Knowledge that Transforms

theory of finance and financial management. New and updated examples help motivate the reader by illustrating the usefulness of theory.

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Professors Alberts and Archer provide a valuable addition to our understanding of capital markets and how resources are allocated to firms of different asset sizes. Their hypothesis is that the cost of equity capital to smaller firms is higher than it is to larger industrial firms. They test their hypothesis by analyzing the variability of returns of 658 industrial firms and attempt to determine whether variability of return is inversely related to asset size. The authors assume that for all firms Ke must equal the sum of the risk-free rate of interest and a risk premium, when risk is defined by four different measures. Two measures of risk use ex post rates of return on book value and two measures of risk employ ex post rates of return on market value. In the first two cases, risk is defined as variability of the firm alone and, in the second two cases, risk is defined as the firm's variability incorporated with the variability of a market portfolio of securities.

Since this comment should be primarily addressed to Professors Bicksler's and Thorp's own research and results, I will not consider those pages that contain the authors' interpretation of prior studies.Professors Bicksler and Thorp study the short-run properties of the optimal growth model via Monte Carlo simulation. This is an interesting idea because of the mathematical difficulty of the problem.

Professor Huntsman's paper is a welcome addition to the growing literature on portfolio theory. Traditional mean-variance analysis, in spite of its obvious simplicity and its ability to explain portfolio diversification, has come under increasing attack, both for the theoretical weaknesses underlying the technique and for its failure to recognize that investors manifestly prefer returns that are positively skewed to those that are not.

It has been a pleasure and a worthwhile experience to serve as President of the Western Finance Association (WFA) and to have had the opportunity to work with its officers and committee chairmen over this past year. In the brief history of the Association, our organization has come a long way in becoming firmly established due to the strong interest of its members and to the dedicated support of its officers and working committees. I wish to take a few moments to publicly acknowledge the services of a few who have given vital support to the activities of the Association during the past twelve months.

In their provocative article that discusses risk as the probability of an investment's worth falling below some specified minimal value, Machol and Lerner observe that by this definition investments may be risky over a short time horizon but not over a long one [5, p. 484], and that a person who could invest in the stock market over a relatively long period of time without needing to withdraw capital during the period could invest with “relatively little worry” [5, p. 488]. The purpose of this comment is to examine the foregoing position rather more closely insofar as the time path of investment values is concerned. To this end, we model the value of an investment in the New York Stock Exchange Index, relative to its initial value, as a Markov chain. We assume that no part of the initial investment or dividends received on it is withdrawn before termination of the process, at which point the entire amount accumulated (which may be less than the initial investment) is realized. Values taken from a record of annual percentage changes in the New York Stock Exchange Index over the period 1940–1968 are then used to define a representative matrix of transition probabilities which describes the manner in which investment values can change from one period to the next. The probability distributions of relative investment values over differing lengths of time for which the investment may be held are then investigated using the Markov chain model.

Professor Stevenson in his comment of my recent paper on odd-lot trading introduces some interesting additional odd-lot data for both the NYSE and the ASE. His criticisms, however, in my opinion are without merit. First, in my paper I state clearly that due to limited financial resources the study concentrates only on the NYSE. Therefore, the odd-lot participation rate on the ASE is not calculated. Although Stevenson's ASE data are interesting, I fail to see how these ASE data raise serious doubts as to whether my statistical significant results between NYSE odd-lot purchases and sales and ASE odd-lot purchases and sales have any meaningful interpretation. Secondly, Stevenson quarrels with my observation in regard to odd-lot short sales that “odd-lotters have become somewhat more speculative and perhaps more sophisticated in the 1960's.” Again I fail to see why a shifting of marketplaces by small investors, which he mentions but does not prove, makes my statement less valid. Thirdly, because of the nature of the distributed lag model, I do not see why I cannot conclude that “results suggest that odd-lot and round-lot volume did affect each other.” Of course his interpretation that odd-lot and round-lot volume tend to move together is also valid. Finally, my data are through 1967, and the two odd-lot dealer firms did not merge until 1970. In short, as Stevenson's criticisms are all minor, he is making a mountain out of a molehill.

In [2]Sethi and Thompson illustrated the applications of the maximum principle to solve simple dynamic cash balance problems. In Section IV of that paper, we introduced the idea of penalty function to solve the cash balance problem with bounded state variables arising out of disallowing overdrafts and short selling. This resulted in the adjoint equations containing terms in the state variables x(t) and y(t). We then stated the need for solving a two-point boundary value problem.