The vector of equilibrium aggregate market values (or per share prices) of a given set of risk assets trading in purely competitive markets of individually risk-averse investors has been derived in earlier work under certain simplifying assumptions, including the absence of taxes and transactions costs and a single (uniform) holding period for the assessment of uncertain outcomes (See [8], [9], [10], [15], and [12]). The other critical assumptions in these studies were (a) the existence of a riskless asset available for holding or borrowing at a fixed, exogenously determined interest rate, (b) an assumption that all investors act in terms of identical joint probability distributions over end-of-period outcomes, and (c) the acceptance of a mean-variance criterion for portfolio decisions.

This article examines some aspects of the portfolio selection problem when the “no-easy-money-condition” holds and the investor is constrained to stay solvent. The possible presence of a non-capital income is also taken into consideration.

Not all racetrack bettors want to handicap. Some prefer to buy expert opinion on the winners of each race and on the day's best bet. Investors are not very different. Many like to avoid the analysis required to choose among investment funds and so seek some summary measure that will rank funds relative to one another and indicate which one is the best bet. Two measures suggested in work by John Lintner [2], William Sharpe [6], Jack Treynor[7], and Henry Latané and Donald Tuttle [1] rank investment funds considering both return and risk. These measures not only can be taken to provide an indication of the best bet among investment fund managements but also offer a guide to building a portfolio.

Risk continues to be a widely discussed topic within the field of finance. Academicians add risk variables to their quantitative models, while financial practitioners include risk considerations in their qualitative deliberations. In both contexts, risk — together with some measure of profit or return — generally comprise a dual or composite criteria for investment decision-making purposes. Whereas the decisionmaking situation can be described as ex ante, this article deals with risk in an ex post context. In particular, it reports an investigation of alternative risk-return measures which are designed to rank and evaluate the ex post performance of investment portfolios. Section I reviews three composite measures of performance and examines their interrelationships. A fourth alternative measure is also suggested. In Section II, the measures are used to rank the portfolio performance of a sample of mutual funds. Some difficulties in making performance comparisons of these funds against the market are discussed in Section III. The final section briefly explores the implications of the study and suggests areas for subsequent research.

In this article, we shall discuss several of the alternative definitions of risk that have been proposed from time to time. We shall show that one definition — risk is the probability of loss — leads to a formulation of the investment decision problem as a chance constrained problem. Three different strategies are then proposed by which an investor can reduce risk. It is our belief that professional investors utilize all three strategies and that risk, in many such cases, is not a substantial constraint on investor behavior.

This article has two goals. The first is to contrast two widely used definitions of risk aversion. The second is to establish the feasibility of plunging behavior, in the sense that a possibly large number of risk averse investors will not be diversifiers.

In retrospect, writing about “risk and valuation” is somewhat akin to killing Hydra, the mythical, many-headed creature which would grow two heads whenever one was cut off. It is only fair to admit at the outset that I cannot lay claim to have slain the beast. As a matter of fact, by the time the reader finishes the article, he may have concluded that the beast has more heads than ever!