Single Payment Interest

A single payment loan is a speciﬁed amount borrowed at the beginning of the loan period; it is repaid plus interest at the end of the period. The interest on this type of loan is referred to as single payment interest and is paid at the same time the loan is repaid.

Suppose you borrow $100,000 for six months, with an interest rate of 6% for this six-month period. After six months you repay the $100,000 plus $6,000 interest. The cost of the ﬁnancing is $6,000 for the six months. To put this cost on an annual basis we need to consider the compounding effect for the second six months in the year. In our example, r = 6% and t = 2, and the effective annual rate is:

EAR = (1 + 0.06) 2 − 1 = 0.1236 or 12.36% per year Borrowing under these terms for one year means that you pay interest of

12.36% on the $100,000, or $12,360. Discount Interest

A discount loan is a loan in which the proceeds—the funds you have available to use—are a portion of the stated loan amount. The interest is paid up front, deducted from the funds at the beginning of the loan.

MANAGING WORKING CAPITAL

Discount interest is the difference between the amount that must be paid back and the amount that you have available to use.

Suppose you borrow $200,000 from a bank for four months. If this is a discount loan with a rate of 5%, you have available for your use 95% of the loan amount, or $190,000. You use the $190,000 for the period of the loan and then pay the full $200,000 at the end of the loan period. The $10,000 difference is the interest on the loan—the discount interest. You effectively pay $10,000/$190,000 = 0.0526 or 5.26% for the use of the $190,000 for four months.

The effective annual rate of a discount loan is calculated in the same way as the EAR of a single payment loan once you have ﬁgured out r, the effective cost for the period. In this example, r = 0.0526 and there are three four-month periods in a year. The EAR is:

EAR = (1 + 0.0526) 3 − 1 = 1.1662 − 1 = 0.1662 or 16.62% per year The only difference in calculation between the single payment and

the discount loan cost is how we determine r: For the single payment we determine r using the end-of-period interest, whereas for the discount loan we use the amount of the discount relative to the funds available for use.

Suppose you are given only the discount rate and the time to matu- rity for the loan. You can still determine the effective annual rate with- out knowing the amount of the loan. The rate per period is:

costs funds available

r = ---------------------------------------

If there is a 1% discount (the cost) we have use of 99% of the funds (funds available). If there is a 5% discount, we have use of 95% of the funds. And so on. If we let d represent the discount rate, the rate per period is:

d r = ------------

(21-3)

For example, if there is a 1% discount, the funds available are 99% of the face value of the loan and the rate per period is:

r = 0.01 ----------- = 0.0101 or 1.01% per period

Management of Short-Term Financing

EXHIBIT 21.2 Effective Cost per Period of Loans for Alternative Discount

Percentages

If the discount rate is 5%, = -----------

0.05 r

= 0.0526 or 5.26%per period

0.95 which is what we ﬁgured out earlier by using the comparison of the dis-

count and the funds available. The greater the discount rate, the higher the effective cost per period. We can see this in Exhibit 21.2, where the effective cost is plotted against the discount percentage. For larger discounts, the effective cost is larger than the discount percentage. As another example, a 15% discount results in an effective cost of r = 0.15/0.85 = 17.65% per period.

Single Payment Interest