《投资学原理 Investments》课程教学资源（阅读资料）How to Use Security Analysis to Improve Portfolio Selection.pdf_大学文库

Jack L, Treynor and Fischer black How to Use Security Analysis to Improve Portfolio selection It has been argued convincingly in a series of papers on the Capital Asset Pricing Model that, in the absence of insight generating expect tions different from the market consensus the investor should hold a replica of the market portfolio. 1 A number of empirical papers have demonstrated that portfolios of more than 50-100 randomly selected securities tend to correlate very highly with the market portfolio, so that, as a practical matter, replicas are relatively easy to obtain. If the investor has no special insights, therefore, he has no need of the elaborate balancing algorithms of Markowitz and Sharpe. 2 On the other hand if he has special insights, he will get little, if any, help from the portfolio-balancing literature on how to translate these insights into the expected returns, variances, and covariances the algorithms require What was needed, it seemed to us, was exploration of the link etween conventional subjective, judgmental, work of the security analyst, on one hand--rough cut and not very quantitativeand the essentially objective, statistical approach to portfolio selection of Markowitz and his successors on the other The void between these two bodies of ideas was made manifest b our inability to answer to our own satisfaction the following kinds of questions: Where practical is it desirable to so balance a portfolio be- tween long positions in securities considered underpriced and short positions in securities considered overpriced that market risk is com- pletely eliminated (i.e, hedged)? Or should one strive to diversify a portfolio so completely that only market risk remains? As this implies, in the highly diversified portfolio market sensitivity in individual secu- rities seems to contribute directly to market sensitivity in the overall portfolio, whereas other sources of return variability in individual securities seem to average out. Does this mean that the latter sources w Editor, Financial Analysts Journal, sor of finance, University Chicago; and executive director, Center for Re in Security Prices lliam F. Shar a Theory of Market equi of Risk, Journal of Finance 19, no. 3 964):425-42; John Lintner, "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets, " Review of Economics and Statistics 57, no. 1( February 1954): 13-37; and Jack L. Tre Toward a Theory of the Market values of Risky Assets"(u 2. Harry Markowitz, Portfolio Selection: Efficient Diversification of In- vestments(New York: John Wiley Sons, 1959; New Haven, Conn. Yale Uni- versity Press, 1970);and William Sharpe, "A Simplified Model Portfolio Analysis, Management Science 9 (January 1963 ): 277-93 Cone doal25m313024153M

Jack L. Treynor and Fischer Black" How to Use Security Analysis to Improve Portfolio Selection It has been argued convincingly in a series of papers on the Capital Asset Pricing Model that, in the absence of insight generating expectations different from the market consensus, the investor should hold a replica of the market portfolio.' A number of empirical papers have demonstrated that portfolios of more than 50-100 randomly selected securities tend to correlate very highly with the market portfolio, so that, as a practical matter, replicas are relatively easy to obtain. If the investor has no special insights, therefore, he has no need of the elaborate balancing algorithms of Markowitz and Sharpe.2 On the other hand if he has special insights, he will get little, if any, help from the portfolio-balancing literature on how to translate these insights into the expected returns, variances, and covariances the algorithms require as inputs. What was needed, it seemed to us, was exploration of the link between conventional subjective, judgmental, work of the security analyst, on one hand-rough cut and not very quantitative-and the essentially objective, statistical approach to portfolio selection of Markowitz and his successors, on the other. The void between these two bodies of ideas was made manifest by our inability to answer to our own satisfaction the following kinds of questions: Where practical is it desirable to so balance a portfolio between long positions in securities considered underpriced and short positions in securities considered overpriced that market risk is completely eliminated (i.e., hedged)? Or should one strive to diversify a portfolio so completely that only market risk remains? As this implies, in the highly diversified portfolio market sensitivity in individual securities seems to contribute directly to market sensitivity in the overall portfolio, whereas other sources of return variability in individual securities seem to average out. Does this mean that the latter sources * Editor, Financial A nalysts Journal; professor of finance, University of Chicago; and executive director, Center for Research in Security Prices. 1. William F. Sharpe, "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk," Journal of Finance 19, no. 3 (September 1964): 425-42; John Lintner, "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets," Review of Economics and Statistics 57, no. 1 (February 1954): 13-37; and Jack L. Treynor's paper, "Toward a Theory of the Market Values of Risky Assets" (unpublished, 1961). 2. Harry Markowitz, Portfolio Selection: Efficient Diversification of Investments (New York: John Wiley & Sons, 1959; New Haven, Conn.: Yale University Press, 1970); and William Sharpe, "A Simplified Model for Portfolio Analysis," Management Science 9 (January 1963): 277-93. 66 This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:47:55 AM All use subject to JSTOR Terms and Conditions

