Question

A small business owner contributes $3000 at the end of each quarter to a retirement account that earns 8% compounded quarterly.

(a) How long will it be until the account is worth $150,000? (Round your answer UP to the nearest quarter.)

quarters

(b) Suppose when the account reaches $150,000, the business owner increases the contributions to $7000 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.)

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Answer 1

The future value (F) of an annuity is given by F = (P/r)[(1+r)ⁿ-1], where P is the periodic payment , r is the rate per period and n is the number of periods.

(a). Here, P = $ 3000, r = 8/400 = 0.02 and F = $ 150000. Therefore, 150000 = (3000/0.02)[(1.02)ⁿ-1] = 150000[(1.02)ⁿ-1] or, [(1.02)ⁿ-1] = 1 or, (1.02)ⁿ = 2.

Now, on taking log of both the sides, we get n log 1.02 = log 2 so that n = log2/log 1.02 = 0.301029995/0.008600171762 = 35.00279978 , say 35 ( on rounding off to the nearest quarter).

Thus, it will take 35 quarters for the small business owner for making the retirement account worth $ 150000.

(b). 15 years are equal to 60 quarters. This means that

(i). The amount of $ 150000 will continue to earn interest @ 8 % compounded quarterly. The formula for maturity value (F) of an initial deposit (P), after n years, where interest rate is r % and interest is compounded t times per year is F = P(1+r/100t)^nt. Here, P = $ 150000, r/100t = 0.02, t = 4 and n = 15. Therefore, F = 150000(1.02)⁶⁰ = 150000* 2.281030788 = $ 342154.62( on rounding off to the nearest cent).

(ii). There will be another annuity of $ 7000 per quarter for 60 quarters. Then F = (7000/0.02)[(1.02)⁶⁰ -1] = 350000*2.281030788   = $ 798360.78( on rounding off to the nearest cent).

Thus, total value of the account will be = $ 342154.62 + $ 798360.78 = $ 1140515.40 after 15 more years.