Question

Using the appropriate interest table, provide the solution to each of the following four questions by computing the unknowns.

(a) What is the amount of the payments that Ned Winslow must make at the end of each of 8 years to accumulate a fund of \(90,000 by the end of the eighth year, if the fund earns 8% interest, compounded annually?

(b) Robert Hitchcock is 40 years old today and he wishes to accumulate \)500,000 by his sixty-fifth birthday so he can retire to his summer place on Lake Hopatcong. He wishes to accumulate this amount by making equal deposits on his fortieth through his sixty-fourth birthday. What annual deposit must Robert make if the fund will earn 8% interest compounded annually?

(c) Diane Ross has \(20,000 to invest today at 9% to pay a debt of \)47,347. How many years will it take her to accumulate enough to liquidate the debt?

(d) Cindy Houston has a \(27,600 debt that she wishes to repay 4 years from today; she has \)19,553 that she intends to invest for the 4 years. What rate of interest will she need to earn annually in order to accumulate enough to pay the debt?

Short answer

The amount paid each year in part A is $8,461.

The amount paid each year in part B is $6,839.

The number of years in part C will be 10 years.

The interest rate in part D is 9%.

Step-by-step solution

Computation amount paid each year

$$\begin{gathered} \mathbf{Amount}\mathbf{}\mathbf{Paid}\mathbf{}\mathbf{each}\mathbf{}\mathbf{year}\mathbf{=}\frac{\mathbf{Required}\mathbf{}\mathbf{amount}}{\mathbf{FV}\mathbf{}\mathbf{factor}\mathbf{}\mathbf{of}\mathbf{}\mathbf{ordinary}\mathbf{}\mathbf{annuity}} \\ \mathbf{=}\frac{\mathbf{90}\mathbf{,}\mathbf{000}}{\mathbf{10}\mathbf{.}\mathbf{63663}} \\ \mathbf{=}\mathbf{\$}\mathbf{8}\mathbf{,}\mathbf{461} \end{gathered}$$

Computation amount paid each year

$$\begin{gathered} \mathbf{Amount}\mathbf{}\mathbf{paid}\mathbf{}\mathbf{each}\mathbf{}\mathbf{year}\mathbf{=}\frac{\mathbf{Required}\mathbf{}\mathbf{Amount}\mathbf{}\mathbf{on}\mathbf{}\mathbf{Birthday}}{\mathbf{FV}\mathbf{}\mathbf{Factor}\mathbf{}\mathbf{of}\mathbf{}\mathbf{ordinary}\mathbf{}\mathbf{annuity}} \\ \mathbf{=}\frac{\mathbf{500}\mathbf{,}\mathbf{000}}{\mathbf{73}\mathbf{.}\mathbf{10594}} \\ \mathbf{=}\mathbf{\$}\mathbf{6}\mathbf{,}\mathbf{839} \end{gathered}$$

Calculation of the number of periods.

$$\begin{gathered} \mathbf{Present}\mathbf{}\mathbf{Value}\mathbf{}\mathbf{Factor}\mathbf{=}\frac{\mathbf{Present}\mathbf{}\mathbf{Value}}{\mathbf{Future}\mathbf{}\mathbf{Value}} \\ \mathbf{=}\frac{\mathbf{20}\mathbf{,}\mathbf{000}}{\mathbf{47}\mathbf{,}\mathbf{347}} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{42241} \end{gathered}$$

The PV factor is 0.42241 at 9% will be for 10 periods.

Calculation of the interest rate

$$\begin{gathered} \mathbf{Present}\mathbf{}\mathbf{Value}\mathbf{}\mathbf{Factor}\mathbf{=}\frac{\mathbf{Present}\mathbf{}\mathbf{Value}}{\mathbf{Future}\mathbf{}\mathbf{Value}} \\ \mathbf{=}\frac{\mathbf{19}\mathbf{,}\mathbf{553}}{\mathbf{27}\mathbf{,}\mathbf{600}} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{70844} \end{gathered}$$

The present value factor is 0.70844 for 4 periods; the interest rate is 9%.