Chapter 7 Appendix Stochastic Dominance

What can be added to the happiness of a man who is in health, out of debt, and has a clear conscience? - Mark Twain

Outline Introduction Efficiency revisited First-degree stochastic dominance Second-degree stochastic dominance Stochastic dominance and utility

Introduction Stochastic dominance is an alternative technique employed in the portfolio construction process Stochastic “denotes the process of selecting from among a group of theoretically possible alternatives those elements or factors whose combination will most closely approximate a desired result” Stochastic models are not always exact Stochastic models are useful shorthand representations of complicated processes

Efficiency Revisited Portfolios are efficient is they are not dominated by other portfolios Portfolios are inefficient if at least one other portfolio dominates them Rational investors prefer efficient investments

First-Degree Stochastic Dominance Cumulative distribution A will be preferred over cumulative distribution B if every value of distribution A lies below or on distribution B, provided the distributions are not identical The distribution lines do not cross

Second-Degree Stochastic Dominance Alternative A is preferred to Alternative B if the cumulative probability of B minus the cumulative probability of A is always non-negative SSD can be a significant aid in reducing the security universe to a workable number of efficient alternatives

Stochastic Dominance and Utility Introduction Stochastic dominance and mean return Higher orders of stochastic dominance Practical problems with stochastic dominance

Introduction Regardless of how much risk a person can tolerate, the FSD criterion is appropriate Both the conservative investor and the gambler will prefer a first-degree stochastic dominant investment over an FSD inefficient alternative Investors who are risk averse can use SSD to weed out inefficient alternatives

Stochastic Dominance and Mean Return Alternative A is FSD efficient over Alternative B if the expected return of A is no less than the expected return of B If alternatives are ranked by both geometric mean and level of stochastic dominance, no FSD-efficient portfolio can have a higher geometric mean return than an SSD-efficient portfolio

Higher Orders of Stochastic Dominance For third-degree stochastic dominance: The investor is risk averse The investor’s degree of risk aversion declines as wealth increases

Practical Problems With Stochastic Dominance FSD frequently fails to reduce the security universe very much SSD is a much more powerful screening tool than FSD It is difficult to calculate higher than third-degree stochastic dominance