FRM schweser part 1 book 4 2013

FRM PART I BOOK 4:VALUATIONAND RISK MODELS VaRMethods 34:Measures ofFinancialRisk 35:QuantifyingVolatilityinVaRModels 36: PuttingVaRtoWork 38: TheBlack-Scholes-Merton Model 39: TheGreek

FOR THE HRM1 EXAM

FRM PART I BOOK 4:

VALUATIONAND RISK MODELS

VaRMethods

34:Measures ofFinancialRisk

35:QuantifyingVolatilityinVaRModels

36: PuttingVaRtoWork

38: TheBlack-Scholes-Merton Model

39: TheGreek Letters

40:Prices,DiscountFactors,andArbitrage

41:Spot,Forward,and ParRates

42:Returns,Spreads,andYields

43:One-FactorRisk Metrics andHedges

44: Multi-Factor RiskMetricsandHedges

45: Empirical ApproachestoRiskMetricsand Hedges

46:Country Risk Models

47:External and Internal Ratings

48:Loan Portfolios and Expected Loss

49: UnexpectedLoss

50: Operational Risk

51 :StressTesting

52:Principlesfor SoundStressTestingPracticesandSupervision

SELF-TEST:VALUATIONANDRISKMODELS

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READING ASSIGNMENTS AND AIM

STATEMENTS

Thefollowingmaterialis a reviewofthe Valuation and Risk Modelsprinciples designedto

address theAIMstatements setforthbytheGlobalAssociationofRiskProfessionals.

READING ASSIGNMENTS

Kevin Dowd,Measuring MarketRisk,2ndEdition(WestSussex,England:JohnWiley&

Sons, 2005).

34.“MeasuresofFinancialRisk,”Chapter2

LindaAllen,JacobBoudoukhandAnthonySaunders, UnderstandingMarket,Credit and

OperationalRisk:The ValueatRiskApproach

35- “QuantifyingVolatilityinVaRModels,”Chapter2

36 “Putting VaRtoWork,”Chapter3

JohnHull,Options,Futures,andOtherDerivatives,8th Edition(NewYork:Pearson

38.“TheBlack-Scholes-MertonModel,”Chapter14

39.“The GreekLetters,”Chapter18

BruceTuckman,FixedIncomeSecurities,3rdEdition(Hoboken,NJ: JohnWiley&

Sons,2011)

40.“Prices, Discount Factors,andArbitrage,”Chapter1

41 “Spot,Forward,andParRates,”Chapter2

42.“Returns,Spreads andYields,”Chapter3

43.“One-FactorRisk MetricsandHedges,” Chapter 4

44.“Multi-FactorRiskMetricsandHedges,” Chapter 5

45.“EmpiricalApproachestoRiskMetricsandHedges,”Chapter 6

Caouette,Altman,Narayanan,andNimmo,Managing CreditRisk,2ndEdition

(NewYork:JohnWiley&Sons,2008)

(page125)

(page141)

(page159)

(page175)(page192)

(page205)

(page216)

46 “Country RiskModels,”Chapter23

Arnaud de Servigny and OlivierRenault,Measuring and Managing CreditRisk(NewYork:McGraw-Hill, 2004).

47 “External and Internal Ratings,” Chapter2

MichaelOng,Internal CreditRiskModels:CapitalAllocationandPerformance

Measurement(London:RiskBooks,2003).

48.“LoanPortfoliosandExpectedLoss,”Chapter 4

49.“UnexpectedLoss,”Chapter5

JohnHull,RiskManagementandFinancialInstitutions,2nd Edition(Boston:Pearson

Prentice Hall,2010).

(page223)

(page233)

(page243)

50 OperationalRisk,”Chapter 18 (page249)

PhilippeJorion, Value-at-Risk: The New BenchmarkforManaging Financial Risk,

3rdEdition.(NewYork: McGrawHill, 2007)

51.“Stress Testing,” Chapter 14 (page262)

52.“Principlesfor Sound Stress TestingPracticesand Supervision”(BaselCommitteeon

BankingSupervisionPublication,Jan2009). (page271)

Book 4Reading Assignmentsand AIMStatements

AIM STATEMENTS

34 Measures of Financial Risk

Candidates,aftercompletingchisreading, shouldbe ableto:

1 Describe themean-varianceframework andtheefficientfrontier,(page23)

2 Explain the limitations of themean-varianceframework withrespect to

assumptions about thereturn distributions,(page25)

3 DefinetheValue-at-Risk(VaR)measureofrisk,describe assumptions aboutreturn

distributions andholdingperiod, andexplainthe limitations of VaR.(page 26)

4 Define the propertiesofacoherentriskmeasureandexplainthe meaningofeach

property,(page27)

5- Explain why VaRisnot acoherent riskmeasure,(page28)

6 Explainand calculateexpected shortfall(ES),and compareandcontrastVaR and

ES (page28)

7 Describespectralriskmeasures,andexplain howVaRand ESarespecialcasesof

spectral riskmeasures,(page29)

8 Describe howtheresults ofscenarioanalysiscanbeinterpretedascoherent risk

measures,(page29)

35 QuantifyingVolatilityin VaR Models

Candidates,aftercompletingthis reading, shouldbeableto:

1 Explainhowasset returndistributionstendtodeviate from thenormaldistribution

(page35)

2 Explain potentialreasonsfortheexistenceof frit tailsina returndistribution and

describe the implications fat tails haveonanalysisofreturn distributions,(page35)

3 Distinguishbetweenconditional and unconditionaldistributions,(page35)

4 Describethe implications ofregimeswitchingonquantifying volatility, (page37)

5 .Explain thevariousapproaches forestimating VaR.(page38)

. ' 6 Compare,contrastand calculate parametric andnon-parametricapproaches for

estimating conditional volatility,including:

• Historicalstandard deviation

9 Explain longhorizon volatility/VaRand the process ofmeanreversionaccordingto

anAR(l)model,(page49)

36 PuttingVaRtoWork

Candidates,aftercompletingthisreading, should be ableto:

1. Explain andgiveexamples of linear and non-linearderivatives,(page56)

2 Explain bowtocalculateVaRforlinearderivatives,(page58)

3 Describethedelta-normal approachtocalculatingVaR fornon-linear derivatives

(page58)

4 Describe the limitationsof the delta-normalmethod,(page58)

5 Explain the fullrevaluationmethodforcomputingVaR.(page62)

6 Comparedelta-normaland fullrevaluationapproaches, (page62)

7 Explain structural MonteCarlo,stresstesting andscenario analysismethodsforcomputingVaR,identifying strengthsand weaknessesof each approach, (page62)

8 Describe theimplicationsof correlation breakdown forscenarioanalysis, (page62)

9 Describeworst-casescenario(WCS)analysis and compareWCStoVaR (page65)

37 BinomialTrees

Candidates,aftercompletingthisreading,should be ableto:

1. Calculate the valueofaEuropean callor putoption using theone-stepand

two-stepbinomialmodel, (page70)

2 Calculatethe valueofanAmericancallorputoption usinga two-stepbinomial

3 Describe how volatilityiscapturedinthebinomialmodel,(page77)

4 Describe how the binomial model value convergesastimeperiodsareadded

(page80)

5 Explainhow the binomial modelcanbealteredtoprice optionson:stocks withdividends,stockindices, currencies,andfutures,(page77)

38 TheBlack-Scholes-MertonModel

Candidates,aftercompletingthisreading, should be ableto:

1. Explainthelognormalpropertyofstockprices,the distribution ofratesofreturn,andthecalculationofexpectedreturn,(page87)

2 Compute the realizedreturnand historicalvolatility ofa stock,(page87)

3 List anddescribe the assumptions underlying the Black-Scholes-Merton optionpricingmodel,(page90)

4 Computethe valueofaEuropeanoption usingtheBlack-Scholes-Merton modelon

anondividend payingstock,(page91)

• • 5- Identify rhecomplications involvingthevaluationofwarrants,(page97)

6.- Defineimpliedvolatilities and describe howto computeimpliedvolatilitiesfrom

.market pricesofoptions usingthe Black-Scholes-Mertonmodel,(page97)

7 Explainhowdividendsaffect the earlydecisionfor American call andputoptions.(page96)

8 Compute thevalueofaEuropeanoption usingthe BlackjScholes-Mertonmodelon

adividend payingstock,(page93)

9 UseBlacksApproximationto computethevalue ofan Americancalloption ona

dividend-payingstock,(page96)

39 TheGreekLetters

Candidates,after completing this reading, should be ableto:

1 Describe and assesstherisksassociatedwith nakedandcoveredoption positions.(page105)

Back 4 Reading Assignments and AJM Statements

8 Explain howtoimplement andmaintainagamma neutralposition,(page113)

9 Describe the relationshipbetweendelta,theta, andgamma,(page113)

10. Describehowhedgingactivitiestake placein practice, and describe howscenario

analysiscanbeusedtoformulateexpectedgains and losses withoption positions

(page119)

