accommodateanytypeofposition;

•Noneedforcovariancematrices,nocursesofdimensionality;

•Usedatathatare(often)readilyavailable;

•Arecapableofconsiderablerefinementandpotentialimprovementif

wecombinethemwithparametric“add-ons“tomakethemsemi­

parametric.

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♦4.A/DofNon-parametricMethods

>Disadvantages

•Verydependentonthehistoricaldataset;

•Subjecttoghosteffect;

•Ifourdataperiodwasunusuallyquiet,non-parametricmethodswill

oftenproduceVaRorESestimatesthataretoolowfortheriskwe

actuallyfacing,viceversa;

•Havedifficulty(actslowly)handlingsh+fe(permanentriskchange)that

takeplaceduringoursampleperiod;

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♦4.A/DofNon-parametricMethods

•Havedifficultyhandlingextremevalue

/Ifourdatasetincorporatesextremelossesthatareunlikelytorecur,

theselossescandominatenon-parametricriskestimateseven

thoughwedon'texpectthemtorecur;

JMakenoallowanceforplausibleeventsthatmightoccur,butdid

notactuallyoccur,inoursampleperiod.

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♦4.A/DofNon-parametricMethods

>ProblemsfromLongWindow

•Thelongerthewindow:

/Thegreatertheproblemswithageddata;

«Thelongertheperiodoverwhichresultswillbedistortedby

unlikely-to-recurpastevents,andthelongerwewillhavetowaitfo『

/Themorethenewsincurrentmarketobservationsislikelytobe

drownedoutbyolderobservations;

/Thegreaterthepotentialfordata-<olleetioA-problems.

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♦5.CoherentRiskMeasures

>Acoherentriskmeasureisaweightedaverageofthequantilesofour

lossdistribution.

1

0=I0(P)P

0

•①(p)=weighingfunctionspecifiedbytheuser.

>ExponentialWeightingFunction

-(i-)/

J：thedegreeofourrisk-aversion

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♦5.CoherentRiskMeasures

jEstimatingexponentialspectralriskmeasuresasaweightedaverageof

VaRs(=0.05)

ConfidencelevelWeight

aVaR<P(a)xaVaR

(a)ct)(a)

10%-1.281600.0000

20%-0.841600.0000

30%-0.524400.0000

40%-0.25330.00010.0000

50%00.00090.0000

60%0.25330.00670.0017

70%0.52440.04960.0260

80%0.84160.36630.3083

90%1.28162.70673.4689

Riskmeasure=mean(0(a)timesaVaR)0.4226

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♦5.CoherentRiskMeasures

>Theestimatedoeseventuallyconvergetothetruevalueasngetslarge.

Estimatesofexponentialspectralcoherentrisk

measureasafunctionofthenumberoftailslices

Estimateofexponential

Numberoftailslices

spectralriskmeasure

100.4227

501.3739

1001.5853

5001.7896

10001.8197

50001.8461

10,0001.8498

50,0001.8529

100,0001.8533

500,0001.8536

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Semi-parametric

Approaches

VaRandotherRiskMeasures

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♦l.Age-weightedHistoricalSimulation

>OnereturnobservationwillaffecteachoftheFieKW^-ebsewatieRS-inourP/L

series.Butafternperiodshavepassed,theobservationwillfalloutofthe

datasetusedtocalculatethecurrentHSP/Lseries,andwillthereafterhave

noeffectonP/L.

>Thisweightingstructurehasanumberofproblems.

•Oneproblemisthatit

samplepeHodthesameweight.

•Theequal-weightapproachcanalsomakeriskestimatesunresponsive

tomajorevents.

•Theequal-weightstructurealsopresumesthateachobservationinthe

sampleperiodisequallylikelyandindependentoftheothersovertime.

However,this'iid'assumptionisunrealistic.

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♦l.Age-weightedHistoricalSimulation

•Itisalsohardtojustifywhyanobservationshouldhaveaweightthat

suddenlygoestozerowhenitreachesagen.

•Ghosteffects

/wecanhaveaVaRthatisundulyhigh(orlow)becauseofasmall

clusterofhighlossobservations,orevenjustasinglehighloss,and

themeasuredVaRwillcontinuetobehigh(orlow)untilndaysorso

havepassedandtheobservationhasfallenoutofthesampleperiod.

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♦l.Age-weightedHistoricalSimulation

>Boudoukh,RichardsonandWhitelaw(BRW:1998)

•w⑴istheprobabilityweightgiventoanobservation1dayold.

•A入closeto1indicatesaslowrateofdecay,anda入farawayfrom1

indicatesahighrateofdecay.

A3(x)1A2(JO1入313]

|J4M3M21

入1（1—入|

3⑴+入3⑴+,・，+入吁1(x)(])=1T3。）=一二J

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♦l.Age-weightedHistoricalSimulation

>Majorattractions

•ItprovidesanicegeneralizationoftraditionalHS,becausewecan

regardtraditional屋asewithzerodecay,or入11.

•AlargelosseventwillreceiveahigherweightthanundertraditionalHSZ

andtheresultingnext-dayVaRwouldbehigherthanitwouldotherwise

havebeen.

•Helpstoreducedistortionscausedbyeventsthatareunlikelytorecur,

andhelpstoreduce

/Asanobservationages,itsprobabilityweightgraduallyfallsandits

influencediminishesgraduallyovertime.Whenitfinallyfallsoutof

thesampleperiod,itsweightwillfallfrom入MQ)tozero,insteadof

from1/ntozero.

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♦l.Age-weightedHistoricalSimulation

>Majorattractions

■Age-weightingallowsus

observation,soweneverthrowpotentiallyvaluableinformationaway.

