Risk, Return, Skewness and Preference

Abstract

This paper considers choice between individual projects and shows that when the choice set includes arbitrary distributions, then any assumed relationship between expected utility theory and general moment preferences for individual decision makers is theoretically unsound. In particular, a risk averse investor with any common utility function may, when choosing between two positive return opportunities, prefer the project simultaneously having a lower mean, higher variance, and lower positive skewness. Moreover, the decision maker can prefer opportunities with higher variance even when the opportunities are continuous, unimodal, and arbitrarily visually and statistically close to the normal distribution in shape. Our conclusions hold for any decision maker with a utility function whose derivatives alternate in sign being strictly positive or negative (i.e., we exclude the uninteresting cases of quadratic and cubic utilities). The method of analysis is based upon the theory of Tchebychev systems of functions which deals with the expected value of [utility] functions of stochastic variables with known moments. Although we focus on the first three moments, the results, as presented here, apply to all higher moments as well. It is also shown that there can be extremely large deviations between the certainty equivalents of distributions having the same moments, so this result is also pertinent to practical decision analysts as well. The paper demonstrates that the properties of utility functions have implications which are much more subtle than previously recognized for evaluating distributions in terms of their moments.moment ordering, CAPM, rational decision making, counter examples

Best Bounds on Measures of Risk and Probability of Ruin for Alpha Unimodal Random Variables When There Is Limited Moment Information

“… when X is unimodal with a known mode and a know range and finite set of moments was presented by Brockett and Cox [8], and Brockett, Cox, and Witt [9] and used in Brockett and Kahane [10] and Brockett and Garven [11]. Their development was based on the theory of Chebychev systems of functions [12] coupled with Kemperman's [13] "transformation of moments" technique.…”

Delivering Left-Skewed Portfolio Payoff Distributions in the Presence of Transaction Costs

“…Pensioners may be entirely reliant on their savings to support themselves, and thus a small payoff could be disastrous. The authors of [3] discovered cases where risk averse investors prefer negative skewness. These investors cannot be represented by either the Merton utility function (see [4]) or the functions from prospect theory (for typical parameter values).…”

“…2 In fact, the literature tends to ignore the distribution of actual outcomes that eventuate from the optimal investment strategy. 3 Traditionally, an investor's utility function is based on psychological experiments and perceived preferences. This is then used to derive a functional form for the utility function and a portfolio optimisation problem is solved based on this measure.…”

“…2 Interestingly, in the context of static portfolio management, [8] also question the wisdom of "traditional optimisation" for some investors, leverage-averse in their case. 3 Exceptions to this include [9], [10]. The work done on the distribution builder in the first, enables subjects to build their desired pension distribution subject to a budget constraint.…”

A Note on Skewness Seeking: An Experimental Analysis

“…This implies that the truncation of the Taylor series may lead to incorrect results since the neglected part might be larger than the part kept, as shown in several examples in Brockett and Kahane (1992). Scott and Horvath (1980) and Ingersoll (1987) show that for a strictly risk averse investor (U > 0 and U < 0) with a strictly consistent direction of the third derivative (U (x) is positive, negative, or zero for all x) U > 0 must hold.…”