Summary
We examine portfolio asset management under safety constraints that control the probability that the portfolio return falls under a given reference level. We extend previous results of Roy (1952) and Kataoka (1963) that have been proved in a one-period setting to both multiperiod discrete-time and continuous-time models. Basic examples illustrate the results.
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Portfolio Management with Safety Criteria in Complete Financial Markets
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I. INTRODUCTIONFrom the seminal work of Markowitz (1952), the mean-variance criteria are widely used by portfolio managers. Its generalization, based on the expected utility maximization of gains, has become also a standard criterion for portfolio optimization. In both cases, investors are supposed to prefer the more to the less and are risk averse. The mean-variance approach is justified under one of the following assumptions: (1) The asset returns are normally distributed and the investors' utilities are exponential; and (2) The utility functions are quadratic and the return distributions are characterized by their two first expectations. Moreover, as it has been proved in Markowitz and Levy (1979), the mean-variance approach is approximatively robust even when the assumptions (1) and (2) are not verified. For example, the quadratic approximations are often good local approximations of non-quadratic utility functions, when the asset returns distributions are not too asymmetric. Other criteria for portfolio choice are based on the geometric return, or the first four moments of returns (in particular, the skewness and the kurtozis).In this paper, we examine portfolio optimization based on safety criteria: the investors focus on unfavourable events and want to limit the probability to get low portfolio returns. For example, the guaranteed funds managers deal with insured payments at maturity. They must...See the full content of this document
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