428 The Journal of finance meters of this distribution--its expected value and standard deviation This can be represented by a total utility function of the form where Ew indicates expected future wealth and ow the predicted standard deviation of the possible divergence of actual future wealth from Ew Investors are assumed to prefer a higher expected future wealth to a lower value, ceteris paribus(dU/dEw>0). Moreover, they exhibit risk-aversion, choosing an investment offering a lower value of wto one with a greater level, given the level of Ew(dU/dow<O) sumptions imply that indifference curves relating Ew and rard-slopin To simplify the analysis, we assume that an investor has decided to commit a given amount(Wi)of his present wealth to investment. Letting Wt be his terminal wealth and R the rate of return on his investment R Wt=R Wi+ Wi This relationship makes it possible to express the investor's utility in terms of R, since terminal wealth is directly related to the rate of return U Figure 2 summarizes the model of investor preferences in a family of indifference curves; successive curves indicate higher levels of utility as one moves down and/or to the right. 8. Under certain conditions the mean-vaniance unsatisfactory predictions of behavior. Markowitz semi-variance(the average of the squared deviations be preferable; in light of the formidable computational problems, variance and standard deviation 9. While only these characteristics are required for the analysis, it is generally assumed that the curves have the property of diminishing marginal rates of substitution between e and o, as do those in our diagrams. 10. Such indifference to maximize expected utility and that his total utility ed by a Inction of R with decreasing marginal utility. Both Markowit pre a derivation. A similar approach is used by Donald E. Farrar in The lnvestment Inder Uncertainty ( Prentice-Hall, 1962). Unfortunately Farrar makes an error derivation; he appeals to the von- Neumann-Morgenstern cardinal utility axioms to trans- form a function of the form E(U=a+ bER -cEr2-ca22 into one of the form E(U)=k1EB一k2012 That such a transformation is not consistent with the axioms can readily be seen in this ves in the er, on2 pla no lie on both a line and a non-linear curve (with a monotonic derivative). Thus the two functions must imply different orderings among alternative choices in at least some instance