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, the remainder $ ( 1 − w )A to be held in cash which is riskless but offers no return. For definiteness, let r ‒ be the rate of return expected on this stock and the standard deviation of this return be σ r . Equation (3) tells us that the market (here confined to cash and one stock only) offers the investor opportunities to vary his over-all rate of return y ‒ ( or investment income = y ‒ A o ) and over-all risk σ y (or σ y A o ) as he may wish along the solid “market opportunity line” in Figure I. (Both his expected return and his risk are increased as he increases his proportionate investment w in risk assets, as shown by (lb) and (2a). The “terms of trade” offered him in this (limited) market between his over-all expected return and risk is given by the slope coefficient ( r ‒ / σ r ), which is the reciprocal of the coefficient of variation on the one-available risk asset. This reciprocal of the coefficient of variation of the rate of return on the stock is thus the “market price of risk” in this simple situation. In choosing where on the market opportunity line he prefers to be, the investor will increase his risk investment w (and reduce cash) as long as his indifference curves are flatter than (and hence cut through) the market opportunity line—in other words, as long as his personal “preference-rate” of substitution requires less incremental expected return per unit of added overall risk than the market offers. He stops increasing w when this (favorable) inequality no longer holds (i.e., at the usual “tangency point”), or when all his assets are invested in stocks (if the inequality is still favorable at that extreme point). The introduction of savings deposits raises the intercept of the “market opportunity line” to r* (from zero), and it reduces its slope to ( r ‒ − r ∗ ) / σ r (from r − ⁡ / σ r ). (See Figure I.) Note also that the “market price of risk” is still the reciprocal of the coefficient of variation, but now it is this ratio based on the available excess return (over the riskless rate r*). The allowance of borrowing simply enables the investor to lever his portfolio if he wishes so that his optimal w may be > 1; graphically, as in Figure I, the introduction of borrowing in this way means that the “market opportunity line” extends indefinitely in the northwest direction.8 With this additional freedom, the optimal decisions are found exactly as in Case I, except that the investor thinks in terms of “excess return” x rather than the gross return r. Case III. The Choice of One Stock Among Many to Hold Along with Savings Deposits (or Debt). Suppose now our investor has knowledge of several stocks, but for some reason can invest in only one of them. He must (a) choose which stock to put his “risk money” in, and (b) how much to invest in it (holding the remainder of his assets in savings deposits, or financing some of his holdings with debt. These decisions in this new situation can be followed in Figure II. Investment Choices Among Stocks, Mutual Funds or Portfolio Mixes of Stock It is clear from the previous discussion that these decisions can (optimally) be made in sequence (and do not need to be made simultaneously) Moreover, the choice of “which stock” should precede his choice of “how much,” and the best stock to invest in is clearly the one with the highest θ ratio which measures expected excess return per unit risk. This is true because the different stocks present the investor with different “market opportunity lines” (equation 3′) fanning out from the intercept r* with different slopes equal to their respective θ ratios. For any possible scale of investment w in risky assets, the investor will clearly be better of if he puts his “risk money” in the stock with the highest ratio. In this way, he gets maximum return y ‒ on his total capital (i.e. total stock plus savings deposits less debt) in relation to his over-all risk σ y , regardless of the scale of his investment w-and being a “risk averter,” this is precisely what he wants (because this is equivalent to getting the same over-all return with less over-all risk). Then, after having found the “best stock,” he ignores all others (in this mutually exclusive case), and using its market opportunity line, he proceeds to decide how much to invest in it (and how much to keep in savings deposits, or how much to borrow) just as in Case II. Case IV. The Choice of One Mutual Fund Among Many to Hold Along with Savings Deposits (or Debt)with Savings Deposits (or Debt). Suppose now that for some reason the investor cannot (or will not) hold individual stocks, but knows of several mutual funds. He desires to invest in only one fund and hold the rest of his assets in riskless form. His best pair of decision “which” and “how much” are found sequentially exactly as in Case III. He first ranks the θ ratios of the different funds, picks the one with the largest ratio, and, ignoring the rest finally decides the best fraction w of his assets to “put at risk” exactly as before. Case V. Choice of Possible Portfolios of Stocks. This last hypothetical case provides all the essentials of the present situation with which we are fundamentally concerned. For mutual funds are simply managed portfolios of securities. Apart from “loads,” management fees and operating expenses, the expected return r ‒ , standard deviation σ r , and hence the θ ratio of each fund, are simply appropriately weighted averages of the returns and risks of the component securities in its portfolio. The mutually exclusive choices of mutual funds in case IV were thus really choices among portfolios of assets; and if the investor considers the desirability of different mixtures or portfolios of securities to hold in his own name, his choice is necessarily a mutually exclusive one. In deciding which portfolio of stock to hold, the investor will thus use his judgments (probability distributions) regarding the prospects of each candidate stock (and their covariances or correlations of outcomes), and then in effect examine