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Risk-adjusted probability measures in portfolio optimization with coherent measures of risk

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  • Miller, Naomi
  • Ruszczynski, Andrzej

Abstract

We consider the problem of optimizing a portfolio of n assets, whose returns are described by a joint discrete distribution. We formulate the mean-risk model, using as risk functionals the semideviation, deviation from quantile, and spectral risk measures. Using the modern theory of measures of risk, we derive an equivalent representation of the portfolio problem as a zero-sum matrix game, and we provide ways to solve it by convex optimization techniques. In this way, we reconstruct new probability measures which constitute part of the saddle point of the game. These risk-adjusted measures always exist, irrespective of the completeness of the market. We provide an illustrative example, in which we derive these measures in a universe of 200 assets and we use them to evaluate the market portfolio and optimal risk-averse portfolios.

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  • Miller, Naomi & Ruszczynski, Andrzej, 2008. "Risk-adjusted probability measures in portfolio optimization with coherent measures of risk," European Journal of Operational Research, Elsevier, vol. 191(1), pages 193-206, November.
    • Handle: RePEc:eee:ejores:v:191:y:2008:i:1:p:193-206
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    1. Andrzej Ruszczyński & Alexander Shapiro, 2006. "Optimization of Convex Risk Functions," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 433-452, August.
    2. Ogryczak, Wlodzimierz & Ruszczynski, Andrzej, 1999. "From stochastic dominance to mean-risk models: Semideviations as risk measures," European Journal of Operational Research, Elsevier, vol. 116(1), pages 33-50, July.
    3. Andrzej Ruszczynski & Robert J. Vanderbei, 2003. "Frontiers of Stochastically Nondominated Portfolios," Econometrica, Econometric Society, vol. 71(4), pages 1287-1297, July.
    4. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    5. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    6. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    7. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    8. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    9. Hiroshi Konno & Hiroaki Yamazaki, 1991. "Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market," Management Science, INFORMS, vol. 37(5), pages 519-531, May.
    10. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
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    Cited by:

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    4. Libo Yin & Liyan Han, 2013. "Options strategies for international portfolios with overall risk management via multi-stage stochastic programming," Annals of Operations Research, Springer, vol. 206(1), pages 557-576, July.
    5. Borgonovo, Emanuele & Gatti, Stefano, 2013. "Risk analysis with contractual default. Does covenant breach matter?," European Journal of Operational Research, Elsevier, vol. 230(2), pages 431-443.
    6. Zsolt Bihary & Péter Csóka & Dávid Zoltán Szabó, 2020. "Spectral risk measure of holding stocks in the long run," Annals of Operations Research, Springer, vol. 295(1), pages 75-89, December.
    7. Constantin Zopounidis & Michael Doumpos, 2013. "Rejoinder on: Multicriteria decision systems for financial problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 282-286, July.
    8. Massimo Pappalardo & Giandomenico Mastroeni & Mauro Passacantando, 2016. "Merit functions: a bridge between optimization and equilibria," Annals of Operations Research, Springer, vol. 240(1), pages 271-299, May.
    9. Sungyong Choi & Andrzej Ruszczyński & Yao Zhao, 2011. "A Multiproduct Risk-Averse Newsvendor with Law-Invariant Coherent Measures of Risk," Operations Research, INFORMS, vol. 59(2), pages 346-364, April.
    10. Balbás, Alejandro & Balbás, Beatriz & Heras, Antonio, 2011. "Stable solutions for optimal reinsurance problems involving risk measures," European Journal of Operational Research, Elsevier, vol. 214(3), pages 796-804, November.
    11. Pichler, Alois & Schlotter, Ruben, 2020. "Entropy based risk measures," European Journal of Operational Research, Elsevier, vol. 285(1), pages 223-236.
    12. Balbás, Alejandro & Balbás, Beatriz & Heras, Antonio, 2010. "Stability of the optimal reinsurance with respect to the risk measure," DEE - Working Papers. Business Economics. WB wb100201, Universidad Carlos III de Madrid. Departamento de Economía de la Empresa.
    13. Lizyayev, Andrey & Ruszczyński, Andrzej, 2012. "Tractable Almost Stochastic Dominance," European Journal of Operational Research, Elsevier, vol. 218(2), pages 448-455.
    14. Gaobo Li & Yanxia Lu & Yeol Je Cho, 2019. "Viscosity extragradient method with Armijo linesearch rule for pseudomonotone equilibrium problem and fixed point problem in Hilbert spaces," Indian Journal of Pure and Applied Mathematics, Springer, vol. 50(4), pages 903-921, December.
    15. Borgonovo, E. & Cappelli, V. & Maccheroni, F. & Marinacci, M., 2018. "Risk analysis and decision theory: A bridge," European Journal of Operational Research, Elsevier, vol. 264(1), pages 280-293.
    16. Włodzimierz Ogryczak & Tomasz Śliwiński, 2011. "On solving the dual for portfolio selection by optimizing Conditional Value at Risk," Computational Optimization and Applications, Springer, vol. 50(3), pages 591-595, December.
    17. Balbás, Alejandro & Balbás, Beatriz & Balbás, Raquel, 2010. "CAPM and APT-like models with risk measures," Journal of Banking & Finance, Elsevier, vol. 34(6), pages 1166-1174, June.
    18. Akhter Mohiuddin Rather, 2012. "Portfolio selection using mean-risk model and mean-risk diversification model," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 14(3), pages 324-342.
    19. Bigi, Giancarlo & Castellani, Marco & Pappalardo, Massimo & Passacantando, Mauro, 2013. "Existence and solution methods for equilibria," European Journal of Operational Research, Elsevier, vol. 227(1), pages 1-11.
    20. Collado, Ricardo & Meisel, Stephan & Priekule, Laura, 2017. "Risk-averse stochastic path detection," European Journal of Operational Research, Elsevier, vol. 260(1), pages 195-211.
    21. Jareño, Francisco & Navarro, Eliseo, 2010. "Stock interest rate risk and inflation shocks," European Journal of Operational Research, Elsevier, vol. 201(2), pages 337-348, March.
    22. Giancarlo Bigi & Mauro Passacantando, 2015. "Descent and Penalization Techniques for Equilibrium Problems with Nonlinear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 804-818, March.

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