Abstract There are three relatively simple methods in common use for determining the rate of interest in installment payment plans. These are known as the Constant Ratio, Series of Payments and Interest at End methods. They have been developed from independent assumption, apparently without any thought that they might be approximations to the more complicated Compound Interest method. In a paper in the American Mathematical Monthly it is shown algebraically that the above methods are approximations to the Compound Interest Method. In order to obtain a better understanding, let us assume temporarily that we know the rate and form a schedule of the payment of the debt. The magnitudes of the rates for the approximate formulas may be compared by means of the areas of the trapezoids. The rate is inversely proportional to the area. The areas for the series of payments, constant ratio, and interest at end methods are in order. This solution is often approximated by means of interpolation in tables. It is the purpose however to compare the compound interest method with the above methods and to present a simple but more accurate approximate formula which can be used without knowledge of the theory of finance.

Abstract The purpose of this article is to apply modifications of Newton's method for approximating the root of an equation to the solution of bond problems which require very accurate determination of the interest rate. Repeated applications of Newton's formula will give any desired degree of accuracy. However, it is not practical to apply Newton's formula more than once since the value obtained by the first application will usually give a value of the interest rate for which the corresponding function can not be found in the table. It is desirable to find a formula which will give in a single application greater accuracy than Newton's formula and which has the advantage of using tabular values obtained from the original estimation of the interest rate. has been developed in several papers in mathematical journals but its value in solving bond problems has not received sufficient attention. The use of Newton's method is explained in an article by N. Lecher.

Abstract 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.

Abstract The article highlights that in problems in the mathematics of finance where it may be desired to find an unknown rate of interest, a uniformity of approach to approximation formulas of considerable accuracy can be achieved for the three fundamental functions and their inverse functions, and also for the bond function, by means of the concept of the total interest growth over the term of the investment, or indebtedness. The article presents various methods to calculate the rate of interest. The total interest, present value of annuity, amount of annuity is first calculated. By means of these relations, the article is able to make substitutions in the usual formulas, obtaining what may be termed indirect formulas, which give rise to series that converge more rapidly than the series obtained by expansion of the original formulas. The new formulas are used as the bases for approximations to the interest rate. The article also presents a table to show the formulas for the calculation of present value of annuity or installment payment problems.