金融机构管理Chap009

9-1Overview

Thischapterdiscussesamarketvalue-basedmodelforassessingandmanaginginterestraterisk:DurationComputationofdurationEconomicinterpretationImmunizationusingduration*Problemsinapplyingduration9-2PriceSensitivityandMaturityIngeneral,thelongerthetermtomaturity,thegreaterthesensitivitytointerestratechangesExample:Supposethezerocouponyieldcurveisflatat12%.BondApays$1790.85infiveyears.BondBpays$3207.14intenyears,andbotharecurrentlypricedat$1000.9-3Examplecontinued...BondA:P=$1000=$1790.85/(1.06)10BondB:P=$1000=$3207.14/(1.06)20Nowsupposetheannualinterestrateincreasesby1%(0.5%semiannually).BondA:P=$1762.34/(1.065)10=$954.03BondB:P=$3105.84/(1.065)20=$910.18Thelongermaturitybondhasthegreaterdropinpricebecausethepaymentisdiscountedagreaternumberoftimes.9-4CouponEffectBondswithidenticalmaturitieswillresponddifferentlytointerestratechangeswhenthecouponsdiffer.Thisiseasilyunderstoodbyrecognizingthatcouponbondsconsistofabundleof“zero-coupon”bonds.Withhighercoupons,moreofthebond’svalueisgeneratedbycashflowswhichtakeplacesoonerintime.Consequently,itislesssensitivetochangesinR.9-5RemarksonPrecedingSlidesIngeneral,longermaturitybondsexperiencegreaterpricechangesinresponsetoanychangeinthediscountrateTherangeofpricesisgreaterwhenthecouponislowerA6%bondwillhavealargerchangeinpriceinresponsetoa2%changethanan8%bondThe6%bondhasgreaterinterestraterisk9-6ExtremeExamplesWithEqualMaturitiesConsidertwoten-yearmaturityinstruments:Aten-yearzerocouponbondAtwo-cashflow“bond”thatpays$999.99almostimmediatelyandonepennytenyearshenceSmallchangesinyieldwillhavealargeeffectonthevalueofthezerobutalmostnoimpactonthehypotheticalbondMostbondsarebetweentheseextremesThehigherthecouponrate,themoresimilarthebondistoourhypotheticalbondwithhighervalueofcashflowsarrivingsooner9-7DurationDurationWeightedaveragetimetomaturityusingtherelativepresentvaluesofthecashflowsasweightsCombinestheeffectsofdifferencesincouponratesanddifferencesinmaturityBasedonelasticityofbondpricewithrespecttointerestrateTheunitsofdurationareyears9-8MacaulayDurationMacaulayDurationWhere D=Macaulayduration(inyears) t=numberofperiodsinthefuture

CFt=cashflowtobedeliveredintperiods N=time-to-maturity

DFt=discountfactor9-9DurationSincetheprice(P)ofthebondequalsthesumofthepresentvaluesofallitscashflows,wecanstatethedurationformulaanotherway:

Noticetheweightscorrespondtotherelativepresentvaluesofthecashflows9-10SemiannualCashFlowsItisimportanttoseethatwemustexpresstinyears,andthepresentvaluesarecomputedusingtheappropriateperiodic

interestrate.Forsemiannualcashflows,Macaulayduration,Disequalto:9-11DurationofZero-couponBondForazero-couponbond,Macaulaydurationequalsmaturitysince100%ofitspresentvalueisgeneratedbythepaymentofthefacevalue,atmaturityForallotherbonds,duration<maturity9-12ComputingdurationConsidera2-year,8%couponbond,withafacevalueof$1,000andyield-to-maturityof12%Couponsarepaidsemi-annuallyTherefore,eachcouponpaymentis$40andtheperperiodYTMis(1/2)×12%=6%PresentvalueofeachcashflowequalsCFt÷(1+0.06)twheretistheperiodnumber9-13

Durationof2-year,8%bond:

Facevalue=$1,000,YTM=12%9-14SpecialCaseMaturityofaconsol:M=.Durationofaconsol:D=1+1/R9-15FeaturesofDurationDurationandmaturityDincreaseswithM,butatadecreasingrateDurationandyield-to-maturityDdecreasesasyieldincreasesDurationandcouponinterestDdecreasesascouponincreases9-16EconomicInterpretationDurationisameasureofinterestratesensitivityorelasticityofaliabilityorasset: [ΔP/P][ΔR/(1+R)]=-DOrequivalently,ΔP/P=-D[ΔR/(1+R)]=-MD×ΔRwhereMDismodifiedduration9-17EconomicInterpretationToestimatethechangeinprice,wecanrewritethisas:

