Question

A small business owner contributes $2000 at the end of each quarter to a retirement account that earns 4% compounded quarterly.

(a) How long will it be until the account is worth $150,000? (Round your answer UP to the nearest quarter.)
quarters the asnwer is not 56
(b) Suppose when the account reaches $150,000, the business owner increases the contributions to $8000 at the end of each quarter. What will the total value of the account be after 15 more years? (Round your answer to the nearest dollar.)
$ the answer is not 907891

Answer 1

(a).The formula for the future value (F) of an annuity is F = P[ (1+r)ⁿ-1]/r, where P is periodic payment, r is the interest rate per period and n is the no. of periods. Here, P = $ 2000, r = 4/400 = 0.01. Let the required no. of periods be n. Then 150000 = (2000/0.01)[(1.01)ⁿ-1] or, [(1.01)ⁿ-1] = 150000/200000 = 0.75 or, (1.01)ⁿ = 1.75. Now, on taking log of both the sides, we get n log 1.01 = log 1.75 so that n = log 1.75/log 1.01 = 0.243038048/0.004321373783 = 56 ( on rounding off to the nearest whole number.

(b).If the business owner increases the contributions to $8000 at the end of each quarter, then afer 15 years, F = (8000/0.01) [(1.01)^15*4 -1] = 800000*[(1.01)⁶⁰ -1] = 800000* 0.816696699= $ 653357.36( on rounding off to the nearest cent).

Further, the $ 150000 is also assumed to earn interest at 4 % compounded quarterly. After 60 quarters, this amount will increase to 150000(1.01)⁶⁰ = 150000*1.816696699=$ 272504.50( on rounding off to the nearest cent).

Hence the total value of the account after 15 more years will be $ 653357.36 + $ 272504.50 = $ 925861.86 , say $ 925862(on rounding off to the nearest dollar).