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Chapter 7 Appendix Stochastic Dominance 1 What can be added to the happiness of a man who is in health, out of debt, and has a clear conscience? - Mark Twain 2 Outline Introduction Efficiency revisited First-degree stochastic dominance Second-degree stochastic dominance Stochastic dominance and utility 3 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 4 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 5 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 6 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 nonnegative • SSD can be a significant aid in reducing the security universe to a workable number of efficient alternatives 7 Stochastic Dominance and Utility Introduction Stochastic dominance and mean return Higher orders of stochastic dominance Practical problems with stochastic dominance 8 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 9 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 FSDefficient portfolio can have a higher geometric mean return than an SSD-efficient portfolio 10 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 11 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 thirddegree stochastic dominance 12