Portfolio Selection The variance of a weighted sum is a;V(X)+2 ∑ If we use the fact that the variance of R is out then Let R, be the return on the i"security. Let u: be the expected value of R; oi, be the covariance between R and r,(thus ou is the variance of R.). Let X be the percentage of the investor,s assets which are al located to the i"security. The yield (R)on the portfolio as a whole is R=∑RX The R(and consequently R)are considered to be random variables The Xi are not random variables, but are fixed by the investor. Since the X: are percentages we have >:=1. In our analysis we will ex clude negative values of the X (i.e, short sales); therefore X:>0for all讠. The return(R)on the portfolio as a whole is a weighted sum of ran dom variables(where the investor can choose the weights). From our discussion of such weighted sums we see that the expected return E from the portfolio as a whole is E and the variance is σ;XX concerning these variables. In general we would expect that the he had probability beliefs atters, he would possess a system of probability beliefs We cannot expect the nt matters that have been carefully considered he will base his actions upon these probability beliefs--even though the This paper does not consider the difficult question of how investors do( or should) form
Portfolio Selection The variance of a weighted sum is If we use the fact that the variance of Ri is uii then Let Ri be the return on the iN"security. Let pi be the expected vaIue of Ri; uij, be the covariance between Ri and Rj (thus uii is the variance of Ri). Let Xi be the percentage of the investor's assets which are allocated to the ithsecurity. The yield (R) on the portfolio as a whole is The Ri (and consequently R) are considered to be random variables.' The Xi are not random variables, but are fixed by the investor. Since the Xi are percentages we have ZXi = 1. In our analysis we will exclude negative values of the Xi (i.e., short sales); therefore Xi > 0 for all i. The return (R) on the portfolio as a whole is a weighted sum of random variables (where the investor can choose the weights). From our discussion of such weighted sums we see that the expected return E from the portfolio as a whole is and the variance is 7. I.e., we assume that the investor does (and should) act as if he had probability beliefs concerning these variables. In general we ~vould expect that the investor could tell us, for any two events (A and B), whether he personally considered A more likely than B, B more likely than A, or both equally likely. If the investor were consistent in his opinions on such matters, he would possess a system of probability beliefs. We cannot expect the investor to be consistent in every detail. We can, however, expect his probability beliefs to be roughly consistent on important matters that have been carefully considered. We should also expect that he will base his actions upon these probability beliefs-even though they be in part subjective. This paper does not consider the difficult question of how investors do (or should) form their probability beliefs
The journal of finance For fixed probability beliefs(μ,σ动 the investor has a choice of vari ous combinations of E and V depending on his choice of portfolio X XN. Suppose that the set of all obtainable(E, v) combina tions were as in Figure 1. The e-v rule states that the investor would or should )want to select one of those portfolios which give rise to the (e, V) combinations indicated as efficient in the figure; i.e., those with minimum V for given E or more and maximum E for given V or less. There are techniques by which we can compute the set of efficient portfolios and efficient(E, v) combinations associated with given u E,V combinations officient FIG. 1 and oi. We will not present these techniques here We will,however, illustrate geometrically the nature of the efficient surfaces for cases in which N(the number of available securities) is small. The calculation of efficient surfaces might possibly be of practi se. Perhaps there are ways, by combining statistical techniques and the judgment of experts, to form reasonable probability beliefs a. We could use these beliefs to compute the attainable efficient combinations of(E, v). The investor, being informed of what(E, n) combinations were attainable, could state which he desired. We could hen find the portfolio which gave this desired combination
82 The Journal of Finance For fixed probability beliefs (pi, oij) the investor has a choice of various combinations of E and V depending on his choice of portfolio XI, . . . ,XN.Suppose that the set of all obtainable (E, V) combinations were as in Figure 1.The E-V rule states that the investor would (or should) want to select one of those portfolios which give rise to the (E, V) combinations indicated as efficient in the figure; i.e., those with minimum V for given E or more and maximum E for given V or less. There are techniques by which we can compute the set of efficient portfolios and efficient (E, V) combinations associated with given pi attainable E, V combinations and oij. We will not present these techniques here. We will, however, illustrate geometrically the nature of the efficient surfaces for cases in which N (the number of available securities) is small. The calculation of efficient surfaces might possibly be of practical use. Perhaps there are ways, by combining statistical techniques and the judgment of experts, to form reasonable probability beliefs (pi, aij).We could use these beliefs to compute the attainable efficient combinations of (E, V). The investor, being informed of what (E, V) combinations were attainable, could state which he desired. We could then find the portfolio which gave this desired combination