Question

Compound Interest Which of the given interest rates and compounding periods would provide the better investment?

(a)$5\frac{1}{8}\%$per year, compounded semiannually.

(b)$5\%$per year, compounded continuously.

Short answer

(a) provides the better investment.

Step-by-step solution

Step 1. Given.

Given interest rates and compounding periods are:

(a)$5\frac{1}{8}\%$ per year, compounded semiannually.

(b) 5% per year, compounded continuously.

Step 2. To determine.

We have to find; which of the given interest rates and compounding periods would provide the better investment.

Step 3. Calculation.

We’ll compare the corresponding annual percentage yield to check; which of the given interest rates and compounding periods would provide the best investment.

For (a);

$r=\mathrm{interest}\mathrm{rate}=5\frac{1}{8}\%=5.125\%=0.05125$.

$n=\mathrm{Number}\mathrm{of}\mathrm{times}\mathrm{compounded}\mathrm{in}\mathrm{a}\mathrm{year}=2$

Because, compounded semiannually means twice in a year.

So, theannual percentage yield for (a) is:

$$\begin{gathered} APY={\left(1+\frac{r}{n}\right)}^n-1 \\ ={\left(1+\frac{0.05125}{2}\right)}^2-1 \\ =0.051906640625 \\ =5.1906640625\% \\ \approx 5.19\% \end{gathered}$$

For (b);

$r=\mathrm{interest}\mathrm{rate}=5\%=0.05$.

In this case the interest rate is compounded continuously.

So, theannual percentage yield for (b) is:

$$\begin{gathered} APY=e^r-1 \\ =e^{0.05}-1 \\ =0.051271096376 \\ =5.1271096376\% \\ \approx 5.13\% \end{gathered}$$

We see that, 5.19% is the highest annual percentage yield. It corresponds to (a).

Hence, (a) provides the better investment.