Security Analysis to Improve Portfoli of variability are unimportant in portfolio selection? When balancing risk against expected return in selection of individual securities, what risk and what return are relevant? Will increasing the number of securities analyzed improve the diversification of the optimal port- folio? Is any measure of the contribution of security analysis to port- folio performance invariant with respect to both levering and turnover? How do analysts opinions enter in security selection? Is there an simple way to characterize the quality of security analysis that will tell us when one analyst can be expected to make a greater contribu tion to a portfolio than another? What role, if any, does confidence in an analysts forecasts have in portfolio selection? This paper offers answers to these questions. The paper has a normative favor. We offer no apologies for this. In some cases, institutional practice and, in some cases, law are short sighted; in all cases they reflect what is by anybody's standard an old fashioned idea of what the about. If we tried to develop a body of theory which reflected some of the constraints imposed institutionally and legally, it would inevitably on an idealized world in which there are no restrictions on borrowing, or on selling securities short; in which the interest rate on loans is equal to the interest rate on short-term assets such as savings accounts; and in which there are no taxes, We expect that the major conclusions derived from the model will largely be valid, however, even with the constraints and frictions of the real world Those that are not valid can usually be modified to fit the constraints that actually exist Certain recent research has suggested that professional invest ment managers really have not been very successful, but we make the assumption that security analysis, properly used, can improve portfolio performance. This paper is directed toward finding a way to make the best possible use of the information provided by security The basic fact from which we build is one that a number of writers have recognized-namely, that there is a high degree of co- prices. Perhaps the simplest mo ability among securities is Sharpe's Diagonal Model. As Sharpe sees it, The major characteristic of the Diagonal Model is the assumption that e returns of various securities are related only through common elationships with some basic underlying factor... This model has two irtues: it is one of the simplest which can be constructed without assuming away the existence of interrelationships among securities and there is considerable evidence that it can capture a large part of such interrelationships. This paper takes Sharpe's Diagonal Model as its 3. Michael Jensen. "The Performance of Mutual Funds in the Period 1945- 1964,"Journal of Finance 23(May 1968): 389-41 Cone doal25m313024153M

67 Security Analysis to Improve Portfolio of variability are unimportant in portfolio selection? When balancing risk against expected return in selection of individual securities, what risk and what return are relevant? Will increasing the number of securities analyzed improve the diversification of the optimal portfolio? Is any measure of the contribution of security analysis to portfolio performance invariant with respect to both levering and turnover? How do analysts' opinions enter in security selection? Is there any simple way to characterize the quality of security analysis that will tell us when one analyst can be expected to make a greater contribution to a portfolio than another? What role, if any, does confidence in an analyst's forecasts have in portfolio selection? This paper offers answers to these questions. The paper has a normative flavor. We offer no apologies for this. In some cases, institutional practice and, in some cases, law are shortsighted; in all cases they reflect what is by anybody's standard an oldfashioned idea of what the investment management business is all about. If we tried to develop a body of theory which reflected some of the constraints imposed institutionally and legally, it would inevitably be a theory with a very short life expectancy. Our model is based on an idealized world in which there are no restrictions on borrowing, or on selling securities short; in which the interest rate on loans is equal to the interest rate on short-term assets such as savings accounts; and in which there are no taxes. We expect that the major conclusions derived from the model will largely be valid, however, even with the constraints and frictions of the real world. Those that are not valid can usually be modified to fit the constraints that actually exist. Certain recent research has suggested that professional investment managers really have not been very successful,' but we make the assumption that security analysis, properly used, can improve portfolio performance. This paper is directed toward finding a way to make the best possible use of the information provided by security analysts. The basic fact from which we build is one that a number of writers have recognized-namely, that there is a high degree of comovement among security prices. Perhaps the simplest model of covariability among securities is Sharpe's Diagonal Model. As Sharpe sees it, "The major characteristic of the Diagonal Model is the assumption that the returns of various securities are related only through common relationships with some basic underlying factor. . . . This model has two virtues: it is one of the simplest which can be constructed without assuming away the existence of interrelationships among securities and there is considerable evidence that it can capture a large part of such interrelationships."4 This paper takes Sharpe's Diagonal Model as its 3. Michael Jensen, "The Performance of Mutual Funds in the Period 1945- 1964," Journal of Finiance 23 (May 1968): 389-416. 4. See Sharpe, n. 2. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:47:55 AM All use subject to JSTOR Terms and Conditions

