Chapter 16, Problem 14QP

Costs of Borrowing. A bank offers your firm a revolving credit arrangement for up to $75 million at an interest rate of 1.65 percent per quarter. The bank also requires you to maintain a compensating balance of 4 percent against the unused portion of the credit line, to be deposited in a noninterest-bearing account. Assume you have a short-term investment account at the bank that pays .49 percent per quarter, and assume that the bank uses compound interest on its revolving credit loans.

a.    What is your effective annual interest rate (an opportunity cost) on the revolving credit arrangement if your firm does not use it during the year?

b.    What is your effective annual interest rate on the lending arrangement if you borrow $40 million immediately and repay it in one year?

c.    What is your effective annual interest rate if you borrow $75 million immediately and repay it in one year?

Summary Introduction

To determine: The effective annual interest rate.

Introduction:

Borrowing cost:

The aggregate value of debt (inclusive of interest and other payments) is termed as borrowing cost. Compensating balance is the amount kept by a company in a bank with lower interest or as non-bearing accounts. It is a part of the loan agreement.

Answer to Problem 14QP

The effective annual rate of interest is 1.97%.

Explanation of Solution

Given information:

A bank offers a loan to Person X’s firm Y. The cost of borrowing on a line of credit agreement is $75 million, the interest rate is 1.65% per quarter, and the required compensating balance is 4% against the portion, which is being unused. The firm Y has a short-term investment account at the bank, which pays 0.49% per quarter.

The interest rate is on a compound interest basis on its revolving credit loans by the bank.

The formula to calculate the effective interest rate:

$\text{Effective}\text{annual}\text{rate}={\left(\text{1+Interest on short-term investments}\right)}^n-1$

Where, ‘n’ is the number of months or year and the given number of months is 12.

Compute the effective annual interest rate:

$\begin{array}{c}\text{Effective}\text{annual}\text{rate}={\left(\text{1+Interest on short-term investments}\right)}^n-1\\ ={\left(\text{1 +}\text{.0049}\right)}^4-1\\ =1.0197-1\\ =0.0197\text{or}1.97\%\end{array}$

Hence, the effective annual rate is 1.97%.

Summary Introduction

To determine: The effective annual interest rate on the loan for $40 million and if it is repayable within a year.

Introduction:

Borrowing cost:

The aggregate value of debt (inclusive of interest and other payments) is termed as borrowing cost. Compensating balance is the amount kept by a company in a bank with lower interest or as non-bearing accounts. It is a part of the loan agreement.

Answer to Problem 14QP

The effective annual interest rate is 6.76%.

Explanation of Solution

Given information:

Firm Y borrows $40 million and repays it within a year; the required compensating balance is 4% against the unused portion of the credit line, firm Y has a short-term investment account at the bank, which pays 0.49% per quarter.

To calculate the effective annual interest rate, the loan’s interest is divided by the total loan value. The interest amount is a part of the opportunity cost of the compensating balance of the loan amount. The compensating balance is determined as the unused portion of the credit balance.

Formulae:

The formula to calculate the opportunity cost:

$\text{Opportunity}\text{cost}=\text{Compensating balance}\times \text{Potential interest rate }$

The formula to calculate the effective interest rate:

$\text{Effective}\text{annual}\text{rate}=\frac{\text{Opportunity cost}+\text{}\text{Interest paid }}{\text{Amount borrowed}}$

Compute the opportunity cost:

$\text{Opportunity}\text{cost = Compensating balance }\times \text{ Potential interest rate }$

$\begin{array}{c}\text{Opportunity cost }=\left(\begin{array}{l}\left(\begin{array}{l}\text{0}\text{.05}\times \left(\text{Bank loan}-\text{Amount}\text{borrowed}\right)\\ \times {\left(\text{1}+\text{}\text{Interest rate}\right)}^n\end{array}\right)\\ -\text{ 0}\text{.05}\times \left(\text{Bank loan}-\text{Amount}\text{borrowed}\right)\end{array}\right)\\ =\left(\begin{array}{l}0.05\times \left(\$75,000,000-40,000,000\right)\\ \times {\left(1.0049\right)}^4-\text{0}.05\times \left(\$75,000,000-40,000,000\right)\end{array}\right)\\ \text{Opportunity cost}=\text{\$34,400}\end{array}$

Hence, the opportunity cost is $34,400.

Compute the interest paid to the bank:

$\text{Interest cost =Amount borrowed }\times {\left(\text{1+}\text{}\text{interest}\text{rate}\right)}^4-\text{Amount borrowed}$

$\begin{array}{c}\text{Interest cost}=\text{ \$40,000,000}\times {\left(\text{1}\text{.0165}\right)}^4-\text{\$40,000,000}\\ =\left(\text{\$40,000,000}\times 1.0676\right)-\text{\$40,000,000}\\ =\$42,704,000-\text{\$40,000,000}\\ =\text{\$2,704,000}\end{array}$

Hence, the interest cost is $2,704,000.

Compute the effective rate of interest:

$\begin{array}{c}\text{Effective}\text{annual}\text{rate}=\frac{\text{Opportunity cost}+\text{}\text{Interest paid }}{\text{Amount borrowed}}\\ =\frac{\$34,400+\$2,704,000}{\$40,000,000}\\ =0.\text{0684, or 6}\text{.846\%}\end{array}$

Hence, the effective annual interest rate is 6.846%.

Summary Introduction

To determine: The effective annual interest rate on the loan for $75 million.

Introduction:

Borrowing cost:

The aggregate value of debt (inclusive of interest and other payments) is termed as borrowing cost. Compensating balance is the amount kept by a company in a bank with lower interest or as non-bearing accounts. It is a part of the loan agreement.

Answer to Problem 14QP

The effective annual interest rate is 6.76%.

Explanation of Solution

Given information:

Firm Y borrows $75 million and repays it within one year.

Note: The compensating balance is applied only to a portion of the credit line, which is unused. Therefore, the effective annual rate of the loan on the full credit line is calculated.

The formula to calculate the effective interest rate:

$\text{Effective}\text{annual}\text{rate}={\left(\text{1+Interest on short-term investments}\right)}^n-1$

Where,

‘n’ is number of months or year and the given number of months is 12.

Compute the effective annual interest rate:

$\text{Effective}\text{annual}\text{rate}={\left(\text{1+Interest on short-term investments}\right)}^n-1$

$\begin{array}{c}\text{Effective}\text{annual}\text{rate}={\left(\text{1 +}\text{.0165}\right)}^4-1\\ ={\left(1.0165\right)}^4-1\\ =1.067-1\\ =0.0676\text{or}6.76\%\end{array}$

Hence, the effective annual rate is 6.76%.