Chapter14RiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20121MarketRiskVaR:HistoricalSimulationApproach

HistoricalSimulation

Collectdataonthedailymovementsinallmarketvariables.ThefirstsimulationtrialassumesthatthepercentagechangesinallmarketvariablesareasonthefirstdayThesecondsimulationtrialassumesthatthepercentagechangesinallmarketvariablesareasontheseconddayandsoonRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20122HistoricalSimulationcontinuedSupposeweusendaysofhistoricaldatawithtodaybeingdaynLetvibethevalueofavariableondayiTherearen-1simulationtrialsTheithtrialassumesthatthevalueofthemarketvariabletomorrow(i.e.,ondayn+1)isRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20123Example:PortfolioonSept25,2008(Table14.1,page304)RiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20124IndexAmountInvested($000s)DJIA4,000FTSE1003,000CAC401,000Nikkei2252,000Total10,000U.S.DollarEquivalentofStockIndices(Table14.2,page305)RiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20125DayDateDJIAFTSECAC40Nikkei0Aug7,200611,219.3811,131.846,373.89131.771Aug8,200611,173.5911,096.286,378.16134.382Aug9,200611,076.1811,185.356,474.04135.943Aug10,200611,124.3711,016.716,357.49135.44….……………..…………499Sep24,200810,825.179,438.586,033.93114.26500Sep25,200811,022.069,599.906,200.40112.82Scenarios(Table14.3,page305)RiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20126ScenarioNumberDJIAFTSECACNikkeiPortfolioValueLoss110,977.089,569.236,204.55115.0510,014.334-14.334210,925.979,676.966,293.60114.1310,027.481-27,481311,070.019,455.166,088.77112.409,946.73653,264….………..…..…..…..…..49910,831.439,383.496,051.94113.859,857.465142.53550011,222.539,763.976,371.45111.4010,126.439-126.439Losses(Table14.4,page307)RiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20127ScenarioNumberLoss($000s)494477.841339345.435349282.204329277.041487253.385227217.974131205.256One-day99%VaR=$253,385Accuracy(page308) Supposethatxistheqthquantileofthelossdistributionwhenitisestimatedfromnobservations.Thestandarderrorofxis wheref(x)isanestimateoftheprobabilitydensityofthelossattheqthquantilecalculatedbyassumingaprobabilitydistributionforthelossRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20128Example14.1(page308)Weestimatethe0.01-quantilefrom500observationsas$25millionWeestimatef(x)byapproximatingtheactualempiricaldistributionwithanormaldistributionmeanzeroandstandarddeviation$10millionThe0.01quantileoftheapproximatingdistributionisNORMINV(0.01,0,10)=23.26andthevalueoff(x)isNORMDIST(23.26,0,10,FALSE)=0.0027TheestimateofthestandarderroristhereforeRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20129Extension1LetweightsassignedtoobservationsdeclineexponentiallyaswegobackintimeRankobservationsfromworsttobestStartingatworstobservationsumweightsuntiltherequiredquantileisreachedRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull201210Applicationto4-IndexPortfolio

l=0.995(Table14.5,page311)RiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull201211ScenarioNumberLoss($000s)WeightCumulativeWeight494477.8410.005280.00528339345.4350.002430.00771349282.2040.002550.01027329277.0410.002310.01258487253.3850.005100.01768227217.9740.001390.01906131205.2560.000860.01992One-day99%VaR=$282,204Extension2UseavolatilityupdatingschemeandadjustthepercentagechangeobservedondayiforamarketvariableforthedifferencesbetweenvolatilityondayiandcurrentvolatilityValueofmarketvariableunderithscenarioesRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull201212Volatilities(%perDay)EstimatedforNextDayin4-IndexExample(Table14.6,page312)RiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull201213DayDateDJIAFTSECAC40Nikkei0Aug7,20061.111.421.401.381Aug8,20061.081.381.361.432Aug9,20061.071.351.361.413Aug10,20061.041.361.391.37….…………………………499Sep24,20082.213.283.111.61500Sep25,20082.193.213.091.59VolatilityAdjustedLosses(Table14.7,page313)RiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull201214ScenarioNumberLoss($000s)1311,082.969494715.512227687.72098661.221329602.968339546.54074492.764Extension3Supposethereare500dailychangesCalculatea95%confidenceintervalforVaRbysampling500,000timeswithreplacementfromdailychangestoobtain1000setsofchangesover500daysCalcuateVaRforeachsetandcalculateaconfidenceintervalThisisknownasthebootstrapmethodRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull201215ComputationalIssuesToavoidrevaluingacompleteportfolio500timesadelta/gammaapproximationissometimesusedWhenaderivativedependononlyoneunderlyingvariable,SRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull201216ExtremeValueTheory(page314)Extremevaluetheorycanbeusedtoinvestigatethepropertiesoftherighttailoftheempiricaldistributionofavariablex.(Ifweinterestedinthelefttailweconsiderthevariable–x.)WefirstchoosealevelusomewhatintherighttailofthedistributionWethenuseGnedenko’sresultwhichshowsthatforawideclassofdistributionsasuincreasestheprobabilitydistributionthatvliesbetweenuandu+yconditionalthatitisgreaterthanutendstoageneralizedParetodistributionRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull201217GeneralizedParetoDistributionRiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull201218Thishastwoparametersx(theshapeparameter)andb(thescaleparameter)ThecumulativedistributionisMaximumLikelihoodEstimator(Equation14.7,page316)RiskManagementandFinancialInstitutions3e,Chapter14,Copyright?JohnC.Hull20