3.货币时间价值与利率期限结构 清华大学绝版金融工程课件

1CHAPTERTWO:TimeValueofMoneyandTermStructureofInterest2Yes!istheexpectedrateofreturn,i.e.,themeanofthediscountratesfordifferenttermsrLetNo!isthediscountratethatcannotbeusedforsolongperiod?01)1(PrCPVnttttnttttntttrCrC11)1()1(0)1(10ntttrCPNPVrDiscountedCashFlowFormula3TermStructureofInterestRates•Ourobjectiveistovaluerisklesscashflows.•Giventherichsetoffixed-incomesecuritiestradedinthemarket,theirpricesprovidetheinformationneededtovaluerisklesscashflowsathand.4FormsofInterestRates•Inthismarket,thisinformationonthetimevalueofmoneyisgiveninseveraldifferentforms:–Spotinterestrates–Priceofdiscountbonds(e.g.,zero-couponbondsandSTRIPS)–Pricesofcouponbonds–Yield-to-maturity(anaverageofspotinterestrates)–Forwardinterestrates•Theforminwhichthisinformationisexpresseddependsontheparticularmarket.5DeterminationofInterestRate1.Capitalproductionability——themorethecapital’sexpectedreturn,thehighertheinterestratesandviceversa.2.Uncertaintyofcapitalproductionability——themoretheuncertainty,thehighertheriskpremiumrequiredandthehighertheinterestratesandviceversa.3.Timepreferenceofconsumption——thestrongerpreferencetocurrentconsumption,thehighertheriskpremiumrequiredandthehighertheinterestratesandviceversa.4.Riskaversion——themoretheriskaversion,thehighertheriskpremiumrequiredandthelowertherisk-freeinterestrates.•Fourbasicfactors6TheoryofRealInterestRates•Realinterestratesaredeterminedbysupplyanddemandoffundsintheeconomy.•3factorsindeterminingrealinterestrates:–Aggregateendowments–Aggregateinvestmentopportunities–Aggregatepreferencesfordifferentconsumptionpath7•Considerarepresentativeinvestor:–Hasendowmentof(e0,e1)–Facesabondmarketwithinterestrater.8•Hemaximizeshisutilityoverhisconsumptionnowandlater:Wherebisthebondholding,u’0andu”0010011max()()..(1)ucucstcebcerb9•Theoptimalityconditionis0110000100'()(1)'()()'()()11(1/1)'()'()ucrucorforccdcuccucdcrucucc0000()1(1/1)[]'()cucdcruccRelativeriskaversioncoefficientThus,therealinterestrateisgivenby10NonlineartechnologyTime0Time101(,)cc2u1u0e1e-(1+RC)Investmentopportunityset21uubb(1+r)11LineartechnologyTime0Time101(,)ccb(1+r)b2u1u0e1e21uu-(1+RC)12•Moregenerally,considerconsumptiongrowatrandomrate.Investorsmaximizetheirexpectedutilityovermanyperiods.•Whereishisholdingsofdiscountbonds,isfutureendowments,isfutureconsumption,bothcanbeuncertain.00012max[()].....(1)1....TtttTttttEucstcebbbcerbtT1(,......)Tbb1(,......)Tee1(,......)Tcc13TheBenchmarkofInterest—YieldtoMaturity(YTM)YTMvarieswithdifferentfinancialinstruments,becausetheexposureoffinancialinstrumentsarequitedifferentandtherequiredriskpremiumsdifferfromeachother.Risk-freeinterestvarieswithterms.It’scalledthetermstructureofinterests.—Risk-freeinterests??No!Yes!14•Nominalandrealinterestrates—nominalinterestrate=realinterestrate+inflation—realinterestrate=puretimevalue+riskpremium•Compoundinterest—interestearnedoninterestalreadyearned)1ln(*rr—Continuouslycompoundingr—simplerateofreturnannuallym—timesofinterestpaymentsannuallyrm1—compoundingrateofinterestpaymentsannuallymrrmmm111mLetrrmemmrlim111Continuouslycompounding15FinancialRisksandRisk-freeSecurity—Defaultrisk—Liquidityrisk—Purchasepowerrisk—Interestrisk—Foreignexchangerisk—Othermarketrisks•Basicfinancialrisks:—Risk-freesecurity:•Substituteinreality:Treasury16—TreasuryYieldCurve•Treasuryyieldcurveusuallyhasthreeforms:upward,flatanddownward.ntrt,,1,•Zero-couponratesset—Billsarezerocouponwhilenotesandbondshavecoupons.—Zero-couponratessetcanbeobtainedbyconversion.17•Conversionexample:Treasurymaturityparcouponratecurrentprice1year10%0A2years1,0001,000910.50982.10%83.950.91050.91010001r22221)1(1100%83.91100)1(1001000110010.982rrr%08.112rB18—ShapesofYieldCurveupwardflatdownward•Sometheoriesfortheshapesofyieldcurve—Unbiasedexpectationstheory—Liquiditypreferencetheory—Marketsegmenttheory—Preferredhabitattheory19115$%)151(100$F.Oneyear’srisk-freerateForwardInterest—Thereisano-dividendstockanditsexpectedreturnis15%.ThecurrentpriceisAminicase:100$0S%5fr.Whatisoneyear’sforwardpriceofthisstock??20PositionImmediateCashFlowCashFlowintheFutureReplicatingStockUsingrisk-freebondandforwardcontractSupposeforwardpriceF=$106pershareShortsell$100risk-freebondShortsellonestockforwardat$106pershareBuyonestockat$100pershare+$100$1050106–S1$100S1NetCashFlow0$1ArbitrageStockforwardprice=$105pershare21Forwardpriceofariskyassetisnottheexpectationofthefuturespotpriceoftheasset.Proposition!22TheForwardPriceforaTradedAsset•Theforwardpriceforatradedassetwithoutinterimincomeis:F=SerT•Theforwardpriceforatradedassetwithdeterministicdividendrateis:F=Se(r-q)T•Theaboveequationcanbeobtainedthroughthefollowingarbitragestrategy:–Buyspote-qToftheassetandreinvestincomefromtheassetintheasset.–Shortaforwardcontractononeunitoftheasset.23TheForwardPriceforaTradedAsset•Theholdingoftheassetgrowsatrateqsothate-qTxeqT,orexactlyoneunitoftheasset,isheldattimeT.Underthetermsoftheforwardcontract,theassetissoldforFattimeT