SECURITY PRICES, RISK, AND MAXIMAL GAINS FROM DIVERSIFICATION*

Abstract

Sections II and III of this paper set forth the simple logic which leads directly to the determination of explicit equilibrium prices of risk assets traded in competitive markets under idealized conditions. These equilibrium valuations of individual risk assets are shown to be simply, explicitly and linearly related to their respective expected returns, variances and covariances. The total risk on a given security is the sum of the variance of its own dollar return over the holding period and the combined covariance of its return with that of all other securities. This total risk on each security is “priced up” by multiplying by a “market price of dollar risk” which is common to all securities in the market. The expected dollar return on any security less this adjustment for its risk gives its certainty-equivalent dollar return and the market price of each security is simply the capital value of this certainty-equivalent return using the risk-free interest rate. In this paper, these relationships are shown to hold rigorously even when investors differ in their probability judgments and in other respects.1 It turns out, however, that the “market price of risk” involved in determining the market values of individual securities within a portfolio of risk assets is not equal to the ratio of the expected return on the optimal portfolio of risk assets to the standard deviation of this portfolio return, i.e. r ‒ / σ r . This is true even though this ratio of return to risk on an optimal portfolio is the “price of risk” which is relevant to the (more frequently discussed) decision of how much of an investor's funds should be held in cash (or another riskless asset) and how much should be “put at risk.” Moreover, the value of an individual security within a portfolio is not simply and linearly related to the standard deviation of its return. Rather, the equilibrium value of a security with a given expected return will be lower in proportion to any increase in its variances and covariances, other things equal. Although the general presumption in the literature has been that “risk premiums” on securities should vary linearly with their risk as measured by the standard deviation of their return,2 it thus turns out that the relevant measure of the risk of an individual security within a portfolio of risk assets is given by its return-variance and covariance (with other securities) Since these results (recently presented in technical form and detail elsewhere3) may seem particularly surprising to readers of Professor Sharpe's recent paper in this Journal4 which tends to confirm the traditional positions, its seems desirable to present a simple exposition of the essential logic of the issues involved at this time. As shown below, these results follow directly from the behavior of an individual maximizing risk-averse investor when there is a risk-free asset to hold and his probability judgments are normally distributed.5 Section II traces the investor's responses through a short series of simplified situations, starting with his choice between cash and a single risky asset, and winding up with the optimal selection of a whole portfolio of risky investments and a riskless asset with positive yield or debt, which is assumed to be available as desired (at the same riskless interest rate) to “lever” the portfolio of risk assets. In the next Section we then assume that all probability judgments pertain to end-of-period dollar values (or dollar returns). With this substitution, the conclusions stated at the outset regarding the equilibrium prices of risky stocks, the market price of risk, and the proper measure of risk, all follow easily from the preceding results. Other things being equal, stock values will always vary directly with both the intercept and the correlation coefficient—and will always vary inversely with the residual variance (or “standard error of estimate”)—of their regression on either an external index of business conditions or the composite market performance of the entire group of stocks composing the market. In either type of regression, changes in the slope coefficient will, in general, involve both an “income effect” and a “risk effect” which tend to affect stock values in opposite directions; in theory, one effect will necessarily dominate the other only if one introduces further restrictive assumptions in advance. The simplest and most plausible assumption under which slopes and values will necessarily be related inversely is that expected returns are independent of the slope (while risks increase with slope). Stocks whose returns are independent of general business conditions (or the general level of the stock market) must sell at a price low enough to make their expected rate of return greater than the pure rate of interest, whenever (as always) there is any uncertainty of regarding what their return will be. The same conclusion applies to the price and weighted average expected rate of return of all stocks which are positively (but less than perfectly) correlated with the general market. Apart from negatively correlated stocks, all the gains from diversification come from “averaging over” the independent components of the returns and risks of individual stocks. Among positively correlated stocks, there would be no gains from diversification if independent variations were absent. No possible degree or manner of diversification will be sufficient to eliminate all the risks of holding common stocks which exist apart from the risks due to swings in economic activity (or the general stock market) This is true because, in reality, there will always be at least some residual or independent uncertainty regarding what the actual return (or end-of-period price) of every “risky” security will be even if the general level of business and the stock market is in a given state. In most cases this uncertainty will be relatively substantial. The best possible diversification merely minimizes the risks due to this residual uncertainty for any given level of return. Even if general business conditions and stock market level were perfectly predictable (so that there were no risks on either score), there would still be risks in holding any diversified portfolio of common stocks. The object of diversification is to produce the best portfolio—the one with the most favorable combination of risk and expected return—and even for investors who are “risk-averters,” this “best portfolio” will never be the one (in Markowitz' “efficient set”) with the lowest attainable risk. Common stocks will, of course, nevertheless be held because the general level of all stock prices will always be low enough to make the expected rates of return high enough to be attractive, in spite of these optimal remaining independent risks and the risks of general business conditions (and general stock market fluctuations), and in spite of the availability of investments offering riskless positive returns. Section VI provides some useful empirical benchmarks on the extent of the “residual uncertainties” involved in leading individual stocks and (professionally) diversified portfolios. Regressions of the annual rates of return on 301 large industrial companies were regressed on the corresponding returns of the S & P 425 Industrials Index; the average residual variance was over 8% (more than twice the average riskfree return over the period) and the regression “explained” less than half the total variance in the returns of 188 of the 301 stocks. The power and limitations of diversification to reduce risks and improve investment performance are indicated by regressions of 70 large mutual funds on the Index: 80% of the funds had a higher ratio of mean return to risk than did the index, but over 85% nevertheless had conditional standard errors of estimate (residual risk) greater than the risk-free return (taken to be 4%). This section considers the investment choices of an individual investor in a simple sequence of situations. In choosing between any two different possible investment positions, we assume that this investor will prefer the one which gives him the largest expected return if the risks involved in the two investment positions are the same; and we also assume that if expected returns are the same, he will choose the investment position which involves less “risk” as measured by the standard deviation of the return on his total investment holdings. In other words, our investor is a “risk-averter,” like most investors in common stocks.6 As Tobin has shown,7 these two assumptions imply that the investor's “indifference curves” are concave upward when expected return is plotted on the vertical axis and standard deviation of the horizontal axis: as the risk of his investment position increases, even larger increments of expected return are required to make our investor feel “as well off.” These difference curves are illustrated by the sets of dashed curves in Figure I. For simplicity, we will also assume that our investor's probability judgments (over the uncertain outcomes of holdings risk assets) can be represented by the “normal” distribution of statistical theory. He can invest any part of his capital in any one (or, later, any combination of) common stock(s), all of which are traded in a single purely competitive market at given prices which do not depend on his own transactions (“he is a little fish in the big puddle”). For simplicity, we will also ignore transaction costs and taxes, and assume that all transactions are made at discrete points in time. The return on any stock is, of course, the sum of the cash dividend received plus the change in its market price during the holding period. Investment Choices Involving A Single Stock Case I. The Choice Between Holding Cash and a Single Common Stock. Suppose our investor, for some reason, is considering only the simple question: what fraction w of his capital $A to invest in some single common stock,

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