A person wants to establish an annuity for retirement purpos

A person wants to establish an annuity for retirement purposes. He wants to make quarterly deposits for 20 years so that he can then make quarterly withdrawals of $5,000 for 10 years. The annuity earns 7.32% interest compounded quarterly. How much will have to be in the account at the time he retires? How much should be deposited each quarter for 20 years in order to accumulate the required amount? What is the total amount of interest earned during the 30-year period?

Solution

36 (A). The annuity payment formula is P = Ar/[ 1-(1+r)^-n] where P is the periodic payment, r is the rate of interest per period , n is the number of eriods and A is the present value. Hete P = 5000, n = 10*4 = 40 and r = (7.32/100)*1/4 = 0.00183 . Then we have 5000 = A*0.00183/[ 1- (1.00183)^-40] = 0.00183 A/( 1- 1/1.07587373) = 0.00183 A/(1- 0.929477105) = 0.00183 A/0.070522895 so that A = 5000*(0.070522895)/ 0.00183 = 352.614475/ 0.00183 = $ 192685.51 ( on rounding off to the nearest cent). Thus, there will have to be $ 192685.51 in the account by the time the person retires.

(B) The formula for the future value of annuity is F = P[{(1+r)ⁿ -1}/r] *(1+r) , where P, n, r are as above. Here F = $ 192685.51, n = 20*4 = 80, r = 0.00183. Then 192685.51 = P [ {(1.00183)⁸⁰-1}/0.00183]* 1.00183 = P[ (1.157504283-1)/0.00183]* 1.00183 = P [(0.157504283)/0.00183]*1.00183 = ( 86.06791421)*(1.00183)P = 86.22541849 P. Therefore P = 192685.51/86.22541849 = $2234.67 ( on rounding off to the nearest cent). Thus, $ 2234.67 has to be deposited every quarter for 20 years in order to have $ 192685.51 in the account at the end of 20 years.

( C ). The person has deposited $ 2234.67 * 80 = $ 178773.60 in his account and he receives $ 5000*40 = 200000. Therefore, the interest earned during the 30 year period is $ 200000- $ 178773.60 = $ 21226.40