Chapter 3.6, Problem 15E

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 $\$70,198.23$ .

Given information:

It is given that $R=\$450,r=5.25\%,t=10,k=12$ .

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, $R=\$450,r=5.25\%,t=10,k=12$

First, calculate the interest rate $i$ .

  $\begin{array}{c}i=\frac{r}{k}\\ =\frac{0.0525}{12}\\ =0.004375\end{array}$

Now, calculate the number of payments $n$ .

  $\begin{array}{c}n=kt\\ =12\left(10\right)\\ =120\end{array}$

Substitute $R$ for $\$450$ , $i$ for $0.004375$ , and $n$ for $120$ in $FV=R\frac{{\left(1+i\right)}^n-1}{i}$ .

  $\begin{array}{c}FV=450\frac{{\left(1+0.004375\right)}^{120}-1}{0.004375}\\ =450\frac{{\left(1.004375\right)}^{120}-1}{0.004375}\\ =\$70,198.23\end{array}$

Hence, future value $FV$ accumulated in an annuity after investing periodic payment $\$450$ for $10$ years at an annual interest rate $0.0525$ with payments made and interest credited $12$ times per year. is $\$70,198.23$ .