Question

Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date.

Find the present value of $10,000 if interest is paid at a rate of 9% per year, compounded semiannually, for 3 years.

Short answer

The present value is $ 7678.96.

Step-by-step solution

Step 1. Given.

$r=\mathrm{interest}\mathrm{rate}=9\%=0.09$.

$t=\mathrm{number}\mathrm{of}\mathrm{years}=3$.

$n=\mathrm{The}\mathrm{number}\mathrm{of}\mathrm{times}\mathrm{compounded}\mathrm{in}\mathrm{a}\mathrm{year}=2$.

Because, interest is compounded semiannually.

$A\left(t\right)=\mathrm{Amount}\mathrm{after}t\mathrm{years}=\$10000$.

Step 2. To determine.

We have to find the present value.

Step 3. Calculation.

Let, $P=$Present value.

Using compound interest formula:$A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$

Then we plug the values and find P.

$$\begin{gathered} A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt} \\ 10000=P{\left(1+\frac{0.09}{2}\right)}^{2\left(3\right)} \\ 10000=P{\left(1.045\right)}^6 \\ P=\frac{10000}{{\left(1.045\right)}^6} \\ P=7678.95738278 \\ P\approx 7678.96 \end{gathered}$$

Hence, the present value is $ 7678.96.