In
finance, marginal conditional stochastic dominance is a condition under which a portfolio can be improved in the eyes of all
risk-averse
In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more ce ...
investors by incrementally moving funds out of one asset (or one sub-group of the portfolio's assets) and into another. Each risk-averse investor is assumed to maximize the expected value of an increasing, concave
von Neumann-Morgenstern utility function. All such investors prefer portfolio B over portfolio A if the portfolio
return of B is
second-order stochastically dominant over that of A; roughly speaking this means that the
density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
of A's return can be formed from that of B's return by pushing some of the probability mass of B's return to the left (which is disliked by all increasing utility functions) and then spreading out some of the density mass (which is disliked by all concave utility functions).
If a portfolio A is marginally conditionally stochastically dominated by some incrementally different portfolio B, then it is said to be inefficient in the sense that it is not the optimal portfolio for anyone. Note that this context of portfolio optimization is not limited to situations in which
mean-variance analysis applies.
The presence of marginal conditional stochastic dominance is sufficient, but not necessary, for a portfolio to be inefficient. This is because marginal conditional stochastic dominance only considers incremental portfolio changes involving two sub-groups of assets — one whose holdings are decreased and one whose holdings are increased. It is possible for an inefficient portfolio to not be second-order stochastically dominated by any such one-for-one shift of funds, and yet to by dominated by a shift of funds involving three or more sub-groups of assets.
Testing
Yitzhaki and Mayshar presented a
linear programming-based approach to testing for portfolio inefficiency which works even when the necessary conditional of marginal conditional stochastic dominance is not met. Other similar tests have also been developed.
[Post, T., and Versijp, P., "Multivariate tests for stochastic dominance efficiency of a given portfolio," '']Journal of Financial and Quantitative Analysis
The ''Journal of Financial and Quantitative Analysis'' is a peer-reviewed bimonthly academic journal published by the Michael G. Foster School of Business at the University of Washington in cooperation with the W. P. Carey School of Business at ...
'' 42(2), 2007, 489-516.
References
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Mathematical finance