Chapter 3.6, Problem 13E

Solution

To find: The future value $FV$ accumulated in an annuity after investing periodic payment $R$ for $t$ years at an annual interest rate $r$ with payments made and interest credited $k$ times per year.

The future value is $\$18,483.38$ .

Given information:

It is given that $R=\$500,r=7\%,t=6,k=5$ .

Concept Used:

Future Value of an Annuity

The future value $FV$ of an annuity consisting of $n$ equal periodic payments of $R$ dollars at an interest rate $i$ per compounding period (payment interval) is $FV=R\frac{{\left(1+i\right)}^n-1}{i}$

Calculation:

Here, $R=\$500,r=7\%,t=6,k=5$

First, calculate the interest rate $i$ .

  $\begin{array}{c}i=\frac{r}{k}\\ =\frac{0.07}{5}\\ =0.014\end{array}$

Now, calculate the number of payments $n$ .

  $\begin{array}{c}n=kt\\ =5\left(6\right)\\ =30\end{array}$

Substitute $R$ for $\$500$ , $i$ for $0.014$ , and $n$ for $30$ in $FV=R\frac{{\left(1+i\right)}^n-1}{i}$ .

  $\begin{array}{c}FV=500\frac{{\left(1+0.014\right)}^{30}-1}{0.014}\\ =500\frac{{\left(1.014\right)}^{30}-1}{0.014}\\ =\$18,483.38\end{array}$

Hence, future value $FV$ accumulated in an annuity after investing periodic payment $\$500$ for $6$ years at an annual interest rate $0.07$ with payments made and interest credited $5$ times per year. is $\$18,483.38$ .