math annuities

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Independent Study Unit-Annuities- Part- 1

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Example - 1 a) How much you need to invest now at 8.3%/a compounded annually to provide $500 per year for the next 10 years?

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Compounding Period 0 1 2 3 8 9 10 Payment $0 $500 $500 $500 $500 $500 $500 Present Value of each Payment 500 (1+0.083)1 500 (1+0.083)2 500 (1+0.083)3 . . . . 500 (1+0.083)8 500 (1+0.083)9 500 (1+0.083)10 i %per compounding period a)

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PV= A (1+i )n PV= 500 (1.083)n PV= 500 (1.083)2 PV= 500 (1.083)3 PV= 500 (1.083)8 PV= 500 (1.083)9 PV= 500 (1.083)10 a= 500 (1.083) = 500 * 1.083-1 r= 1 (1.083) =1.083-1 n= 10 S10 = 500*1.083-1 + 500*1.083-2 + 500*1.083-3 + 500*1.083-4 + … + 500* 1.083-9 + 500* 1.083-10 PV = a(rn-1) (r-1) S10 = 500*1.083-1 [(1.083-1)10-1] 1.083-1 -1 S10 = $3310.11

A sum of $3310.11 invested now would provide a payment of $500 each of the next 10 years : 

A sum of $3310.11 invested now would provide a payment of $500 each of the next 10 years

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b) How much would you need to invest now to provide n regular payments of $R if the money is invented at a rate of i% per compounding period?

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Compounding Period 0 1 2 3 n-3 n-2 n Payment $0 $R $R $R $R $R $R Present Value of each Payment R (1+i)1 R (1+i)2 R (1+i)3 . . . . R (1+i)n-3 R (1+i)n-2 R (1+i)n i %per compounding period

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PV= A (1+i )n PV1= 500 (1.083)n PV2= 500 (1.083)2 PV3= 500 (1.083)3 PVn= R (1+i)n Sn = R*(1+i)3-1 + R* (1+i) -2 + R*(1+i) -3 + ... + R*(1+i)-n Sn = a(rn-1) (r-1) Sn = R*(1+i) -1 [1+i] –n -1 ] [1+i] –n -1 . . . a= R* (1+i)-1 r = (1+i)-1 Sn = R*(1+i) -1 [1+i] –n -1 ] [1+i] –n -1 1+i 1+i Sn = R[(1+i)-n -1] 1-[1+i] Sn = R[(1+i)-n -1] -i Sn = R*( ) 1-(1+i)-n -i

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The Present Value of an annuity in which $R is paid at the end of the each n regular intervals earning i% compound interest per interval is PV = R*( ) 1-(1+i)-n -i