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DISCOUNT INTEREST

The interest rate on a bank loan is often calculated on a discount basis. Similarly, when companies issue commercial paper, they also usually quote the interest rate as a discount. With a discount interest loan, the bank deducts the interest up front. For example, suppose that you borrow $100,000 on a discount basis for 1 year at 12 percent. In this case the bank hands you $100,000 less 12 percent, or $88,000. Then at the end of the year you repay the bank the $100,000 face value of the loan. This is equivalent to paying interest of $12,000 on a loan of $88,000. The effective interest rate on such a loan is therefore $12,000/$88,000 = .1364, or 13.64 percent.

Now suppose that you borrow $100,000 on a discount basis for 1 month at 12 percent. In this case the bank deducts 1 percent up-front interest and hands you Face value of loan (1 Ј quoted annual interest rate ) number of periods in the year = $100,000 (1 Ј .12) = $99,000 12

At the end of the month you repay the bank the $100,000 face value of the loan, so you are effectively paying interest of $1,000 on a loan of $99,000. The monthly interest rate on such a loan is $1,000/$99,000 = 1.01 percent and the compound, or effective, annual interest rate on this loan is 1.010112 Ј 1 = .1282, or 12.82 percent. The effective interest rate is higher than on the simple interest rate loan because the interest is paid at the beginning of the month rather than the end.

The general formula for the equivalent compound interest rate on a discount interest loan is _ 1 _m Effective annual rate on a discount loan = Ј 1 1 Ј quoted annual interest rate m

where the quoted annual interest rate is stated as a fraction (.12 in our example) and m is the number of periods in the year (12 in our example).

INTEREST WITH COMPENSATING BALANCES

Bank loans often require the firm to maintain some amount of money on balance at the bank. This is called a compensating balance. For example, a firm might have to maintain a balance of 20 percent of the amount of the loan. In other words, if the firm borrows $100,000, it gets to use only $80,000, because $20,000 (20 percent of $100,000) must be left on deposit in the bank.

If the compensating balance does not pay interest (or pays a below-market rate of interest), the actual interest rate on the loan is higher than the stated rate. The reason is that the borrower must pay interest on the full amount borrowed but has access to only part of the funds. For example, we calculated above that a firm borrowing $100,000 for 1 month at 12 percent simple interest must pay interest at the end of the month of $1,000. If the firm gets the use of only $80,000, the effective monthly interest rate is $1,000/$80,000 = .0125, or 1.25 percent. This is equivalent to a compound annual interest rate of 1.012512 Ј 1 = .1608, or 16.08 percent.

In general, the compound annual interest rate on a loan with compensating balances is Effective annual rate on a = (1 + actual interest paid )m Ј 1 loan with compensating balances borrowed funds available where m is the number of periods in the year (again 12 in our example).