contract, rate of 9.0 per cent was charged instead One family in the study, in which the husband's Savings possible by shopping for credit. Many The failure to shop for credit was, in all probabil- used cars. Six of the families could have saved more than $200 each. While shopping for a car, the prospective buyer Some implications for home economists and sources from which we may obtain credit, including We must learn to be on guard against various tion about costs is usually given reluctantly and, We can also exert our pressure for legislation rate. Reprinted from The Accounting Review Vol. XXX, No. 4, October, 1955 R INTEREST RATES CHARGED ON INSTALLMENT PURCHASES' RALPH R. BOTTS AND FRED L. GARLOCK Agricultural Economists, U. S. Department of Agriculture ETAIL SELLING on the installment plan is a widely accepted practice. Yet few buyers know the interestrate equivalent of the carrying charges they pay for installment credit. That it is usually relatively high no one denies. This situation has led to suggestions from some quarters that sellers be required to specify on their sales contracts the interest-rate equivalent of their service charges, so that buyers will be better able to determine whether they are paying "too much" for the privilege of using installment credit. By the end of 1950, 12 states had passed laws to control installment financing in various ways; but no state requires that the finance charge be stated as an annual interest rate. Perhaps one reason why state laws have not required the posting of equivalent annual interest rates is that several methods of computing rates are in use and all are apparently recognized as valid. This creates a confusing situation. It is desirable, therefore, to review some of the more commonly used methods of computing equivalent rates and try to determine whether certain of them should be favored over others. The following problem is taken as a basis for discussion: An article advertised for $200 cash may be bought on time for $50 down and $20 a month for 8 months. What The authors are indebted to Hugh E. Stelson, Michigan State College, Ralph W. Snyder, of Geo. S. Olive & Co. and Wm. H. Rowe, of the U.S. Dept. of Agriculture, for helpful suggestions and criticisms. For further discussion, see articles by W. P. Mors, in the January and April issues, 1943, and in the October, 1950, and January, 1951 issues of the Journal of Business. Also see article by Johnson and Gregory, in the April, 1953 issue of the same Journal. Eight methods of computing the rate equivalent of the service charge in the illustrative problem may be considered. These methods may be grouped into two broad categories, as shown in table 1. First is a group of four methods which apparently were developed from various practices of sellers or lenders in crediting the service charge on installment contracts to income. These are shown under the heading "Accounting Approach." Second is another group of four methods, all of which are derived by methods of computThey appear under the heading "Presenting the present value of future payments. Value Approach." For further discussion, see the Installment Mathematics Handbook, M. V. Ayres (Ronald Press, New York, 1946). TABLE 1 BALANCES OUTSTANDING, BY MONTHS, AND ANNUAL RATE EQUIVALENTS OF THE CARRYING CHARGE! 1 Cash price of article $200. On time, it may be bought for $50 down and $20 a month for 8 months. Carrying charge is equal to $10 or $50+(8×$20)-$200. Finance charge distributed over the 8 months at rate of $1.25 per month; therefore principal is reduced by only $18.75 per month. The finance charge of $10 is added to cash price ($200) less the down payment ($50), making the beginning balance $160. The latter figure is also 8X$20. This plan is also known as the "Series-of-payments" plan. Monthly rate of 1.457 per cent applied to outstanding balance. Interest for first month is $2.19 (or 0.01457 X$150), leaving $17.81 for reduction of principal. Finance charge of $10 is deducted from first payment of $20, leaving $10 for reduction of principal; therefore the balance at beginning of second month is $150-$10 or $140. The sum of the digits from 1 to 8 (the number of payments) is 36; therefore 8/36 of the finance charge, or $2.22, is taken from first payment, leaving only $17.78 to be applied toward principal. During the second month, 7/36 of the $10 or $1.94 is earned, leaving $18.06 for reduction of principal. 7 All the finance charge ($10) is taken out of last payment, leaving only $10 for reduction of principal. Finance charge (of $10) divided by one-twelfth of total shown just above annual rate, except for present-value plan (see footnote 9). For example, one-twelfth of $710=$59.17 and $10÷$59.17-0.169 or 16.9 per cent for the Priority Plan. This rate (0.175) is for the small-loan plan. It is 12×0.01457. It may also be obtained by dividing $10 by onetwelfth of $686.44. The rate for the present-value plan is 0.190. The only difference between the small-loan and present-value plans is that under the latter the monthly rate (0.01457) is converted to an effective annual rate, that is, (1.01457)-1=0.1896 or approximately 19 per cent. As shown, the monthly balances are the same under both plans. THE ACCOUNTING APPROACH The group of methods described by the heading "Accounting Approach" includes the following: The priority, or yield-minimum, plan, under which the finance charge is considered to be deducted from the first payment. If this charge is more than the periodic payment, the excess is deducted from the second payment before the remainder is applied toward reduction of principal. The constant-ratio, or uniform, plan, under which the finance charge is considered as being divided equally over the installment period; that is, the seller or lender credits his income account with an equal part of each installment. The direct-ratio, or 12/78, plan, under which the finance charge is considered as being apportioned over the installment term in decreasing amounts; that is, the seller or lender credits a decreasing proportion of the periodic payment to his income account. The residuary, yield-maximum, or Merchant's, plan, under which the finance charge is considered • Snyder's Essential Business Mathematics (1947 edition), pp. 182-183; also Cassidy and Robusto's Business Mathematics (1952 edition), p. 98. to be deducted from the final payment or, if it amounts to more than the periodic payment, any excess is deducted from the next-to-last payment. Under this plan, the seller or lender therefore credits his income account out of the last installment(s). By each of these methods, the service charge is considered to be $10 in the illustrative problem. In each case, the account balance of the purchaser is considered to start at $150 and to be reduced monthly during the term of the contract as shown in table 1. However, the amounts by which the account balances are reduced vary according to the method used in taking the payments into income, as shown in table 2. From the total of the account balances (table 1) and the service charge, the rates according to the four methods described above may be computed as follows: 150.00 18.75 1.25 20 131.25 18.75 1.25 20 112.50 18.75 1.25 20 8 18.75 18.75 1.25 20 Priority Direct-ratio method |