金融风险Durationmodel课件

17四月2024金融风险DurationmodelIntroductionFirstdevelopedin1938byFrederickMacaulayTakingintoaccountbothleverageandtimingofcashflowofassetsandliabilitiesMoreaccuracyininterestratemeasurementBetterforinterestrateriskimmunizationRegulatoryrequirement2ReflectionontheShortcomingofMaturitymodel:IgnoringcouponeffectMaturitymodeltriestotakeadvantageofthematurityeffectonbondvalueanduseitsmaturityasanindicatorofitsinterestratesensitivity.Butstrictlyspeaking,itisagoodcaseonlywhenthebondgeneratesnocoupon,i.e.,itisazerocouponbond.Couponeffectisignoredinmaturitymodel.Morebondspaycoupons,andcouponeffectmustbetakenintoaccount.3ShortcomingofMaturitymodel:

IgnoringcouponeffectBondswithidenticalmaturitiesbutdifferentcouponpaymentsrespondsdifferentlytointerestratechanges.Couponeffectdoesexist.Withhighercoupons,moreofthebond’svalueisgeneratedbycashflowswhichtakeplacesoonerintime.Consequently,lesssensitivetochangesinR.So,maturitycannotservewellasanaccuratemeasureofinterestratesensitivityofcouponbonds.4Maturityeffectvs.Couponeffectonbondvalue

(InterestRateSensitivityof6%CouponBond)5Maturityeffectvs.Couponeffectonbondvalue

(InterestRateSensitivityof8%CouponBond)6RemarksonPrecedingSlidesThelongermaturitybondsexperiencegreaterpricechangesinresponsetoanychangeinthediscountrate.(Maturityeffect)Therangeofpricesisgreaterwhenthecouponislower.(Couponeffect)The6%bondshowsgreaterchangesinpriceinresponsetoa2%changethanthe8%bond.Thefirstbondishasgreaterinterestraterisk.7Exploringtheambiguityofmaturity

Whatdoesmaturityofabondexactlymeaninacouponbondcase?Anambiguousterm!Inthecaseofcouponbonds,thematurityofabondisnotthematurityofallthecashflowsgeneratedinthebond,onlythatofthelastpayment,thelastcouponplusfacevalue,whileothercashflowshaveshorter“maturities”.t:0123412years

CF:(—$931)$40$40$40$10400.5

1.5MaturitiesofCFs8DefinitionandCalculationofduration

N

D=Σt×wt

t=1

CFt×DFtCFt/(1+R)tPVt

wt===

P

P

P

NN

P

=ΣPVt=ΣCFt×DFt;DFt=1/(1+R)t

t=1t=1Duration:Theaveragelifeofabond,ormoretechnically,theweighted-averagetimetomaturityofallcashflows,usingrelativepresentvaluesofcashflowsasweights.

Forbetterandeasierunderstanding,takeacouponBondasabundleoraportfolioof“zero-coupon”bonds.9Theintegratedformula

D====NΣCFt×DFt

t=1NΣt×CFt×DFt

t=1NΣPVtt=1NΣPVt×

tt=1NΣPVt×

tt=1PNΣ(PVt/

P)×

tt=110ComputingdurationConsidera2-year,8%couponbond,withafacevalueof$1,000andyield-to-maturityof12%.Couponsarepaidsemi-annually.Therefore,eachcouponpaymentis$40andtheperperiodYTMis(1/2)×12%=6%.PresentvalueofeachcashflowequalsCFt÷(1+0.06)twheretistheperiodnumber.t:012341year2years

CF:(—$931)$40$40$40$104011Durationof2-year,8%bond:

Facevalue=$1,000,YTM=12%12

Time:00.511.52ys

CF:(—$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)13DurationofZero-couponBondForazerocouponbond,duration=maturitysince100%ofitspresentvalueisgeneratedbythepaymentofthefacevalue,atmaturity.Forallotherbonds:duration<maturitySo,maturitymodelcanserveasaspecialcaseofdurationmodelonlywheninthecaseofzerocouponbonds.14Durationofaconsolbond(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=13.5years15InterpretingdurationAtoolofinterestrateriskmanagement(measurement)forfixedincomeportfolioMeasurethesensitivityofaportfoliotointerestratechangeItisasimplesummarystatisticoftheeffectiveaveragematurityoftheportfolioItisalsothefirstorderderivativeofthebondpricewithrespecttointerestrateDurationGapandIRriskimmunizationstrategy16Effectiveaveragematurity

--InterpretdurationasatimeconceptTheambiguityof“maturity”:lifeofthecontract,lifeofthelastcashflowTheweight:marketvalueTheweightedaverage,lifeofallthecashflowsinvolvedinthecontractTimeconceptworkingasasensitivitymeasure:thelongertimethecashflowsareexposedininterestraterisk(Thelargerthedurationis),themoresensitivethecontract(bond)istointerestratechangeForbetterandeasierunderstanding,takeacouponBondasabundleoraportfolioofzero-couponbonds.17Thefirstorderderivative

--Interpretdurationasriskmeasureconcept

(Interestsensitivity,orelasticity)TCFt

P

=Σ

t=1(1+R)tDp

Tt·CFtDP

=－Σ=－dRt=1(1+R)t+11+R

dP

/P

=－DdR/(1+R)

