Two-stage portfolio optimization with higher-order conditional measures of risk

We describe a study of application of novel risk modeling and optimization techniques to daily portfolio management. In the first part of the study, we develop and compare specialized methods for scenario generation and scenario tree construction. In the second part, we construct a two-stage stochastic programming problem with conditional measures of risk, which is used to re-balance the portfolio on a rolling horizon basis, with transaction costs included in the model. In the third part, we present an extensive simulation study on real-world data of several versions of the methodology. We show that two-stage models outperform single-stage models in terms of long-term performance. We also show that using high-order risk measures is superior to first-order measures.

[1]  M. Shirosaki.  Another proof of the defect relation for moving targets , 1991 .

[2]  D. Tasche,et al.  Expected Shortfall: a natural coherent alternative to Value at Risk , 2001, cond-mat/0105191.

[3]  J. F. Benders.  Partitioning procedures for solving mixed-variables programming problems , 1962 .

[4]  M. Chapman Findlay,et al.  Stochastic dominance : an approach to decision-making under risk , 1978 .

[5]  George L. Nemhauser,et al.  Handbooks in operations research and management science , 1989 .

[6]  A. McNeil,et al.  Copulas and credit models , 2001 .

[7]  T. Ralphs,et al.  Decomposition Methods , 2010 .

[8]  H. Brachinger,et al.  Risk as a primitive: A survey of measures of perceived risk , 1997 .

[9]  Sovan Mitra,et al.  Risk Measures in Quantitative Finance , 2009, 0904.0870.

[10]  M. Frittelli,et al.  RISK MEASURES AND CAPITAL REQUIREMENTS FOR PROCESSES , 2006 .

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

[12]  Andrzej Ruszczynski,et al.  Risk-averse dynamic programming for Markov decision processes , 2010, Math. Program..

[13]  Andrzej Ruszczynski,et al.  Decomposition methods in stochastic programming , 1997, Math. Program..

[14]  Christian Küchler,et al.  On Stability of Multistage Stochastic Programs , 2008, SIAM J. Optim..

[15]  Ying Xiong.  Nonlinear Optimization , 2014 .

[16]  Evan L. Porteus,et al.  Temporal Resolution of Uncertainty and Dynamic Choice Theory , 1978 .

[17]  Darinka Dentcheva,et al.  Kusuoka representation of higher order dual risk measures , 2010, Ann. Oper. Res..

[18]  M. Rothschild,et al.  Increasing risk: I. A definition , 1970 .

[19]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[20]  Brian Everitt,et al.  Cluster analysis , 1974 .

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

[22]  R. Rockafellar,et al.  Conditional Value-at-Risk for General Loss Distributions , 2001 .

[23]  P. Sopp.  Cluster analysis. , 1996, Veterinary immunology and immunopathology.

[24]  H. Föllmer,et al.  Convex risk measures and the dynamics of their penalty functions , 2006 .

[25]  Roy van der Weide,et al.  GO-GARCH: a multivariate generalized orthogonal GARCH model , 2002 .

[26]  H. Levy,et al.  Efficiency analysis of choices involving risk , 1969 .

[27]  Jacques F. Benders,et al.  Partitioning procedures for solving mixed-variables programming problems , 2005, Comput. Manag. Sci..

[28]  Leon Cooper,et al.  Heuristic Methods for Location-Allocation Problems , 1964 .

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

[30]  Ronald Hochreiter,et al.  Financial scenario generation for stochastic multi-stage decision processes as facility location problems , 2007, Ann. Oper. Res..

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

[32]  Akihiko Inoue.  On the worst conditional expectation , 2001 .

[33]  Sergei Vassilvitskii,et al.  k-means++: the advantages of careful seeding , 2007, SODA '07.

[34]  W. Ogryczak,et al.  LP solvable models for portfolio optimization: a classification and computational comparison , 2003 .

[35]  Naomi Miller,et al.  Risk-Averse Two-Stage Stochastic Linear Programming: Modeling and Decomposition , 2011, Oper. Res..

[36]  T. Koopmans.  Stationary Ordinal Utility and Impatience , 1960 .

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

[38]  Masaaki Kijima,et al.  Mean-risk analysis of risk aversion and wealth effects on optimal portfolios with multiple investment opportunities , 1993, Ann. Oper. Res..

[39]  M. Rothschild,et al.  Asset Pricing with a Factor Arch Covariance Structure: Empirical Estimates for Treasury Bills , 1988 .

[40]  A. Ruszczynski,et al.  Nonlinear Optimization , 2006 .

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

[42]  R. Tyrrell Rockafellar.  Conjugate Duality and Optimization , 1974 .

[43]  G. Pflug,et al.  Modeling, Measuring and Managing Risk , 2008 .

[44]  F. Delbaen,et al.  Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes , 2004, math/0410453.

[45]  Charles Goodhart,et al.  The Basel Committee on Banking Supervision , 2011 .

[46]  Naomi Liora Miller,et al.  MEAN-RISK PORTFOLIO OPTIMIZATION PROBLEMS WITH RISK-ADJUSTED MEASURES , 2008 .

[47]  David Heath,et al.  Coherent multiperiod risk adjusted values and Bellman’s principle , 2007, Ann. Oper. Res..

[48]  Alexander Shapiro,et al.  Lectures on Stochastic Programming: Modeling and Theory , 2009 .

[49]  Jacques Janssen,et al.  Clayton copula and mixture decomposition , 2005 .

[50]  Timo Terasvirta,et al.  Multivariate GARCH Models , 2008 .

[51]  J Figueira,et al.  Stochastic Programming , 1998, J. Oper. Res. Soc..

[52]  Werner Römisch,et al.  Scenario tree modeling for multistage stochastic programs , 2009, Math. Program..

[53]  Frank Riedel,et al.  Dynamic Coherent Risk Measures , 2003 .

[54]  Spiridon I. Penev,et al.  Multistage optimization of option portfolio using higher order coherent risk measures , 2013, Eur. J. Oper. Res..

[55]  Jerzy A. Filar,et al.  Time Consistent Dynamic Risk Measures , 2006, Math. Methods Oper. Res..

[56]  Georg Ch. Pflug,et al.  Scenario tree generation for multiperiod financial optimization by optimal discretization , 2001, Math. Program..

[57]  Alexander Shapiro,et al.  Conditional Risk Mappings , 2005, Math. Oper. Res..

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