Question

What is the accumulated value of periodic deposits of $40 at the beginning of every six months for 18 years if the interest rate is 4.62% compounded semi-annually?_________________Round to the nearest cent

Answer 1

We are asked to determine the future value for an annuity that is paid at the beginning of every six months with an interest rate of 4.62% compounded semi-annually (every six months). To do that we will use the following formula:

$FV_{\text{due}}=PMT(\frac{(1+i)^n-1}{i})(1+i)$

Where:

$\begin{gathered} \text{PMT}=\text{payments in each time period} \\ i=\text{ interest rate in decimal form} \\ n=\text{ time periods} \end{gathered}$

The PMT is the $40 payments every six months and the interest rate "i" in decimal form is:

$i=\frac{4.62}{100}=0.0462$

And the time period "n" is the number of six months in a year, since every year there are two periods of six months, in 18 years we have:

$n=18\text{years}\frac{2periods}{1year}=36periods$

Replacing the values we get:

$FV_{\text{due}}=(40)(\frac{(1+0.0462)^{36}-1}{0.0462})(1+0.0462)$

Now we solve the operations:

$FV_{\text{due}}=(40)(\frac{5.08-1}{0.0462})(1.0462)$

Solving the operations we get:

$FV_{\text{due}}=5168.41$

Therefore, the accumulated value is $5168.41