Reviewed by Charles E. Boynton IV University of Wisconsin-Madison
This book is a collection of reprints of articles, many from actuarial journals, reflecting the historical development of approxi-mation procedures for implicit unknown interest rates in annuity and bond problems. The book is prefaced by an extremely useful interpretive review of that history by the editors.
Today, inexpensive handheld calculators permit anyone capable of pushing a few buttons to approximate the implied unknown interest rate that equates an initial payment amount with one or more repayment amounts, for problems in economics, finance, and accounting. The solution to implicit unknown interest problems in these fields bear such names as the discount rate, the internal rate of return, the yield to maturity, and the effective interest rate.
Why should anyone in these fields be interested in the history of such approximation methods and in original readings in the development of such approximation methods when it is now possible to enjoy excellent approximations in blissful ignorance of the actual solution technique employed by one’s black (plastic) box and its transistors or chips? The reason, of course, is that the process of conceptualization in one field may be both of interest in itself and of use to those engaged in similar conceptualization in another field. For example, in the final selection of the book, the editors point out that in a capital budgeting context, the exact yield to a premium or discount bond problem is the internal rate of return, the conventional approximate yield is similar to the accounting rate of return on average investment, and the current yield is the reciprocal of the payback period of the investment proposal.
The history of interest approximations has involved issues dealing with the concept of return to capital and issues dealing with ap-proximation accuracy. The Sutton (1875) selection is interesting both for its humorous comments on the floating of risky foreign bonds and for its discussion of mean investment period in an analysis of sinking funds. The Hardy (1921) selection, itself a reprint of a 1890 paper, comments on what would now be called risk free return and risk premium in analyzing the income of an insurance company. History also involves people; The DeMorgan (1859) selection discusses the seventeenth century developers John Newton and Michael Dary.
The historical review provided by the editors identifies the principal approaches to the implicit interest problem as applied to annuities and bonds, the advantages and disadvantages of the approaches, and the principal authors involved in the development of (or attack upon) a given approach. A non-actuary will wish to start with that review. Notation is always a problem in dealing with a mathematical history, and appended to their historical review, the editors have rearranged the various formulas of the authors into standard forms that permit the different issues emphasized by the authors to be more readily identified. In the case of band approxi-mations, the standard form involved recognition of an implied functional relationship, namely, the fractional amount of a discount to subtract from face to estimate average principal. Recognition of the functional relationship reduces the approach of different authors effectively to different estimates of the function. The mathe-matical behavior of the function is then analyzed to provide insight into the conditions under which given approaches lead to better or worse approximations.
The authors included in the anthology are Bizley (1962), Cantelli (1907), Craig (1929), DeMorgan (1859), Evans (1944-46), Hardy (1882), Hardy (1921), Hawawini and Vora (1980), Henderson (1907-08), the editors of the Journal of the Institute of Actuaries (1924), Karpin (1967), “M’ (1855), McLauchlan (1874), Newling (1903), Spencer (1904), Steffensen (1916), Sutton (1875), Todhunter (1897), Webb (1930), and Worger (1967).
The editors have chosen not to add new page numbers to the reprints, which will make the book more difficult to use and cite. There is an error in the table of contents: the seventh selection, Hardy (1882), is not listed. This would have been more easily noticed if new page numbers had been added.
