Expected Utility Theory, Optimal Portfolios, and Polyhedral Coherent Risk Measures *
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SYSTEMS ANALYSIS EXPECTED UTILITY THEORY, OPTIMAL PORTFOLIOS, AND POLYHEDRAL COHERENT RISK MEASURES1
UDC 519.21
V. S. Kirilyuk
Abstract. Searching for optimal solutions in terms of expected utility theory is reduced to minimizing a risk measure. The technique of polyhedral coherent risk measures is used to reduce the search for optimal portfolio solutions in the obtained problems to the appropriate linear programming problems. Keywords: expected utility theory, polyhedral coherent risk measure, conditional value-at-risk, spectral risk measure, portfolio optimization. INTRODUCTION The present paper continues the study [1]. We will describe the use of polyhedral coherent risk measures (PCRM) to find optimal portfolio solutions for problems caused by the application of expected utility theory. We will relate this theory and risk measures. In particular, we will analyze it for expected utility theory (for some properties of utility function) and for dual theory of choice. We will reduce searching for optimal solutions for such problems with continuously distributed random variables (r.v.) to minimizing the appropriate spectral risk measure. We will analyze discrete approximations of the problems of minimization of spectral risk measures in which continuous r.v. are approximated by discrete distributions with a finite set of scenario events. We will show how the minimization of the spectral risk measure of a portfolio for discretely distributed r.v. is reduced to the corresponding linear programming (LP) problem. We will consider the expected utility in the Gilboa–Schmeidler robust statement. The problems of searching for optimal portfolio solutions in this statement for finite discretely distributed r.v. will also be reduced to the corresponding LP problems. This will be done for the case of the polyhedral set of uncertainty with respect to scenario probabilities and piecewise linear utility function, as well as for the case of smooth strictly concave utility function and fuzzy scenario probabilities. 1. ELEMENTS OF THE EXPECTED UTILITY THEORY In financial applications, it is customary to make decisions based on reward–risk ratio, where decisions are evaluated by means of some reward and risk functions. In classical Markowitz models, these were average return and variance [2] (semivariance [3]); nowadays, more modern measures are used for risk evaluation (see, for example, [1]). It is implied that by choosing the optimal decision on reward–risk ratio, investor implicitly maximizes some utility function. 1
The study was partially supported by the International project within the frameworks of the cooperation with the International Institute of Applied Systems Analysis (IIASA), order of the Presidium of the NAS of Ukraine No. 212 of 2/28/2012.
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2014, pp. 63–72. Original article submitted May 21, 2013. 874
1060-0396/14/5006-0874
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