The Journal of Business starting point; we accept without change the form of the Diagonal Model and most of Sharpes assu Use of the Diagonal Model for portfolio selection implies de- parture from equilibrium in the sense of all investors having the same information (and appraising it similarly )-as, for example, is assumed in some versions of the Capital Asset Pricing Model. The viewpoint in this paper is that of an individual investor who is attempting to trade profitably on the difference between his expectations and those of a monolithic market so large in relation to his own trading that market prices are unaffected it. Throughout, we ignore the costs of buying and selling. This makes it possible for us to treat the portfolio-selection problem as a single-period problem(implicitly assuming a one-period utility function as given), in the tradition of Markowitz, Sharpe, et al We believe that these costs are often substantial and, if incorporated into this analysis, would modify certain of our results substantially DEFINITIONS on short-term risk-free assets over that interval A regression of the excess return on a security against the market's excess return gives two regression factors. The first is the market or“beta.” of the securit le error the second should be zero. We define the explained return on the security over a given time interval to be its market sensitivity times the market's excess return over the interval We define the independent return to be the excess return minus the explained return. The independent return, because of the of regression, is statistically independent of the markets excess re turn. Our model assumes that the "independent returns of different securities are almost, but not quite, statistically independent. The"risk premium"on the ith security is equal to the securitys market sensitivity times the market's expected excess return. Symbols for these concepts re defined as r= riskless rate of return x turn on the ith security, yi=excess return on the ith security, ym= excess return on the market b i= market sensitivity of the ith security, bi ym= explained, or systematic, return on ith security, zi= independent return on ith Let E[ and var I represent the expectation and variance, respec tively, of the variable in brackets. Then define Cone doal25m313024153M

68 The Journal of Business starting point; we accept without change the form of the Diagonal Model and most of Sharpe's assumptions. Use of the Diagonal Model for portfolio selection implies departure from equilibrium in the sense of all investors having the same information (and appraising it similarly)-as, for example, is assumed in some versions of the Capital Asset Pricing Model. The viewpoint in this paper is that of an individual investor who is attempting to trade profitably on the difference between his expectations and those of a monolithic market so large in relation to his own trading that market prices are unaffected by it. Throughout, we ignore the costs of buying and selling. This makes it possible for us to treat the portfolio-selection problem as a single-period problem (implicitly assuming a one-period utility function as given), in the tradition of Markowitz, Sharpe, et al. We believe that these costs are often substantial and, if incorporated into this analysis, would modify certain of our results substantially. D E F I N I T I O N S Following Lintner, we define the excess return on a security for a given time interval as the actual return on the security less the interest paid on short-term risk-free assets over that interval. A regression of the excess return on a security against the market's excess return gives two regression factors. The first is the market sensitivity, or "beta," of the security; and, except for sample error, the second should be zero. We define the explained return on the security over a given time interval to be its market sensitivity times the market's excess return over the interval. We define the independent return to be the excess return minus the explained return. The independent return, because of the properties of regression, is statistically independent of the market's excess return. Our model assumes that the "independent" returns of different securities are almost, but not quite, statistically independent. The "risk premium" on the ith security is equal to the security's market sensitivity times the market's expected excess return. Symbols for these concepts are defined as: r riskless rate of return, xi return on the ith security, y- excess return on the ith security, Ym excess return on the market, bi market sensitivity of the ith security, biym explained, or systematic, return on ith security, zi independent return on ith security. Let E [ ] and var [ ] represent the expectation and variance, respectively, of the variable in brackets. Then define This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:47:55 AM All use subject to JSTOR Terms and Conditions

