Question

A 20-year-old student wants to save $5 a day for her retirement. Every day she places $5 in a drawer. At the end of each year, she invests the accumulated savings ($1,825) in a brokerage account with an expected annual return of 8%. 1. If she keeps saving in this manner, how much will she have accumulated at age 65? 2. If a 40-year-old investor began saving in this manner, how much would he have at age 65? 3. How much would the 40-year-old investor have to save each year to accumulate the same amount at 65 as the 20-year-old investor? please solve using a financial calculator.

Answer 1

(a) The solution to this problem involves certain assumptions. First, even though $ 5 is added to the savings every day, for the convenience of calculation one can assume that the savings start earning money only at the end of each year. This essentially means that $ 5 deposited every day does not earn any interest until its the end of the year and the total deposit for that year has reached its annual value of $ 1825. Hence, the first $ 5, the second $ 5 and so on will start earning interest only at the end of year 1 and continue doing so till the student reaches 65 years of age. Similarly, for the subsequent deposits.

Savings Tenure = 65 - 20 = 45 years

Annual Savings = $ 1825 and Annual Returns = 8 %

Therefore, Accumulated Value at Retirement = 1825 x (1.08)^(44) + 1825 x (1.08)^(43) + ...........+ 1825 x (1.08)^(2) + 1825 x (1.08) + 1825 = [{(1.08)^(45) - 1} / {1.08 - 1}] x 1825 = $ 705372.8 ~ $ 705373

Financial Calculator Solution:

- Input PMT = 1825, I/Y = 8, N = 45 and PV = 0

- CMPT -> FV

- FV = $ 705372.8

(b) Savings Tenure = 65 - 40 = 25 years

Accumulated Value at Retirement = 1825 x (1.08)^(24) + 1825 x (1.08)^(23) + ......+ 1825 x (1.08) + 1825 = [{(1.08)^(25) - 1} / {1.08 - 1} ] x 1825 = $ 133418.34 ~ $ 133418

Financial Calculator Solution:

- Input PMT = 1825, I/Y = 8, N = 25 and PV = 0

- CMPT -> FV

- FV = $ 133418.34

(c) Required Value of Accumulation = $ 705372.8

Let the required annual deposits be $ K

Therefore, 705372.8 = k x (1.08)^(24) + k x (1.08)^(23) + ..... + k x (1.08) + k = k x [{(1.08)^(25) - 1} / {1.08 - 1}]

k = 705372.8 / 73.106 ~ $ 9648.64

Financial Calculator Solution:

- Input FV = 705372.8, I/Y = 8, N = 25 and PV = 0

- CMPT -> PMT

- PMT = $ 9648.64