11 Describe howportfolioinsurancecanbe createdthroughoptioninstrumentsand

stock indexfutures,(page120)

40 Prices,Discount Factors,andArbitrage

Candidates,aftercompleting thisreading,should be ableto:

1 Define discount factor anduseadiscount functionto compute presentand future

values,(page128)

2 Definethe “lawofoneprice,”explainit usinganarbitrageargument,anddescribe

how it can beappliedtobond pricing,(page130)

3 Identify thecomponentsofaU.S Treasury couponbond, andcompare and

contrasrthestructure toTreasurySTRIPS,including thedifferencebetween

P-STRIPSand C-STRIPS.(page132)

-4 Constructareplicating portfoliousingmultiple fixedincome securitiestomatch

the cash flowsofagiven fixedincome security,(page133)

5 Identify arbitrageopportunitiesfor fixedincome securities withcertaincash flows

(page130)

6 Differentiate between “clean” and“dirty”bond pricing and explaintheimplications

of accrued interestwith respect tobond pricing, (page134)

7 Describethe commonday-countconventionsusedinbondpricing,(page134)

41.Spot,Forward,and Par Rates

Candidates,after completingthisreading,should be ableto:

1. Calculateand describe theimpactof differentcompoundingfrequencieson-a

bondsvalue,(page141) - '

-2 Calculatediscount factorsgiven interestrateswaprates,(page142)

3 Computespot ratesgivendiscountfactors,(page144)

4 Define and interprettheforwardrate,andcomputeforwardratesgivenspot rates.

(page146)

5 Define parrateand describethe equationfortheparrateofa bond,(page148)

6 Interpret therelationship betweenspot,forward and parrates,(page149)

7 Assesstheimpactofmaturityonthepriceofabondandthereturnsgeneratedby

bonds,(page151)

8 Define the“flattening”and“steepening” ofrate curvesandconstructahypothetical

traderoreflectexpectations thatacurvewillflattenor steepen,(page151)

42 Returns,Spreads,andYields

Candidates,aftercompletingthis reading, should be ableto:

1. Distinguishbetweengross andnetrealizedreturns,and calculate the realizedreturn

forabond overaholdingperiodincludingreinvestments,(page159)

2 Define and interpret thespreadofabond,andexplainhowaspreadisderivedfrom

abond priceanda term structureofrates,(page161)

3 Define, interpret,andapplyabond’s yield-to-maturity(YTM)tobond pricing

(page161)

4 Computeabond’s YTM givenabondstructureandprice,(page161)

5 Calculate thepriceofanannuityandaperpetuity, (page165)

6 Explainthe relationship berweenspot ratesandYTM (page166)

7 Definethe coupon effect and explain the relationship between couponrate, YTM,

and bondprices,(page167)

S Explain the decomposition oi P&Lfor abondintoseparatefactors includingcarryroll-down,ratechangeandspreadchangeeffects,(page168)

9 Identifythemostcommon assumptionsin carry roll-downscenarios,including

realizedforwards,unchangedterm structure,and unchangedyields,(page169)

43 One-FactorRiskMetricsand Hedges

Candidates,aftercompletingthisreading, shouldbe ableto:

1. Describeaninterestratefactorandidentifycommonexamplesofinterestrate

6 Define,compute,and interpret theconvexityofafixed incomesecuritygivena

changeinyieldand theresultingchangein price,(page181)

7 Explainthe process ofcalculatingtheeffective durationand convexityofaportfolio

offixedincomesecurities,(page183)

8 Explainthe impact of negative convexityonthehedgingof fixedincome securities.(page184)

9 Constructabarbellportfoliotomatch thecostand durationofagiven bulletinvestment, andexplaintheadvantagesanddisadvantagesof bulletversusbarbellportfolios.-(page185)

44 Multi-FactorRisk Metrics andHedges

Candidates,after completing thisreading,should beableto: .

1 Describeandassessthe major weaknessattributabletosingle-factor approaches

whenhedging portfoliosorimplementingassetliability techniques, (page192)

2 Define keyrateexposures and knowthe characteristicsof keyrateexposure factors

_ including partial‘01sandforward-bucket‘01s.(page193)

3 Describe key-rateshift analysis, (page193)

4 Define, calculate,and interpret keyrate‘01and keyrate duration,(page194)

5 Describe the keyrateexposuretechniqueinmulti-factorhedgingapplicationsandsummarizeitsadvantagesanddisadvantages, (page195)

6 Calculatethekeyrateexposures foragivensecurity,andcomputetheappropriatehedgingpositionsgivenaspecific keyrateexposureprofile,(page195)

7- Describe therelationshipbetween keyrates,partial‘01s and forward-bucket‘01s,

andcalculatetheforward-bucket‘01forashiftin in one or morebuckets

Book4Reading Assignmentsand AIMStatements

45 EmpiricalApproachestoRiskMetrics andHedges

Candidates,aftercompletingthis reading,shouldbeableto:

I Explainthe drawbackstousingaDVO1-neutral hedge forabondposition

(page205)

2. Describearegression hedge andexplainhowitimprovesonastandard

DV01-neutralhedge, (page206)

3- Calculate the regressionhedge adjustmentfactor,beta,(page207)

4 Calculatethefacevalueofanoffsettingpositionneededtocarryout aregression

hedge,(page207)

5 Calculatetheface value ofmultipleoffsettingswappositionsneededtocarryouta

two-variable regressionhedge, (page208)

6 Compareandcontrastbetween levelandchange regressions,(page209)

7- Describeprincipalcomponentanalysisand explain howitisappliedinconstructing

ahedging portfolio,(page209)

46 CountryRisk Models

Candidates,aftercompletingthis reading, shouldbeableto:

1 Defineand differentiatebetweencountryriskandtransfer riskanddescribesomeof

thefactors that mightleadto each,(page216)

2 Describecountryriskinahistoricalcontext,(page216)

3 Identifyand describesomeofthe majorriskfactors thatarerelevantforsovereign

riskanalysis, (page217)

4 Compare andcontrast corporateandsovereignhistorical defaultrate patterns.

(page218)

5 Explainapproaches for andchallengesin assessingcountry risk,(page218)

6 Describehowcountryrisk ratingsareusedinlendingandinvestmentdecisions

(page219)

7 Describesomeof thechallengesin countryrisk analysis,(page219)

47 External and InternalRatings

Candidates,aftercompletingthisreading,shouldbe ableto:

1 Describeexternalraring scales,theratingprocess, andthelinkbetweenratingsand

default,(page223)

2 Describe theimpactoftime horizon, economiccycle, industry, andgeographyon

external ratings,(page225)

3 Review the results andexplanationof the impact of ratings changesonbondand

stockprices,(page226)

4 Compareexternalandinternalratingsapproaches, (page226)

5 Explainandcompare thethrough-the-cycleandat-the-pointinternalratings

approaches,(page227)

6 Defineandexplainaratings transitionmatrixanditselements,(page224)

7 Describetheprocess for andissueswithbuilding, calibratingand backtestingan

internalratingsystem,(page227)

8 Identifyand describe thebiasesthat may affectaratingsystem,(page228)

48 Loan PortfoliosandExpected Loss

Candidates,after completing this reading, should be ableto:

I Describe the objectives ofmeasuring credit riskforabanks loanportfolio

(page233)

2 Define,calculate and interpret the expectedlossforanindividual credirinstrument

(page236)

3 Distinguish between loan and bond portfolios, (page233)

4 Explainhowacreditdowngradeorloan defaultaffects thereturnofaloan

(page234)

5 Distinguish betweenexpectedand unexpectedloss,(page234)

6 Define exposures, adjusted exposures,commitments, covenants,andoutstandings:

• Explainhow drawnandundrawnportionsofa commitmentaffect exposure

* Explainhowcovenantsimpactexposures

(page234)

7 Define usage given default and howitimpactsexpectedandunexpectedloss.(page236)

8 Explain theconceptofcreditoptionality.(page236)

9 Describethe process of parameterizing credit risk models anditschallenges

(page237)

49 Unexpected Loss

Candidates,aftercompletingthis reading, should be ableto:

1 Explainthe objective forquantifying bochexpected and unexpectedloss,(page243)

2 Describefactorscontributingtoexpected and unexpectedloss,(page243)

3 Define,calculate and interpret the unexpected loss ofan asset,(page244)

4 Explaintherelationshipbetween economic capital, expectedloss andunexpected

loss,(page245)

50 OperationalRisk

Candidates,aftercompletingthis reading, shouldbeableto:

1 Calculatetheregulatorycapitalusingthe basic indicatorapproachandthestandardizedapproach,(page250)

2 Explainthe Basel Committee’s requirements for the advancedmeasurement

approach(AMA)and theirsevencategories of operationalrisk,(page250)

