18Lecture 2 The time value of money

FinancialMathematics李社环上海财经大学金融学院保险系（E-mail:[e-mail]）2009年9月Lecture2ThetimevalueofmoneyWhatwe’llknowconceptssimpleinterestcompoundinterestEffectiverateofinterestDiscountinterestratepresentvalueAccumulationvalueDiscountfactortimevalueofmoneyProcessDiscountingAccumulatingPayattention!simpleinterest&compoundinterestinterestrate&discountratepresentvalue&AccumulationvalueWe’llanswerthefollowingquestionsWhatisthevalueofacashflow?Fortwocashflows,howcanweknowoneismorevaluedthantheother?$1inthisyear=$1innextyear?ADOLLARTODAYISWORTHMORETHANADOLLARINTHEFUTUREFactorsaffectingmoneyvalueGainofinvestmentPriceinflationChangeofexchangeratesOtherrisks,like,DefaultLiquidateandsoon.TimevalueHowmuchonedollartodaycanearnduringtheyear，inaeconomicactivity?interestinterestrateComposingofinterestrateInterestrate=risk-freerate+riskpremiumHowmuchriskpremium?In2000,intheFortune1000companies,Aaaratedcorporationspaidanaveragespreadof1.6%overrisk-freeBondsandBaaratedcorporationspaidanaveragespreadof2.3%.GiventhetotalvolumeofcorporatebondsisUSD7.8trillion,thisprovidesanestimatedrangeofUSD125billiontoUSD180billionforriskpremiumtoinvestorsinUScorporatebonds.(SwissRe,ThepictureofART,sigma,No.1/2003)MarketcapitalizationUSDbnAriskpremiumlowestimatein%Briskpremiumhighestimatein%CriskpremiumlowestimateUSDbnD=A*BriskpremiumhighestimateUSDbnD=A*CEquitymarket&equityriskpremium109954.3%7.4%473814Bondmarketvalue&Bondriskpremium77951.6%(Aaa)2.3%(Baa)125179VariouscostsofrisktakingforFortune1000corporationsin2000Esp.(iftheonlyriskispriceinflation):nominalrate=realrate+inflationrate2formsofinterestratesinpracticeSimpleinterest：simpleinterestcannotearnfurtherinterest.i—interestrate;C—capital;n—term-C（Ci）（Ci）（Ci）C+nCi012n-1n（Ci）meansinterestofCipaidnominallynotactuallySimpleInterestRateAninvestmentof$1underanannualsimpleinterestrateofiwillhavevalue$(1+it)attimetcompoundinterest:interestcanalsoearninterest.-CCiCiCiCi+C012n-1ncompoundinterestQuestionandthinkingAninvestorinitialsatimedepositofsavingthatpaysinterestrateof2.80%pa,withatermof3yearsandamountofprinciple￥10000,intheCommercialBankofChina.(a)Howmuchwillhereceiveontheredemptiondate?(b)Issimpleorcompoundinterestused?OnJune10,1998,ourgovernmentissuedakindofcredit-bookedTreasurebonds(凭证式国债)withatermof3yearsandacouponof7.11%pa.Investorsgotinterestannuallyfromthoseappointedbanksbytheircreditbooksandgotthecapitalontheredemptiondate.Didtheygetsimpleorcompoundinterest?ValueonredemptionSymbolsAn——theamountwhichwillbereceivedbytheinvestoriftheaccountisclosedafternyearsA0=C——theinitialprinciplei—ainterestrateWhatisAn?Simpleinterest?yearsCapital+interestinterest1A1=C+Ci=C（1+i）Ci2A2=C+2Ci=C（1+2i）2Ci………………nAn=C+nCi=C（1+ni）nCiThefuturevalueof$1todayislineartothelengthoftimeundersimpleinterestratevalueof$1undersimpleinterestrate10%11.11.21.31.41.51.60123456time(yearn)dollarvalueAnUndercompoundinterestrate?yearsCapital+interestinterest1A1=C+Ci=C（1+i）Ci2A2=C(1+i)(1+i)=C(1+i)2C[(1+i)2–1]………………nAn=C(1+i)n-1(1+i)=C(1+i)nC[(1+i)n–1]Aninvestmentof$1atayearlycompoundinterestrateofiwillhavevalue$(1+i)tattimet(inyears)Comparisonofsimpleandcompoundratevalueof$1undersimpleinterestrate10%11.11.21.31.41.51.60123456time(yearn)dollarvalueAnsimplecompoundNotenCiC[(1+i)n–1]=nCiIfn1nCi=AccumulationvaluesC(1+ni):theaccumulationvalueatnofinitialcapitalCundersimpleinterestrate.C(1+i)n:theaccumulationvalueatnofinitialcapitalCunderCompoundinterestrate.Or(1+i)nistheaccumulationvalueatnofaunitinitialcapitalunderCompoundinterestrate.Whatdoestheaccumulationvaluemean?Withasimpleinteresti=6%pa,whichofthefollowingswouldprefer?$100atthebeginningofthisyear;$106attheendofthisyear.WithaCompoundinteresti,whichofthefollowingswouldprefer?$Cat0;$C(1+i)natn.Remarks$Cat0isequivalentto$C(1+i)natn;Or$C/(1+i)nat0isequivalentto$Catn;or$1/(1+i)nat0isequivalentto$1atn.$1/(1+i)n$1n0Presentvalues1/(1+ni):thePresentvalueattime0ofaunitofcapitalattimenbysimpleinterest.1/(1+i)n:thePresentvalueattime0ofaunitofcapitalattimenbycompoundinterest.v=1/(1+i)iscalledadiscountfactor.practiceQuestion2.5_P11Presentvalue&accumulationvalueofacashflow(ch.4)Whatisthepresentvalueof{CFti}i=1n?Whatistheaccumulationvalueof{CFti}i=1n?identitiesCi(1+i)n-1+Ci(1+i)n-2+……+Ci(1+i)+Ci+C≡C(1+i)n或者ExamplesExample1.(P6)Example2.Aninvestormakesaninitialinvestmentof$3000andiscreditedwith$158interestattheendofthesecondyear.Whatamountofinteresttheinvestorwillgetfortheinitial$3000afterhalfpast3yearsifthesamecompoundinterestisused?DiscountrateArateofcommercialdiscountisawayofexpressingtheamountoftherepaymentrequiresforaloanusuallyinaperiodt1year.commercialdiscountd-(1-nd)1payment0ntimeforarateofcompounddiscountoreffectivediscountd(1-d)n10nExamplesAnone-monthtreasurebillisissuedatarateofcommercialdiscountof18%pa.Iftherepaymentis$20,000,howmuchwaspaidtothegovernmentinitially?Relationofianddiid1111diExamplesAnInvestorbuysoneyearTreasuryBilldiscountedatarateofdiscountof7%p.a..Findtheeffectiverateofinterestperannumforthetransacti