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) formPortfolio 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 al￾located 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 ex￾clude 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 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 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