3 Explainhowtogeealossdistributionfrom the loss frequency distribution andtheloss severity distributionusingMonteCarlosimulations,(page252)

4 Describe the commondataissuesthatcanintroduceinaccuraciesandbiases in theestimationof lossfrequencyandseveritydistributions,(page253)

5- Describehowtousescenarioanalysis ininstanceswhen thereis scarcedata

(page254)

6 Describe howtoidentifycausal relationshipsand howtouseriskandcontrolselfassessment (RCSA)and key riskindicators(KRIs) tomeasureandmanageoperationalrisks,(page254)

Book 4

ReadingAssignments and AIM Statements

51- StressTesting

Candidates,aftercompletingthis reading, shouldbeableto:

1. Describethepurposes ofstresstesting and theprocess ofimplementinga stress

testingscenario,(page262)

2. Contrast between event-drivenscenariosandportfolio-drivenscenarios,(page263)

3 Identifycommonone-variablesensitivitytests,(page263)

4 DescribetheStandardPortfolio Analysis of Risk(SPAN®)systemformeasuring

portfoliorisk,(page263)

5 Describethedrawbackstoscenarioanalysis,(page264)

6 Explain the differencebetweenunidimensionalandmultidimensionalscenarios

(page264)

7 Compareandcontrastvariousapproachesto scenarioanalysis,(page265)

8 Define and distinguishbetweensensitivity analysis andstresstesting model

9 Explainhow the results ofa stress testcan be usedtoimproveourrisk analysis and

52 Principles forSound StressTestingPracticesand Supervision

Candidates,after completing this reading, should be ableto:

1 Describe the rationale for theuseofstresstestingas ariskmanagementtool

(page271)

2. Describe weaknesses identifiedandrecommendationsforimprovement in:

• Theuseofstresstesting and integration in risk governance

• Stress testing methodologies

• Stress testingscenarios

• Stress testing handlingofspecificrisksand products.

. (page272)

3.- Describestresstesting principles forbankswithin:

• Useofstresstestingand integration in riskgovernance

• Stress testing methodology andscenarioselecrion

• Principlesfor supervisors •

(page272) ' •

*

-EXAM FOCUS

Valueat risk(VaR)wasdevelopedas anefficient,inexpensivemethod todetermine economicrisk exposure of banks withcomplexdiversifiedassethold ings.Inthisreading,wedefineVaR,demonstrateits calculation,discuss howVaRcan be converted tolonger timeperiods, andexaminetheadvantagesanddisadvantagesof the threemainVaR estimation methods For the

method.VaRis oneof GARPsfavoritetestingtopicsand.irappearsin manyassigned readingsthroughouttheFRM PartIand Part0curricula

DEFINING VAR

Valueat risk(VaR) isaprobabilistic methodof measuring thepotentiallossinportfolio

value overagiventimeperiod and foragiven distribution of historicalreturns.VaRisthedollarorpercentagelossinportfolio(asset)valuethatwill beequaledorexceededonly

ATpercentofthetime.In otherwords,thereisanATpercentprobabilitythatthe lossin

portfoliovalue will beequalto Of greaterthantheVaR measure.VaRcanbecalculatedfor anypercentageprobabilityof loss andoveranytimeperiod.A1%, 5%,and 10% VaRwould be denotedas VaR(l%), VaR(5%),andVaR(10%),respectively.The riskmanagerselects the Xpercentprobabilityofinterestand thetimeperiodoverwhichVaRwill bemeasured Generally, thetimeperiod selected(andtheone wewalluse)is oneday

A briefexamplewillhelp solidify theVaRconcept.Assumeariskmanager calculatesthe.daily5%VaRas $10,000.TheVaR(5%) of$10,000indicatesthat thereis a5% chance that

onany given day, theportfoliowillexperiencealossof$10,000or more.Wecould alsosaythat thereis a95% chancethatonanygiven day theportfoliowill experienceeithera

loss lessthan$10,000or again Ifwefurtherassumethaÿthe$10,000lossrepresents 8%

of theportfoliovalue,thenonanygiven day thereis a5% chance that theportfoliowillexperiencealoss of8%or greater,butthereisa95% chance that thelosswill be lessthan8% ora percentagegaingreaterthanzero

CALCULATING VARCalculatingdelta-normal VaRisasimplematterbut requires assuming thatasset returnsconformto astandard normaldistribution Recall thatastandard normaldistributionis

From Figure1, weobservethefollowing: theprobability of observingavaluemorethan

1.28standard deviations below the meanis10%;theprobabilityof observingavaluemore

than1.65standard deviations below themeanis5%;and the probability of observinga

valuemorethan2.33standarddeviationsbelow the mean is 1%.Thus, wehave critical

z-valuesof-1.28, -1.65,and -2.33for1 0%,5%,and1% lower tailprobabilities,

respectively.Wecan nowdefinepercentVaRmathematicallyas:

VaR(X%) =ZX%CT

where:

VaR(X%) =the X%probability' valuearrisk ‘

= the critical z-value basedonthenormal distribution and the selectedX%

probability'

= the standard deviationof dailyreturns on a percentagebasis

Professor'sNote: VaRis aone-tailedtest, so thelevelofsignificanceisentirely

inone tailofthedistribution.Asa result,the criticalvalues will bedifferent

thana two-tailedtestthatusesthesamesignificance level

*X%

a

In ordertocalculateVaR(5%)using this formula,wewould use acriticalz-valueof-1.65

andmultiplyby thestandard deviation ofpercent returns.Theresulting VaRestimate

would bethepercentageloss inassetvalue thatwould only be exceeded 5% of thetime

VaRcan alsobeestimatedonadollarrather thana percentagebasis.TocalculateVaRon a

dollarbasis, wesimply multiplythepercentVaRby theassetvalueasfollows:

VaR(X%)do!krbasis =VaR(X%)dccimdbasls x assetvalue

=(zx%o)x assetvalue

TocalculateVaR(5%)using thisformula,wemultiplyVaR(5%)on a percentagebasisby

thecurrentvalueoftheassetin question This isequivalenttotaking the product ofthe

criticalz-value,the standard deviation of percent returns,andthecurrent assetvalue An

estimateofVaR(5%)onadollar basisisinterpretedasthe dollarloss inassetvaluethat willonlybeexceeded 5% ofthe time.

isconsidering addingtothebanksportfolio.If theassethasadaily standard deviation of

returnsequal to 1.4%and theassethasa currentvalueof$5 3million,calculatethe

VaR (5ÿi>)ionbothapercentageanddollarbasis

VaR(5%)=zÿcr=-1.65(0.014)=-0.0231 = -2.31%

TheVaR(5%) onadollar basisiscalculatedasfollows:

' '

Tiyjs,thereisa §%probabilitythat,ohanygiven day,thelossinvalueonthisparticular;

assetwill egualorexceed _.3f%,or $122,430.

.

Ifanexpectedreturnother thanzero is given,VaRbecomes the expectedreturn minusthequantityof the critical value multiplied by the standard deviation

VaR=[E(R)-zo]

In the exampleabove,theexpectedreturnvalueiszeroand thusignored.Thefollowing

example demonstrates howtoapplyanexpectedreturn to aVaR calculation

if|§fi

m

VAR CONVERSIONS

VaR, ascalculated previously, measuredthe riskofalossinassetvalueover ashort time

period.Risk managersmay,however,beinterested in measuringriskoverlongertime

periods,suchas a mouth, quarter, oi year.VaRcan beconveiled boma1-daybasis to a

longerbasisbymultiplyingthe dailyVaRbythe squarerootofthe number ofdays (/)in

thelongertimeperiod(called the squareroot rule).Forexample,to convert to aweekly

VaR,multiplythedailyVaR by the squarerootof5(he.,five businessdaysin a week).We

cangeneralizethe conversionmethodasfollows:

= VaR(X%)1ÿ).v/j

VaR(X%)J-days

Example:Converting dailyVaRtoothertimebases

Assumethata riskmanagerhascalculatedthe dailyVaR(10%)doj|arbaS!S 6faparticular

asset tobe $12,500.Cÿculate"the;weeklyyihortt|il.yy>semiannuiyandianpuÿ!\ÿf(-fo|(|Kis

asset.Assume*25,P,days peryear and50 weeks peryear

Answer:

ThedailydollarVaRReconvertedto aweekly,monthly,semiannual, andannualdollar

VaR,measurebytmultiplying hy?thesquarerootof anjÿ507jespectively.

