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
Security Analysis to Improve Portfolio ha, with i= 1, n-in selecting among n I sources of return Since the market asset can always be freely bought or sold to acquire n explicit position h,m in the market asset, when we take this into account we have for the excess portfolio return the expression (5) It is obvious that availability of the market asset makes it possible to achieve any desired exposure to market risk, approximately inde- pendently of any decisions regarding desired exposure to independent returns on individual securities. In effect, we then have n+ 1 mutually ndependent securities, where hn+1=hn+∑Ab=ECa),=1,…,n,p+1=Eya6) If we apply these conventions and run our summations from 1 to n+ 1 we have for the mean and variance of the portfolio return, respectively, hnp,02=△h2or2 (7) We take as our objective minimizing o, while holding p, fixed We form the Lagrangian h2o?-2n hiu;-Ap introducing the undetermined multiplier 7, differentiate with respect hi, and set the result equal to zero 2ho2-2NH=0. (9) Solving for h we have h= Au/o2. (10) Substituting this result in equation(7) we have a2 u/o2, 0,2=X2 2/o2 (11) We see from(11) that the value of the multiplier 2 is given by The optimum position h, in the ith security (i= l,.., n)is given by equation(13) Cone doal25m313024153M
71 Security Analysis to Improve Portfolio hi, with i - 1, . . ., n-in selecting among n + 1 sources of return. Since the market asset can always be freely bought or sold to acquire an explicit position hmn in the market asset, when we take this into account we have for the excess portfolio return the expression ?b n (hm + Z hibi) ym + hizi. (5) i=1 i=1 It is obvious that availability of the market asset makes it possible to achieve any desired exposure to market risk, approximately independently of any decisions regarding desired exposure to independent returns on individual securities. In effect, we then have n + 1 mutually independent securities, where n hnw + I hm, + Adhibi, /ui =E(z+), i =1,.., n, /Jn+l =E[ym] (6) i=1 If we apply these conventions and run our summations from 1 to n + 1, we have for the mean and variance of the portfolio return, respectively, n+1 n+1 fJ p hui p2 >ii: h2,O2. (7) We take as our objective minimizing GP2 while holding [p fixed. We form the Lagrangian n+1 n+1 2?2 Zh 2 A (8) i~~~l ~i=l introducing the undetermined multiplier A, differentiate with respect to hi, and set the result equal to zero: 2hi 0-,2 -2 Xui ?. (9) Solving for hi we have hi A/Ot2 (10) Substituting this result in equation (7) we have n+1 n+1 /up A E J i72/0i2,0rp2= X2 E {i 2/c-4j2.(1 i==1 i=1 We see from (11) that the value of the multiplier X is given by X = -2/Jp. (12) The optimum position hi in the ith security (i 1, . . ., n) is given by equation (13): hi = _. i = 1, . . ., n. (13) Up (r* 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