衍生工具与风险管理 Don M. Chance 12

Chapter12:FinancialRiskManagement:TechniquesandApplicationsRiskmanagementisakintoadialysismachine.Ititdoesn’twork,youmighthaveanobleobituary,butyou’redead.BenGolubDerivativesStrategy,October,2000,p.26

D.M.ChanceCh.15:1AnIntroductiontoDerivativesandRiskManagement,6thed.ImportantConceptsinChapter15TheconceptandpracticeofriskmanagementThebenefitsofriskmanagementThedifferencebetweenmarketandcreditriskHowmarketriskismanagedusingdelta,gamma,vega,andValue-at-RiskHowcreditriskismanaged,includingcreditderivativesRisksotherthanmarketandcreditriskD.M.Chance2AnIntroductiontoDerivativesandRiskManagement,6thed.Definitionofriskmanagement:thepracticeofdefiningtherisklevelafirmdesires,identifyingtherisklevelitcurrentlyhas,andusingderivativesorotherfinancialinstrumentstoadjusttheactualriskleveltothedesiredrisklevel.D.M.Chance3AnIntroductiontoDerivativesandRiskManagement,6thed.WhyPracticeRiskManagement?TheImpetusforRiskManagementFirmspracticeriskmanagementforseveralreasons:Interestrates,exchangeratesandstockpricesaremorevolatiletodaythaninthepast.SignificantlossesincurredbyfirmsthatdidnotpracticeriskmanagementImprovementsininformationtechnologyFavorableregulatoryenvironmentSometimeswecallthisactivityfinancialriskmanagement.D.M.Chance4AnIntroductiontoDerivativesandRiskManagement,6thed.WhyPracticeRiskManagement?(continued)TheBenefitsofRiskManagementWhatarethebenefitsofriskmanagement,inlightoftheModigliani-Millerprinciplethatcorporatefinancialdecisionsprovidenovaluebecauseshareholderscanexecutethesetransactionsthemselves?Firmscanpracticeriskmanagementmoreeffectively.Theremaytaxadvantagesfromtheprogressivetaxsystem.Riskmanagementreducesbankruptcycosts.Managersaretryingtoreducetheirownrisk.D.M.Chance5AnIntroductiontoDerivativesandRiskManagement,6thed.WhyPracticeRiskManagement?(continued)TheBenefitsofRiskManagement(continued)Byprotectingafirm’scashflow,itincreasesthelikelihoodthatthefirmwillgenerateenoughcashtoallowittoengageinprofitableinvestments.Somefirmsuseriskmanagementasanexcusetospeculate.Somefirmsbelievethattherearearbitrageopportunitiesinthefinancialmarkets.Note:Thedesiretolowerriskisnotasufficientreasontopracticeriskmanagement.D.M.Chance6AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRiskMarketrisk:theuncertaintyassociatedwithinterestrates,foreignexchangerates,stockprices,orcommodityprices.Example:Adealerwiththefollowingpositions:Afour-yearinterestrateswapwith$10millionnotionalprincipalinwhichitpaysafixedrateandreceivesafloatingrate.A3-yearinterestratecallwith$8millionnotionalprincipal.Thedealerisshortandtheexerciserateis12%.SeeTable15.1,p.546forcurrenttermstructureandforwardrates.Weobtainthecallpriceas$73,745andtheswaprateis11.85%.D.M.Chance7AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)DeltaHedgingWeestimatethedeltabyrepricingtheswapandoptionwithaonebasispointmoveinallspotratesandaveragethepricechange.SeeTable15.2,p.547forestimatedswapandoptiondeltas.Wearelongtheswapsowehaveadeltaof$2,130.5,roundto$2,131.Weareshorttheoptionsowehaveadeltaof-$244.Ouroveralldeltais$1,887.D.M.Chance8AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)DeltaHedging(continued)WeneedaEurodollarfuturespositionthatgains$1,887ifratesmovedownandlosesthatamountifratesmoveup.Thus,werequirealongpositionof$1,887/$25=75.48contracts.Roundto75.Overalldelta:$2,131(fromswap)-$244(fromoption)75(-$25)(fromfutures)=$12(overall)D.M.Chance9AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)GammaHedgingHerewedealwiththeriskoflargepricemoves,whicharenotfullycapturedbythedelta.SeeTable15.3,p.549fortheestimationofswapandoptiongammas.Swapgammais-$12,500,andoptiongammais$5,000.Beingshorttheoption,thetotalgammais-$17,500.Eurodollarfutureshavezerogammasowemustaddanotheroptionpositiontooffsetthegamma.Weassumetheavailabilityofaone-yearcallwithdeltaof$43andgammaof$2,500.D.M.Chance10AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)GammaHedging(continued)Weusex1Eurodollarfuturesandx2oftheone-yearcalls.Theswapandoptionhaveadeltaof$1,887andgammaof-$17,500.Wesolvethefollowingequations:$1,887+x1(-$25)+x2($43)=$0(zerodelta)-$17,500+x1($0)+x2($2,500)=$0(zerogamma)Solvingthesegivesx1=87.52(golong88Eurodollarfutures)andx2=7(golong7times$1,000,000notionalprincipalofone-yearoption)D.M.Chance11AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)VegaHedgingSwaps,futures,andFRAsdonothavevegas.WeestimatethevegasoftheoptionsOnour3-yearoption,ifvolatilityincreases(decreases)by.01,optionwillincrease(decrease)by$42(-$42).Averageis$42.Weareshortthisoption,sovega=-$42.One-yearoptionhasestimatedvegaof$3.50.Overallportfoliohasvegaof($3.50)(7million)-$42=-$17.50.D.M.Chance12AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)VegaHedging(continued)WeaddaEurodollarfuturesoption,whichhasdeltaof-$12.75,gammaof-$500,andvegaof$2.50per$1MM.Solvethefollowingequations$1,887+x1(-$25)+x2($43)+x3(-$12.75)=0(delta)-$17,500+x1($0)+x2($2,500)+x3(-$500)=0(gamma)-$42+x1($0)+x2($3.50)+x3($2.50)=0(vega)Thecoefficientsarethemultiplesof$1,000,000notionalprincipalweneed.Solutionsarex1=86.61,x2=8.09375,x3=5.46875.D.M.Chance13AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)VegaHedging(continued)Anytypeofhedge(delta,delta-gamma,ordelta-gamma-vega)mustbeperiodicallyadjusted.Virtuallyimpossibletohaveaperfecthedge.D.M.Chance14AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)Adollarmeasureoftheminimumlossthatwouldbeexpectedoveragiventimewithagivenprobability.Example:VARof$1millionforonedayat.05meansthatthefirmcouldexpecttoloseatleast$1millionoveraonedayperiod5%ofthetime.Widelyusedbydealersandincreasinglybyendusers.SeeTable15.4,p.553forexampleofdiscreteprobabilitydistributionofchangeinvalue.VARat5%is$3millionloss.See

Figure15.1,p.554forcontinuousdistribution.D.M.Chance15AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)VARcalculationsrequireuseofformulasforexpectedreturnandstandarddeviationofaportfolio:whereE(R1),E(R2)=expectedreturnsofassets1and21,2=standarddeviationsofassets1and2=correlationbetweenassets1and2w1,w2=%ofone’swealthinvestedinasset1or2D.M.Chance16AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)ThreemethodsofestimatingVARAnalyticalmethod:Usesknowledgeoftheparameters(expectedreturnandstandarddeviation)oftheprobabilitydistributionandassumesanormaldistribution.Example:$20millionofS&P500withexpectedreturnof.12andvolatilityof.15and$12millionofNikkei300withexpectedreturnof.105andvolatilityof.18.Correlationis.55.Usingtheaboveformulas,theoverallportfolioexpectedreturnis.1144andvolatilityis.1425.D.M.Chance17AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)ForaweeklyVAR,convertthesetoweeklyfigures.Expectedreturn=.1144/52=.0022Volatility=.1425/Ö52=.0198.Withanormaldistribution,wehaveVAR=.0022-1.65(.0198)=-.0305SotheVARis$32,000,000(.0305)=$976,000.D.M.Chance18AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)Exampleusingoptions:200short12-monthcallsonS&P500,whichhasvolatilityof.15andpriceof$14.21.Totalvalueof$1,421,000.Basedonmonthlydata,expectedreturnis.0095andvolatilityis.0412.Upside5%is.0095+1.65(.0412)=.0775,whichis720(1.0775)=775.80.Optionwouldbeworth775.80-720=55.80solossis55.80-14.21=41.59peroption.Totalloss=200(500)(41.59)=$4.159million.ThisistheVAR.D.M.Chance19AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)Oneassumptionoftenmadeisthattheexpectedreturniszero.Thisisnotlikelytobetrue.Sometimesratherthanusethepreciseoptionpricefromamodel,adeltaisusedtoestimatetheprice.Thismakestheanalyticalmethodbesometimescalledthedelta-normalmethod.Volatilityandcorrelationinformationisnecessary.Seethewebsite,wheredataofthissortareprovidedfree.D.M.Chance20AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)Historicalmethod:Useshistoricalinformationontheuser’sportfoliotoobtainthedistribution.Example:SeeFigure15.2,p.557.Forportfolioof$15million,VARat5%isapproximatelyalossof10%or$15,000,000(.10)=-$1,500,000.Historicalmethodissubjecttolimitationthatthepastholdingsoftheportfoliomaynothavethesamedistributionalpropertiesasthefutureholdings.Italsoislimitedbytheresultsofthechosentimeperiod,whichmightnotberepresentativeofthefuture.D.M.Chance21AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)MonteCarloSimulationmethod:UsesMonteCarlomethod,asdescribedinAppendix14.B,togeneraterandomoutcomesontheportfolio’scomponents.D.M.Chance22AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)AComprehensiveCalculationofVARWedoanexampleofaportfolioof$25millionintheS&P500.Wewanta5%1-dayVARusingeachmethod.WecollectasampleofdailyreturnsontheS&P500forthepastyearandobtainthefollowingparameterestimates:Averagedailyreturn=.0457%anddailystandarddeviation=1.3327%.TheseresultinannualfiguresofD.M.Chance23AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)AComprehensiveCalculationofVAR(continued)Analyticalmethod:Wehave0.0457%-(1.65)1.3327%=-2.1533%.SotheVARis.021533($25,000,000)=$538,325The.21standarddeviationishistoricallyabithigh.Re-estimatingwithastandarddeviationof.15givesusadailystandarddeviationof0.9430.Thenweobtain0.0474%-1.65(0.9430)=-1.5086%andaVARof.015086($25,000,000)=$377,150Areourdatanormallydistributed?ObserveFigure15.3,p.559.D.M.Chance24AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)AComprehensiveCalculationofVAR(continued)Historicalmethod:Herewerankthereturnsfromworsttobest.For253returnsweobtainthe5%worstbyobservingthe.05(253)=12.65worstreturn.Weshallmakeitthe13thworst.Thiswouldbe-2.0969%.Thus,theVARis.020969($25,000,000)=$524,225D.M.Chance25AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)AComprehensiveCalculationofVAR(continued)MonteCarlosimulationmethod:Weshalluseanexpectedreturnof12%,astandarddeviationof15%andanormaldistribution.Wegenerate253randomreturns(thisnumberisarbitraryandshouldactuallybemuchlarger)bythefollowingmethod:whereisastandardnormalrandomnumber.D.M.Chance26AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)AComprehensiveCalculationofVAR(continued)MonteCarlosimulationmethod(continued):Wedothis253times,rankthereturnsfromworsttobestandobtainthe13thworstreturn,whichis-1.3942%.ThentheVARis.013942($25,000,000)=$348,550D.M.Chance27AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)AComprehensiveCalculationofVAR(continued)SoVARisestimatedtobeeither$538,325$377,150$524,225$348,550Keyconsiderations:widerangessuchasthisarecommon,real-worldportfoliosaremorecomplicatedthanthis,expostevaluationshouldbedoneD.M.Chance28AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)BenefitsandCriticismsofVARWidelyusedFacilitatescommunicationwithseniormanagementAcceptableinbankingregulationUsedtoallocatecapitalwithinfirmsUsedinperformanceevaluationD.M.Chance29AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingMarketRisk(continued)Value-at-Risk(VAR)(continued)ExtensionsofVARStresstestingConditionalVAR:expectedloss,giventhatlossexceedsVAR.InS&Pexample,thisis$719,450.CashFlowatRisk(CARorCFAR),whichismoreappropriateforfirmswithassetsthatgeneratecashbutforwhichamarketvaluecannoteasilybeassigned.Example:expectedcashflowof$10millionwithstandarddeviationof$2million.CARat.05wouldbe1.65($2million)=$3.3millionbecauseProb[$10million-$3.3million>ActualCashFlow]=.05.Note:CARisashortfallmeasure.EarningsatRisk(EAR):measuresearningsshortfall.D.M.Chance30AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingCreditRiskCreditriskordefaultriskistheriskthatthecounterpartywillnotpayoffinafinancialtransaction.Creditratingsarewidelyusedtoassesscreditrisk.D.M.Chance31AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingCreditRisk(continued)OptionPricingTheoryandCreditRiskHereweuseoptionpricingtheorytounderstandthenatureofcreditrisk.ConsiderafirmwithassetswithmarketvalueA0,anddebtwithfacevalueofFdueatT.ThemarketvalueofthedebtisB0.ThemarketvalueofthestockisS0.SeeTable15.5,p.563forthepayoffstothesuppliersofcapital.Notehowthestockislikeacalloptionontheassets.ItspayoffisMax(0,AT–F)Usingput-callparity,wehaveP0=S0–A0+F(1+r)-TD.M.Chance32AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingCreditRisk(continued)OptionPricingTheoryandCreditRisk(continued)Thisputisthevalueoflimitedliabilitytothestockholders.Rearranging,weobtainS0=A0+P0–F(1+r)-T.But,bydefinition,S0=A0–B0.Settingtheseequal,weobtainB0=F(1+r)-T–P0.Thus,thebondsubjecttodefaultriskisequivalenttoarisk-freebondandaputoptionwrittenbythebondholderstothestockholders.TheBlack-ScholesformulaadaptedtopricethestockasacallwouldbeUsingB0=A0–S0,wecanobtainthevalueofthebonds.D.M.Chance33AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingCreditRisk(continued)TheCreditRiskofDerivativesCurrentcreditriskistherisktoonepartythattheotherwillbeunabletomakepaymentsthatarecurrentlydue.Potentialcreditriskistherisktoonepartythatthecounterpartywilldefaultinthefuture.Inoptions,onlythebuyerfacescreditrisk.FRAsandswapshavetwo-waycreditriskbutatagivenpointintime,theriskisfacedbyonlyoneofthetwoparties.D.M.Chance34AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingCreditRisk(continued)TheCreditRiskofDerivatives(continued)Potentialcreditriskislargestduringthemiddleofaninterestrateswap’slifebutduetoprincipalrepayment,potentialcreditriskislargestduringthelatterpartofacurrencyswap’slife.Typicallyallpartiespaythesamepriceonaderivative,regardlessoftheircreditstanding.Creditriskismanagedthroughlimitingexposuretoanyoneparty(primarymethod)collateralperiodicmarking-to-market(bydealers)captivederivativessubsidiariesnetting(seenext)D.M.Chance35AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingCreditRisk(continued)NettingNetting:severalsimilarprocessesinwhichtheamountofcashowedbyonepartytotheotherisreducedbytheamountowedbythelattertotheformer.Bilateralnetting:nettingbetweentwoparties.Multilateralnetting:nettingbetweenmorethantwoparties;essentiallythesameasaclearinghouse.Paymentnetting:Onlythenetamountofapaymentowedfromonepartytotheotherispaid.Cross-productnetting:paymentsfromonetypeoftransactionarenettedagainstpaymentsforanothertypeoftransaction.D.M.Chance36AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingCreditRisk(continued)Netting(continued)Nettingbynovation:netvalueoftwoparties’mutualobligationsisreplacedbyasingletransaction;oftenusedinFXmarkets.Closeoutnetting:nettingintheeventofdefault,wherealltransactionsbetweentwopartiesarenettedagainsteachother;seeexampleintext.TheOTCderivativesmarkethasanexcellentrecordofdefault.NotetheHammersmithandFulhamdefaultwhereitwasfoundthatatownhadnolegalauthoritytoengageinswaps.Thetownwasabletogetoutofpayingitslosses.D.M.Chance37AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingCreditRisk(continued)CreditDerivatives:Afamilyofderivativeinstrumentsthathavepayoffscontingentonthecreditqualityofaparticularparty.TypesincludeTotalreturnswaps:SeeFigure15.4,p.570.Creditderivativebuyerpurchasesswapfromcreditderivativesellerinwhichitpaysthetotalreturnonaspecificbond.Ifthatreturnisreducedbysomecreditevent,thislossispassedthroughautomaticallyintheswap.D.M.Chance38AnIntroductiontoDerivativesandRiskManagement,6thed.ManagingCreditRisk(continued)CreditDerivatives(continued)Creditswap:Aswapinwhichthecreditderivativesbuyerpaysaperiodicfeetothecred