VaRcanalso be convertedtodifferentconfidencelevels For example,ariskmanager may

Thisconversion isdone byadjustingthecurrentVaRmeasureby theratiooftheupdated

confidence leveltothecurrentconfidence level

-r-ipsti

./;/C ,;;V

VaRv

THE VAR METHODS

The threemainVaRmethodscanbedividedintotwogroups:linear methodsandfullvaluationmethods

1 Linearmethodsreplaceportfoliopositionswith linear exposuresonthe appropriate riskfactor.Forexample, the linear exposure used for optionpositionswould bedelta while.the linear exposure forbondpositions would be duration.This method is usedwhen

calculatingVaR withthedelta-normalmethod

2 Full valuation methods fully repricetheportfoliofor eachscenarioencounteredovera

historical period,orovera greatnumberofhypotheticalscenariosdeveloped through

historical simulationorMonteCarlo simulation.ComputingVaR using full revaluation

is morecomplexthan linear methods.However,thisapproach willgenerallyleadto

moreaccurateestimatesof riskinthelongrun. _

LinearValuation: The Delta-Normal Valuation Method

The delta-normalapproach beginsbyvaluingthe portfolioatan initialpointas a

relationshiptoaspecificriskfactor,5(consideronlyoneriskfactorexists):

V0 = V(S0)

VaR Methods

Here,A0isthe sensitivityoftheportfoliotochangesintheriskfactor,5.Aswithanylinear

relationship,thebiggest changein thevalueoftheportfoliowillaccompanythebiggest

changeintheriskfactor TheVaRat agiven level ofsignificance,z, canbewritten as:

VaR= [A0| x(zaS0)

where:

zaS0 = VaRs

horizonsthan longer horizons

Consider,forexample,afixedincomeportfolio.The riskfactor impacting the value of

thisportfolioisthe change inyield.TheVaRof thisportfoliowould then be calculatedas

follows:

VaR= modified durationx z xannualized yield volatilityxportfoliovalue

Noticehere thatthevolatilitymeasureappliedis thevolatility of changesin theyield In

future examples,thevolatilitymeasuredusedwillbe the standard deviation ofreturns.

Since thedelta-normal methodisonlyaccuratefor linear exposures, non-linearexposures,

such as convexity,are notadequatelycapturedwiththisVaR method.ByusingaTaylor

series expansion, convexitycanbe accountedfor inafixedincomeportfoliobyusing what

isknown as thedelta-gamma method You willseethis methodinTopic36.Fornow,just

takenotethatcomplexitycanbe addedtothedelta-normalmethodtoincrease its reliability

when measuringnon-linear exposures

Full Valuation:MonteCarlo andHistoricSimulation Methods

The Monte Carlosimulationapproach revaluesaportfolioforalarge numberofriskfactor

values,randomly selected fromanormal distribution Historicalsimulationrevaluesa

portfoliousing actual valuesforriskfactorstakenfrom historicaldata Thesefullvaluation

approachesprovidethemost accurate measurementsbecause they include all nonlinear

relationshipsandotherpotentialcorrelationsthatmaynotbe includedin the linear

valuation models

COMPARINGTHEMETHODS

Thedelta-normal methodisappropriate forlarge portfolioswithoutsignificant option-like

exposures This methodisfastand efficient

Full-valuationmethods,eitherbasedonhistorical dataoron MonteCarlosimulations, are

more timeconsuming and costly.However,theymay be theonlyappropriatemethodsfor

largeportfolioswithsubstantial option-like exposures,awiderrange of riskfactors,ora

longer-term horizon

Delta-Normal Method

The delta-normal method (a.k.a.thevariance-covariancemethodorthe analyticalmethod)

for estimatingVaR requiresthe assumption ofanormal distribution Thisisbecause themethod utilizes theexpectedreturnand standard deviationofreturns.Forexample,in

calculatingadailyVaR, wecalculate the standard deviation of dailyreturnsin thepastandassumeitwillbeapplicabletothefuture.Then,usingtheassetsexpected1-dayreturnandstandard deviation,weestimatethe1-day VaRat the desired levelof significance

The assumption ofnormalityistroublesome because manyassetsexhibit skewedreturndistributions(e.g.,options), andequityreturnsfrequentlyexhibit leptokurtosis(fattails) Whenadistribution has“fattails,”VaRwilltendtounderestimate the loss anditsassociatedprobability.Also knowthat delta-normal VaRiscalculated using the historicalstandarddeviation,whichmaynotbe appropriate if thecomposition oftheportfoliochanges, ifthe estimationperiodcontained unusualevents, orifeconomicconditionshave

changed

Example: Delta-normal VaR

«

1-dayreturnfor a$100,000,(100portfoliois0.00085an.

deviationof daily.tetufnsisj).0,01.1.Calculate daily valueat r

ToIpcate thevalue fora 5%VaR,we usetheAlternative Tablein t

book.We lookthrough the bodyofthe table untilwefindthevaluefor In thiscase, we want5%indie lowertail,which wouldleave45

thatis notinthe tail.Searchingfor0.45,we find the value0.4505(t

Youwill alsofindaCumulativeViableinthe appendix When usinglook directlyforthesignificancelevelof the VaR:Forexample, ifyou

fook for the value in the tablewhichiscldsescto(1-significahceleve0.9500.Youwillfind0.9505,whichliesattheintersectionof 1.6int0.05in the columnheading

Rp— expected1-dayreturn on theportfolio

Vp— value;of theportfolio

z = z-valuecorrespondingwiththedesiredlevelofsignificance

O = standard deviationof1-dayreturns

The imerpretation ofthisVaRisrhatthere isa5%chancetheminimum 1-day lossis

0.0965%, or $96,500.(There is5%probabilitythat thell-daylosrwillexceed$96,500.)

Alternatively,wecould saywe are95%confidentthe1-dayloss willnotexceed$96,500.

Ifyou aregiventhe standarddeviation of annualreturnsand needtocalculateadaily VaR,

the daily standard deviationcan be estimatedastheannual standard deviation divided by

thesquarerootof the number of(trading)days ina year,andsoforth:

CTdaiiy = \f25Q l*7 monthly= Jn

Delta-normalVaRisoften calculatedassuminganexpectedreturnofzeroratherthan the

portfoliosactual expectedreturn.Whenthisis done,‘VaRcanbeadjustedtolongeror

-shorterperiods oftimequiteeasily For example, dailyVaRis-estimatedasannualVaR

divided bythesquare rootof 250(aswhenadjustingthestandarddeviation).

Likewise,the annualVaRisestimated as the daily VaRmultipliedbythesquarerootof

250 If thetrueexpectedreturn isused,VaRfordifferentlengthperiodsmustbe calculated

independently

ProfessorsNote: Assuminga zeroexpectedreturn whenestimating VaRisa

conservativeapproachbecause thecalculatedVaRwill begreater (i.e.,fartheroutin

the tailofthedistribution)thaniftheexpectedreturn isused

Sinceportfoliovaluesarelikelytochangeoverlongtimeperiods,it isoften thecasethat

VaRoverashorttimeperiodiscalculated andthen convertedtoalongerperiod.The Basel

Accord(discussedintheFRM Part IIcurriculum)recommends theuse ofatwo-week

period(10days)

Professor’sNote: For theexam,youwilllikelyberequiredtomakethesetime

conversationcalculationssince VaRisoftencalculatedover ashorttimeframe.

Advantages of the delta-normalVaRmethod includethefollowing:

• Easytoimplement

• Calculationscanbeperformedquickly

• Conducivetoanalysisbecauseriskfactors,correlations,and volatilitiesareidentified

Disadvantagesof the delta-normalmethodinclude thefollowing:

• Theneedtoassumeanormaldistribution

• The methodisunabletoproperlyaccountfordistributions with fartails,rirhrr because

ofunidentifiedtime variation inriskorunidentified riskfactors and/or correlations

• Nonlinearrelationships of option-likepositionsarenotadequatelydescribedbythedelta-normal method VaRismisstated becausetheinstabilityof the option deltasisnot

captured

HistoricalSimulation Method

Thehistorical method for estimating VaRisoften referredtoasthehistorical simulationmethod.Theeasiestwaytocalculatethe5% dailyVaR using the historicalmethodistoaccumulateanumberofpastdailyreturns,rank thereturnsfromhighesttolowest,and

identifythelowest 5%ofreturns.Thehighestofthese lowest 5% ofreturnsisthe 1-day,5%VaR

tail

Example:HistoricalVaRYouhaveaccumulated100dailythereturnsfromhighesttolowest,youidentifythe lowest' — m

Thelowestfivereturns representthe 5% lower tail'of the “d:

returns.Thefijffilowestreturn'(-0.0019)is*the 5‘

59hchanceofadailylossexceeding0.19%,or $1 '

&

AsyouwillseeinTopic35,the historical simulation methodmayweightobservationsand takeanaverageoftwo returns toobtainthehistorical VaR.Eachobservationcanbeviewedashavingaprobabilitydistributionwith50%totheleft and 50%totherightofa

given observation Whenconsideringthe previous example, 5%VaRwith100observationswould take the averageofthe fifth and sixthobservations[i.e., (—0.0011 +—0.0019) /

2=-0.0015] Therefore,the 5% historical VaRin thiscasewould be$150,000.Either

VaR Methods

Advantagesof the historical simulationmethodinclude thefollowing:

• The modelis easy toimplementwhenhistoricaldata isreadily available

• Calculationsaresimpleandcanbeperformedquickly

• Horizonisapositivechoicebasedontheintervals of historicaldata used

• Full valuationofportfolioisbasedonactualprices

• Itis notexposedtomodel risk

• It includes all correlationsasembeddedin marketpricechanges

Disadvantagesof thehistorical simulation method includethe following:

• It maynotbeenoughhistoricaldatafor allassets.

