Chapter 3, Problem 53P

To determine

(a)

The effective interest rate.

Answer to Problem 53P

The Effective Interest rate in this case is 4.06%.

Explanation of Solution

Given information:

• The nominal annual interest rate = 4%
• Compounded every three months

• 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

• 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:

Using the information given in the question, one knows m = 4 since the interest rate is compounding the subperiods on a quarterly basis. We need to find the effective annual interest rate to using the value of the nominal annual interest rate given in the question:

$\begin{array}{l}{{\displaystyle i}}_a={\left(1+\frac{r}{m}\right)}^m-1\\ {{\displaystyle i}}_a={\left(1+\frac{r}{400}\right)}^4-1\\ {{\displaystyle i}}_a={\left(1+\frac{4}{400}\right)}^{12}-1\\ {{\displaystyle i}}_a={\left(1+0.01\right)}^4-1\\ {{\displaystyle i}}_a={\left(1.01\right)}^4-1\\ {{\displaystyle i}}_a=1.0406-1\\ {{\displaystyle i}}_a=0.0406\\ {{\displaystyle i}}_a=4.06\%\end{array}$

Therefore, effective annual interest rate is 4.06%.

To determine

(b)

The effective interest rate.

Answer to Problem 53P

The effective Interest rate in this case is 4.040584%.

Explanation of Solution

Given information:

• The nominal annual interest rate = 4%
• Compounded every daily

• 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

• 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:

Using the information given in the question, one knows m = 4 since the interest rate is compounding the subperiods on a daily basis. We need to find the effective annual interest rate to using the value of the nominal annual interest rate given in the question:

$\begin{array}{l}{{\displaystyle i}}_a={\left(1+\frac{r}{m}\right)}^m-1\\ {{\displaystyle i}}_a={\left(1+\frac{r}{36500}\right)}^{365}-1\\ {{\displaystyle i}}_a={\left(1+\frac{4}{36500}\right)}^{365}-1\\ {{\displaystyle i}}_a={\left(1+0.000109\right)}^{365}-1\\ {{\displaystyle i}}_a={\left(1.000109\right)}^{365}-1\\ {{\displaystyle i}}_a=\text{1}\text{.040584}-1\\ {{\displaystyle i}}_a=0.\text{040584}\\ {{\displaystyle i}}_a=4.0584\%\end{array}$

Therefore, effective annual interest rate is 4.040584%.

To determine

(c)

The effective interest rate.

Answer to Problem 53P

Effective Interest rate in this case is 4.08%.

Explanation of Solution

Given information:

• The nominal interest per month = 4%
• 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:

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

Calculation:

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

$\begin{array}{l}{{\displaystyle i}}_a=e^r-1\\ {{\displaystyle i}}_a=e^{0.04}-1\\ {{\displaystyle i}}_a=1.0408-1\\ {{\displaystyle i}}_a=0.0408\\ {{\displaystyle i}}_a=4.08\%\end{array}$

Therefore, effective interest rate is 4.08%.