Chapter 12.4, Problem 14E

Funding an Annuity A 55-year-old man deposits $50,000 to fund an annuity with an insurance company. The money will be invested at 8% per year, compounded semiannually. He is to draw semiannual payments until he reaches age 65. What is the amount of each payment ?

To determine

To find: The amount of each payment.

Answer to Problem 14E

The amount of each payment is $3,679.09.

Explanation of Solution

Given:

Present Value = $50,000

Interest Rate = 8% per year (or 4% semiannually)

Number of Payments = 10 years (or 20 months)

Formula:

$R=\left[\frac{i{A}_p}{1-{\left(1+i\right)}^{-n}}\right]$

Here,

A_p = Present Value

R = Periodic Payment

i = Interest Rate

n = Number of Years/Payments

Annuity: An annuity is a form of financial contract that are typically used for retirement plans by the insurance companies which assures guaranteed payment to the insurer. It is paid by the insurance company for a certain period of time or on lifetime basis.

Calculation:

The amount of each payment is calculated by multiplying the present value with interest rate and the result is divided by 1 minus, 1 plus interest rate to the power minus number of years.

$\begin{array}{c}R=\left[\frac{i{A}_p}{1-{\left(1+i\right)}^{-n}}\right]\\ =\left[\frac{\left(\frac{8\%}{2}\right)\times \$50,000}{1-{\left(1+\left(\frac{8\%}{2}\right)\right)}^{-10\times 2}}\right]\\ =\left[\frac{4\%\times \$50,000}{1-{\left(1+4\%\right)}^{-20}}\right]\\ =\left[\frac{\$2,000}{1-0.4564}\right]\end{array}$

$\begin{array}{c}=\left[\frac{\$2,000}{0.5436}\right]\\ =\$3,679.0875or\$3,679.09\end{array}$

Therefore, the amount of each payment is $3,679.09.