Chapter 3, Problem 54P

To determine

The best saving option with respect to the effective interest rate.

Answer to Problem 54P

The interest rate of $2\frac{1}{4}\%$ compounded quarterly should be chosen as it gives maximum effective rate of return.

Explanation of Solution

Given information:

• The interest $2\frac{3}{8}\%$ compounded annually
• The interest $2\frac{1}{4}\%$ compounded quarterly
• The interest $2\frac{1}{8}\%$ compounded continuously

• Effective Interest Rate per year is the annual interest rate taking into account the effect of any compounding during the year. It is calculated as

${{\displaystyle i}}_a={\left(1+\frac{r}{m}\right)}^m-1$

where: i_a = effective interest rate

r = nominal interest rate

m = number of compounding subperiods per year

When the interest rate is compounded continuously:

${{\displaystyle i}}_a=e^r-1$

• Nominal Interest Rate per year is the annual interest rate without taking into account the effect of any compounding during the year. It is calculated as:

  $r=\left(m\times i_m\right)$

  where: r = nominal interest rate

  m = number of compounding subperiods per year

  i_m = interest rate during the compounding subperiod

Calculation:

Case 1:

Using the information given in the question, one knows that the interest rate is compounding on an annual basis. We need to find the nominal interest rate to calculate the effective interest rate

$\begin{array}{l}r=\left(m\times i_m\right)\\ r=1\times 2\frac{3}{8}\%\\ r=1\times \frac{19}{8}\\ r=2.375\%\end{array}$

After inserting the value of nominal interest rate as calculated above in the effective interest rate formula:

$\begin{array}{l}{{\displaystyle i}}_a={\left(1+\frac{r}{m}\right)}^m-1\\ {{\displaystyle i}}_a={\left(1+\frac{2.375}{100}\right)}^1-1\\ {{\displaystyle i}}_a={\left(1+0.02375\right)}^1-1\\ {{\displaystyle i}}_a={(}^{1.02375}-1\\ {{\displaystyle i}}_a=1.02375-1\\ {{\displaystyle i}}_a=0.02375\\ {{\displaystyle i}}_a=2.375\%\end{array}$

The effective interest rate is 2.375%

Case 2:

Using the information given in the question, one knows that the m = 4 as the interest rate is compounding on a quarterly basis. We need to find the nominal interest rate to calculate the effective interest rate

$\begin{array}{l}r=\left(m\times i_m\right)\\ r=4\times 2\frac{1}{4}\%\\ r=4\times \frac{9}{4}\\ r=9\%\end{array}$

After inserting the value of nominal interest rate as calculated above in the effective interest rate formula:

$\begin{array}{l}{{\displaystyle i}}_a={\left(1+\frac{r}{m}\right)}^m-1\\ {{\displaystyle i}}_a={\left(1+\frac{9}{400}\right)}^4-1\\ {{\displaystyle i}}_a={\left(1+0.0225\right)}^4-1\\ {{\displaystyle i}}_a={(}^{1.0225}-1\\ {{\displaystyle i}}_a=1.09308-1\\ {{\displaystyle i}}_a=0.09308\\ {{\displaystyle i}}_a=9.31\%\end{array}$

The effective interest rate is 9.31%.

Case 3:

Using the information given in the question, one knows that the interest rate is compounding on a continuous basis. After inserting the value of nominal interest rate as calculated above in the effective interest rate formula:

$\begin{array}{l}{{\displaystyle i}}_a=e^r-1\\ {{\displaystyle i}}_a=e^{\frac{17}{800}}-1\\ {{\displaystyle i}}_a=e^{0.02125}-1\\ {{\displaystyle i}}_a=1.02148-1\\ {{\displaystyle i}}_a=0.02148\\ {{\displaystyle i}}_a=2.15\%\end{array}$

Therefore, effective interest rate is 2.15%.