* Onlyonepathofeventsisused(theactualhistory), which includeschangesin

correlations andvolatilitiesthatmayhave occurred onlyinthathistorical period

* Timevariationofrisk inthepastmaynot representvariation inthefuture

• Themodel maynotrecognizechangesinvolatilityandcorrelationsfrom structural

changes

* Itisslowtoadaptto newvolatilitiesandcorrelationsasolddata carriesthesameweight

asmorerecentdata.However,exponentiallyweightedaverage(EWMA)modelscanbe

usedtoweighrecentobservationsmoreheavily

* Asmall numberof actualobservationsmay leadtoinsufficiently defined distribution

tails

MonteCarlo Simulation Method

The MonteCarlomethodrefersto computersoftware thatgenerateshundreds,thousands,

or evenmillionsofpossibleoutcomesfrom the distributions ofinputsspecifiedby theuser

For example,aportfolio managercouldenter adistributionofpossible1-weekreturnsfor

eachof thehundreds ofstocksinaportfolio On each “run”(thenumberof runs isspecified

bytheuser),thecomputerselectsoneweeklyreturnfrom eachstocksdistribution of

possiblereturnsand calculatesaweightedaverageportfolioreturn.

The severalthousandweightedaverageportfolioreturnswillnaturallyformadistribution,

whichwill approximatethenormal distribution Using the portfolio expectedreturnand

thestandarddeviation,whichare partof theMonteCarlooutput,VaR iscalculatedin the

same way aswith thedelta-normalmethod,

>le:Monte Carlo VaR

Advantagesof theMonteCarlomethod include thefollowing:

• Itisthemostpowerfulmodel

• Itcanaccountfor both linear and nonlinear risks

• Itcanincludetimevariationin risk andcorrelationsbyagingpositionsoverchosenhorizons

• Itisextremely flexible andcan incorporate additional riskfactors easily

• Nearlyunlimited numbers ofscenarios canproduce well-describeddistributions

DisadvantagesoftheMonteCarlomethod include thefollowing:

• Thereis alengthycomputationtime asthenumberof valuationsescalatesquickly

• Itis expensivebecauseof the intellectual andtechnologicalskills required

• Itissubjecttomodel riskof the stochastic processes chosen

4 Itissubjecttosamplingvariationatlower numbersof simulations

Forpracticequestions relatedtoVaRMethods see:

PastFRMExamQuestions:#1-7(page291)

The following is a review of the Valuation and Risk Models principles designed to address the AIM statements

set forth byCARP®.This topic is also covered in:

MEASURES OF FINANCIAL RISK

Topic34

EXAM FOCUS

The assumption regarding the shape of the underlying return distribution is critical in

determininganappropriate risk measure.Themean-varianceframeworkcanonlybeapplied

underthe assumptionofanelliptical distributionsuchasthenormal distribution The valueat

risk(VaR) measurecancalculateriskmeasureswhen thereturndistributionisnon-elliptical,

butthemeasurementisunreliableandnoestimateof theamountoflossisprovided.Expected

shortfallis a morerobust risk measure thatsatisfies alltheproperties ofacoherentrisk measure

with lessrestrictive assumptions.Fortheexam,focusyourattentiononthe calculation ofVaR,

properties of coherent riskmeasures,and theexpectedshortfallmethodology

MEAN-VARIANCE FRAMEWORK

AIM34.1;Describethemean-varianceframework and theefficient frontier

Thetraditionalmean-variancemode! estimates theamountoffinancialriskfor portfolios

intermsoftheportfolio’sexpectedreturn (i.e., mean)and risk(i.e.,standarddeviation

orvariance).Under themean-variance framework, it isnecessaryto assumethatreturn

distributionsfor portfoliosareelliptical distributions Themostcommonlyknownelliptical

probabilitydistributionfunctionisthe normaldistribution

Thenormaldistributionis acontinuousdistributionth.atillustratesallpossibleoutcomes

forrandom variables.Recallthat-thestandard normaldistributionhasameanofzeroand

astandarddeviationofone.Ifreturns arenormallydistributed,approximately 66.7% of

returnswilloccurwithinplusorminus onestandard'deviationof themean.Approximately

95% oftheobservationswill occur withinplusor minus twostandarddeviationsof the

mean Thus,giventhistypeofdistribution, returns are morelikelytooccur closertothe

Portfoliomanagersareconcerned withmeasuring downside riskandthereforeare

particularly interestedin measuringthepossibilityofoutcomes tothe leftorbelow

theexpectedmeanreturn.If thereturndistributionissymmetrical(likethe normal

distribution),then thestandarddeviationis anappropriatemeasureof risk when

determiningtheprobabilitythatanundesirableoutcomewilloccur.

Ifweassume thatreturndistributionsforallriskysecurities arenormallydistributed,

then we can chooseportfoliosbasedontheexpectedreturnsand standard deviations of

allpossiblecombinationsof risky'securities.Figure1below illustrates theconceptofthe

efficient frontier

In theory,all investorsprefersecurities orportfoliosthatlieontheefficient frontier.

Consider portfoliosA, B,and Cin Figure1.Ifyouhadtochoose betweenportfoliosA

andC,whichonewould youpreferand why? SinceportfoliosAand C havethe sameexpectedrerurn,arisk-averseinvestorwould choose theponfoliowiththeleastamountof

risk(whichwould be PortfolioA).Nowifyou hadtochoose between portfolios B andC,

whichonewouldyouchoose andwhy?BecauseportfoliosB and C have thesameamount

ofrisk, arisk-averseinvestor wouldchoose the portfolio with thehigher expectedreturn

(whichwould be PortfolioB).We say thatPortfolio B dominates Portfolio C with respect to

expectedreturn,andthat PortfolioAdominatesPortfolioCwithrespect torisk.Likewise,allportfoliosontheefficient frontier dominate all other portfoliosintheinvestment

universeof riskyassetswithrespect toeitherrisk,return,orboth

Therearean almost unlimited number of combinations of riskyassets totherightandbelow the efficientfrontier.However,in the absenceofarisk-freesecurity,portfoliostotheleft and above the efficientfrontierare notpossible.Therefore,allinvestorswillchoose

someportfolioontheefficient frontier Ifan investoris morerisk-averse,she may choosea

portfolioonthe efficientfrontierclosertoPortfolioA.Ifan investor islessrisk-averse,shewillchooseaportfolioonthe efficientfrontierclosertoPortfolioB

Figure1 :TheEfficientFrontier

Ifwenowassume thatthereisa risk-freesecurity, thenthemean-varianceframeworkisextendedbeyondthe efficientfrontier.Figure2illustratesthat the optimalsetofportfolios

nowlieon astraightline that runsfrom the risk-freesecuritythroughthemarketportfolio,

M.Allinvestorswillnowseekinvestmentsbyholdingsomeportionoftherisk-freesecurityandthe marketportfolio To achieve pointson thelinetothe right of the marketportfolio,

aninvestorwhoisvery aggressivewill borrowmoney (atthe risk-freerate)andinvestinmoreof themarketportfolio.More risk-averse investorswillholdsomecombinationoftherisk-freesecurityand the marketportfoliotoachieve portfoliosonthelinesegmentbetweenthe risk-freesecurityand the marketportfolio

Figure2:TheEfficientFrontier withthe Risk-FreeSecurity

E(R)

EfficientFrontier

M

Rr

TotalRisk<o>

Mean-Variance FrameworkLimitations

AIM34.2:Explainthelimitations of themean-varianceframework withrespectto

assumptions about thereturndistributions

Theuseof the standard deviationas ariskmeasurementisnotappropriate fornon-normal

distributions.Iftheshapeofthe underlyingreturndensity functionis notsymmetrical,

then thestandarddeviationdoesnot capturethe appropriateprobability ofobtaining

undesirablereturn outcomes.

Figure3illustratestwoprobabilitydistributionfunctions.Oneprobabilitydistribution

functionis thenormal distribution witha mean ofzero Theotherprobabilitydistribution

ispositivelyskewed.Thispositivelyskewed distribution hasthe same meanand standard

deviationasthenormaldistribution.Thedegree of skewnessalterstheentiredistribution.

For thepositivelyskeweddistribution, outcomesbelow themean are morelikelyto occur

closertothemean.Clearlynormalityisanimportant assumptionwhen using the

mean-varianceframework.Thus,rhemean-varianceframeworkisunreliablewhen theassumption

of normalityisnot met.