the r ‒ , σ r and θ ratios which are implied by various possible portfolio mixtures of the stocks. The best portfolio for him will be the one with the highest θ ratio. He will distribute any funds he invests in stocks according to the weights used in finding the portfolio with the largest θ; and after these proportionate weights are found, he can then decide “how much” he wants to invest in this best portfolio mix (and how much to put in savings deposits, or borrow) on utility grounds. The analysis so far establishes a conclusion of crucial importance. We saw earlier that the ratio x ‒ / σ x of the expected excess return on the best portfolio to its standard deviation was the price or wage of risk-bearing which would determine how much of his assets an investor would invest in a stock portfolio. But we also saw that the prior question was what was the best portfolio, and that this involved finding the portfolio (or mixture) of stocks which (on the basis of the investor's own judgments of their prospects and risks) would maximize θ. A glance at the equations in (11) now shows that it is the variances and (weighted) covariances of the returns on any individual stock which, given its expected excess return x i _ , will determine the size of its h i ∗ —i.e., the fraction of the whole portfolio which will be invested in the stock.12 Other things the same, more will be put in a given security within a portfolio the higher its expected excess return, and less will be put in the larger its marginal contribution to the risks of the whole portfolio.13 Within portfolios a stock's riskiness thus varies with variances and covariances; within a portfolio its riskiness is not properly measured either directly or simply by the standard deviation of its return. Similarly, the expected excess rate of return which the investor requires per unit of risk for holding individual stock within portfolios is given by the factor λ, which is the ratio of the porifolio x ‒ to the variance of the return on the portfolio σ x 2 . If the product of this return requirement λ with the total risk attributable to holding a given stock within a portfolio—i.e., with the weighted sum of its variances and its covariances on the left side of equation (11)—is not ⩾ its x i ¯ , the stock will not be held (or will be sold short).14 This return requirement to hold stocks within the portfolio is the same for all the stocks within the portfolio, but it is essentially diferent from the price or return per unit of portfolio risk (the θ above) which controls the size of his investment in this best portfolio mix. Earlier failures to distinguish between these two different requirements—used, it will be noted, in different ways—has led to much confusion. With this background we can proceed directly to determine the equilibrium market prices of stocks. Aggregate Value of All Outstanding Shares of Each Security. So far we have assumed that current market prices are given data, and that each investor acts in terms of his own judgments of prospective rates of return, given these prices. But the investor's estimate of the rate of return r i ¯ will be equal to the sum of cash dividends received plus or minus capital gain (i.e., change in market price), expressed as a percentage of the current market price. Suppose now that each investor in a purely competitive frictionless stock market makes his estimates directly in terms of the end-of-period values of each stock (including dividend receipts as well as market price), which we can write H ~ i for each ith stock. Suppose also for the moment that every investor assigns identical sets of means, variances and covariances to their end-of-period values for the stocks available in the market. [Note that while different investors' estimates are the same for each stock (and each pair of stocks), each investor will of course (in general) have different estimates for each different stock.] With this latter simplification, the explicit equilibrium values of each security in the market follow very directly from our preceding analysis. For the assumption of identical probability distributions means that the same percentage holdings of each stock will be optimal for each investor,15 and consequently, when the market is in equilibrium, the set of h i o values given by the solution of the set of equations (11) represent the ratio of the aggregate market value of each ith stock V oi to the aggregate market value of all stocks ( T o = Σ i V oi ) at time zero. If investors have assigned a set of numbers H i ¯ to the expected aggregate market values (and dividend receipt) of the ith stocks in the market at the end of the holding period, a set of numbers σ i ∗ 2 to the variance of these ending valuations, and a set of numbers σ ij ∗ to the covariances of each i,j pair of ending valuations, then the market values V oi for all stocks will have to adjust and readjust until the set of equations (11) is satisfied. For any given H ¯ i , variations of the current value V oi will modify the expected excess rate of return x i ¯ on the stock according to the relation x ¯ i = [ H i ¯ − ( 1 + r ∗ ) V 0 i ] / V 0 i ,(14a) which merely restates our earlier definition of x i ¯ in terms of our present variables. Similarly, for any given σ * 2 and σ ij *, any variations in V oi would modify the variance and covariance of the rates of return according to the relations σ 2 i = σ i ∗ 2 / V 0 i 2 ,(14b) and σ ij = σ ij ∗ / V 0 i V 0 j .(14c) We now simply substitute these relations (14a, b, c) for each stock in the equations in (11), and see that the relation16 H ¯ i − ( 1 + r ∗ ) V 0 i = ( λ / T 0 ) [ σ i ∗ 2 + Σ j ≠ i σ ij ∗ ] ,(15) holds with respect to each ith stock in equilibrium. Consequently, the aggregate value of the stock will be given by V 0 i = ( H i ¯ − W i ) / ( 1 + r ∗ )(16) where W i = ( γ ) [ σ i ∗ 2 + Σ j ≠ i σ ij ∗ ] ,(16a) and γ = λ / T 0 .