The journal of business paper shares with Markowitz the mean-variance approach, implying normal return distributions. There is fairly conclusive evidence that the distribution is not normal, but that its behavior is similar to that of a normal distribution, so the model assumes a normal distribution as an approximation to the actual distribution. The qualitative results of the model should not be affected by this approximation, but the quantitative results should be modified somewhat to reflect the actual distribution However one defines"risk"in terms of the probability distribu tion of portfolio return, the distribution, being approximately normal is virtually determined by its mean and variance. But under the as- sumptions noted here(finite variances and independence) the mean and variance of portfolio return depend only on the means and variances of independent returns for specific securities and on the explained return(and, of course, on the portfolio weights ) On the other hand, risk in the specific security is significant to the investor only as it affects portfolio risk. Hence it is tempting to identify risk in the ith security with the elements in the security that contribute to portfolio variance-the variance e independent return o2(“ specific risk”) and the variance of explained return b,20m("market risk"). In what follows, we will occasionally yield to this temptation Let the fraction of the investor's capital devoted to the ith security be hj. Using symbols defined above, the one-period return on his port hx1-rΔ h4-1 l=1 1 =2h(+A-r△(∑h (3) =rAt+△hy We note that, although are three sources of return on the individual security-the riskless return, the explained return, and the independent return-only two of these are at stake in portfolio selection. Hence- forth we shall ignore the first term in equation (3) Understanding the way in which portfolio mean and variance are nfluenced by selection decisions requires expansion of security return into all its elements. Excess return on the portfolio, expressed in term of the individual securities held. is b ym+ Evidently we have only n degrees of freedom-the portfolio weights Cone doal25m313024153M

70 The Journal of Business paper shares with Markowitz the mean-variance approach, implying normal return distributions. There is fairly conclusive evidence that the distribution is not normal, but that its behavior is similar to that of a normal distribution, so the model assumes a normal distribution as an approximation to the actual distribution. The qualitative results of the model should not be affected by this approximation, but the quantitative results should be modified somewhat to reflect the actual distribution. However one defines "risk" in terms of the probability distribution of portfolio return, the distribution, being approximately normal, is virtually determined by its mean and variance. But under the assumptions noted here (finite variances and independence) the mean and variance of portfolio return depend only on the means and variances of independent returns for specific securities and on the explained return (and, of course, on the portfolio weights). On the other hand, risk in the specific security is significant to the investor only as it affects portfolio risk. Hence it is tempting to identify risk in the ith security with the elements in the security that contribute to portfolio variance-the variance of the independent return 0,2 ("specific risk") and the variance of explained return b 2 a .2 ("market risk"). In what follows, we will occasionally yield to this temptation. Let the fraction of the investor's capital devoted to the ith security be hi. Using symbols defined above, the one-period return on his portfolio is n n Zhixi- rAt (Zhi h 1 n n Z hi (yi + rAt) - rAt (Z h1 ) (3) n rAt +- L h-jyj. i=l We note that, although there are three sources of return on the individual security-the riskless return, the explained return, and the independent return-only two of these are at stake in portfolio selection. Henceforth we shall ignore the first term in equation (3). Understanding the way in which portfolio mean and variance are influenced by selection decisions requires expansion of security return into all its elements. Excess return on the portfolio, expressed in terms of the individual securities held, is n n hibiym, + hizi. (4) Evidently we have only n degrees of freedom-the portfolio weights This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:47:55 AM All use subject to JSTOR Terms and Conditions

The Journal of business In order to obtain an expression for the optimal position hm in the market portfolio, we recall that Am+1= Elm]=pm, from equation (5)in to o together with the definitions of hn+ and substitute these expressions hb;+hm=入m/Om2 (14) Multiplying both members of (10) by b; and summing we have hb2=A2,b;/ (15) which can be substituted in(14 )to give It was apparent in equation(5)that market risk enters the por folio both in the form of an explicit investment in the market portfolio and implicitly in the selection of individual securities, the returns from COV ry with the market. Equation(13)saystake positions in securities 1,..., n purely on the basis of expected independent return and varlance The resulting exposure to market risk is disregarded Equation (16)provides us with an expression for the optimal investment in an explicit market portfolio. This investment is designed to complement the market position accumulated in the course of taking positions in individual securities solely with regard to their independent returns. Under the assumptions of the Diagonal Model, position in the market follows the same rule as position in individual securities; but because market position is accumulated as a by-product of positions in dividual securities, explicit investment in the market as a whole is limited ne difference between the optimal market p tion and the by-product accumulation (which may, of course, be negative, requiring an explicit position in the market that is short, rather than long) Equation(16)suggests that the optimal portfolio can be thought of as two portfolios: (1)a portfolio assembled purely with egard for the means and variances of independent returns of specific d to market risk qui incidental to this regard; and(2)an approximation to the market port folio. Positions in the first portfolio are zero when appraisal premiums are zero. Since the special information on which expected independent eturns are based typically propagates rapidly, becoming fully dis Cone doal25m313024153M

