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1 1/30/13 ModelingCompoundGrowthinExcel Part3:Annui>es RobertMuller CS021ComputersinManagement BostonCollege AnAnnuityisaSequenceofCashFlows A A A A 1/30/13 CS021ComputersinManagement 2 AnAnnuityisaSequenceofCashFlows A A A A Examplesincludemortgages,bonds,rentandloanpayments. 1/30/13 CS021ComputersinManagement 3 1

2 1/30/13 CashFlowsatEndofPeriod(type0) pmt pmt pmt pmt FV(rate,nper,pmt,pv,type) 1/30/13 CS021ComputersinManagement 4 CashFlowsatStartofPeriod(type1) pmt pmt pmt pmt type0isthedefault 1/30/13 CS021ComputersinManagement 5 Annui>esHaveFutureandPresent Values A 1 A 2 A 3 A n 1/30/13 CS021ComputersinManagement 6 2

3 1/30/13 FutureValue(type1) A 1 A 2 A 3 A n 1/30/13 CS021ComputersinManagement 7 FutureValue(type1) A 1 A 2 A 3 A n FV n =A 1 (1+i) n +A 2 (1+i) nx1 + +A n (1+i) 1 1/30/13 CS021ComputersinManagement 8 FutureValue(type1) A A A A FV n =A(1+i) n +A(1+i) nx1 + +A(1+i) 1 1/30/13 CS021ComputersinManagement 9 3

4 1/30/13 ClosedForm FV n =A(1+i) n +A(1+i) nx1 + +A(1+i) 1 Wewanttoeliminate the andsoon ellipsis 1/30/13 CS021ComputersinManagement 10 GeometricSeries Asumoftheform: Σ ar i =ar 1 +ar 2 + +ar n n i=1 1/30/13 CS021ComputersinManagement 11 ClosedformofaGeometricSeries n Σ ar i =ar 1 +ar 2 + +ar n i=1 n (1Xr) Σ ar i =(1Xr)(ar 1 +ar 2 + +ar n ) i=1 1/30/13 CS021ComputersinManagement 12 4

5 1/30/13 (1Xr) ClosedformofaGeometricSeries n (1Xr) Σ ar i =(1Xr)(ar 1 +ar 2 + +ar n ) i=1 n Σ ar i =ar 1 +ar 2 + +ar n i=1 Xar 2 X Xar n Xar n+1 1/30/13 CS021ComputersinManagement 13 ClosedformofaGeometricSeries (1Xr) n Σ ar i =ar 1 +ar 2 + +ar n i=1 Xar 2 X Xar n Xar n+1 =ar ar n+1 a(r n+1 r) = r 1 1/30/13 CS021ComputersinManagement 14 ClosedformofaGeometricSeries n Σ ar i = i=1 a(r n+1 r) r 1 1/30/13 CS021ComputersinManagement 15 5

6 1/30/13 FutureValue Plugging(1+i)inforr: A((1+i) n+1 (1+i)) FV n = (1+i) 1 = A((1+i) n+1 i 1) i =FV(i,n,XA,0,1) 1/30/13 CS021ComputersinManagement 16 Example Problem:Andrew sgrandparentshaveput$500 inabankaccountforhimeveryyearonhis birthday.howmuchwillhehaveaaerthe paymentonhis21 st birthdayifthemoney growsat5peryear? Answer: 1/30/13 CS021ComputersinManagement 17 Example Problem:Andrew sgrandparentshaveput$500 inabankaccountforhimeveryyearonhis birthday.howmuchwillhehaveaaerthe paymentonhis21 st birthdayifthemoney growsat5peryear? 500((1+.05) Answer:= ) /30/13 CS021ComputersinManagement 18 6

7 1/30/13 Example Problem:Andrew sgrandparentshaveput$500 inabankaccountforhimeveryyearonhis birthday.howmuchwillhehaveaaerthe paymentonhis21 st birthdayifthemoney growsat5peryear? Answer: =FV(5,21,X500,0,1)+500 =$19,252 1/30/13 CS021ComputersinManagement 19 AllowingforNonzeroPV A((1+i) n+1 i 1) FV=PV(1+i) n + i =FV(i,n,XA,PV,1) 1/30/13 CS021ComputersinManagement 20 SolvingfortheAnnuityA(pmt) A((1+i) n+1 i 1) FV=PV(1+i) n + i A((1+i) n+1 i 1) FVXPV(1+i) n = i i(fvxpv(1+i) n ) A= (1+i) n+1 i 1 1/30/13 CS021ComputersinManagement 21 7

8 1/30/13 SolvingfortheAnnuityA(pmt) i(fvxpv(1+i) n ) A= (1+i) n+1 i 1 =pmt(rate i,nper n,pv,fv,type) 1/30/13 CS021ComputersinManagement 22 Example Problem:Andrewwouldliketowithdrawafixedamountoncea monthwhileheisinlawschool.howmuchcanhewithdraw assumingthathewantstosavehalftheprincipal?assume thesameinterestratebutcompoundingmonthly. Answer: 1/30/13 CS021ComputersinManagement 23 Example Problem:Andrewwouldliketowithdrawafixedamountoncea monthwhileheisinlawschool.howmuchcanhewithdraw assumingthathewantstosavehalftheprincipal?assume thesameinterestratebutcompoundingmonthly. Answer: 500((1+.05) ) LetX= /30/13 CS021ComputersinManagement 24 8

9 1/30/13 Example Problem:Andrewwouldliketowithdrawafixedamountoncea monthwhileheisinlawschool.howmuchcanhewithdraw assumingthathewantstosavehalftheprincipal?assume thesameinterestratebutcompoundingmonthly. Answer: 500((1+.05) ) LetX=.05.05/12(X/2XX(1+.05/12) 36 ) A= (1+.05/12) 36+1 (.05/12) /30/13 CS021ComputersinManagement 25 Example Problem:Andrewwouldliketowithdrawafixedamountoncea monthwhileheisinlawschool.howmuchcanhewithdraw assumingthathewantstosavehalftheprincipal?assume thesameinterestratebutcompoundingmonthly. Answer: A1=FV(5,21,X500,0,1)+500 =PMT(5/12,36,XA1,A1/2,1) =$327 1/30/13 CS021ComputersinManagement 26 PresentValue A A A A A 1/30/13 CS021ComputersinManagement 27 9

10 1/30/13 PresentValue A 0 A 1 A 2 A 3 A n 1/30/13 CS021ComputersinManagement 28 PresentValue A 0 A 1 A 2 A 3 A n A PV= 0 (1+i)0 + A 1 (1+i) A n (1+i) n 1/30/13 CS021ComputersinManagement 29 PresentValue PV= A + A (1+i) + + A (1+i) n = A i 1 1X (1+i) n =PV(rate i,nper n,a,0,1) 1/30/13 CS021ComputersinManagement 30 10

11 1/30/13 Example Problem:Youwanttobuyahousefor $500,000.You vesavedup20andyourbank isofferingtoloanyoutheremaining80for 15yearsat7annualinterestcompounded monthly.howmuchinterestwillyoupayin the8 th year? Answer:SeetheLoanAmor>za>onspreadsheet. 1/30/13 CS021ComputersinManagement 31 11