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Optimality conditions in portfolio analysis with general deviation measures
 Mathematical Programming
, 2006
"... Correspondence should be addressed to: Stan Uryasev Optimality conditions are derived for problems of minimizing a generalized measure of deviation of a random variable, with special attention to situations where the random variable could be the rate of return from a portfolio of financial instrumen ..."
Abstract

Cited by 16 (7 self)
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Correspondence should be addressed to: Stan Uryasev Optimality conditions are derived for problems of minimizing a generalized measure of deviation of a random variable, with special attention to situations where the random variable could be the rate of return from a portfolio of financial instruments. Generalized measures of deviation go beyond standard deviation in satisfying axioms that do not demand symmetry between ups and downs. The optimality conditions are applied to characterize the generalized “master funds ” which elsewhere have been developed in extending classical portfolio theory beyond the case of standard deviation. The consequences are worked out for deviation based on conditional valueatrisk and its variants, in particular.
Portfolio Analysis with General Deviation Measures.” Research Report
, 2003
"... Generalized measures of deviation, as substitutes for standard deviation, are considered in a framework like that of classical portfolio theory for coping with the uncertainty inherent in achieving rates of return beyond the riskfree rate. Such measures, associated for example with conditional valu ..."
Abstract

Cited by 6 (1 self)
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Generalized measures of deviation, as substitutes for standard deviation, are considered in a framework like that of classical portfolio theory for coping with the uncertainty inherent in achieving rates of return beyond the riskfree rate. Such measures, associated for example with conditional valueatrisk and its variants, can reflect the different attitudes of different classes of investors. They lead nonetheless to generalized onefund theorems as well as to covariance relations which resemble those commonly used in capital asset pricing models (CAPM), but have wider interpretations. A more customized version of portfolio optimization is the aim, rather than the idea that a single “master fund ” might arise from market equilibrium and serve the interests of all investors. The results cover discrete distributions along with continuous distributions, and therefore are applicable in particular to financial models involving finitely many future states, whether introduced directly or for purposes of numerical approximation. Through techniques of convex analysis, they deal rigorously with a number of features that have not been given much attention in this subject, such as solution nonuniqueness, or nonexistence, and a potential lack of differentiability of the deviation expression with respect to the portfolio weights. Moreover they address in detail the previously neglected phenomenon that, if the riskfree rate lies above a certain threshold, a master fund of the usual type will fail to exist and need to be replaced by one of an alternative type, representing a “net short position ” instead of a “net long position ” in the risky instruments.