Your firm is considering the following three alternative bank loans for $1,000,000:

a) 10 percent loan paid at year end with no compensating balance
b) 9 percent loan paid at year end with a 20 percent compensating balance
c) 6 percent loan that is discounted with a 20 percent compensating balance requirement

Assume that you would normally not carry any bank balance that would meet the 20 percent compensating balance requirement. What is the rate of annual interest on each loan?

I don't have a clue how to solve this problem

To find the rate of annual interest on each loan, we need to calculate the effective interest rate for each option. The effective interest rate takes into account the additional costs or benefits associated with the loan terms.

Option a) 10 percent loan paid at year end with no compensating balance:
Since there is no compensating balance requirement, the interest rate remains at 10 percent. This means the rate of annual interest is 10 percent.

Option b) 9 percent loan paid at year end with a 20 percent compensating balance:
In this case, the loan has a compensating balance requirement of 20 percent. This means that 20 percent of the loan amount ($1,000,000) must be kept in the bank as a balance. So, $200,000 ($1,000,000 * 20%) would be held as a compensating balance, while the company receives a loan of $800,000 ($1,000,000 - $200,000). Therefore, to calculate the effective interest rate, we need to factor in the loss of interest on the compensating balance.
The formula to calculate the effective rate on a loan with a compensating balance is:
Effective Interest Rate = (Stated Interest Rate) / (1 - Compensating Balance Rate)
In this case, the effective interest rate for option b is:
Effective Interest Rate = 9% / (1 - 20%)
= 9% / 0.8
= 11.25%
So, the rate of annual interest for option b is 11.25%.

Option c) 6 percent loan that is discounted with a 20 percent compensating balance requirement:
Similar to option b, this loan also has a 20 percent compensating balance requirement. In this case, the loan is discounted, which means the bank deducts the interest upfront and gives the company the remaining amount. To calculate the effective interest rate, we need to consider the interest deduction and the loss of interest on the compensating balance.
The formula to calculate the effective rate on a discounted loan with a compensating balance is:
Effective Interest Rate = (Stated Interest Rate) / (1 - Compensation Balance Rate) * (1 / (1 - Stated Interest Rate))
In this case, the effective interest rate for option c is:
Effective Interest Rate = 6% / (1 - 20%) * (1 / (1 - 6%))
= 6% / 0.8 * 1.06
= 7.875%

So, the rate of annual interest for option c is 7.875%.

To summarize:
- Option a) 10 percent loan paid at year end with no compensating balance has an interest rate of 10%.
- Option b) 9 percent loan paid at year end with a 20 percent compensating balance has an interest rate of 11.25%.
- Option c) 6 percent loan that is discounted with a 20 percent compensating balance requirement has an interest rate of 7.875%.