Tutorial Exercise

A bank pays $6\%$ interest, compounded semiannually. Use the appropriate formula to find how much should be deposited now to yield an annuity payment of $$800$ at the beginning of each six months, for $19$ years.

Step $1$

Since the goal is to receive payments at the beginning of each six months, this is as an annuity due. The following formula can be used to calculate the present value $PV$ of the annuity where $Pmt$ is the amount of each annuity payment, $i$ is the interest rate per period expressed as a decimal, and $n$ is the number of compounding periods.

$PV=Pmt\frac{1-(1+i{)}^{-n}}{i}(1+i)$

The interest rate per period, $i,$ is calculated using the nominal, or annual, rate and the number of periods per year as follows.

$i=\frac{\text{n}\text{o}\text{m}\text{i}\text{n}\text{a}\text{l}\text{r}\text{a}\text{t}\text{e}}{\text{p}\text{e}\text{r}\text{i}\text{o}\text{d}\text{s}\text{p}\text{e}\text{r}\text{y}\text{e}\text{a}\text{r}}$

The rate was given to be $6\%.$ Interest is compounded semiannually, or $2$ times per year. Find the interest rate per period.

$i=\frac{\text{n}\text{o}\text{m}\text{i}\text{n}\text{a}\text{l}\text{r}\text{a}\text{t}\text{e}}{\text{p}\text{e}\text{r}\text{i}\text{o}\text{d}\text{s}\text{p}\text{e}\text{r}\text{y}\text{e}\text{a}\text{r}}$

$=\frac{\%}{2}$

$=$

Your answer cannot $be$ understood $or$ graded. More Information $\frac{?}{?}\%$

The total number of compounding periods will be the number of years annuity payments will be made multiplied by the number of compounding periods per year. There are $2$ compounding periods per year and payments are to be received for $19$ years. Find the total number of compounding periods.

$n=$ number $of$ years $$ number $of$ compounding periods per year

$=19$

$=$