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

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.

[1]  M. Frittelli,et al.  Putting order in risk measures , 2002 .

[2]  G. Calafiore,et al.  Probabilistic and Randomized Methods for Design under Uncertainty , 2006 .

[3]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[4]  Alexander Shapiro,et al.  Optimization of Convex Risk Functions , 2006, Math. Oper. Res..

[5]  Abaxbank,et al.  Spectral Measures of Risk : a Coherent Representation of Subjective Risk Aversion , 2002 .

[6]  Hans-Jakob Lüthi,et al.  Convex risk measures for portfolio optimization and concepts of flexibility , 2005, Math. Program..

[7]  A. Ruszczynski,et al.  Optimization of Risk Measures , 2006 .

[8]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[9]  R. Rockafellar,et al.  Deviation Measures in Risk Analysis and Optimization , 2002 .

[10]  R. Rockafellar,et al.  Generalized Deviations in Risk Analysis , 2004 .

[11]  Wlodzimierz Ogryczak,et al.  On consistency of stochastic dominance and mean–semideviation models , 2001, Math. Program..

[12]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[13]  Maria Grazia Speranza,et al.  On LP Solvable Models for Portfolio Selection , 2003, Informatica.

[14]  S. Kusuoka.  On law invariant coherent risk measures , 2001 .

[15]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[16]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[17]  Wlodzimierz Ogryczak,et al.  From stochastic dominance to mean-risk models: Semideviations as risk measures , 1999, Eur. J. Oper. Res..

[18]  H. Konno,et al.  Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market , 1991 .

[19]  D. Tasche,et al.  On the coherence of expected shortfall , 2001, cond-mat/0104295.

[20]  A. Ruszczynski,et al.  Frontiers of Stochastically Nondominated Portfolios , 2003 .

[21]  G. Pflug.  Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk , 2000 .

[22]  S. Uryasev.  Probabilistic constrained optimization : methodology and applications , 2000 .

[23]  Wlodzimierz Ogryczak,et al.  Dual Stochastic Dominance and Related Mean-Risk Models , 2002, SIAM J. Optim..

[24]  Alejandro Balbás,et al.  CAPM and APT-like models with risk measures , 2010 .