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Investment under duality risk measure. (English) Zbl 1339.91111

Summary: One index satisfies the duality axiom if one agent, who is uniformly more risk-averse than another, accepts a gamble, the latter accepts any less risky gamble under the index. R. J. Aumann and R. Serrano [J. Political Econ. 116, No. 5 (2008; Zbl 1341.91040)] show that only one index defined for so-called gambles satisfies the duality and positive homogeneity axioms. We call it a duality index. This paper extends the definition of duality index to all outcomes including all gambles, and considers a portfolio selection problem in a complete market, in which the agent’s target is to minimize the index of the utility of the relative investment outcome. By linking this problem to a series of Merton’s optimum consumption-like problems, the optimal solution is explicitly derived. It is shown that if the prior benchmark level is too high (which can be verified), then the investment risk will be beyond any agent’s risk tolerance. If the benchmark level is reasonable, then the optimal solution will be the same as that of one of the Merton’s series problems, but with a particular value of absolute risk aversion, which is given by an explicit algebraic equation as a part of the optimal solution. According to our result, it is riskier to achieve the same surplus profit in a stable market than in a less-stable market, which is consistent with the common financial intuition.

MSC:

91G10 Portfolio theory
91A80 Applications of game theory
91B16 Utility theory
91B26 Auctions, bargaining, bidding and selling, and other market models
91B30 Risk theory, insurance (MSC2010)

Citations:

Zbl 1341.91040
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References:

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