Chapter 8, Problem 17P

Your company plans to borrow $\text{\$13}$ million for 12 months, and your banker gives you a stated rate of 24 percent interest. You would like to know the effective rate of interest for the following types of loans. (Each of the following parts stands alone.)

a. Simple 24 percent interest with a 10 percent compensating balance.

b. Discounted interest.

c. An installment loan (12 payments).

d. Discounted interest with a 5 percent compensating balance.

Summary Introduction

To calculate: The effective rate of simple interest payment at 24% with a compensating balance of 10%.

Introduction:

Effective interest rate:

Also termed as annual equivalent rate, it is the rate actually charged on an investment or a loan over a specific time period.

Simple Interest:

It is the interest computed on the original amount of the loan, that is, the principal amount. It is the easiest way of calculating the interest on a loan.

Answer to Problem 17P

The effective rate of simple interest payment at 24% with a 10% compensating balance is 26.67%.

Explanation of Solution

Calculation of the effective rate of interest with a 10% compensating balance:

$\begin{array}{c}\text{Effective Rate of Interest = }\frac{\text{Interest}}{\text{Principal}-\text{Interest}}\text{×}\frac{\text{Days in the year }}{\text{Days Loan is Outstanding}}\\ =\frac{\$3,120,000}{\$13,000,000-\$1,300,000}\times \frac{360}{360}\\ =\frac{\$3,120,000}{\$11,700,000}\times 1\\ =26.67\%\end{array}$

Working Notes:

Calculation of interest:

$\begin{array}{c}\text{Interest}=\text{Borrowings }\times \text{Interest Rate}\\ =\$13,000,000\times 24\%\\ =\$3,120,000\end{array}$

Calculation of the compensating balance:

$\begin{array}{c}\text{Compensating Balance}=\text{Borrowings }\times \text{Interest Rate}\\ =\$13,000,000\times 10\%\\ =\$1,300,000\end{array}$

Summary Introduction

To calculate: The effective interest rate for the discounted interest:

Introduction:

Effective interest rate:

Also termed as annual equivalent rate, it is the rate actually charged on an investment or a loan over a specific time period.

Discounted Interest:

The loan on which the interest owed is deducted up front is termed as discount interest. The amount that the borrower receives is the net amount of interest.

Answer to Problem 17P

The effective rate of discounted interest is 31.58%.

Explanation of Solution

Calculation of the effective rate of discounted interest:

$\begin{array}{c}\text{Effective Rate of Interest = }\frac{\text{Interest}}{\text{Principal }-\text{ Interest}}\text{×}\frac{\text{Days in the year}}{\text{Days Loan is Outstanding}}\\ =\frac{\$3,120,000}{\$13,000,000-\$3,120,000}\times \frac{360}{360}\\ =\frac{\$3,120,000}{\$9,880,000}\times 1\\ =31.58\%\end{array}$

Summary Introduction

To calculate: The effective rate of interest on the installment loan.

Introduction:

Effective interest rate:

Also termed as annual equivalent rate, it is the rate actually charged on an investment or a loan over a specific time period.

Answer to Problem 17P

The effective rate of interest on the installment loan is 44.31%.

Explanation of Solution

Calculation of the effective rate of interest on the installment loan:

$\begin{array}{c}\text{Effective Interest Rate}=\frac{2\times \text{Annual No}\text{. of Payments}\times \text{Interest}}{\left(\text{Total no}\text{. of payments}+1\right)\times \text{Principal}}\\ =\frac{2\times 12\times \$3,120,000}{\left(12+1\right)\times \left(\$13,000,000\right)}\\ =\frac{\$74,880,000}{\$169,000,000}\\ =44.31\%\end{array}$

Summary Introduction

To calculate: The effective rate of discounted interest with a compensating balance of 5%.

Introduction:

Effective interest rate:

Also termed as annual equivalent rate, it is the rate actually charged on an investment or a loan over a specific time period.

Discounted Interest:

The loan on which the interest owed is deducted up front is termed as discount interest. The amount that the borrower receives is the net amount of interest.

Answer to Problem 17P

The effective rate of discounted interest with a 5% compensating balance is 33.80%.

Explanation of Solution

Calculation of the effective rate of discounted interest with a 5% compensating balance:

$\begin{array}{c}\text{Effective Rate = }\frac{\text{Interest}}{\left(\text{Principal}-\text{Interest}-\text{Compensating Balance}\right)}\text{×}\frac{\text{Days in the year}}{\text{Days Loan is Outstanding}}\\ =\frac{\$3,120,000}{(\$13,000,000-\$3,120,000-\$650,000)}\times \frac{360}{360}\\ =\frac{\$3,120,000}{\$9,230,000}\times 1\\ =33.80\%\end{array}$

Working Notes:

Calculation of the compensating balance:

$\begin{array}{c}\text{Compensating Balance}=\text{Borrowings }\times \text{Rate}\\ =\$13,000,000\times 5\%\\ =\$650,000\end{array}$