On The Differences Among "Equivalent" Loan Payment Plans

Throughout its many editions, Principles of Engineering Economy, by Eugene L. Grant and, subsequently, in partnership with W. Grant Ireson and Richard S. Leavenworth, provides an example in which a loan can be repaid using one of four different plans: I (interest payments only until the end of the loan period, at which time the principal is repaid); II (equal principal payments, plus interest on the unpaid balance of the loan); III (equal periodic payments); and IV (single payment of principal and accumulated interest at the end of the loan period). These and similar payment plans appear in a number of other books. Insofar as the lender is concerned, the four payment plans are equivalent. However, unless the borrower’s time value of money is identical to the lender’s interest rate, the four plans are not equivalent for the borrower. Here, we explore differences among the four payment plans from both the borrower’s and the lender’s perspectives. Sensitivity analyses are performed for the various payment plans and conclusions are drawn regarding the plans that maximize the borrower’s after-tax present worth and the lender’s after-tax present worth in an inflationary economy. Interestingly, the payment plan that is emphasized in engineering economy courses and found to be most prevalent in practice, a uniform series of loan payments, does not perform as well as other plans when considered from either the borrower’s or the lender’s perspective. Further, the performance of the fourth plan (a lump-sum payment at the end of the loan period) is radically different from the other plans when the lender’s interest rate is greater than the borrower’s time value of money. From the analysis, it is evident that neither the borrower nor the lender should be indifferent when choosing a payment plan from among those considered. Introduction In the early editions of Principles of Engineering Economy, four plans were presented for repaying a $10,000 loan in 10 years with interest at 6%. Plan I consisted of 10 equal annual interest payments of $600 and a $10,000 payment at the end of 10 years. Plan II consisted of 10 equal annual principal payments of $1,000, plus interest payments on the unpaid principal balance. Plan III, the familiar equal-annual-payment plan, consisted of 10 equal annual payments of $1,358.68. Plan IV consisted of a single payment of $17,908.49 after 10 years. As subsequent editions were published, the interest rate in the example changed to reflect economic conditions at the time of publication. The most recent edition uses a rate of 9%. Accompanying the discussion of the example, the present worth of each plan is computed using a range of interest rates; again, the range of rates changed over the years. The point is made that the four plans are equivalent only at the stated interest rate, a rate which we refer to as the equivalent rate. As we studied the results of the calculations involving a range of interest rates, we noted that the rank ordering of present worth values (from largest to smallest) was Plan IV, Plan I, Plan III, and Plan II when the interest rate was less than the equivalent rate. However, the rank ordering was P ge 15924.2 reversed when the interest rate was greater than the equivalent rate. By definition, all four plans had the same present worth when the interest rate was equal to the equivalent rate. From the lender’s perspective, all four plans are defined to be equivalent. As such, the lender is indifferent as to which plan is used to repay a loan. However, from the perspective of the borrower, the four plans are equivalent only if the borrower’s time value of money (TVOM) equals the lender’s interest rate. Since the borrower wishes to minimize the present worth of loan payments, if the borrower’s TVOM is less than the equivalent rate, the borrower prefers to repay the loan using Plan II; alternatively, if the borrower’s TVOM is greater than the equivalent rate, the borrower prefers to repay the loan using Plan IV. (Notice, in neither scenario is Plan III, the familiar equal-annual payment plan, the preferred plan.) To gain a better understanding of the present worth performance of the four payment plans, we studied the effects of the following: lender’s interest rate; income taxes; depreciation; inflation; and the proportion of investment capital borrowed. Specifically, we considered borrowing $100,000 over a 10-year period with a 40% income tax rate, a borrower’s 9% inflation-free or real before-tax TVOM, a borrower’s 15% real before-tax TVOM, and a borrower’s 9% real after-tax TVOM. Finally, we considered a lender’s 9% real after-tax minimum required return on a loan made over a 10-year period. Before-Tax Analysis with Negligible Inflation In performing before-tax analyses of the payment plans in the absence of inflation, we began by using a 9% lender’s interest rate, which is the same rate used for the lender in the Grant, Ireson, Leavenworth 8 edition, The cash flows for the four plans, based on a 9% lender’s interest rate, are shown in Table 1. Letting the borrower’s TVOM be 9%, we computed the borrower’s present worth for lender’s interest rates ranging from 0% to 15%; Figure 1 contains the results. Since the borrower receives $100,000 from the lender and repays the loan over a 10-year period, the borrower prefers the payment plan with the greatest present worth based on the borrower’s TVOM. As expected, when the lender’s interest rate is less than the borrower’s TVOM, Plan IV maximizes the borrower’s present worth; when the lender’s interest rate is greater than the borrower’s TVOM, Plan II maximizes present worth; and when the two rates are equal, all four plans have the same present worth. Hence, if the lender’s interest rate is 12% and the borrower’s TVOM is 9%, then Plan II maximizes the borrower’s present worth. (Henceforth, we refer to the plan that maximizes present worth as the preferred plan.) Before-Tax Analysis when Inflation Is Considered If the lender provides a fixed rate loan, then the borrower can benefit when inflation occurs. With fixed rate loans, loan payments are made in then-current dollars, not constant dollars. Therefore, the real cost of loan payments decreases with increasing inflation. Specifically, if the borrower’s TVOM is a real 9% per year and inflation is 3% per year, then a combined 12.27% interest rate is used to compute the real present worth of the then-current loan payments. Figure 2 illustrates the effect of a 3% inflation rate on the borrower’s present worth of the loan payments. P ge 15924.3 Table 1. Cash flows for a $100,000 loan at a 9% lender’s fixed interest rate. Figure 1. Present worth values for four payment plans based on a 9% borrower’s TVOM and a lender’s fixed interest rate ranging from 0% to 15%. EOY CF(I) CF(II) CF(III) CF(IV) 0 $100,000.00 $100,000.00 $100,000.00 $100,000.00 1 -$9,000.00 -$19,000.00 -$15,582.01 $0.00 2 -$9,000.00 -$18,100.00 -$15,582.01 $0.00 3 -$9,000.00 -$17,200.00 -$15,582.01 $0.00 4 -$9,000.00 -$16,300.00 -$15,582.01 $0.00 5 -$9,000.00 -$15,400.00 -$15,582.01 $0.00 6 -$9,000.00 -$14,500.00 -$15,582.01 $0.00 7 -$9,000.00 -$13,600.00 -$15,582.01 $0.00 8 -$9,000.00 -$12,700.00 -$15,582.01 $0.00 9 -$9,000.00 -$11,800.00 -$15,582.01 $0.00 10 -$109,000.00 -$10,900.00 -$15,582.01 -$236,736.37 -$60,000 -$40,000 -$20,000 $0 $20,000 $40,000 $60,000 0% 3% 6% 9% 12% 15% P re se n t W o rt h Lender's Interest Rate Plan I Plan II Plan III Plan IV