72 The Journal of Business In order to obtain an expression for the optimal position hm in the market portfolio, we recall that Uj.n+? E[ym] =lun 0r2 n+l varyy] = -l and substitute these expressions together with the definitions of hl+ from equation (5) in (11) to obtain n Z hfbi + hm (14) i= 1 Multiplying both members of (10) by bi and summing we have -hjb.; - E b1j/o-12, (15) which can be substituted in (14) to give hill A F/cX~) Z (16) It was apparent in equation (5) that market risk enters the portfolio both in the form of an explicit investment in the market portfolio and implicitly in the selection of individual securities, the returns from which covary with the market. Equation (13) says "take positions in securities 1, . , n purely on the basis of expected independent return and variance." The resulting exposure to market risk is disregarded. Equation (16) provides us with an expression for the optimal investment in an explicit market portfolio. This investment is designed to complement the market position accumulated in the course of taking positions in individual securities solely with regard to their independent returns. Under the assumptions of the Diagonal Model, position in the market follows the same rule as position in individual securities; but because market position is accumulated as a by-product of positions in individual securities, explicit investment in the market as a whole is limited to making up the difference between the optimal market position and the by-product accumulation (which may, of course, be negative, requiring an explicit position in the market that is short, rather than long). Equation (16) suggests that the optimal portfolio can usefully be thought of as two portfolios: (1) a portfolio assembled purely with regard for the means and variances of independent returns of specific securities and possessing an aggregate exposure to market risk quite incidental to this regard; and (2) an approximation to the market portfolio. Positions in the first portfolio are zero when appraisal premiums are zero. Since the special information on which expected independent returns are based typically propagates rapidly, becoming fully disThis content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:47:55 AM All use subject to JSTOR Terms and Conditions

73 Security Analysis to Improve Portfolio ounted by the market and eliminating the justification for positions based on this information, the first portfolio will tend to experience a ignificant amount of trading. Accordingly, we call it the"active port folio It is clear from equation (10) that changes in the investor's at titude toward risk bearing (2), or in his market expectations(H-m) or in the degree of market risk(om)-which, as we shall see, depends on how well he can forecast the market--have no effect on the propor tions of the active portfolio The Capital Asset Pricing Model suggests that any premium for risk bearing will be associated with market, rather than specific risk. If investors in the aggregate are risk averse, then an investment in the asset-explicit or implicit--offers a premium. We call this particular source of market premium "risk premium"as opposed to"market premium"deriving from the investor's attempts to forecast fluctuations in the general market level. When all the appraisal pre miums are zero, the optimal portfolio is therefore the market portfolio -even if the investor has no power to forecast the market. We shall call a portfolio devoid of specific risk "perfectly diversified. " In other words in our usage"perfect diversification"does not mean the absence of risk. nor does it mean an optimally balanced portfolio, except in the case of In general, a given security may play two different roles simul taneously:(1)A temporary position based entirely on expected inde- pendent return (appraisal premium)and appraisal risk. As price fluctuates and the investor's information changes, the optimum position changes.(2)A position resulting purely from the fact that the security in question constitutes part of the market portfolio. The latter position hanges as market expectations change but is virtually independent of expectations regarding independent return on the security. Hence we call the approximation to the market portfolio employed to achieve the desired level of systematic risk the"passive portfolio. The literal inter pretation of equation (16)is that a desired explicit market position h,n would be achieved by adding positions in individual securities in the proportions in which they are represented in the market as a whole. For example, let the fraction of the market as a whole comprised by the ith security be h wj. Then an explicit market position hm can be achieved by taking positions h,; in the individual securities. These positions are, of course, in addition to positions taken with regard to specific return. Overall positions are then given by combining positions desir for fulfilling the two functions of bearing appraisal and market risk h;=λh bμ/2)+pa2 This is the result one would get by solving the Markowitz formula- Cone doal25m313024153M

