金融风险Duration model（PPT 38）

1、IntroductionCalculatingdurationInterpretingdurationUnderstandingmorefeaturesofdurationUsingdurationforinterestrateriskimmunizationLimitationsofdurationimmunizationFirstdevelopedin1938byFrederickMacaulayTakingintoaccountbothleverageandtimingofcashflowofassetsandliabilitiesMoreaccuracyininterestratemeasurementBetterforinterestrateriskimmunizationRegulatoryrequirementMaturitymodeltriestotakeadvantageofthematurityeffectonbondvalueanduseitsmaturityasanindicatorofitsinterestratesensitivit。

2、y.Butstrictlyspeaking,itisagoodcaseonlywhenthebondgeneratesnocoupon,i.e.,itisazerocouponbond.Couponeffectisignoredinmaturitymodel.Morebondspaycoupons,andcouponeffectmustbetakenintoaccount.Bondswithidenticalmaturitiesbutdifferentcouponpaymentsrespondsdifferentlytointerestratechanges.Couponeffectdoesexist.Withhighercoupons,moreofthebond’svalueisgeneratedbycashflowswhichtakeplacesoonerintime.Consequently,lesssensitivetochangesinR.So,maturitycannotservewellasanaccuratemeasureofinterestratesens。

3、itivityofcouponbonds.(InterestRateSensitivityof6%CouponBond)MarketI.R.8%6%4%PriceRangeM40$802$1,000$1,273$47120$864$1,000$1,163$29910$919$1,000$1,089$170(InterestRateSensitivityof8%CouponBond)MarketI.R.10%8%6%PriceRangeM40$828$1,000$1,231$40320$875$1,000$1,149$27410$923$1,000$1,085$162Thelongermaturitybondsexperiencegreaterpricechangesinresponsetoanychangeinthediscountrate.(Maturityeffect)Therangeofpricesisgreaterwhenthecouponislower.(Couponeffect)•The6%bondshowsgreaterchangesinpriceinresponse。

4、toa2%changethanthe8%bond.Thefirstbondishasgreaterinterestraterisk.Whatdoesmaturityofabondexactlymeaninacouponbondcase?Anambiguousterm!Inthecaseofcouponbonds,thematurityofabondisnotthematurityofallthecashflowsgeneratedinthebond,onlythatofthelastpayment,thelastcouponplusfacevalue,whileothercashflowshaveshorter“maturities”.t:0123412yearsCF:(—$931)$40$40$40$10400.51.5MaturitiesofCFs=Σt×wtt=1CFt×DFtCFt/(1+R)tPVtwt===PPPNNP=ΣPVt=ΣCFt×DFt;DFt=1/(1+R)tt=1t=1Duration:Theaveragelifeofabond,ormoretechnic。

5、ally,theweighted-averagetimetomaturityofallcashflows,usingrelativepresentvaluesofcashflowsasweights.Forbetterandeasierunderstanding,takeacouponBondasabundleoraportfolioof“zero-coupon”bonds.====NΣCFt×DFtt=1NΣt×CFt×DFtt=1NΣPVtt=1NΣPVt×tt=1NΣPVt×tt=1PNΣ(PVt/P)×tt=1Considera2-year,8%couponbond,withafacevalueof$1,000andyield-to-maturityof12%.Couponsarepaidsemi-annually.Therefore,eachcouponpaymentis$40andtheperperiodYTMis(1/2)×12%=6%.PresentvalueofeachcashflowequalsCFt÷(1+0.06)twheretistheperiodnum。

6、ber.t:012341year2yearsCF:(—$931)$40$40$40$1040:Facevalue=$1,000,YTM=12%tyearsCFtPV(CFt)Weight(w)w×years10.54037.7360.0410.02021.04035.6000.0380.03831.54033.5850.0360.05442.01,040823.7770.8851.770P=930.6981.000D=1.883(years):(—$931)$40$40$40$1040PV:37.735.633.6823.8∑=$931Weight:0.0410.0380.0360.885∑=1W×time:0.0200.0380.0541.770∑=1.883ys(Duration)Forazerocouponbond,duration=maturitysince100%ofitspresentvalueisgeneratedbythepaymentofthefacevalue,atmaturity.Forallotherbonds:durationmaturitySo,mat。

7、uritymodelcanserveasaspecialcaseofdurationmodelonlywheninthecaseofzerocouponbonds.(Perpetuities)Consolbondisabondthatpaysafixedcouponeachyearforever.ConsolbondsthatwereissuedbytheBritishgovernmentinthe1890stofinancetheBoerWarsinSouthAfricaarestilloutstanding.Mc=∞(Infinite)Dc=1+1/R(Finite)Prooftobeconductedbythosewhoknowcalculus.Example:ata10%yield,thedurationofaperpetuitythatpays$100onceayearforeverwillequal1.10/0.10=11years,butatan8%yielditwillequal1.08/0.08=13.5yearsAtoolofinterestrater。

8、iskmanagement(measurement)forfixedincomeportfolioMeasurethesensitivityofaportfoliotointerestratechangeItisasimplesummarystatisticoftheeffectiveaveragematurityoftheportfolioItisalsothefirstorderderivativeofthebondpricewithrespecttointerestrateDurationGapandIRriskimmunizationstrategyTheambiguityof“maturity”:lifeofthecontract,lifeofthelastcashflowTheweight:marketvalueTheweightedaverage,lifeofallthecashflowsinvolvedinthecontractTimeconceptworkingasasensitivitymeasure:thelongertimethecashflow。

9、sareexposedininterestraterisk(Thelargerthedurationis),themoresensitivethecontract(bond)istointerestratechangeForbet。