Thiswouldimproveefficiencyandeliminateghosteffects,becausethere

wouldnolongerbeany“jumps"inoursampleresultingfromold

observationsbeingthrownaway.

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♦2.Volatility-weightedHistoricalSimulation

>HullandWhite(HW1998)

•WeadjustthehistoHcalretumstoreflecthowvolatilitytomorrowis

believedtohavechangedfromitspastvalues.

/rti=actualreturnforassetiondayt

Jat>i=volatilityforecastforassetiondayt

/aTi=currentforecastofvolatilityforasseti

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♦2.Volatility-weightedHistoricalSimulation

>Majorattractions

•Ittakesaccountofvolatilitychangesinanaturalanddirectway.

•Itproducesriskestimatesthatareappropriatelyseroitive4G-WTOfrt

volatilityestimates.

•ItallowsustoobtainVaRandESestimatesthatcanexceedthe

maximumlossinourhistoricaldataset.

/Inrecentperiodsofhighvolatility,historicalreturnsarescaled

upwards,andtheHSP/LseriesusedintheHWprocedurewillhave

valuesthatexceedactualhistoricallosses.

•ProducessuperiorVaRestimatestotheBRWone.

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♦3.Correlation-weightedhistoricalsimulation

>Correlation-weightedhistoricalsimulation

•Correlation-weightingisalittlemoreinvolvedthanvolatility-weighting.

•Toseetheprinciplesinvolved,supposeforthesakeofargumentthatwe

havealreadymadeanyvolatility-basedadjustmentstoourHSreturns

alongHull-Whitelines,butalsowishtoadjustthosereturnstoreflect

changesincorrelations.

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♦4•Filteredhistoricalsimulation

,Filteredhistoricalsimulation(FHS)

•CombineshistoricalsimulationmodelwithGARCHorAGARCHmodel.

>Thestepsareasfollows:

•Firstly,usethehistoricalreturntofindanysurpriseandthusreproduce

volatilitywithGARCHorAGARCHmodel.

•Secondly,thesevolatilityforecastsarethendividedintotherealized

returnstoproduceasetofstandardizedreturns,whichisLED..

•Thethirdstageinvolvesbootstrappingfromthesetofstandardized

returns.

•Finally,eachofthesesimulatedreturnsgivesusapossibleend-of-

tomorrowportfoliovalue,andacorrespondingpossibleloss,andwe

taketheVaRtobethelosscorrespondingtoourchosenconfidence

level.

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♦4•Filteredhistoricalsimulation

>Majorattractions

•Combinethenon-parametricattractionsofHSwithasophisticated(eg,

GARCH)treatmentofvolatility,andsotakeaccountofchangingmarket

•Itisfest,evenforlargeportfolios

estimatesthatcanexceedthemaximumhistoricallossinousdataset.

•Itmaintainsthecoirelationstructureinourreturn

•Itcanbemodifiedtotakeaccountofautocorrelationsinassetreturns

•ItcanbemodifiedtoproduceestimatesofVaRorESconfidence

intervals.

•ThereisevidencethatFHSworkswell.

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Extremevalue

VaRandotherRiskMeasures

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♦l.Introduction

“Thefitteddistributionwilltendtoaccommodatethemorecentral

observations,ratherthantheextremeobservations,whicharemuch

sparser.

>Theestimationoftherisksassociatedwithlowfrequencyeventswithlimited

dataisinevitablyproblematic.

>Extreme-valuetheory(EVT):

•Centraltendencystatisticsaregovernedbycentrallimittheorems,but

centrallimittheoremsdonotapplytoextremes.Instead,extremesare

governedbyextreme-valuetheorems.

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♦2.GeneralizedExtremeValueDistribution

>SupposewehavearandomlossvariableXzandweassumetobeginwith

thatXisindependentandidenticallydistributed(iid)fromsomeunknown

distribution.ConsiderasampleofsizendrawnfromF(x)zandletthe

maximumofthissamplebeMnIfnislarge,wecanregardMnasanextreme

value.

>Underrelativelygeneralconditions,thecelebratedFisher-Tippetttheorem

thentellsusthatasngetslarge,thedistributionofextremes(i.e.zMn

convergestothefollowinggeneralizedextreme-value(GEV)distribution:

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♦2.GeneralizedExtremeValueDistribution

>Thisdistributionhasthreeparameters.

x-U—x

exp[-(1+m丁)刃,"0

F(x)=Ix°_

exp[-exp(-----=0

r“thelocationparameterofthelimitingdistribution,whichisameasureofthe

centraltendencyofMn.

r,thescaleparameterofthelimitingdistribution,whichisameasureofthe

dispersionofMn.

r,thetailindex,givesanindicationoftheshape(orheaviness)ofthetailofthe

limitingdistribution.

•When5>0:Frechetdistribution,heavytails,I次et-dist,Paretodist.

•When5=0:Gumbeldistribution,lighttails,likenormalorlognormaldist.

■When5<0:Weibulldistribution,verylighttails,notusefulformodelling

financialreturns.

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♦2.GeneralizedExtremeValueDistribution

三

S

U

E

>

三

q一

R

q

o

」

d

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♦2.GeneralizedExtremeValueDistribution

>HowdowechoosebetweentheGumbelandtheFrechet?

•WechoosetheEVdistributiontowhichtheextremesfromtheparent

distributionwilltend.

•Wecouldtestthesignificanceofthetailindex,andwemightchoose

theGumbelifthetailindexwasinsignificantandtheFrechetotherwise.

•Giventhedangersofmodelrisk,theestimatedriskmeasureincreases

withthetailindex,asaferoptionisalwaystochoosetheFrechet.

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♦2.GeneralizedExtremeValueDistribution

>EstimationofEVP