Question

You have just turned 25 years old and decide to start saving for your retirement. You plan to save $5,000 at the end of each year (so the first deposit will be one year from now), and will make the last deposit when you retire when you turn 65 (that is, 40 deposits in total. Suppose your pension fund earns 8% per year on your retirement savings.

a) How much will you have saved for retirement by the time you have turned 65? [Hint: It helps to draw a time line! One way to solve this is to calculate the present value of the total savings first and then proceed to calculate the future value of the total savings at the age of 65 ]

b) Suppose that you will live till the age of 90 (that is, 25 withdrawals in total and the first one happens just after you turn 66). How much can you spend annually during your retirement? Assume the funds you have not yet spent continue to earn 8\% annually. [By the way, (only) if you did not answer Part a, then assume you saved \$1,000,0 as total saving at the end of 65)

Answer 1

a) Duration of annual deposit = 40 years

Annual deposit = 5,000

Interest rate = 8%

First deposit made will be saved for 39 years whose future value would be 5,000 * 1.08^³⁹

Second deposit made will be saved for 38 years whose future value would be 5,000 * 1.08^³⁸

.......

40th deposit made will be saved for 0 years whose future value would be 5,000 * 1.08^⁰

Amount of money at the end of 65 years = [5,000 * 1.08^³⁹] + [5,000 * 1.08^³⁸] + ...... + [5,000 * 1.08^⁰] which makes a G.P. whose sum can be calculated using formula: [a * (1 - r^ⁿ)] / [1 - r]

where a = [5,000 * 1.08^³⁹]

r (ratio of two consecutive terms) = (1 / 1.08) = 0.9259

n = 40

Sum of G.P. = [5,000 * 1.08^³⁹ * (1 - 0.9259^⁴⁰)] / (1 - 0.9259) = 1,295,283.88

You will have 1,295,283.88 at the end of age 65

b) Let say you deposit X every year from age 66 to 90

Present value of money spent in age 66 = [X / 1.08^¹]

Present value of money spent in age 67 = [X / 1.08^²]

.....

Present value of money spent in age 90 = [X / 1.08^²⁵]

Sum of present value is 1,295,283.88 which is equal to [X / 1.08^¹] + [X / 1.08^²] + ..... + [X / 1.08^²⁵] whose sum can be calculated using above G.P. formula which is 10.764X = 1,295,283.88

X = 120,334.8