Chapter 4, Problem 93P

To determine

(a)

Effective annual interest rate.

Answer to Problem 93P

Effective annual interest rate is 12.36%.

Explanation of Solution

Given information:

Loan amount: $5,000

Interest rate: 12% per annum, compounded semi-annually.

Loan amount is $5,000 and Rate of Interest is 12% compounded semi-annually.

Effective annual interest rate is calculated as periodic interest rate expressed in terms of a year.

Effective Interest rate is calculated using the following formula:

$\begin{array}{l}{\text{E=(1+i/n)}}^{\text{n}}\text{-1}\\ \\ \text{Where;}\\ \\ \text{i=Annual interest rate}\\ \\ \text{n=Number of periods}\\ \\ \text{In the given scenario, }\\ \\ \text{i = 6\% since interest is compounded semi-annually}\\ \\ \text{n = 2 since interest is compounded monthly}\\ \\ \text{Substituting values in the above equation, we get,}\\ \\ \text{E = (}1+6\%{)}^2-1\\ \\ \text{E = 12}\text{.36\%}\end{array}$

Substituting the above values in the equation we get effective interest rate as 12.36%.

Conclusion:

Hence the Effective annual interest rate is calculated.

To determine

(b)

Amount of semi-annual payment.

Answer to Problem 93P

Amount of semi-annual payment is $ 1442.96.

Explanation of Solution

Given information:

Loan amount: $5,000

Interest rate: 12% per annum, compounded semi-annually.

Duration: 4 Periodic repayments.

The periodic payment to be made is calculated as per the following formula:

$\begin{array}{l}P=(R/(1-{(}^1)L\\ \\ Where;\\ \text{P=Periodic Repayment}\\ \\ \text{R=Periodic Rate of Interest }\\ \\ \text{N= Duration of the Loan}\\ \\ \text{L=Loan Amount}\\ \\ \text{P = }\left(\text{0}\text{.0618 / (1 - }\left({\left(1+0.0618\right)}^{-4}\right)\right)\times \text{5,000}\\ \\ \text{P = \$ 1448}\text{.91}\end{array}$

Loan amount is $5,000, Interest rate per period is 12.36% per annum, compounded semi-annually, and Duration is 4 repayments.

Substituting values in the above equation, we get the periodic payment as $1,448.91.

Conclusion:

Hence the amount of monthly payment is calculated.

To determine

(c)

Total interest paid.

Answer to Problem 93P

Total interest paid is $795.64.

Explanation of Solution

Given information:

Loan amount: $5,000

Interest rate: 12% per annum, compounded semi-annually.

Duration: 4 Periodic repayments.

The periodic payment to be made is calculated as per the following formula:

$\begin{array}{l}P=(R/(1-{(}^1)L\\ \\ Where;\\ \text{P=Periodic Repayment}\\ \\ \text{R=Periodic Rate of Interest }\\ \\ \text{N= Duration of the Loan}\\ \\ \text{L=Loan Amount}\\ \\ \text{P = }\left(\text{0}\text{.0618 / (1 - }\left({\left(1+0.0618\right)}^{-4}\right)\right)\times \text{5,000}\\ \\ \text{P = \$ 1448}\text{.91}\end{array}$

Loan amount is $5,000, Interest rate per period is 12% per annum, compounded semi-annually, and Duration is 4 repayments.

Substituting values in the above equation, we get the periodic payment as $1,448.91.

Total interest paid is calculated using the following formula:

$\begin{array}{l}\text{I = }\left(\text{P x N}\right)\text{ – L}\hfill \\ \hfill \\ \text{Where:}\hfill \\ \hfill \\ \text{I = Interest paid}\hfill \\ \hfill \\ \text{P = Periodic repayment}\hfill \\ \hfill \\ \text{N = Number of repayments}\hfill \\ \hfill \\ \text{L = Original Loan amount}\hfill \\ \hfill \\ \text{Substituting values,in the above equation, we get;}\hfill \\ \hfill \\ \text{I = }\left(\text{\$1,448}\text{.91 x 4}\right)\text{ - \$5,000}\hfill \\ \hfill \\ \text{I = \$ 795}\text{.64}\hfill \end{array}$

Hence the total interest paid is $795.64.

Conclusion:

Hence the total interest paid is calculated.