Effective Cost of Borrowing The effective cost of borrowing is the cost of financing, considering both

Effective Cost of Borrowing The effective cost of borrowing is the cost of financing, considering both

direct and indirect costs. This effective cost is the cost of funds for a given period, the duration of time over which interest is paid and the end of which compounding (if there is compounding) is calculated. For example, if we borrow $1,000 today and must repay it plus $50 after three months, the period is three months and the cost of funds expressed in percent is $50/$1,000 = 5% for the period of the loan. Suppose we borrow $1,000 today and must repay it at the end of the year, plus 5% each quarter, with interest compounded at the end of each quarter. The “period” is three months and the amount repaid after one year is $1,000 compounded four periods at 5% per period or $1,000

× (1 + 0.05) 4 = $1,215.51. To compare alternative forms of financing that may have different

terms, including different periods, we convert the effective cost of financing into a common unit of time—by convention, one year. Return- ing to our last example, suppose we use this same financing arrange- ment throughout a year. At the end of the first three months we borrow the amount we owe—the $1,050—for three more months. At the end of the second three months, we owe $1,050 plus the interest on $1,050:

Amount owed = $1,050 + (0.05)$1,050 = $1,102.50 Borrowing once again for three more months, at the end of the second

three-month period we owe $1,102.50(1.05) = $1,157.63. Borrowing again for the last three months, at the end of the year we owe $1,157.63 (1.05) = $1,215.51.

If we borrow $1,000 for one year and pay 5% interest every three months, we pay $1,215.51 − $1,000.00 = $215.51 interest. Comparing the $215.51 interest with the amount we borrowed, the effective cost over the year is:

Effective cost of borrowing = -------------------------- = 0.2155 or 21.55% per year

By stepping through this example with interest compounded every three months, we see the effect of compounding: 5% for three months compounded over one year translates into 21.55% for one year. We can look at this another way by combining our compounding into one step:

Interest = $1,000 1.05 ( ) 1.05 ( ) 1.05 ( ) 1.05 ( ) – $1,000

 Balance due after one year

Principal

MANAGING WORKING CAPITAL

The effective annual rate (EAR) on the borrowing is the ratio of the interest paid in one year to the principal, the amount borrowed:

interest EAR = -----------------------

principal

Substituting the interest on the borrowing for a year,

4 EAR = ------------------------------------------------------------------------ $1,000 1 ( + 0.05 ) – $1,000

which we can break into two fractions,

4 EAR = ----------------------------------------------- $1,000 1 ( + 0.05 ) – ------------------ $1,000

$1,000 Simplifying the fractions, we arrive at the formula for the effective

annual rate: EAR = (1 + 0.05) 4 − 1 = 0.2155 or 21.55% per year Designating the rate per compounding period r and the number of

compounding periods within a year t, we have in general terms:

(21-1) Now let’s see why the effective cost, as expressed by EAR, is the true

EAR = (1 + r) t −1

cost of borrowing. Annual Percentage Rate

The costs of borrowing are often stated on an annualized basis by mul- tiplying the rate per compounding period by the number of compound- ing periods in a year. This is done partly because of custom and partly to simplify matters. The annual percentage rate (APR) is the annualized cost of financing (or lending, if you are on the other side of the transac- tion) without considering the compounding of interest. The APR is the product of the rate per period, r, and the number of periods in a year, t:

(21-2) This APR is also referred to as the nominal rate or the stated rate. If

APR = r ×t

$50 interest is paid every three months on a loan of $1,000, the APR is 5% times 4 = 20%.

Management of Short-Term Financing

The APR is simple to compute, but it is not very useful for compar- ing costs of alternative financing arrangements since it ignores com- pounding. For example, if 1% interest is paid each month, the APR is:

APR = 0.01(12) = 0.1200 or 12% per year

But each month this loan’s effective annual cost with compounding is: EAR = (1 + 0.01) 12 − 1 = 0.1268 or 12.68% per year The APR understates the effective cost of financing if interest is com-

pounded each period during the year. The costs of short-term financing, however, are not always straight- forward. The costs of short-term financing may be direct, such as inter- est or commitment fees, or indirect, such as discount interest and com- pensating balances (we explain these terms later in the chapter). Managers must understand how to calculate the cost of financing for the alternative ways in which these costs may be stated so that we can compare them. In the remainder of this section, we demonstrate how to calculate EAR for a variety of loans.