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Stochastic dominance for law invariant preferences: The happy story of elliptical distributions

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  • Matteo Del Vigna

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    (Dipartimento di Matematica per le Decisioni - Universita' degli Studi di Firenze)

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    Abstract

    We study the connections between stochastic dominance and law invariant preferences. Whenever the functional that represents preferences depends only on the law of the random variable, we shall look for conditions that imply a ranking of distributions. In analogy with the Expected Utility paradigm, we prove that functional dominance leads to first order stochastic dominance. We analyze in details the case of Dual Theory of Choice and Cumulative Prospect Theory, including all its distinctive features such as S-shaped value function, reversed S-shaped probability distortions and loss aversion. These cases can be seen as special examples of a more general scheme. We find necessary and sufficient conditions that imply preferences to depend only on the mean and variance of the lottery. Our main result is a characterization of such distributions that imply Mean-Variance preferences, namely the elliptical ones. In particular, we prove that under mild assumptions over the reference wealth, the prospect value of a portfolio depends only on its mean and variance if and only if the random assets' return are elliptically distributed. The analysis is of particular relevance for optimal portfolio choice, mutual fund separation and Capital Asset Pricing equilibria.

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    Bibliographic Info

    Paper provided by Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa in its series Working Papers - Mathematical Economics with number 2012-08.

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    Length: 26 pages
    Date of creation: Oct 2012
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    Handle: RePEc:flo:wpaper:2012-08

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    Keywords: Stochastic dominance; Cumulative Prospect Theory; Elliptical distributions; Mean-Variance analysis.;

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    1. Kahneman, Daniel & Tversky, Amos, 1979. "Prospect Theory: An Analysis of Decision under Risk," Econometrica, Econometric Society, vol. 47(2), pages 263-91, March.
    2. Levy, Haim & Wiener, Zvi, 1998. "Stochastic Dominance and Prospect Dominance with Subjective Weighting Functions," Journal of Risk and Uncertainty, Springer, vol. 16(2), pages 147-63, May-June.
    3. Dana, Rose-Anne & Carlier, Guillaume, 2006. "Law invariant concave utility functions and optimization problems with monotonicity and comonotonicity constraints," Economics Papers from University Paris Dauphine 123456789/5392, Paris Dauphine University.
    4. Fortin, Ines & Hlouskova, Jaroslava, 2010. "Optimal Asset Allocation Under Linear Loss Aversion," Economics Series 257, Institute for Advanced Studies.
    5. Baucells Alib├ęs Manel & Heukamp Franz H., 2007. "Stochastic Dominance and Cumulative Prospect Theory," Working Papers 201061, Fundacion BBVA / BBVA Foundation.
    6. Carlier Guillaume & Dana Rose-Anne, 2006. "Law invariant concave utility functions and optimization problems with monotonicity and comonotonicity constraints," Statistics & Risk Modeling, De Gruyter, vol. 24(1/2006), pages 26, July.
    7. Galichon, Alfred & Henry, Marc, 2012. "Dual theory of choice with multivariate risks," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1501-1516.
    8. Bernard, Carole & Ghossoub, Mario, 2009. "Static Portfolio Choice under Cumulative Prospect Theory," MPRA Paper 15446, University Library of Munich, Germany.
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