Quantile Portfolio Optimization Under Risk Measure Constraints
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Quantile Portfolio Optimization Under Risk Measure Constraints Luis D. Cahuich · Daniel Hernández-Hernández
© Springer Science+Business Media New York 2013
Abstract This paper analyzes the problem of optimal portfolio choice with budget and risk constraints. The problem is formulated in terms of quantile functions and the risk is quantified through a large family of coherent risk measures. The solution is obtained analyzing the problem without constraints using Lagrange multipliers, getting a unique solution to the optimization problem. Keywords Quantile function · Portfolio optimization · Risk quantification 1 Introduction The problems of portfolio selection and risk quantification of financial positions have been extensively studied in mathematics finance, and are among the two most important problems in the financial industry. An interesting problem arise when an agent is dealing with the issue of choosing an optimal portfolio and, at the same time, he wants to limit his risk. In this work a solution to this problem is proposed, putting together ideas from portfolio choice and risk measures, motivated by recent results of He and Zhou [7] and Gundel and Weber [5]. One of the most studied models in portfolio selection theory is the expected utility maximization, initiated with the fundamental contributions of Merton [16] and Samuelson [19]. After these works, several models have been proposed as alternatives; among all these approaches we shall mention some of the most representative, without attempting to give an exhaustive description. For instance, in 1971 Yaari L.D. Cahuich BBVA Bancomer, México City, 03339, Mexico e-mail: [email protected] D. Hernández-Hernández () Centro de Investigación en Matemáticas, Apartado Postal 402, Guanajuato, Gto. 36000, Mexico e-mail: [email protected]
Appl Math Optim
proposed the dual theory of choice, and later Lopes [15] analyzed the SP/A model, Kahneman and Tversky’s prospect theory is due to these authors in their papers from 1979 and 1992. A brief summary of the previous models and some others can be found in the recent work of He and Zhou [7], where they propose a general framework of portfolio selection; in a recent paper He, Jin and Zhou [6] present related results with the problem studied in this note. In this framework the decision variable is taken as the quantile function of the terminal payoff, a distinctive characteristic from other models. We shall use this framework to formulate our model, including another component in the set of restrictions, limiting the risk assumed by the agent. The restrictions of risk have been also incorporated in the analysis of insurance portfolios by De Franco and Tankov [4]. The standard risk measure for financial practitioners is the Value at Risk (VaR). It is well known that this measure presents some inconsistencies; for instance, it can penalize diversification and encourage concentration in some cases. A class of measures that do not present these deficiencies are the coherent risk measures introduced originally by Artzner, De
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