Chapter 12.4, Problem 4E

Annuity Find the amount of an annuity that consists of 24 monthly payments of $500 each into an account that pays 8% interest per year, compounded monthly.

To determine

To find: The amount of annuity for the given data.

Answer to Problem 4E

The amount of annuity is $12,966.59.

Explanation of Solution

Given:

Monthly Payments = $500

Interest Rate = 8% per year (or 0.6667% per month)

Number of Payments = 24 payments

Formula:

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

Here,

A_j = Future Value of Annuity

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 annuity is calculated by adding 1 plus the interest rate with the power of number of years and the result is subtracted by 1, the whole result is divided by interest rate and the end value is multiplied with period payment.

$\begin{array}{c}{A}_f=R\times \left[\frac{{\left(1+i\right)}^n-1}{i}\right]\\ =\$500\times \left[\frac{{\left(1+\left(\frac{8\%}{12}\right)\right)}^{24}-1}{\left(\frac{8\%}{12}\right)}\right]\\ =\$500\times \left[\frac{{\left(1+0.6667\%\right)}^{24}-1}{0.6667\%}\right]\\ =\$500\times \left[\frac{1.172888-1}{0.6667\%}\right]\end{array}$

$\begin{array}{l}=\$500\times \left[\frac{0.172888}{0.6667\%}\right]\\ =\left[\$500\times 25.933190\right]\\ =\$12,966.5949or\$12,966.59\end{array}$

Therefore, the amount of annuity is $12,966.59.