Figure 3:NormalDistributionvs.Positively-Skewed Distribution

Positive-Skew

NormalDistribution

+

F=0

AIM34.3:Definethe Value-at-Risk(VaR) measureof risk,describeassumptions

Valueat risk(VaR)isinterpretedastheworstpossible lossundernormalconditionsover

aspecified period Anotherwaytodefine VaRisas an estimateof themaximum loss that

canoccurwithagivenconfidencelevel.Ifananalystsays,“foragiven month,theVaR

is $1 millionat a95%levelofconfidence,”then this translates to mean“undernormalconditions, in 95%of the months(19 outof20months), we expect the fundtoeitherearn

aprofitorlosenomorethan $1 million.” Analystsmay alsouseotherstandardconfidencelevels(e.g.,90%and99%).Recall that delta-normalVaRcanbecomputedusing the

followingexpression: [p- (z)(cr)].

A majorlimitationoftheVaRmeasureforrisk isthattwoarbitraryparametersareusedin

rhe calculation——theconfidence level andtheholding period.Theconfidence level indicatesthelikelihoodorprobability thatwewill obtainavaluegreaterthanorequaltoVaR Theholding periodcan be any pre-determinedtimeperiod measured in days,weeks, months, or

years

Figure 4illustratesVaRparametersforaconfidencelevelof95% and 99%.Asyou can see,the levelof riskisdependentonthedegree of confidencechosen.VaR increaseswhentheconfidence levelincreases.Inaddition,VaRwill increaseat anincreasingrare as theconfidence levelincreases

Figure 4:VaRMeasurementsforaNormal Distribution

95% VaR

99% VaR

Profit/Loss

-2.33 -1.65Thesecondarbitraryparameter istheholdingperiod VaR willincreasewithincreasesintheholding period Therate atwhichVaR increases isdeterminedinpartby themeanofthedistribution.Ifthereturndistribution hasa mean,p,equalto 0,then VaRriseswiththesquarerootof theholdingperiod(i.e.,the squarerootoftime).If thereturndistribution

CrossReferencetoGARP Assigned Reading- Dowd, Chapter 2

VaRestimates are alsosubjecttobothmodelriskandimplementation risk Model riskis

theriskoferrorsresulting fromincorrectassumptionsusedinthe model Implementation

riskisthe riskoferrorsresultingfromtheimplementationofthemodel

Anothermajorlimitationof theVaR measureisthatitdoesnottelltheinvestortheamount

ormagnitudeof theactualloss.VaRonlyprovidesthemaximum value wc canlosefora

givenconfidence level Twodifferentreturndistributionsmayhave thesameVaR,butvery

different risk exposures.Apracticalexampleof how thiscanbea seriousproblemiswhen

aportfolio managerissellingout-of-the-moneyoptions.Foramajoriryof therime,rhe

options willhaveapositivereturn and,therefore,theexpectedreturnispositive.However,

in theunfavorableeventthat theoptions expirein-the-money, theresultinglosscanbevery

large.Thus,different strategiesfocusingonloweringVaRcanbeverymisleadingsincethe

magnitudeof the lossisnotcalculated

Tosummarize,VaRmeasurementsworkwellwithellipticalreturn distributions,suchas

the normal distribution VaRisalso abletocalculate the riskfornon-normaldistributions;

however,VaRestimates may beunreliablein thiscase.Limitations inimplementingthe

VaRmodelfordeterminingrisk resultfromtheunderlyingreturn distribution,arbitrary

confidencelevel,arbitraryholdingperiod,andtheinabilitytocalculate themagnitudeof

losses ThemeasureofVaR also violates the coherent riskmeasurepropertyof subadditivity

whenthereturndistributionisnotelliptical.Thispropertyisfurther explainedin thenext

AIM

COHERENT RISK MEASURES

AIM34.4:Define the properties ofacoherentriskmeasureandexplainthe

meaning of eachproperty

Inordertoproperlymeasure risk, one mustfirst clearly define whatismeantbyameasure

of risk Ifweallowi?tobea setof randomeventsandp(R)tobetherisk measurefor the

randomevents,then coherentrisk measuresshould exhibitthefollowingproperties:

1. Monotonicity:aportfoliowithgreaterfuturereturnswilllikely havelessrisk:

4 Translationinvariance:the risk ofaportfolioisdependentontheassetswithinthe

portfolio:for allconstants c,p(c+ R)=p(R)—c

Thefirst, third,andfourthproperties are morestraightforwardproperties of well-behaved

distributions.Monotonicityinfers that ifarandom futurevalue isalwaysgreaterthana

randomfuturevalueRvthenthe riskofthereturndistributionforRlisless than therisk

ofthereturndistributionfor i?2.Positivehomogeneitysuggeststhattheriskofaposition

isproportionalto itssize Positivehomogeneityshould holdaslongasthe securityisin a

liquid market Translationinvarianceimpliesthat theadditionofasureamountreducestheriskatthesamerateasthe cash neededtomakethe position acceptable.

Subadditiviryisthemostimportantpropertyforacoherentriskmeasure Theproperty

ofsubadditivitystatesthataportfoliomadeup ofsub-portfolioswillhaveequalorlessrisk than thesumof the risks of each individual subportfolio.Thisassumesthat whenindividual risksarecombined, theremaybesomediversification benefitsor,in theworst

case, nodiversification benefits andnogreaterrisk Thisimpliesgroupingoradding risksdoesnotincreasethe overallaggregateriskamount.

EXPECTED SHORTFALL

AIM34.5: Explainwhy VaRisnotacoherentriskmeasure

AIM34.6:Explain and calculate expected shortfall(ES),and compare andcontrast

VaR and ES

Valueatriskistheminimumpercent loss,equalto apre-specifiedworst casequantile

return (typically the5th percentilereturn).Expectedshortfall(ES)isthe expected lossgiventhatthe portfolio returnalreadyliesbelow thepre-specifiedworstcasequantile

return (Le„belowthe5th percentilereturn).In otherwords,expectedshortfallis the

knownasconditionalVaRorexpected tail loss(ETL).

For example,assumean investor is interested inknowingthe5%VaR(the5%VaR isequivalenttothe5thpercentilereturn)forafund.Further,assumethe5th percentilereturnfor the fund equals-20% Therefore,5% of thetime,thefundearnsa returnlessthan-20% Thevalueatrisk is—20%.However, VaRdoesnotprovide goodinformation

regarding theexpectedsizeoftheloss ifthe fundperformsin thelower5% ofthepossibleoutcomes.That questionisanswered bytheexpected shortfallamount,whichisthe

expectedvalueofallreturnsfallingbelowthe5th percentilereturn (i.e.,below-20%).

Therefore,expected shortfallwill equalalargerlossthantheVaR.Inaddition,unlikeVaR,

EShas theabilitytosatisfy thepropertyof subadditivity

TheES method providesan estimateof howlargeofalossisexpected ifanunfavorable

eventoccurs.VaRdidnotprovideany estimateof the magnitudeoflosses,only theprobability that they mightoccur.Thepropertyof subadditivity undertheESframeworkisalsobeneficialineliminatinganotherproblemforVaR.Whenadjusting both theholding

periodandconfidence levelatthesamerime, anESsurfacecurveshowingtheinteractions

ofbothadjustmentsisconvex.Thisimpliesthatthe ESmethodis moreappropriate thantheVaRmethodinsolving portfoliooptimization problems

Cross Reference to GARPAssigned Reading—Dowd, Chapter 2 However,ESisamoreappropriateriskmeasurethan VaRforthefollowingreasons:

• ESsatisfiesallof thepropertiesof coherentriskmeasurementsincludingsubadditivity

VaR onlysatisfiestheseproperties for normal distributions

• TheportfoliorisksurfaceforESisconvexbecause thepropertyof subadditivityis met.

