The article represents a yield formula for calculating irregular installment payments. A fundamental problem in the mathematics of finance is the present valuation of future payments. When they take the form of an annuity of C every period for "n" periods, we have the concise formula P= C(1 -&mul;)/I, which may be expanded in an elementary series. If there is only one future payment, the formula is even simpler, P = Cμ n , and the series is quite as elementary. There are cases in which payments are not all for the same amount. If the variations follow some law, there is still a formula to be had but it becomes more complex. In this fall increasing, decreasing, and deferred annuities, bonds and serial bonds and "balloon note" and "drop payment" installment finance deals. The author proposes to derive an approximation formula for the rate of interest in the general case of future repayments of a present indebtedness, whether they be many or one, equal or unequal and when such a formula, has been obtained, one shall find that it includes both the annuity and single payment formulas as special cases.

The ultimate point of reference in any mathematical development in the field of installment finance is or at least ought to be the compound interest method. It is not implied that pure actuarial theory should be introduced into practical day-to-day usage in installment transactions. But it is asserted that the mathematical relation between pure theory and the various practical short-cuts and simplified methods in use should be clearly established in the literature on the subject. Several of the common formulas for determining the rate of interest in installment payment plans have been shown to be approximations to the actuarially determined rate. The present paper relates the so called "Rule of 78" method of figuring rebates in pay-offs, to the theoretically equitable rebate found by the compound interest method. The "Rule of 78" method has that name because on a 12-payment loan the sum of the integers from 1 to 12 equals 78. It is also known as the "Sum of Digits" method.