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Copyright 1996 2005byNeilD Pearson Allrightsreserved Delta Normal Variance Covariance MethodforanEquityPortfolio NeilD Pearson Copyright 1996 2005byNeilD Pearson Allrightsreserved SummaryofDelta Normalmethod Assume Normaldistributionsformarketfactorsportfoliovalueisgivenbylinear delta approximation changesinportfoliovalueareNormallydistributedUsemathematicsof multivariate NormaldistributiontodeterminedistributionofchangesinportfoliovalueFromdistribution density computeorreadoffvalue at risk Copyright 1996 2005byNeilD Pearson Allrightsreserved FocusofDelta Normalmethod Oncewedeterminethedistributionofchangesinportfoliovalue wecandeterminethe cutoff thatleavesprobabilitypinleft handtail Copyright 1996 2005byNeilD Pearson Allrightsreserved Equityportfolioexample 110millioninwell diversifiedportfoliooflargecap U S equities perfectlycorrelatedwithS P500500FT SE100indexfuturescontractsShort200S P500contracts600writtenFT SE100indexcalloptionswithstrikelevelof5 875800writtenS P500indexcallswithstrikelevelof1 100Netvalue 110million valueofwrittenoptions USD101 485 220 Copyright 1996 2005byNeilD Pearson Allrightsreserved Indexandexchangeratelevels Indexes S P500isat1 097 6FT SE100isat5 862 3Exchangerate USD1 6271 GBP Copyright 1996 2005byNeilD Pearson Allrightsreserved Portfoliovalueasfunctionof2indexes Inthisfigure theexchangerateisheldfixedatlevelofUSD1 6271 GBP Copyright 1996 2005byNeilD Pearson Allrightsreserved Port valueasfunctionofS P500 Copyright 1996 2005byNeilD Pearson Allrightsreserved Port valueasfunctionofFT SE100 Copyright 1996 2005byNeilD Pearson Allrightsreserved Riskmapping Computingportfoliostandarddeviationcanbedifficultrisksofsomeinstruments e g options changeaslevelofstockpriceschangeActualportfoliosoftenincludemanypositionstherecanbe100 sor1 000 sofdifferentpositionstoestimatecovariancematrixofthousandsofinstrumentsisnotfeasible Individualstocksare mapped topositionsinalimitednumberofmarketindexes Mapping theportfolio Complicatedactualportfolio toocomplicatedtodirectlycomputeVaR Simplerportfolio map VaRofsimplerportfolio ComputeVaR Mapping isthekeystep Itinvolvesreplacingtheactualcomplicatedportfoliowithasimplerportfolioofapproximatelythesamerisk forwhichwecanactuallycomputetheVaR Ofcourse acrucialissueisthesenseinwhichtheriskisapproximatelythesame Copyright 1996 2005byNeilD Pearson Allrightsreserved Mappingequityportfolio mainidea Fornow first1 2ofthislecture wethinkofportfolioasbeingexposedtotwomarketfactors USDvalueofportfoliounderlyingS PUSDvalueofportfoliounderlyingFT SE100Approach map S Poptions futurestopositionsinindex map FT SEoptions futurestopositionsinindexcomputeVaRof mapped position WhenIwriteorsay positioninindex thisisjustshorthandforapositionintheportfolioofstocksthatunderliestheindex Youcan tactuallybuy selltheindexitself Thisexampleillustratesmainideaofmappingoptions butthetreatmentofexchangerateriskisincorrect Wewillpresentacorrectanalysisinthesecondhalfofthelecture Mapping theportfolio Complicatedactualportfolioconsistingof5instruments SimplerportfolioconsistingofpositionsinS P FT SE map VaRofsimplerportfolio ComputeVaR Mapping isthekeystep Itinvolvesreplacingtheactualcomplicatedportfoliowithasimplerportfolioofapproximatelythesamerisk forwhichwecanactuallycomputetheVaR Ofcourse acrucialissueisthesenseinwhichtheriskisapproximatelythesame Copyright 1996 2005byNeilD Pearson Allrightsreserved Mappingoptions Optionsandothercomplicatedinstrumentsarehandledbyaprocedureknownas riskmapping optionsaremappedtodelta equivalentspotpositionse g optionwithdelta 1 2ismappedto1 2sharewetalkaboutthisinrestoflectureIndividualstocksaremappedtopositionsinalimitednumberofmarketindexeswedon tconsiderthissecondissue ifyouareinterested seechapter7ofRiskBudgeting PortfolioProblem SolvingwithValue at Risk byN D Pearson Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingtheU