Suppose that between the ages of 28 and 39 you contribute 30

Suppose that between the ages of 28 and 39, you contribute $3000 per year to a 401(k) and your employer matches the contribution dollar for dollar on your behalf. The interest rate is 8.75% compounded annually. (a) What is the value of the 401(k), rounded to the nearest dollar, after 11 years? (b) Suppose that after 11 years of working for this firm, you move on to a new job. However, you keep your accumulated retirement funds in the 401(k). How much money, to the nearest dollar, will you have in the plan when you reach age 65? (c) What is the difference between the amount of money you will have accumulated in the 401(k) and the amount you contributed to the plan?

Solution

(a).The formula for future value (F) of annuity is F = P[(1+r)ⁿ-1]/r , where P is the periodic payment, r is the rate of interest per period and n is the number of periods. Here, P = $3000, r = 8.75% = 0.0875, n = 11. Then F = 3000[((1.0875)¹¹-1]/0.0875 = 3000*1.516065379/0.0875 = 51979.38 = $ 51979( on rounding off to the nearest dollar). Thus, the value of the 401(k), rounded to the nearest dollar, after 11 years , is $ 51979.

(b). The formula for compound interest is A = P(1+r)ⁿ , where P is the principal/initial amount, r is the rate of interest, A is the maturity amount and n is the number of years. Here, P = $ 51979, r = 0.0875 and n = 65-39 = 26. Then A = 51979(1.0875)²⁶ = 51979*8.854436063= 460244.73 = $460245 ( on rounding off to the nearest dollar). Thus, the amount of money in 401(k0 will grow to $460245 by the age of 65 years.

(c ). The difference between the amount of money accumulated in 401(k) and the amount contributed is:

(i).after 11 years- $ 51979 -$ 11*3000 = $18979.

(ii). after 11+26 = 37 years-$460245- $33000 = $ 427245.