Knowledge that Transforms

such investors can be dichotomized into two steps.'6 First the investor picks the optimal ratios of risky securities in his portfolio. Then, he decides on the optimal amounts of this composite risky security and the certain security. latter problem is just like that already studied by Arrow,17 so it is known that in this case all the risky securities will be inferior or superior 18 as U/U' increases or decreases with increasing wealth. When the utility function is of the form, U= bZa, V is linear homogeneous of degree a. Since, in this case, the slopes of the indifference surfaces along rays from the origin will be constant, all securities (including the certain one) will change in the same proportion as net worth. 16This result has been obtained by Tobin, Sharpe, and Lintner, when the investor's preferences depend only on the mean and standard deviation. For arbitrary probability distributions this means the U must be quadratic. quadratic is a special case of the family given above. Tobin's paper and Lintner's have been cited previously. Sharpe's is to be found in W. Sharpe, Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, Journal of Finance (Sept. 1964). 17 See the introduction and K. Arrow, Comment, on James S. Duesenberry, The Portfolio Approach to Demand for Money and Other Assets. 18 Securities are taken to be superior or inferior as the absolute value of their quantity increases or decreases with net worth.