TABLE 6

DIRECT-RATIO METHOD: ANNUAL (Nominal) Interest RateS ACCORDING TO (1) NUMBER OF TIMES
SERVICE CHARGE IS CONTAINED IN AMOUNT Borrowed, and (2) NUMBER OF MONTHS
IN INSTALLMENT PERIOD!

This particular table can be used only when payments are to be made monthly.
'Number of times the amount borrowed is of the service charge.

[From the Miami Business Review, May 1957]

THE COST OF CONSUMER INSTALLMENT CREDIT

(By Dr. Joseph Mayer, professor of economics and chairman of the department and Dr. R. E. Glos, dean of the School of Business Administration, University of Miami)

Recent figures released by the Federal Reserve Board estimate that on April 30 millions of people in these United States owed some $312 billion classified as consumer installment credit. This type of debt, once viewed with strong disapproval and thought of mainly as a device for circumventing legal interest rates, is now generally recognized by consumers as a convenient method of payment. The conventional type loan involves a lump-sum repayment at the end of a certain number of months or years. Since wages and salaries are received periodically, monthly or oftener, it is more convenient to pay for certain consumer goods in installments, at the same time that other bills are being paid, out of current income.

Much has been written in recent years about the value of installment credit to mass merchandising and also about the potential dangers inherent in the huge sum presently owed. Surprisingly, the literature on this subject contains only occasional references to the cost involved. Actually, consumers are paying somewhere in the neighborhood of $5 billion a year in interest and other charges connected with the extension of installment credit.

Everyone knows that it costs money to borrow money. Banking and financing services, like other services, have to be paid for if the consumer wishes to make use of them, such as the charges now regularly paid to the banks for the maintenance of checking accounts. The bookkeeping involved in calculating, recording, and policing installment payments is naturally more elaborate and costly than that involved in receiving lump-sum payments at simple interest. However, the calculation of the added costs has, in many instances, been regarded as a trade secret of the finance companies or as too complicated for the average consumer to understand. Financial institutions should advise consumers making use of installment credit how much of the payment is principal and how much interest cost if for no other purpose than for proper income tax accounting.

Consumers have, of course, a right to know the relative costs for servicing a debt by one credit agency under a certain plan of installment payments as compared with the charges of another agency with a different plan. The family that will shop grocery and clothing stores for the best buy should do the same thing when goods on an installment basis are purchased. In the past, the layman has frequently found comparison of servicing costs difficult if not impossible, and this situation has not entirely disappeared. It is gratifying that many consumers today insist on finding out what the costs will be of one plan or another. At the same time, cerdit agencies, because of keener competition among themselves, are endeavoring to package their payment plans in more attractive and understandable forms.

There are still some individuals whose only criterion of whether or not to make a purchase on the installment plan is whether the amount of the downpayment can be met and whether the family budget can stand the added monthly payments. There are still some situations in which it is impossible to determine the cost of installment credit in that the goods are quoted solely as so much down and so much per month. By and large, however, the average consumer can compute the effective interest rate on money borrowed or purchases made on the installment basis even though he may have no more than a rudimentary knowledge of mathematics.

EFFECTIVE INTEREST VERSUS SIMPLE INTEREST

The first step in an understanding of the cost of installment credit is to distinguish between simple interest and the effective interest rate. The average consumer need not master any financial mathematics to appreciate the difference between a payment of 8 percent simple interest on a $100 loan for a year and the effective interest charge on a $100 loan, at the same 8 percent apparent rate, paid in monthly installments of $9 throughout a year. Although $8 is the cost in either case, in the first instance the borrower had the use of $100 for 12 months, but with installment payments he had the use of $100 for 1 month only and after that a decreasing sum as successive payments of $9 were made.

This basic relationship between the effective and the simple interest rate in ordinary installment credit is easy to understand. Effective rates are about twice the simple rates for uniform payments on a loan at regular time intervals. This is because the money borrowed and paid off in regular installments is in fact available to the borrower only about half the time as compared with the full-time use to the borrower of the traditional loan. Expressed otherwise, the borrower has the use, over the full term of the installment loan, of only about half the money borrowed and thus his effective interest rate is nearly twice what it is in lump-sum repayment at the end of the term.

The question may be raised, why is the usable loan not exactly one-half of the sum borrowed? To understand this, take a loan of $120, without interest, payable in $10 installments throughout a year. On a monthly payment basis, the borrower has the use of $120 the first month only, then $110 the second month, $100 the third, and so on down until the loan is paid. If we add these monthly loans together ($120+110+100+90+80+70+60+50+40+30+20+10) and divide by 12 to get the usable loan, the result is $65, instead of $120, which is a fraction more than half the loan.