S equitypositions PortfoliodeltawithrespecttoS P500is4 864Loosely theportfolio actslike 4 864indexesHowdidwecomputethisdelta Copyright 1996 2005byNeilD Pearson Allrightsreserved Computingthedelta S Pcashposition 110millioncashpositionislike110 000 000 1097 6 100 218 7indexesThatis Or Copyright 1996 2005byNeilD Pearson Allrightsreserved DeltaofS Pfuturesposition Through cost of carry formulaF Se r d T t 1pointchangeinindex 1 01391changeinfutures multiplieris250 so1pointchangeinindexresultsin253 48changeinvalue cashflow offuturespositionthereare 200S P500indexfuturescombiningallthis futuresdeltais 50 965 Parametersr 0 05 d 0 014 andT t 0 3836 Copyright 1996 2005byNeilD Pearson Allrightsreserved Computingthedelta S Poptionsposition Optionsposition optiondelta peroption is55 825ona perindex basis deltais0 55825multiplieris100thereare 800S P500indexoptionsDeltaofoptionspositionis 44 660 Copyright 1996 2005byNeilD Pearson Allrightsreserved Combiningthe3U S equitypositions Thenetdeltaisjustthesum deltaof cash position 100 219deltaoffuturesposition 50 965deltaofoptionsposition 44 660Adding DwithrespecttoS P500is4 864 PositionismappedtoaninvestmentofUSD4 864 1097 6 USD5 339millionintheindex Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingtheU K equitypositions PortfolioDwithrespecttoFT SE100isGBP1 734 Loosely theportfolio actslike 1 734indexesHowdidwecomputethisdelta Copyright 1996 2005byNeilD Pearson Allrightsreserved Computingthedelta FT SEfuturesposition Through cost of carry formulaF Se r d T t 1pointchangeinindex 1 0131changeinfutures multiplieris10 so1pointchangeinindexresultsin10 131changeinvalue cashflow offuturespositionthereare500FT SE100indexfuturesAdding deltaoffuturespositionis5 066 Parametersr 0 05 d 0 016 andT t 0 3836 Copyright 1996 2005byNeilD Pearson Allrightsreserved Computingthedelta FT SEoptionsposition Optionsposition optiondelta peroption is5 553ona perindex basis deltais0 5553multiplieris10thereare 600S P500indexoptionsDeltaofoptionspositionis 3 332pounds Copyright 1996 2005byNeilD Pearson Allrightsreserved Combiningthe2U K equitypositions Thenetdeltaisjustthesum deltaoffuturesposition 5 066deltaofoptionsposition 3 332Adding DwithrespecttoFT SE100is1 734 Positionismappedto GBP1 734 5 862 GBP10 165millioninindexUSD1 6271 1 734 5 862 16 541millioninindex Copyright 1996 2005byNeilD Pearson Allrightsreserved Mappedportfolio Tosummarize theportfolioisequivalentto USD5 339millionintheportfoliothatunderliestheS P500indexUSD16 541millionintheportfoliothatunderliestheFT SE100index Copyright 1996 2005byNeilD Pearson Allrightsreserved Parametersfortheexample Std dev ofmonthlyratesofreturnontheportfoliounderlyingtheS P500indexiss1 0 061 6 1 Std dev ofmonthlyratesofreturninthedollarvalueoftheportfoliounderlyingtheFT SE100indexiss2 0 065 6 5 Correlationisr 0 55Expectedratesofchangeinindexesarem1 0 01 1 andm2 0 0125 1 25 permonth respectively U S portfoliopaysdividendsatrateof1 4 12 0 1167 permonth Copyright 1996 2005byNeilD Pearson Allrightsreserved Otherparametersandassumptions Holdingperiod 1monthProbabilityisp 0 05 5 Monthlyreturnson2indexesarenormallydistributedimpliesportfolioreturnsarenormallydistributeddistributioncompletelydescribedbymeanandvariance Copyright 1996 2005byNeilD Pearson Allrightsreserved ExpectedchangeinportfoliovalueDV ExpectedchangeinportfoliovalueiswhereX1 5 339isUSDamountinS PX2 16 541isUSDamountinFT SEm1 0 01m2 0 0125D 110 0 014 12 isthedividendstobereceivedduringthenextmonthWehaveE DV 0 388million Copyright 1996 2005byNeilD Pearson Allrightsreserved Varianceandstd deviationofchangeinportfoliovalue ThevarianceofthechangeiswhereX1 5 339isUSDamountinS PX2 16 541isUSDamountinFT SEs1 0 061 s2 0 065 r 0 55Wehavevar DV USD1 647millionstd dev DV USD1 283million Copyright 1996 2005byNeilD Pearson Allrightsreserved Densityfunctionofchangesinport value Copyright 1996 2005byNeilD