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67 obviously banks use a 360 day year to boost their

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Unformatted text preview: earnings. The effective interest rate on a loan depends on how frequently interest must be paid—the more frequently, the higher the effective rate. We demonstrate this point with two time lines, one for interest paid once a year and one for quarterly payments: Interest Paid Annually: 0 0.25 0.5 0.75 10,000 0 0 1.0 0 1,200.00 10,000.00 11,200.00 The borrower gets $10,000 at t 0 and pays $11,200 at t 1. With a financial calculator, enter N 1, PV 10000, PMT 0, and FV 11200, and then press I to get the effective cost of the loan, 12 percent. Interest Paid Quarterly: 0 10,000 0.25 299.18 0.5 0.75 299.18 1.0 302.47 299.18 10,000.00 10,299.18 Note that the third quarter has 92 days. We enter the data in the cash flow register of a financial calculator (being sure to use the / key to enter 299.18), and we find the periodic rate to be 2.9999 percent. The effective annual rate is 12.55 percent: Effective annual rate, quarterly (1 0.029999)4 1 12.55%. Had the loan called for interest to be paid monthly, the effective rate would have been 12.68 percent, and if interest had been paid daily, the rate would have been 12.75 percent. These rates would be higher if the bank used a 360-day year. In these examples, we assumed that the loan matured in one year but that interest was paid at various times during the year. The rates we calculated would have been exactly the same as the ones above even if the loan had matured on each interest payment date. In other words, the effective rate on a monthly payment loan would be 12.68 percent regardless of whether it matured after one month, six months, one year, or 10 years, providing the stated rate remains at 12 percent. Discount Interest In a discount interest loan, the bank deducts the interest in advance (discounts the loan). Thus, the borrower receives less than the face value of the loan. On a oneyear, $10,000 loan with a 12 percent (nominal) rate, discount basis, the interest is $10,000(0.12) $1,200. Therefore, the borrower obtains the use of only $10,000 $1,200 $8,800. If the loan were for less than a year, the interest charge (the discount) would be lower; in our example, it would be $600 if the loan were for six months, hence the amount received would be $9,400. The effective rate on a discount loan is always higher than the rate on an otherwise similar simple interest loan. To illustrate, consider the situation for a discounted 12 percent loan for one year: 27-20 Chapter 27 Providing and Obtaining Credit Discount Interest, Paid Annually: 0 0.25 0.5 0.75 1.0 10,000 1,200 8,800 0 0 0 10,000.00 With a financial calculator, enter N 1, PV 8800, PMT 0, and FV 10000, and then press I to get the effective cost of the loan, 13.64 percent.6 If a discount loan matures in less than a year, say, after one quarter, we have this situation: Discount Interest, One Quarter: 0 10,000 300 0.25 0.5 0.75 1.0 0 0 0 10,000 9,700 Enter N 1, PV 9700, PMT 0, and FV 10000, and then press I to find the periodic rate, 3.092784 percent per quarter, which corresponds to an effective annual rate of 12.96 percent. Thus, shortening the period of a discount loan lowers the effective rate of interest. Effects of Compensating Balances If the bank requires a compensating balance, and if the amount of the required balance exceeds the amount the firm would normally hold on deposit, then the excess must be deducted at t 0 and then added back when the loan matures. This has the effect of raising the effective rate on the loan. To illustrate, here is the setup for a one-year discount loan, with a 20 percent compensating balance that the firm would not otherwise hold on deposit: Discount Interest, Paid Annually, With 20 Percent Compensating Balance: 0 0.25 0.5 0.75 1.0 10,000 1,200 2,000 6,800 0 0 0 10,000 2,000 8,000 Note that the bank initially gives, and the borrower gets, $10,000 at time 0. However, the bank takes out the $1,200 of interest in advance, and the company must 6Note that the firm actually receives less than the face amount of the loan: Funds received Face amount of loan (1.0 Nominal interest rate). We can solve for the face amount as follows: Face amount of loan 1.0 Funds received . Nominal rate (decimal) Therefore, if the borrowing firm actually requires $10,000 of cash, it must borrow $11,363.64: Face value $10,000 1.0 0.12 $10,000 0.88 $11,363.64. Now, the borrower will receive $11,363.64 0.12($11,363.64) $10,000. Increasing the face value of the loan does not change the effective rate of 13.64 percent on the $10,000 of usable funds. The Cost of Bank Loans 27-21 leave $2,000 in the bank as a compensating balance, hence the borrower’s effective net cash flow at t 0 is $6,800. At t 1, the borrower must repay the $10,000, but $2,000 is already in the bank (the compensating balance), so the company must repay a net amount of $8,000. With a financial calculator, enter N 1, PV 6800, PMT 0, and FV 8000, and then press I to get the effective cost of the discount loan with a compensating balance, 17.65 percent. Installment Loans: Add-On Interest Lenders typically charge add-on interest on automobile and other types of i...
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