The reason for the $65 rather than $60 may be more apparent when the loan is viewed in another light. The following table summarizes the months for which the full loan was available.

TABLE I.-Number of months full loan is available when repaid with equal monthly installments

Number of months of

use of full loan

612

12

Certain months paired

January $120_

February $110+Dec. $10-$120_

March $100+Nov. $20-$120_.

April $90+Oct. $30-$120_

May $80+Sept. $40-$120.
June $70+Aug. $50-$120.
July $60----

1

1

1

1

1

1

1/2

seen that each of the By adding January to July, which represents

If we pair 10 of the 12 months as indicated, it is five pairs represents a full use of the loan for a month. these five pairs, we get 6 months of full use of the loan. half a month of full use (the same as a full month of half use), is left over. Thus instead of 12 months full use of the loan, we have 6 months out of 12 of such use.

Dividing 12 by 61⁄2 we arrive at the usable percentage of the total loan, which is 0.5417. Shown below, in the second column of table II, are selected percentages of the total loan, based on the indicated payments required to liquidate the loan, that yield the usable sum. It will be noted that as the number of payments increases, the percentages begin to approach the 50 percent figure that might have been expected from a snap conclusion on the average amount of a loan available to the borrower when regular installment payments are made.

TABLE II.-Percentages and multipliers to compute usable loan and effective interest for selected payments

The figures in the third column of table II are the reverse or the reciprocal of the percentages in the second column. Thus for 0.625 or five-eights we have 8/5 or 1.60. The significance of these reciprocals is that they represent multi

pliers through which the effective interest rate is secured from the simple interest rate for each number of payments indicated.

For example, to get the effective rate from a simple rate of 6 percent, with four payments to liquidate the loan, the 6 percent is multiplied by 1.60. The result is 9.6 percent as the effective rate. For 12 payments, the multiplier is 1.85, which gives an 11.1 percent effective rate compared with the 6 percent simple rate. It will be noted here that, as the number of payments increases, the effective interest rate gets closer to twice the simple interest rate, a conclusion previously stated as a first approximation.

COST OF INSTALLMENT CREDIT

With this basic understanding of the difference between effective interest and simple interest, we are ready to compute the cost of installment credit. To do so, certain preliminary calculations may be needed in connection with which a specific case will be cited.

Assume that a $300 refrigerator is offered for sale on an installment basis for a down payment of $50 and payments of $14 a month for 24 months. Since the total payments amount to $336, which is $36 more than the cash price, and since this sum is $18 a year (2 years are involved in repayment), it might be assumed that the financing charge was 6 percent a year ($300 x 6 percent=$18). However, to arrive at the correct cost of this installment credit, the following steps should be taken:

1. Determine the amount of credit that is actually being extended. When a cash loan is secured this computation is obvious. When buying on an installment basis, the amount may need to be determined. In the illustration given, since $50 in cash was required as a down payment, the amount of the credit extended was $250 ($300-$50).

2. Next determine the overall cost of the credit. This is the total amount to be paid over the life of the loan in relationship to the total amount of credit extended at time of loan or purchase. Again referring to the illustration, the cost of the credit is $86. Twenty-four payments at $14 each equal $336 while the credit extended was only $250.

3. A nominal rate, without regard to the time factor, can now be computed by dividing the cost of the loan by the amount of credit extended. If $86 (the cost of the loan) is divided by $250 (the amount of credit extended) a nominal rate of 34.4 percent will result.

4. The next step is to reduce or increase the nominal rate, computed without regard to the time factor, to 1 year, which is the base of the simple interest rate. This is an easy computation, as the nominal rate can be divided by the number of monthly payments and then multiplied by 12 or otherwise adjusted to a yearly basis. In the illustration, since 2 years are involved, it is obvious that a 1-year simple interest rate would be one-half of the 34.4 percent or 17.2 percent per year.

5. The last calculation requires the conversion of the simple rate into an effective interest rate. This may be done through rather complicated mathematics, but the pinpoint accuracy thus obtained would be of little use to the layman. There would be but minor and somewhat insignificant differences from the answers secured by way of the multipliers listed in table II, the simple formula for which is as follows:

Effective interest rate simple interest rate x double the number of installments

number of installments plus 1

For the purchase of the refrigerator, 17.2 percent X 48 divided by 25 provides an answer of 33 percent, which is the really effective percentage cost on money borrowed to make this installment purchase.

For those who prefer symbols, this formula can be expressed as

r (2n)
E=
n+1

when E-effective rate, r=simple interest rate, and n=number of installments. Another version of this same calculation, which eliminates some of the intermediate steps, is