The Rationale of the Mean-Standard Deviation Analysis, Skewness Preference, and the Demand for Money

Abstract

This chapter discusses the rationale of the mean-standard deviation analysis, skewness preference, and the demand for money. In January 1969, issue of the Review of Economic Studies, Karl Borch and Martin Feldstein separately criticized the widely used mean-variance analysis of portfolio selection. Borch contended that any system of upward sloping mean-standard deviation (E-S) indifference curves can be shown to be inconsistent with the basic axiom of choice under uncertainty. He pointed out that one can always pick any two points on an upward sloping indifference curve to represent two Bernoulli distributions with probability (1—p) of gaining the same amount $x in both the cases and probability p of gaining $y1 in the case of one distribution, and $y2 in the case of the second distribution; with y2 > y1, if the second distribution represents the point to the northeast of the other point. Then, by the dominance axiom, the second distribution must be preferred to the first, which contradicts the basic meaning of indifference curves. Feldstein, by using a log utility function and a lognormal distribution for investment outcome, showed that E-S indifference curves for a risk-averter need not be convex downwards, though upward sloping. They would change from convex to concave, once the standard deviation of the outcome exceeds the mean multiplied by . On the face of it, this seems to suggest that risk aversion might decrease as risk is increased beyond a certain extent.