Abstract
The equivalence between partial moments and stochastic dominance dates back to Bawa [1] and Fishburn [2]. We present a test for first, second, and third degree stochastic dominance between two variables using Lower Partial Moments. The results uphold Hadar and Russell’s [3] original conclusions about the odd moments of preferred prospects. We recall Nawrocki’s [4] research comparing Mean/Variance portfolios against the continuum of risk-averse investors using Lower Partial Moments. The excess skewness of the LPM portfolios clearly demonstrates the preference of positive skewness for risk-averse investors. Finally, we provide an algorithm for efficiently determining stochastic dominance efficient sets among large numbers of variables.
Highlights
Stochastic dominance (SD) is a very powerful risk analysis tool
This paper proposes an integration of stochastic dominance analysis and lower partial moment analysis by defining a stochastic dominance (SD) test via the Lower Partial Moments (LPM) of the investment’s probability distribution
In order to generalize further, one would have to expand the analysis into an Upper Partial Moment/Lower Partial Moment (UPM/LPM) framework, capable of incorporating the often observed four-fold pattern of risk behavior identified in prospect theory and expected utility theory such as the UPM/LPM optimization model described by Viole and Nawrocki [21] [22] and Cumova and Nawrocki [23]
Summary
Stochastic dominance (SD) is a very powerful risk analysis tool. It converts the probability distribution of an investment into a cumulative probability curve. Bey [5] proposes a mean-semivariance algorithm to approximate the second degree stochastic dominance efficient portfolio sets. Since stochastic dominance is an analysis of cumulative distribution functions; only below target deviations are considered over the interval [−∞, target] with the target encompassing all values of X This “for all X” target condition is directly responsible for the generalization to all possible risk averse utility assumptions. The LPM Degree 2 used in Equation (5) is equivalent to the semivariance statistic These equations must be used from every target in the return distribution to generalize for all risk-averse investor types. TSD takes less time to run than FSD per our routines, the explanation of which follows
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