Abstract The yield formulas for irregular installment payments can be used to find the yield or effective rate of interest on an issue of serial bonds. Unequal retirement installments or intervals of payment, deferred first coupons, even maturity premium requirements, with any or all of these the general installment payment formulas will produce a reasonable approximation, usually very close and sufficient for an amortization schedule, to the yield rate. If, however, the serial issue is such that every bond by itself is a simple and complete bond, that is, with no initial coupon irregularities and no premium due on maturity, it is possible to derive a yield formula but which can be used by considering only the maturity payments and ignoring the coupon payments. This will be so regardless of inequalities in amounts and intervals of the serial maturities themselves. The article presents such a formula. The shortening of the summation process can be carried to its limit, and is most easily demonstrated, with a perfectly regular issue, but even with an issue having irregularities, short cuts are still available.

Abstract 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.

Abstract In the July 1952, issue of the "Accounting Review,"under the title "Some Notes on the Bond Yield Problem," the author discussed the application of a modified or improved Newton's correction formula to the problem of finding the accurate yield on a bond, the improved formula having been briefly set forth in an earlier paper by Hugh E. Stelson,' who had drawn it from one of the mathematical journals. This article includes the continuation of numbers of formulas and equations herein from where they stopped in the preceding article. With serial bonds, the problem of finding the initial approximation to the yield rate is more acute than it is with a single bond. This article also discusses the matter appearing in the thirteen pages of Mr. FitzGerald's article in the "Accounting Review," for January, 1953, on accounting for variations in gross profit.

Abstract In a recent article accountant Hugh K. Stelson found the yield on a bond by use of a modified or improved correction formula, apparently published only a few years ago. The new formula is an extension of the long-known Newton's formula for finding the root of an equation by successive corrections to progressively closer. First point, The initial trial rate should meet two standards: it should be as close as possible to the true rate, and it should be a tabular rate, so that the labor of computing can be avoided. If a bond table is also available, use of the formula is still easier. The table should have a rate interval no larger than half percent. The benefits of an initial error no larger than .000625 lie not so much in the digits "625" as in the three zeros. For there is a rule of thumb for Newton's formula in one of his texts on the theory of equations to the effect that if h begins with k zeros, the computation should be carried to 2k decimal places. The foregoing discussion is applicable to short-term bonds, up to say 30 or 40 periods.

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.