Question

CHAPTER THREE The Time Value of Money 87 a) How much interest will the saver carn if the intcrest is left to accumulatc? b) How much interest will the saver carn if the interest is withdrawn cach yeatr? c) Why arc the answers to (a) and (b) different? 2. At the end of cach ycar a self-cmployed person deposits $1,500 in a retirement account that carns 7 percent annually a) How much will be in the account when the individual retires at the age of 65 if the contributions start when the person is 45 ycars old? b) How much additional moncy will be in the account if the individual stops making the contribution at age 65 but defers retirement until age 70? c) How much additional moncy will be in the account if the individual continucs making the contribution but defers retirement until age 70? d) Compare the answers to (b) and (c). What is the effect of continuing the contribu- 2 tions? How much is the difference between the two answers? A saver wants $100,000 after ten years and believes that it is possible to earn an an- nual rate of 8 percent on invested funds. a) What amount must be invested cach ycar to accumulate $100,000 if (1) the pay- ments are made at the beginning of cach year or (2) they are made at the end of

Answer 1

(a) End of Year Contributions = $ 1500, Interest Rate = 7 % annually,

Contributions begin when the person is 45 years and continues till the age of 65. This implies a total deposit tenure of 20 years

Total Accumulation in the Retirement Account = 1500 x (1.07)^(20) + 1500 x (1.07)^(19) + 1500 x (1.07)^(18) + ....... + 1500 x (1.07) + 1500 = 1500 x [{1.07}^(21) - 1] / [{1.07} - 1] = $ 67297.77

(b) Accumulated Retirement Account Value at the age of 70 = Accumulation Value at 65 x (1.07)^(5) = 67297.77 x (1.07)^(5) = $ 94388.6

Additional Money Earned = 94388.6 - 67297.77 = $ 27090.83

(c) Total Accumulation if Retirement is pushed to 70 years of age = 1500 x (1.07)^(25) + 1500 x (1.07)^(24) + ......+ 1500 x (1.07) + 1500 = $ 103014.7

Additional Amount Earned = 103014.7 - 67297.77 = $ 35716.94

(d) As is observable, the additional amount accumulated is higher in case (c) as compared to case(b). This means that continuing the contributions is beneficial to the investor as additional contributions are also compounded at the 7% interest rate.

Difference between the two accumulations = 35716.94 - 27090.83 = $ 8626.109 ~ $ 8626.11