A Note on the Implications of Quadratic Utility for Portfolio Theory

The shortcomings of a quadratic utility function are so serious and so widely known that by now one might assume that it would simply have been dropped from consideration. Arrow [1] and Pratt [6] have shown that such a function implies ever increasing absolute risk aversion, that is, reduced risk taking as wealth increases, which contradicts everyday experience. Moreover, the assumption of quadratic utility also implies ultimate satiation with respect to risk taking. This function has a well-defined maximum beyond which the marginal utility of money declines, and as a result the range of admissable returns must be restricted. Wippern [12] has focused attention on the second of the above two shortcomings. Using a rather ingenious device, based on the Sharpe-Lintner market model [8 and 5], Wippern has measured empirically the admissable range of returns implied by the quadratic utility function. Since his empirical findings imply that returns beyond as little as 1.3 standard deviations from the expected return provide negative marginal utility to investors, Wippern concludes that the Sharpe-Lintner market model, and/or the mean-variance portfolio theory upon which it is based, have “inconsistent and implausible properties.”