Chapter 18, Problem 14QP

a)

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 the company in a bank with lower interest (or as non-bearing accounts). It is a part of a loan agreement.

Answer to Problem 14QP

The effective annual rate of interest is 4.266%.

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 $70 million, interest rate 1.9% per quarter, and the required compensating balance is 4% against the portion which is unused. Person X has a short-term investment account at the bank, which pays 1.05% 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 quarters in a year.

Compute the effective annual interest rate:

$\begin{array}{c}\text{Effective}\text{annual}\text{rate}={\left(\text{1+Interest on short-term investments}\right)}^n-1\\ \text{ = }{\left(\text{1 +}\text{.0105}\right)}^4-1\\ =1.042-1\\ =0.042\text{or}4.266\%\end{array}$

Hence, the effective annual rate is 4.266%.

b)

Summary Introduction

To determine: The effective annual interest rate on the loan for $45 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 the company in a bank with lower interest (or as non-bearing accounts). It is a part of a loan agreement.

Answer to Problem 14QP

The effective annual interest rate is 7.895%.

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 $70 million, interest rate 1.9% per quarter, and the required compensating balance is 4% against the portion which is unused. Person X has a short-term investment account at the bank, which pays 1.05% per quarter. The interest rate is on a compound interest basis on its revolving credit loans by the bank.

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:

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

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:

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

$\begin{array}{c}\text{Opportunity cost }=\left(\begin{array}{l}\left(\begin{array}{l}\text{.04}\times \left(\text{Bank loan }-\text{Amount}\text{borrowed}\right)\\ \times {\left(\text{1 +}\text{}\text{Interest rate}\right)}^n\end{array}\right)\\ -\text{ 0}\text{.04}\times \left(\text{Bank loan }-\text{Amount}\text{borrowed}\right)\end{array}\right)\\ =\left(\begin{array}{l}0.04\times \left(\$70,000,000-45,000,000\right)\\ \times {\left(1.0085\right)}^4-\text{ 0}.04\times \left(\$70,000,000-45,000,000\right)\end{array}\right)\\ \text{Opportunity cost }=\text{ \$34,435}\text{.96}\end{array}$

Hence, the opportunity cost is $34,435.96

Compute the interest paid to the bank:

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

$\begin{array}{c}\text{Interest cost}=\text{ \$45,000,000}\times {\left(\text{1}\text{.019}\right)}^4-\text{\$45,000,000}\\ =\left(\text{\$45,000,000}\times 1.078\right)-\text{\$45,000,000}\\ =\$48,518,710-\text{\$45,000,000}\\ =\text{\$3,518,710}\end{array}$

Hence, interest cost is $3,518,710.

Compute the effective rate of interest:

$\begin{array}{c}\text{Effective}\text{annual}\text{rate}=\frac{\text{Opportunity cost}+\text{}\text{Interest paid }}{\text{Amount borrowed}}\\ =\frac{\$34,435.96+\$3,518,710}{\$45,000,000}\\ =0.\text{07895, or 7}\text{.895\%}\end{array}$

Hence, the effective annual interest rate is 7.895%.

c)

Summary Introduction

To determine: The effective annual interest rate on the loan for $70 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 the company in a bank with lower interest (or as non-bearing accounts). It is a part of a loan agreement.

Answer to Problem 14QP

The effective annual interest rate is 7.81%.

Explanation of Solution

Given information:

Firm Y borrows $70 million and repays it within one year

Note: The compensating balance is applied only to the portion of 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 the number of quarter in a year.

Compute the effective annual interest rate:

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

$\begin{array}{c}\text{Effective}\text{annual}\text{rate}={\left(\text{1 +0}\text{.019}\right)}^4-1\\ ={\left(1.019\right)}^4-1\\ =1.07819-1\\ =0.0781\text{or}7.81\%\end{array}$

Hence, the effective annual rate is 7.81%.