Thus,ESismoreappropriate for solvingportfoliooptimizationproblemsthantheVaR

method

• ES givesan estimateof the magnitude ofaloss for unfavorableevents.VaRprovidesno

estimateof how largealossmay be

• EShas lessrestrictiveassumptionsregardingrisk/returndecision rules

AIM 34.7:Describespectralriskmeasures,andexplainhowVaRand ESarespecial

casesofspectralriskmeasures

A moregeneralriskmeasurethaneither VaRorESisknownasthe riskspectrumorrisk

aversionfunction Theriskspectrummeasurestheweightedaverages ofthereturnquantiles

_from thelossdistributions.ES isaspecialcaseof this riskspectrum measure.When

modelingthe EScase,theweightingfunctionisset to[1 /(1—confidencelevel)]for rail

losses Allotherquantileswill have aweightofzero

VaRisalsoaspecialcaseofspectralriskmeasuremodels.Theweightingfunction withVaR

assignsaprobabilityofoneto the eventthatthe/-valueequalsthelevelofsignificance(i.e.,

p=a),andaprobabilityofzerotoall othereventswhere p*a Thus,theESmeasure

placesequalweightsontaillosses whileVaR placesnoweighton taillosses

In orderforarisk measuretobecoherent,itmustgivehigherlossesatleast thesame

weightaslowerlosses.‘Inthe EScase,all lossesaregiventhesameweight.Thissuggeststhat

investorsarerisk-neutral withrespect tolosses Thisiscontradictorytothecommonnotion

thatinvestorsarerisk-averse IntheVaRcase,only theJossassociatedwitha/-valueequal

• toaisgivenanyweight.Greaterlossesare givennoweightatall Thisimplies thatinvestors

arerisk-seekers.Thus, theESandVaRmeasures areinadequatein thattheweighting

function is nor consistentwithrisk aversion

SCENARIO ANALYSIS

AIM34.8:Describehow the resultsofscenarioanalysiscanbeinterpretedas

coherent riskmeasures

Theresultsofscenarioanalysiscanbeinterpretedascoherentriskmeasuresby first

assigningprobabilitiestoasetof lossoutcomes.These lossescanbe thought ofastail

drawingsof the relevant distribution function The expected shortfallforthe distribution

canthenbecomputed byfindingthe arithmetic average of the losses.Therefore,the

outcomesofscenarioanalysismustbe coherent riskmeasurements,because ESisacoherent

Scenarioanalysiscanalso be appliedin situationswhere thereare numerousdistribution

functionsinvolved.It canbeshown thattheES,thehighestESfroma setofcomparable

expectedshortfalls basedondifferent distributionfunctions, andthe highest expected

shortfallfroma setof highest lossesareall coherentriskmeasures.For example,assume

youareconsideringa setofnlossoutcomes outofafamilyof distribution functions The

F.Sisobtainedfrom eachdistributionfunction.If thereis a setofmcomparable expectedshortfalls, that eachhaveadifferentcorrespondingloss distributionfunction,then themaximumoftheseexpected shortfallsis acoherentriskmeasure Thus,incaseswheren=1,

the ESisthe sameasthe probablemaximumloss becausethereisonlyonetail lossineachscenario.Ifmequalsone,then thehighest expected loss fromasinglescenarioanalysisisa

coherenrmeasure.Incaseswherem is greaterthanone,thehighestexpectedofm worst case

outcomesis acoherent riskmeasure

Cross Reference toGARPAssigned Reading—Dowd, Chapter 2

AIM34.1

Thetraditionalmean-variancemodelestimatestheamountof financialriskforportfolios

intermsof theportfolios expectedreturn(mean)and risk(standarddeviationor variance)

Anecessaryassumptionforthis modelisthatreturndistributions fortheportfoliosare

ellipticaldistributions

Theefficient frontieristhesetofportfoliosthat dominate allotherportfoliosinthe

investmentuniverseofriskyassetswithrespect torisk andreturn.Whenarisk-freesecurity

is introduced,theoptimalsetof portfoliosconsistsofalinefromtherisk-freesecuritythat

istangent totheefficient frontieratthe marketportfolio

AIM34.2

Themean-varianceframeworkisunreliable when theunderlyingreturndistributionis not

normalorelliptical.Thestandard deviationisnot an accuratemeasureof risk and does

returndensityfunctionisnotsymmetrical

AIM34.3

Valueatrisk(VaR) isariskmeasurementthatdetermines theprobabilityofanoccurrence

inthe left-hand tailofa returndistributionat agiven confidencelevel.VaRisdefinedas:

[p-(z)(o)].The underlyingreturn distribution,arbitrarychoiceofconfidence levels and

holdingperiods, and the inabilitytocalculatethemagnitudeof losses resultinlimitations'

inimplementingtheVaRmodel when determiningrisk

Subadditivity, themostimportantpropertyforacoherentriskmeasure, statesthata

portfolio made up ofsub-portfolioswill have equalorless risk thanthesumof the risks of

each individualsub-portfolio.VaR violatesthepropertyofsubadditivity

Expected shortfallis a moreaccuraterisk measurethanVaRforthefollowingreasons:

• ESsatisfies allthe propertiesof coherent riskmeasurementsincluding subadditivity

• Theportfoliorisk surface for ESis convex sincethepropertyof subadditivityis met.

Thus,ESis moreappropriate forsolving portfoliooptimizationproblemstitantheVaRmethod

• ES givesan estimateofthemagnitudeofalossfor unfavorableevents.VaRprovidesno

estimateofhowlargealoss maybe

• EShas lessrestrictiveassumptionsregardingrisk/return decision rules

AIM34.7

ESis aspecialcaseof the riskspectrum measurewhere theweightingfunctionisset to

1 /(1—confidencelevel)for tail losses that all have an equalweight,andallotherquantiles

haveaweightofzero.The VaRisaspecialcasewhereonlyasingle quantileis measured,

and theweightingfunctionisset to onewhen />-valueequalsthelevelofsignificance,andallotherquantiles haveaweightofzero

AIM34.8

Theoutcomesofscenarioanalysisarecoherentriskmeasurements,becauseexpected

shortfallisacoherent riskmeasurement.TheES for the distributioncan be computedby

findingdiearithmetic average of the lossesforvarious scenariolossoutcomes.

CrossReference roGARPAssigned Reading-Dowd,Chapter2

CONCEPT CHECKERS

The mean-varianceframeworkisinappropriate for measuring risk when the

underlyingreturndistribution:

A isnormal

B iselliptical

C hasakurrosis equal tothree

D ispositively skewed

Assumeaninvestor is veryrisk-averseandiscreatingaportfoliobasedonthe

mean-variance modelandtherisk-freeasset.Theinvestorwillmostlikely choosean

investmentonthe:

A left-hand side of the efficientfrontier

B right-handsideof the efficient frontier

C linesegmentconnecting the risk-freerate tothemarketportfolio

D linesegmentextendingtotherightof themarketportfolio

p(X+ Y)<p(X)+p(Y) isthemathematicalequationforwhichpropertyofa

coherentriskmeasure?

Whichofthefollowingisnot areasonthatexpected shortfall(ES)is amore

appropriateriskmeasurethan valueatrisk(VaR)?

A For normaldistributions,only ES satisfiesalltheproperties of coherent risk

measurements.

B Fornon-ellipticaldistributions,the portfolio risksurfaceformed byholding

period and confidence levelis more convexforES

C ESgivesanestimateofthemagnitudeofaloss

D EShaslessrestrictiveassumptionsregardingrisk/returndecision rulesthan VaR

Iftheweighting functioninthegeneralriskspectrummeasureisset to

1/(1-confidencelevel)foralltaillosses, thentheriskspectrumis aspecialcaseof

Self-TestQuestions:41(page284)

PastFRM Exam Questions:#8-9(page292)

CONCEPT CHECKER ANSWERS

i. D The mean-varianceframeworkisonlyappropriatewhentheunderlyingdistribution is

elliptical.The normal distribution is aspecialcase ofellipticaldistributionswhereskewness is

equalto zero and kurtosis isequalto three.If thereis any skewness,the distribution function

will notbe symmetrical, and standarddeviation will not be an appropriateriskmeasure.

2. C Under themean-variance framework,whenarisk-freesecurity isincludedin theanalysis,

theoptimalsetof portfolioslies on astraight linethat runs fromtherisk-free security tothemarketportfolio Allinvestorswill holdsomeportion oftherisk-freesecurityand themarket

portfolio.Morerisk-averseinvestorswill holdsomecombinationofthe risk-free security and the marketportfoliotoachieve portfolioson the line segment between therisk-freesecurity andthe market portfolio

3 B Thepropertyof subadditivitystates that aportfoliomadeupof sub-portfolioswill have

equalorlessrisk than the sumofthe risks of each individualsub-portfolio

4 A VaR andESboth satisfyallthe properties of coherent riskmeasuresfornormaldistributions

However,only ES satisfies allthe properties ofcoherentrisk measures when the assumption

ofnormalityis not met.

5 C Expected shortfallis aspecialcase of theriskspectrum measure that isfoundby setting the

weightingfunction to 1 / (1—confidencelevel)fortail losses that all have anequal weight

The following is a review of the Valuation and Risk Models principles designed to address the AIM statements

set forth byGARP®.This topic is also covered in:

MODELS

Topic 35

EXAM FOCUS

Obtaininganaccurateestimateofan asset’svalue that is atrisk of losshinges greatlyonthe

measurementoftheassetsvolatility(orpossibledeviation in valueover acertain timeperiod)

Asset valuecanbeevaluatingusinganormaldistribution; however,deviationsfrom normality

willcreatechallengesfor therisk manager inmeasuring both volatility and valueatrisk(VaR).

