Question

Benjamin invested an amount of $12,000.00 in a mutual fund. After 4 years and 6 months the accumulated value of his investment was $13,407.58. What is the nominal interest rate of the investment if interest is compounded semi-annually?__________%

Answer 1

Given the following parameters

$\begin{gathered} PV\Rightarrow\text{Present value}\Rightarrow12000.00 \\ T\Rightarrow\text{time}\Rightarrow4\text{years and 6 month} \\ FV\Rightarrow\text{Future Value}\Rightarrow13407.58 \\ n=2 \end{gathered}$

To calculate the nominal rate, we will have to calculate the interest rate and the compounded period, the following formula will be used to find the interest rate and the compounded period.

$\begin{gathered} i=(\frac{FV}{PV})^{\frac{1}{n}}-1 \\ m=\frac{n}{t} \end{gathered}$

To find the value of the interest, we have

$\begin{gathered} i=(\frac{13407.58}{12000.00})^{\frac{1}{2}}-1 \\ i=(1.117298333)^{\frac{1}{2}}-1 \\ i=1.057023336-1=0.05702336 \end{gathered}$

To find the compounded period we will have that

$\begin{gathered} m=\frac{n}{t} \\ n=2 \\ t=4.5 \\ m=\frac{2}{4.5} \\ m=0.4444444444 \end{gathered}$

Thus, the nominal rate formula is given as;

$\begin{gathered} j=m\times i \\ \end{gathered}$

Substitute for m and i to find the nominal rate

$\begin{gathered} i=0.05702336 \\ m=0.4444444444 \\ j=0.05702336\times0.4444444444 \\ j=0.02534371556\approx0.0253 \end{gathered}$

The nominal rate in percentage is

$\begin{gathered} j=0.0253\times100\text{ \%} \\ j=2.53\text{ \%} \end{gathered}$

Hence, the nominal rate of the investment if interest is compounded semi-annually is 2.53%