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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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