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 $100,000 if interest is paid at a rate of 8% per year, compounded monthly, for 5 years.

Short answer

The present value is $67121.04.

Step-by-step solution

Step 1. Given.

$r=\mathrm{interest}\mathrm{rate}=8\%=0.08$.

$t=\mathrm{number}\mathrm{of}\mathrm{years}=5$.

$n=\mathrm{The}\mathrm{number}\mathrm{of}\mathrm{times}\mathrm{compounded}\mathrm{in}\mathrm{a}\mathrm{year}=12.$

Because, interest is compounded monthly.

$A\left(t\right)=\mathrm{Amount}\mathrm{after}t\mathrm{years}=\$100,000$.

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.

$$\begin{gathered} A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt} \\ 100000=P{\left(1+\frac{0.08}{12}\right)}^{12\left(5\right)} \\ 100000=P{\left(1.00666666667\right)}^{60} \\ P=\frac{100000}{{\left(1.00666666667\right)}^{60}} \\ P=67121.0444296 \\ P\approx 67121.04 \end{gathered}$$

Hence, the present value is $67121.04.