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Duality Mappings For The Theory of Risk Aversion with Vector Outcomes

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  • Sudhir A. Shah

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    Abstract

    The Author considera a decision-making environment with an outcome space that is a convex and compact subset of a vector space belonging to a general class of such spaces. Given this outcome space,he defines general classes of (a) risk averse von Neumann-Morgenstern utility functions defined over the outcome space, (b) multi-valued mappings that yield the certainty equivalent outcomes corresponding to a lottery, (c) multi-valued mappings that yield the risk premia corresponding to a lottery, and (d) multi-valued mappings that yield the acceptance set of lotteries corresponding to an outcome. Their duality results establish that the usual mappings that generate (b), (c) and (d) from (a) are bijective.They apply these results to the problem of computing the value of financial assets to a risk averse decision-maker and show that this value will always be less than the arbitrage-free valuation.[CDS WP NO 160]

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    Bibliographic Info

    Paper provided by eSocialSciences in its series Working Papers with number id:2085.

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    Date of creation: Jun 2009
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    Handle: RePEc:ess:wpaper:id:2085

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    Keywords: Risk aversion; vector outcomes; certainty equivalence; risk premia; acceptance set;

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    1. Kihlstrom, Richard E & Mirman, Leonard J, 1981. "Constant, Increasing and Decreasing Risk Aversion with Many Commodities," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 48(2), pages 271-80, April.
    2. Grant, Simon & Kajii, Atsushi & Polak, Ben, 1992. "Many good choice Axioms: When can many-good lotteries be treated as money lotteries?," Journal of Economic Theory, Elsevier, Elsevier, vol. 56(2), pages 313-337, April.
    3. Juan Martínez-Legaz & John Quah, 2007. "A contribution to duality theory, applied to the measurement of risk aversion," Economic Theory, Springer, Springer, vol. 30(2), pages 337-362, February.
    4. Grant, Simon & Kajii, Atsushi & Polak, Ben, 1992. "Many good risks: An interpretation of multivariate risk and risk aversion without the Independence axiom," Journal of Economic Theory, Elsevier, Elsevier, vol. 56(2), pages 338-351, April.
    5. Kihlstrom, Richard E. & Mirman, Leonard J., 1974. "Risk aversion with many commodities," Journal of Economic Theory, Elsevier, Elsevier, vol. 8(3), pages 361-388, July.
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