ΔP=-D[ΔR/(1+R)]P=-(MD)×(ΔR)×(P)NotethedirectlinearrelationshipbetweenΔPand-D9-18DollarDurationDollardurationequalsmodifieddurationtimespriceDollarduration=MD×PriceUsingdollarduration,wecancomputethechangeinpriceas ΔP=-Dollarduration×ΔR

9-19Semi-annualCouponPaymentsWithsemi-annualcouponpaymentsthepercentagechangeinpriceis

ΔP/P=-D[ΔR/(1+(R/2)]9-20ImmunizationMatchingthematurityofanassetinvestmentwithafuturepayoutresponsibilitydoesnotnecessarilyeliminateinterestrateriskMatchingdurationswillimmunizeagainstchangesininterestrates9-21AnExampleConsiderthreeloanplans,allofwhichhavematuritiesof2years.Theloanamountis$1,000andthecurrentinterestrateis3%.Loan#1isatwo-paymentloanwithtwoequalpaymentsof$522.61each.Loan#2isstructuredasa3%annualcouponbond.Loan#3isadiscountloan,whichhasasinglepaymentof$1,060.90.9-22DurationasIndexofInterestRateRisk9-23BalanceSheetImmunizationDurationisameasureoftheinterestrateriskexposureforanFIIfthedurationsofliabilitiesandassetsarenotmatched,thenthereisariskthatadversechangesintheinterestratewillincreasethepresentvalueoftheliabilitiesmorethanthepresentvalueofassetsisincreased9-24DurationGapSupposethata2-yearcouponbondistheonlyloanasset(A)ofanFI.A2-yearcertificateofdepositistheonlyliability(L).Ifthedurationofthecouponbondis1.8years,then: Maturitygap:MA-ML=2-2=0,but DurationGap:DA-DL=1.8-2.0=-0.2Deposithasgreaterinterestratesensitivitythanthebond,soDGAPisnegativeFIexposedtorisinginterestrates9-25ImmunizingtheBalanceSheetofanFI

DurationGap:Fromthebalancesheet,E=A-L.Therefore,DE=DA-DL.Inthesamemannerusedtodeterminethechangeinbondprices,wecanfindthechangeinvalueofequityusingduration.

DE=[-DAA+DLL]DR/(1+R)or

DE=-[DA-DLk]A(DR/(1+R))9-26DurationandImmunizingTheformulashows3effects:LeverageadjustedD-GapThesizeoftheFIThesizeoftheinterestrateshock9-27AnExampleSupposeDA=5years,DL=3yearsandratesareexpectedtorisefrom10%to11%.(Rateschangeby1%).Also,A=100,L=90andE=10.FindchangeinE.

DE=-[DA-DLk]A[DR/(1+R)]=-[5-3(90/100)]100[.01/1.1]=-$2.09.Methodsofimmunizingbalancesheet.AdjustDA,DLork.9-28ImmunizationandRegulatoryConcerns

RegulatorssettargetratiosforanFI’scapital(networth):Capital(Networth)ratio=E/AIftargetistoset(E/A)=0:DA=DLBut,tosetE=0:DA=kDL9-29LimitationsofDurationImmunizingtheentirebalancesheetneednotbecostlyDurationcanbeemployedincombinationwithhedgepositionstoimmunizeImmunizationisadynamicprocesssincedurationdependsoninstantaneousRLargeinterestratechangeeffectsnotaccuratelycapturedConvexityMorecomplexifnonparallelshiftinyieldcurve9-30ConvexityThedurationmeasureisalinearapproximationofanon-linearfunction.IftherearelargechangesinR,theapproximationismuchlessaccurate.Allfixed-incomesecuritiesareconvex.Convexityisdesirable,butgreaterconvexitycauseslargererrorsintheduration-basedestimateofpricechanges.9-31*ConvexityThosewhoarefamiliarwithcalculusmayrecognizethatdurationinvolvesonlythefirstderivativeofthepricefunction.WecanimproveontheestimateusingaTaylorexpansion.Inpractice,theexpansionrarelygoesbeyondsecondorder(usingthesecondderivative).Thissecondorderexpansionistheconvexityadjustment.9-32*ModifiedDuration&Convexity

DP/P=-D[DR/(1+R)]+(1/2)CX(DR)2or DP/P=-MDDR+(1/2)CX(DR)2

WhereMDimpliesmodifieddurationandCXisameasureofthecurvatureeffect.CX=Scalingfactor×[capitall