Mathematics of finance.

Abstract

To the student of pure mathematics the term mathematics of finance often seems somewhat of a misnomer since, in solving the problems usually presented in textbooks under this title, the types of mathematical operations involved are very few and very elementary. Indeed, in a first course in the mathematics of finance the development of the most important formulas usually involves no greater difficulties than those encountered in the study of geometric progressions. Whether it is because of this seeming simplicity or because of a tendency to limit the problems to the very simplest kinds, the usual presentation has shown a decided lack of generality and flexibility in many of the formulas and their applications. Since no new mathematical principles are involved, a student who can develop and understand the simplerappearing formulas should be able to develop easily the more general formulas, which are much more useful. And no student should use important formulas whose derivation and meaning, and hence possibilities and limitations, he does not understand. There is a marked preference in many places in mathematics for presenting general definitions and formulas first, with the special cases following naturally from them. Tn trigonometry, for instance, the main importance of the trigonometric functions of an angle is emphasized by presenting first the general definitions of these functions; then the definitions of the functions of an acute angle in terms of the elements of a right triangle follow naturally as special cases. Up to the present time, textbooks in the mathematics of finance have not followed this plan of presentation. The foregoing considerations, plus years of experience in teaching the subject, sometimes with the more general formulas presented first and sometimes with the limited formulas presented first, have caused the author to feel the need of such a presentation as is attempted here. As everyone in this field of work is aware, the major problem is the thorough understanding of annuities and complete facility in their evaluation. The late Professor Glover, whose valuable and comprehensive tables* for use in problems in the field of finance are well known, often remarked that "few teachers of the subject realize the power and facility to be gained from a thorough appreciation of the double superscript notation in annuity formulas." The method of presentation emphasizes the point that very few fundamental formulas are necessary for handling financial problems if these formulas are thoroughly understood and appreciated. Mathematical forms are of inestimable value, as evidenced by their use in solving ordinary quadratic equations, in performing integration in the calculus, in classifying differential equations for solution, in handling many problems connected with infinite series, and in numerous other places familiar only to the accomplished mathematician. Moreover, these forms, if thoroughly mastered, far from reducing the subject to a mere substituting in formulas, reduce the laborious detail that is necessary without them and bring to the subject much significance and effectiveness otherwise unappreciated. Any method of presentation is likely to involve a choice of forms, and usually it is possible to make choices which will emphasize the fundamentals. It is the author's experience that the method of presentation in this text does contribute to an understanding of these fundamentals. In Part One of this text two distinct contributions are aimed at: (1) a somewhat different approach to the study of annuities and (2) a presentation of some of the more recent methods in building-and-loan-association practice. The treatment of annuities is different in two essential respects: (1) the general annuity formulas are presented first, with the so-called simpler annuity formulas following readily as special cases, and (2) the double superscript notation is used in the annuity symbols. The treatment of loans by the direct-reduction plan, which in many places has largely replaced previous methods of building-and-loan-association practice, has been given detailed discussion and illustration. It is usual in many courses in the mathematics of finance to devote several lessons to a discussion of the simpler forms of life insurance. Part Two of this text is designed to meet this need, without going into theorems on probability and other technical details which are necessary to the student specializing in actuarial theory. Basic principles enabling the student to set up the formulas used in the simpler forms of life insurance follow readily the background furnished in Part One. The examples and exercises have been solved by the use of tables in which the interest and annuity forms are given to eight decimal places, and their logarithms to seven decimal places. If tables with fewer decimal places are used, there will, of course, be slight variations from the results given in the text. The author wishes to express his appreciation of the valuable suggestions and criticisms offered by the several members of the staff here at the University of Michigan who have taught the preliminary edition of this text. Mrs. Raiford has given most valuable assistance in reading and checking much of the work done in preparation for publication. In spite of all this help, probably some errors still remain, and to anyone reporting such errors the author will be most grateful.