(16b) The aggregate market value of any ith stock is thus equal to the certainty equivalent ( H i ¯ − W i ) of its value at the end of the period, discounted at the risk-free interest rate. This certainty equivalent, in turn, is equal to its end-of-period expected value H i ¯ less an adjustment W i to allow for the market effect of its total risks. These risks, as shown by the bracket in (16a), are given by the sum of the variance of its end-of-period value and the total of its corresponding covariances with all other stocks; and the adjustment term W i is the product of these total risks with the “market price of dollar risk.” This market price of dollar risk, in turn, is the same for all companies in the market in equilibrium because it appears as a common term in the equation (15) which must be valid simultaneously for all stocks in the market.17 Also, it can be shown that 18, the market price of dollar risk, is equal to (A) the sum (over all stocks and investors) of the expected excess of end-of-period values over current values raised by the riskless rate, to (B) the corresponding aggregate dollar variance of all portfolios combined. In the first paragraph of this paper, the corresponding conclusions regarding the aggregate market values of risk assets in equilibrium were stated in terms of dollar returns over the holding period, rather than in terms of the expected end-of-period values H i ¯ just used. The strict equivalent of the two forms of our results is readily seen by noting that for any possible V oi the expected dollar return R i ¯ ≡ H i ¯ − V oi so that (16) can be rewritten as. V 0 i = ( R i ¯ − W i ) / r ∗ (16′) while the adjustment W i given in (16a) is not affected at all.19 The market price of dollar risk is the same in each case, and the risks inherently involve the variances of return on the given security so that they cannot be linear in the standard deviation of the company's own return, as so widely thought. Prices of Individual Shares. The preceding results for the aggregate valuation of all the shares of a company's stock when the market is in equilibrium can readily be adapted to show the equilibrium price per share. If we let N i be the number of shares of the ith stock outstanding, P li ¯ be its expected price (before dividend payment) at the end of the holding period, and P oi its current equilibrium price, we have H i ¯ = N i P li ¯ and V oi = N i P oi . Similarly, if ( var ) i is the variance per share—i.e., the variances of the random P li —we have the aggregate σ i ∗ 2 = N i 2 ( var ) i , and correspondingly the aggregate σ ij ∗ = N i N j ( cov ) ij where ( cov ) ij represents the per share covariance. Direct substitution in (16) gives us the desired relationship after dividing through by a common factor N i : ( 1 + r ∗ ) P 0 i ¯ = P 1 i − γ [ N i ( var ) i + Σ j ≠ i N j ( cov ) ij ] .(17) The “market price of dollar risk,” γ, is the same on a per share basis as it was in the equation for the valuation of all the company's outstanding stock. But it should be especially noted that in the equation for price per share, the variances and covariances of the uncertain end-of-period prices per share are weighted by the number of shares outstanding. This weighting of per share variances and covariances is required precisely because the variance and covariance of aggregate valuations of a company's stock are independent of stock splits.20 Share Prices When Investor's Judgments Differ. To this Point, We have assumed for simplicity that all investors assign the same probability distribution to the end-of-period values of each stock (though these common investor judgments were different for different stock). It can readily be shown, however, that all the conclusions reached, both for aggregate valuations of a company's total equity and for prices per share, still hold with no change other than the substitutions of weighted averages for expected end-of-period values, and for the variances and covariances. Since equation (17) was derived directly from the equilibrium conditions for an individual investor shown in equations (11), each K'th investor will be in equilibrium if the market price is such that equation (17) holds in terms of his own judgmental data (indicated by adding K as a subscript) We must consequently find prices P oi for each i'th security so that the following equation is satisfied for each K'th investor simultaneously: P ¯ 1 i ( K ) − ( 1 + r ∗ ) P 0 i = γ K [ N i ( K ) ( var ) i ( K ) + Σ j ≠ i N j ( K ) ( cov ) ij ( K ) ] ,(17a) where γK is equal21 to the ratio of (a) the aggregate effected excess dollar return on the K'th investor's entire portfolio—which we will write as A K —to (b) the dollar variance of the end-of-period value of his whole portfolio—which we will B K . Using γ K = A K / B K , and letting [ ] K represent the entire bracket on the right hand of (17a), we have B K [ P ¯ 1 iK − ( 1 + r ∗ ) P 0 i ] = A K [ ] K .(17b) Summing over all investors in the market,22 we have for each stock Σ K B K P ¯ 1 i ( K ) − ( 1 + r ∗ ) P 0 i Σ K B K = Σ K A K [ ] K ,(18) which reduces23 to ( 1 + r ∗ ) P 0 i = Σ K v K P ¯ 1 i ( K ) − γ Σ K u K [ ] K ,(19) where v K = B K / Σ K B K and u k = A K / Σ K A K . Current price per share is thus equal to the discounted value (at the riskless rate r*) of a weighted average of the individual investor's expectation of end-of-period price (including dividend receipts) less the product of the market price of dollar risk γ with a weighted average of the total contribution of the i'th stock to the individual investor's portfolio. Note that the weights attached to expected future values are proportional to the dollar-variances of different investors' entire portfolios, while the weights attached to the i'th stock's own contribution to each portfolio's variance—the [ ] K term—are proportional to the expected excess dollar returns on the different investor's portfolios. But the market price of risk γ is identical to that in the “homogeneous expectations” case—i.e., the ratio of the aggregate expected excess dollar returns (over-all stocks and all investors) to the aggregate dollar variance of all stock in all portfolios combined. Regressions on an External Index. Markowitz'24 has suggested that investor