Pearson Allrightsreserved Value at risk Criticalvalueequalto1 645std dev sbelowmeanleaves5 oftheprobabilityinleft handtailCallthisthe5percentquantileValueatrisk 1 734millionorVaR 1 734 101 485 1 709 ofvalueofportfolio Copyright 1996 2005byNeilD Pearson Allrightsreserved Value at risk Asonpreviousslide the5percentquantileisIfwewantadifferentquantile weuseadifferentconstantk 5 quantilethenk 1 652 5 quantilethenk 1 961 quantilethenk 2 33 k Copyright 1996 2005byNeilD Pearson Allrightsreserved Effectsofholdingperiod probabilityp Whatwouldhappenifwechoseasmallerprobability say1 Whatwouldhappenifwechoseashorterholdingperiod say1day Copyright 1996 2005byNeilD Pearson Allrightsreserved Problemswithwhatwejustdid Whataresomeproblemswithwhatwejustdid Copyright 1996 2005byNeilD Pearson Allrightsreserved Port valueasfunctionof2indexes InthisfiguretheexchangerateisheldfixedatlevelofUSD1 6271 GBP Copyright 1996 2005byNeilD Pearson Allrightsreserved PortfoliovalueasfunctionofS P500 Copyright 1996 2005byNeilD Pearson Allrightsreserved Exchangerateriskoffutures Supposeyoubuy golong 500FT SE100indexfuturescontractswhathappensiftheexchangeratechanges buttheFT SEdoesnot whatistheexchangerateriskofthisposition Concludethatthemappingwejustcompletedisincorrect Copyright 1996 2005byNeilD Pearson Allrightsreserved Mappinganequityportfolio consideringexchangeraterisk Todotheanalysiscorrectly wemustconsiderexchangeraterisktobeaseparatemarketfactorWeusethreeriskfactors S1 S P500S2 FT SE100e exchangerateWeneedtoknowexposures delta s withrespecttotheseriskfactors Thisisthecorrectmappingoftheportfolio Copyright 1996 2005byNeilD Pearson Allrightsreserved Equityportfolio justtoremindyou 110millioninwell diversifiedportfoliooflargecap U S equities perfectlycorrelatedwithS P500500FT SE100indexfuturescontractsShort200S P500contracts600writtenFT SE100indexcalloptionswithstrikelevelof5 875800writtenS P500indexcallswithstrikelevelof1 100Netvalue 110million valueofwrittenoptions USD101 485 220 Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingU S equitypositions ThemappingoftheU S equitypositionsisnotaffectedbytheexchangerateriskPortfoliodeltawithrespecttoS P500isstill4 864Thisisjustwhatwehadbefore Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingtheFT SEoptionsposition USDvalueis Wehave Thisresultisintuitive GBPoptionDtoUSD Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingtheFT SEoptionsposition FortheexchangeratedeltaThisresultisalsointuitive exchangerateexposureisjusttheinvestmentintheposition Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingtheFT SE100futuresposition Forthedelta softhefuturesposition weneedandTocomputetheseweneedanexpressionfortheresettlementpayments Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingtheFT SE100futuresposition TheUSDvalueoftheresettlementpaymentsis Contracts multiplier Initialfuturesprice GBP Newfuturesprice GBP Exchangerate Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingtheFT SE100futuresposition Thefirstdelta withrespecttoS2 Toseewherethiscamefrom doesn tdependonS2 PartthatdependsonS2islike5 000timesS2 multipliedbyfuturevaluefactorexp convertedtoUSD Thisisreallyjustanequityposition convertedtoUSD Evaluatederivativeate 1 6271 Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingtheFT SE100futuresposition Theseconddelta withrespecttoe Toseewherethiscamefrom Ifderivativesareevaluatedatinitialvalues rhsiszero Copyright 1996 2005byNeilD Pearson Allrightsreserved MappingtheFT SE100futuresposition Copyingfrompreviousslide Ifderivativesareevaluatedatinitialvalues right handsideiszeroIntuitively thereisnoinvestmentinthefutures sotherearenoGBPexposedtoexchangeraterisk Copyright 1996 2005byNeilD Pearson Allrightsreserved Delta s Puttingthisalltogether thedelta sareTheinterpretationis Copyright 1996 2005byNeilD Pearson Allrightsreserved Delta s Inpercentagetermsor substitutingvaluesforvariable