In this topic,wewill discussissueswith volatilityestimation anddifferent weighting methods

_thatcanbe usedtodetermineVaR.The advantages, disadvantages, andunderlyingassumptions

of thevariousmethodologieswill also be discussed Fortheexam,understand why deviations

from normality occurandhaveageneral understandingoftheapproachestomeasuringVaR

(parametricand nonparanietric)

AIM35.1:Explain howasset returndistributionstend todeviatefrom the normal

distribution

AIM35.2: Explainpotentialreasonsfor theexistenceoffat tailsin areturn

distributionand describe theimplicationsfat tails haveon analysisofreturn

distributions.

AIM35.3: Distinguishbetweenconditional and unconditional distributions

Threecommondeviations fromnormalitythatareproblematicinmodelingriskresultfrom

asset returnsthatare fat-tailed, skewed,orunstable,

Fat-tailedrefersto adistributionwithahigher probability ofobservationsoccurring in

the tails relativetothe normal distribution Asillustrated inFigure1,thereisalarger

probability ofanobservation occurringfurtherawayfrom,themeanof the distribution

The firsttwo moments (meanandvariance)of thedistributionsaresimilarforthefat-tailed

andnormaldistribution.However, inaddition tothegreatermass inthetails,thereisalso a

greaterprobabilitymassaroundthe meanfor the fat-tailed distribution.Furthermore,there

isless probabilitymassinthe intermediate range(around+/-onestandarddeviation)for

thefat-taileddistribution compared tothe normal distribution

Figure1:Illustrationof Fat-Tailedand NormalDistributions

Adistributionisskewed when the distributionisnotsymmetrical.Ariskmanager is moreconcernedwhen thereisahigherprobabilityofalargenegativereturnthanalargeposirive

return.Thisisreferredtoasleft-skewed and isillustrated in Figure2.

Figure2:Left-SkewedandNormalDistributions

In modelingrisk, anumberofassumptionsarenecessary.Iftheparametersofthe modelare

unstable, theyare not constantbutvary over time.For example, ifinterestrates, inflation,

andmarketpremiums arechangingover time,thiswill affectthevolatility ofthereturnsgoing forward

DEVIATIONS FROMTHENORMAL DISTRIBUTION

Thephenomenon of “fat tails”ismostlikely the result of the volatility and/orthe meanofthe distributionchangingovertime.If themeanand standard deviationare thesameforasset returnsforany givenday, thedistributionofreturnsisreferredto as an unconditionaldistributionofasset returns.However,different marketor economicconditionsmay causethemeanand varianceofthereturndistributiontochangeovertime.Insuchcases,the

Cross Reference to GARPAssigned Reading Allen et al.,Chapter2

reflectedinstockprices, it isnotlikelythat thefirstmoments orconditionalmeansof the

distributionvaryenoughtomakeadifferenceover time.

The second possible explanation for “fat tails”isthatthesecondmomentorvolatilityis

time-varying.This explanationismuch morelikely given observedchangesin interestrate

volatility (e.g.,prior to amuch-anticipated Federal Reserveannouncement).Increased

marketuncertaintyfollowing significant politicalor economiceventsresultsinincreased

volatilityofreturndistributions

MARKET REGIMESANDCONDITIONAL DISTRIBUTIONS

AIM35.4:Describetheimplications of regimeswitchingonquantifyingvolatility

Aregime-switching volatilitymodel assumesdifferent marketregimes existwithhighor

low volatility.Theconditional distributionsofreturnsarealways normalwitha constant

mean but eitherhaveahighorlow volatility Figure 3 illustratesahypo

thetieaTregime-switchingmodelforinterestratevolariliry Notethat thetrueinterestrarevolatilitydepicted

by the solidlineiseither13 basis pointsper day(bp/day)or6bp/day.The actual observed

returnsdeviate aroundthehigh volatility 13bp/day levelandthe low volatility 6bp/day

In thisexample,the unconditionaldistributionis notnormally distributed.However,

assuming time-varying volatility, theinterestratedistributionsareconditionally normally

distributed

Theprobability of large deviations from normalityoccurring are muchless likelyunder the

regime-switchingmodel.Forexample,the interestratevolatilityinFigure3rangesfrom

5.7bp/dayto13.6bp/daywithanoverallmeanof8.52bp/day.However,rhe 13.6bp/day

hasadifferenceof only 0.6bp/day from the conditionalhigh volatility level comparedto a

5.08bp/day difference from the unconditional distribution.Thiswould resultinafrit-tailed

unconditional distribution.Theregime-switchingmodelcapturesthe conditional normality'

andmayresolvethefat-tail problemandotherdeviations from normality

Figure3: ActualConditionalReturnVolatility Under MarketRegimes

Cross Reference to GARPAssigned Reading—Alien et al Chapter 2

Ifwe assumethatvolatilityvarieswith timeandthatasset returns areconditionallynormallydistributed,thenwe maybeabletotoleratethe fat-tail issue.Inthenextsection

wedemonstrate howtoestimateconditionalmeansandvariances However,despite efforts

to moreaccuratelymodelfinancialdata, extreme eventsdo still occur Themodel(or

distribution)usedmaynot capturetheseextreme movements.Forexample,valueatrisk

(VaR)models aretypicallyutilizedtomodel the risk levelapparentinassetprices.VaRassumesasset returnsfollowanormal distribution,butaswehave just discussed,asset

returndistributions tendtoexhibitfattails.Asa result,VaRmayunderestimatetheactuallossamount.

However, some tools existthatservetocomplement VaR by examining thedata in the tail

ofthedistribution For example,stresstestingand scenario analysiscan examine extreme eventsby testing how hypothetical and/orpastfinancial shockswill impactVaR.Also,

extremevalue theory(EVT)canbeapplied toexamine justthe tail ofthedistributionandsomeclassesofEVTapplya separatedistributiontothe tail Despitenotbeingable

toaccuratelycapture eventsin the tail, VaR isstill usefulfor approximatingtherisklevelinherentinfinancial-assets

VALUEAT RISKAIM35.5:Explain the variousapproachesfor estimating VaR

AIM35.6: Compare,contrastandcalculate parametric andnon-parametric

approaches for estimatingconditionalvolatility,including:

• Historical standard deviation

-A valueatrisk (VaR)methodfor estimating riskistypicallyeitherahistorical-based•

approachoranimplied-volatility-based approach Under the historical-based approach,theshapeof theconditionaldistribution isestimated basedonhistoricaltime seriesdata.Historical-basedapproaches typically fallintothreesub-categories:parametric,nonparametric,and hybrid

1. Theparametricapproach requires specificassumptionsregardingrheasset returnsdistribution.A parametricmodeltypicallyassumes asset returns arenormallyor

lognormallydistributedwith time-varyingvolatility.Themost commonexample ofthe parametricmethodinestimating future volatilityisbasedoncalculatinghistoricalvarianceorstandarddeviationusing“meansquared deviation.” For example,the

followingequation isusedtoestimatefuturevariance basedon awindowofthe Kmost

Topic 35

Cross Reference to GARPAssignedReading-Allen et al.,Chapter2

Ifweassumeasset returnsfollowarandomwalk,the meanreturniszero.Alternatively,

ananalyst mayassumeaconditionalmeandifferentfromzeroandavolatilityfora

specificperiod oftime

Professor’sNote:The delta-normal methodisanexampleofaparametric

approach

_ _ r •

'

>gK=100 (an estimationwindowusing themost recent 100*sscc returns),

ailyconditionalmean asset return, is estimatedTodje-15bp/dayg v

2 Thenonparametricapproachisless restrictive inthat therearenounderlying

assumptions of theasset returnsdistribution Themostcommonnonparametric

approach models volatility using thehistoricalsimulation method

3 Asthename suggests,the hybrid approach combines techniques of both parametric

andnonparametricmethodstoestimatevolatility using historical data

Theimplied-volatility-based approachuses derivativepricingmodels suchasthe

Black-Scholes-Mertonoption pricingmodel toestimateanimplied volatilitybasedon current

marketdataratherthan historical data

PARAMETRIC APPROACHESFORVAR

TheRiskMetrics®[i.e.,exponentiallyweighted moving average(EWMA) model]and

GARCH approachesareboth exponential smoothing weighting methods.RiskMetrics®is

actuallyaspecialcaseoftheGARCH approach.Bothexponentialsmoothingmethodsare

similartothe historicalstandarddeviationapproachbecauseall three methods:

• Areparametric

• Attemptto estimateconditional volatility

• Userecenthistorical data

• Applya setof weightsto pastsquaredreturns.

Professor’sNote:TheRiskMetrics®approachis just anEWMAmodel that

uses apre-specified decayfactorfordailydata(6.94) andmonthlydata

(0.97)

The onlymajordifference between thehistoricalstandard deviationapproachand the

twoexponential smoothingapproachesiswithrespect totheweights placedonhistorical

returnsthatareusedtoestimatefuturevolatility.The historical standarddeviationapproach

assumes allKreturnsin thewindowareequallyweighted.Conversely,theexponential