73 Security Analysis to Improve Portfolio counted by the market and eliminating the justification for positions based on this information, the first portfolio will tend to experience a significant amount of trading. Accordingly, we call it the "active portfolio." It is clear from equation (10) that changes in the investor's attitude' toward risk bearing (X), or in his market expectations (do), or in the degree of market risk (-m,2)-which, as we shall see, depends on how well he can forecast the market-have no effect on the proportions of the active portfolio. The Capital Asset Pricing Model suggests that any premium for risk bearing will be associated with market, rather than specific risk. If investors in the aggregate are risk averse, then an investment in the market asset-explicit or implicit-offers a premium. We call this particular source of market premium "risk premium"-as opposed to "market premium" deriving from the investor's attempts to forecast fluctuations in the general market level. When all the appraisal premiums are zero, the optimal portfolio is therefore the market portfolio -even if the investor has no power to forecast the market. We shall call a portfolio devoid of specific risk "perfectly diversified." In other words, in our usage "perfect diversification" does not mean the absence of risk, nor does it mean an optimally balanced portfolio, except in the case of zero appraisal premiums. In general, a given security may play two different roles simultaneously: (1) A temporary position based entirely on expected independent return (appraisal premium) and appraisal risk. As price fluctuates and the investor's information changes, the optimum position changes. (2) A position resulting purely from the fact that the security in question constitutes part of the market portfolio. The latter position changes as market expectations change but is virtually independent of expectations regarding independent return on the security. Hence we call the approximation to the market portfolio employed to achieve the desired level of systematic risk the "passive portfolio." The literal interpretation of equation (16) is that a desired explicit market position hMrb would be achieved by adding positions in individual securities in the proportions in which they are represented in the market as a whole. For example, let the fraction of the market as a whole comprised by the ith security be h?,. Then an explicit market position h.n can be achieved by taking positions h,,h ,, in the individual securities. These positions are, of course, in addition to positions taken with regard to specific return. Overall positions are then given by combining positions desired for fulfilling the two functions of bearing appraisal and market risk: n hi A -h miZ)nl J12 blL,/Oi2) + /UTi,-2 (17) This is the result one would get by solving the Markowitz formulaThis content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:47:55 AM All use subject to JSTOR Terms and Conditions

74 The Journal of Business tion, under the assumptions of the Diagonal Model, in the absence of constraints. But it is not a solution of much practical interest, because approximations to the market portfolio add very little additional specific risk while being vastly cheaper to acquire than an exact pro rata replica of the market. A practical interpretation of equation (16) is that portfolio selection can be thought of as a three-stage process, in which the first stage is selection of an active portfolio to maximize the appraisal ratio, the second is blending the active portfolio with a suitable replica of the market portfolio to maximize the Sharpe ratio, and the third entails scaling positions in the combined portfolio up or down through lending or borrowing while preserving their proportions. Because the investor's attitude toward risk bearing comes into play at the third stage, and only at the third stage, a second-stage definition of "goodness" that disregards differences in attitude toward risk bearing from one investor to another is possible.5 THE SHARPE AND APPRAISAL R A T I O S From equation (10) we have, for the optimal holdings, hi - X/(r/, where, for the optimal second-stage portfolio, we have n+1 Iu A, - 2/ Oi2 n+1 (18) O-p2 - X2 E ki2/'i 2. i= How good is the resulting portfolio? A relationship between expected excess return and variance of return is obtained by forming n+1 2 AX2 ( -i 2/i2) E (19) X2 n+1 X2 ZzH2/ O.2 *= 1 The resulting ratio is essentially the square of a measure of goodness proposed by William Sharpe;6 we shall call it the Sharpe ratio. It is obviously independent of scale. The right hand expression readily partitions into two terms, one of which depends only on market forecasting and the other of which depends only on forecasting independent returns for specific securities: 5. See, for example, William Sharpe, "Mutual Fund Performance," Journal of Business 39 (January 1966): 119-38. % 6. Ibid. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:47:55 AM All use subject to JSTOR Terms and Conditions