The field of investment analysis provides an example of a situation in which individuals or corporations make inferences and decisions in the face of uncertainty about future events. The uncertainty concerns future security prices and related variables, and it is necessary to take account of this uncertainty when modeling inferential or decision-making problems relating to investment analysis. Since probability can be thought of as the mathematical language of uncertainty, formal models for decision making under uncertainty require probabilistic inputs. In financial decision making, this is illustrated by the models that have been developed for the portfolio selection problem; such models generally require the assessment of probability distributions (or at least some summary measures of probability distributions) for future prices or returns of the securities that are being considered for inclusion in the portfolio (e.g., see Markowitz [11] and Sharpe [19]).

There exists a relatively large body of evidence that is consistent with the proposition that the market for securities (in particular, the New York Stock Exchange) is an efficient market in the sense that market prices react instantaneously and unbiasedly to new information and, therefore, market prices fully reflect all publicly available information. To what extent do accounting numbers reflect the kinds of information reflected in market prices? One might not, of course, expect accounting numbers to reflect all events reflected in current market prices. For example, if an economically significant piece of legislation is under discussion in, say, the United States Senate, then the expected effects (if any) of this legislation may be impounded in current market prices. One should not, however, expect these effects (if any) to be reflected in currently issued accounting numbers because of the nature of accepted accounting procedures. Yet, in general, over a period of time, there may be a systematic correspondence between some types of events reflected in market prices and accounting numbers. That is, over time, there may be a correlation between the information impounded in market prices and that impounded in accounting numbers.

Some recent research into the short-term behavior of speculative prices has combined the methodology of information theory with the analysis of aggregate market data on the daily proportions of securities advancing, declining, and remaining unchanged in price. For example, Theil and Leenders [8] employed the measure of average information inaccuracy to analyze aggregate data on the Amsterdam Stock Exchange and concluded that there was a strong dependence between the proportions of prices advancing, declining, and remaining unchanged today, and the respective changes tomorrow. The evidence of dependence on the Amsterdam Exchange prompted Fama [4] to apply similar methodology on the NYSE data, for the period 6/2/52 to 10/29/62, in search of similar patterns. Professor Fama, after proper adjustment for nonstationarity in the time series, concluded that the evidence of dependence on the NYSE was meager and unreliable for forecasting short-term security prices. However, Dryden [3], utilizing the same methodology on data from the London Stock Exchange, found significant dependence between the proportions advancing, declining, and remaining unchanged today, and the respective changes tomorrow. Dryden's results yielded a higher dependence for the London Exchange than for the Amsterdam and New York Exchanges, and were later verified by the straightforward application of a Markovian approach.

One of the major problems in finance is that of combining the separate costs of debt and equity into an appropriate cutoff rate for new investment; this problem is particularly acute when the firm is changing its capital structure. Solutions to this problem which have been proposed include various types of both marginal costing and average costing.

In response to the suggestions of the editorial and reviewing staff of this journal, some additional explanation and extensions of the model presented in an earlier paper [4] seem desirable at this time. In that paper the investor in securities was assumed to have a utility function that depended on the first n moments of the statistical distribution of returns rather than just on the mean and variance. When the borrowing-lending possibility was introduced as in the Sharpe-Lintner model, the investor's perceived risk premium could be expressed in the higher moments' dimensions as well as in terms of the variance.

Most large companies develop formal or informal agreements with banks to cover anticipated seasonal or temporary cash needs and/or to provide assurance of the availability of funds against unanticipated cash requirements. We address the problem of determining the optimal limits of available funds a company should maintain under a revolving credit agreement. Typically a company will negotiate a legal commitment with a bank or group of banks for a specified period of time, usually one to three years, in which the bank agrees to extend credit up to a specified maximum amount. During the duration of the commitment, the bank must lend money to the company whenever the company wishes to borrow, provided the amount of money borrowed does not exceed the maximum amount noted in the agreement. The company must not be in default of any of the restrictive covenants of the agreement, such as working capital limits, compensating balances, limits on other indebtedness, etc. Although the agreement itself provides for intermediate-term financing, the agreement often takes the form of short-term (30-60-90 days) renewable notes.

It is likely that warrants will be used increasingly in the future by corporations both as means of sweetening other securities such as preferred stocks and bonds, and as securities in their own right. Many authors have analyzed the characteristics of warrants from the point of view of investors but attempts to develop a procedure for determining the cost to a corporation of issuing warrants have been lacking. We will find it convenient to measure the yield to a corporation of issuing warrants instead of common stock, rather than determining the percentage cost of warrants analogous to the cost of common stock equity, preferred stock, and interest-bearing debt.

Within the past decade, considerable progress has been made in measuring ex post portfolio performance. The two parameter risk-return dimension of investment performance as pioneered by Markowitz has been reduced to a single parameter which incorporates measures of both risk and return. Several different but related one-parameter measures of performance have been developed, notably by Sharpe [8], Treynor [11], and Jensen [3], and are commonly referred to as composite performance measures. Theoretically, the composite measures allow portfolios with different risks and returns to be compared directly.

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 Wu, in his recent article [8], investigated the trading activities of odd-lotters and their market impact. The purpose of this note is to update the data examined by Wu since recent trends are extremely significant and to question the use and interpretation of some of Wu's original data. In addition, some errors in Wu's article will be discussed.

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.