Question

2. You have $26,000 and want to have $35,100 in 6 years. The interest is compounded monthly.What named interest rate do you need to meet your goal? Be sure to show the equation you used to find the solution and show all your work. Round the interest rate to the nearest hundredth of a percent.

Answer 1

We have an investment that is compounded monthly.

The initial value PV is 26,000 and the final value FV is 35,100.

The numberof periods is n=6 years and th numbers of subperiods in a year is m=12, as it is compounded monthly.

Our unknown is the nominal annual interest rate (r).

We can relate all this variables as:

$FV=PV=(1+\frac{r}{m})^{n\cdot m}$

If we rearrange we get:

$\begin{gathered} \frac{FV}{PV}=(1+\frac{r}{m})^{n\cdot m} \\ \sqrt[nm]{\frac{FV}{PV}}=1+\frac{r}{m} \\ \frac{r}{m}=\sqrt[nm]{\frac{FV}{PV}}-1 \\ r=m\cdot(\sqrt[nm]{\frac{FV}{PV}}-1) \end{gathered}$

Then, we can find the value of r replacing with the known values as:

$\begin{gathered} r=12\cdot(\sqrt[6\cdot12]{\frac{35100}{26000}}-1) \\ r=12\cdot(\sqrt[72]{1.35}-1) \\ r\approx12(1.0042-1) \\ r\approx12\cdot0.0042 \\ r\approx0.0501 \\ r=5.01\% \end{gathered}$

Answer: the annual nominal interest rate is 5.01%