InterestElasticity,orsensitivityofabondisdefinedasthepercentagechangeinthepriceofabondforanygivenchangeininterestrates.18ModifiedDurationdP/P

dPdRD=－=－DdR/(1+R)P

1+RLetMD=D/(1+R),whereMDismodifiedduration.Then:

dP/P=-D[dR/(1+R)]=-MD×dR

(inpercentage)Toestimatethechangeinprice,rewritethisas: dP=-D[dR/(1+R)]P=-(MD)×(dR)×(P)(indollarvalue)NotethedirectlinearrelationshipbetweendPand-D.19AnexampleConsidera6-yearEurobond(facevalue$1000andcouponpaidannually)withan8%couponand8%yield.ItsdurationswasD=4.993years.(SeeTextbookPage197)Supposethatyieldsweretorisebyonebpfrom8%to8.01%,howmuchwilltheEurobondloseitsvalue?dP/P=-D[dR/(1+R)]dP=-D[dR/(1+R)]P=-(4.993)×[0.0001/1.08]×1000=-0.462So,thedurationmodelpredictsthatthepriceofthebondwouldfallto$999.538aftertheincreaseinyieldby1bp.IfMDisgiven,thecalculationiseveneasier.NoticeprinterrorinTextbook20320RelationshipbetweendPand-D.-DYieldchangesinpercentage(dR/(1+R))PricechangeinpercentagedP/P21UnderstandingmorefeaturesofdurationDurationandmaturity:DincreaseswithM,butatadecreasingrate.Durationandyield-to-maturity:Ddecreasesasyieldincreases.Durationandcouponinterest:Ddecreasesascouponincreases22DurationofaportfolioDurationofportfolioofassets(liabilities)equalsweightedaverageofdurationsofindividualcomponentsoftheportfolio,withtheweightsbeingtheirvaluesrelativetotheentireportfoliovalue.23DurationGapInthecaseofa2-year,8%couponbond(Couponsarepaidsemi-annually),withafacevalueof$1,000andyield-to-maturityof12%.Supposethebondistheonlyloanasset(L)ofanFI,fundedbya2-yearcertificateofdeposit(D).Maturitygap:ML-MD=2-2=0DurationGap:DL-DD=1.883-2.0=-0.117Deposithasgreaterinterestratesensitivitythantheloan,soDGAPisnegative.FIexposedtodeclininginterestrates.24Immunizingthe

BalanceSheetofanFIDurationgapvs.Leverage-adjusteddurationgapFromthebalancesheet,E=A-L.Therefore,DE=DA-DL.Inthesamemannertodeterminethechangeinbondprices,wecanfindthechangeinvalueofequityusingduration.DA/A=-DA[

DR/(1+R)];DL/L=-DL[

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

-DLk]×A×[DR/(1+R)]Letk=L/A,theleverageratiooftheFI.25DurationandImmunizationDE=-[DA

-DLk]×A×[DR/(1+R)]Theformulashows3effectsofinterestratechangesonthemarketvalueofanFI’sEquityornetworth(DE):LeverageadjustedDurationGap=DA

-DLkThesizeoftheFI=AThesizeoftheinterestrateshock=DR/(1+R)26Anexample:SupposeDA=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(millions)27MethodsofimmunizingbalancesheetIfDA=DL,DE=-[DA-DLk]A[DR/(1+R)]=-[5-5(90/100)]100[.01/1.1]=-$0.45(millions)So,DA=DLmatchdoesnotworkforaleveragedFIimmunization.Importantly,DA=DLkDE=-[DA-DLk]A[DR/(1+R)]=[0]A[DR/(1+R)]=0ToachieveDA=DLk,adjustDA,DLork.

28Immunizationand

RegulatoryConcernsAsweknowfrompreviousslides,FI’sshareholdersfocusonE,andtargetonE=0,theimmunizationstrategyisDA=kDL.Whileregulatorsfocuson(E/A),becausetheysettargetratiosforabank’scapital(networth):Capital(Networth)ratio=E/A.Iftargetistoset(E/A)=0,ratherthanE=0,theimmunizationstrategyisDA=DL,

ratherthanDA=kDLProoftobeconductedbyyourself.Akeyclueisgivenas:(E/A)=(1/A)E–(E/A2)

A=…?29*LimitationsofDurationImmunizingtheentirebalancesheetthroughdurationmatchcanbecostlyinrestructuring.

However,easingfactorsincludeGrowthofpurchasedfunds,assetsecuritization,andloansalesmarket;takinghedgingpoisitionsinthemarketsforderivatives.ImmunizationisadynamicprocesssincedurationdependsoninstantaneousR.Largeinterestratechangeeffectsnotaccuratelycaptured.Convexity:Non-linearrelationshipbetweenbondpriceandinterestratechange(yieldcurve)Theproblemoftheflatyieldcurveassumption.Morecomplexifnonparallelshiftinyieldcurve.Theproblemofdefaultrisk

…….302.5betacoefficientofstocksStockvaluationmodelsCalculatingbetacoefficientTakingbetaasasensitivitymeasureExtensiontomulti-factorpricingmodel31Thecapitalassetpricingformula

E(ri)

=

rf+βi

[E(rM)–rf

]Theriskpremiumofanindividualassetisproportionaltoitsβcoefficientandtheriskpremiumofthemarketportfolio

βi=σiM/σM232MorethanonefactorinAPTAPTPricingEquation

Ei=λ0+λ1βi1+λ2βi2+…+λkβik

λ0:risk-freerateβ:sensitivecoefficientλi:riskpremiumofcommonfactorICommonfactorsGDP,Inflationrate,Interestrate,Oilprice…33ImportantTermsDurationInterest