Robust Portfolio Selection Problems

In this paper we show how to formulate and solve robust portfolio selection problems. The objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters. We introduce "uncertainty structures" for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second-order cone programs and, therefore, the computational effort required to solve them is comparable to that required for solving convex quadratic programs. Moreover, we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures employed to estimate the market parameters. Finally, we demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence.

[1]  W. Sharpe.  CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* , 1964 .

[2]  J. Lintner.  THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS , 1965 .

[3]  J. Mossin.  EQUILIBRIUM IN A CAPITAL ASSET MARKET , 1966 .

[4]  Vijay S. Bawa,et al.  The effect of estimation risk on optimal portfolio choice , 1976 .

[5]  Peter A. Frost,et al.  An Empirical Bayes Approach to Efficient Portfolio Selection , 1986, Journal of Financial and Quantitative Analysis.

[6]  C. O'Connor.  An introduction to multivariate statistical analysis: 2nd edn. by T. W. Anderson. 675 pp. Wiley, New York (1984) , 1987 .

[7]  P. Frost,et al.  For better performance , 1988 .

[8]  Robert B. Litterman,et al.  Asset Allocation , 1991 .

[9]  Mark Broadie,et al.  Computing efficient frontiers using estimated parameters , 1993, Ann. Oper. Res..

[10]  V. K. Chopra.  Improving Optimization , 1993 .

[11]  V. K. Chopra,et al.  Massaging mean-variance inputs: returns from alternative global investment strategies in the 1980s , 1993 .

[12]  W. Ziemba,et al.  The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice , 1993 .

[13]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[14]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[15]  Laurent El Ghaoui,et al.  Robust Solutions to Least-Squares Problems with Uncertain Data , 1997, SIAM J. Matrix Anal. Appl..

[16]  Laurent El Ghaoui,et al.  Robust Solutions to Uncertain Semidefinite Programs , 1998, SIAM J. Optim..

[17]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[18]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[19]  Stephen P. Boyd,et al.  Applications of Second-order Cone , 1998 .

[20]  W. Ziemba,et al.  Worldwide asset and liability modeling , 1998 .

[21]  Arkadi Nemirovski,et al.  Robust solutions of uncertain linear programs , 1999, Oper. Res. Lett..

[22]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[23]  Helmut Mausser,et al.  Beyond VaR: from measuring risk to managing risk , 1999, Proceedings of the IEEE/IAFE 1999 Conference on Computational Intelligence for Financial Engineering (CIFEr) (IEEE Cat. No.99TH8408).

[24]  David E. Booth,et al.  Analysis of Incomplete Multivariate Data , 2000, Technometrics.

[25]  Hans Frenk,et al.  High performance optimization , 2000 .

[26]  Arkadi Nemirovski,et al.  Robust Modeling of Multi-Stage Portfolio Problems , 2000 .

[27]  D. Bertsimas,et al.  Moment Problems and Semidefinite Optimization , 2000 .

[28]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[29]  Helmut Mausser,et al.  Credit risk optimization with Conditional Value-at-Risk criterion , 2001, Math. Program..

[30]  T. Sargent,et al.  Robust Control and Model Uncertainty , 2001 .

[31]  Laurent El Ghaoui,et al.  Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach , 2003, Oper. Res..

[32]  B. Halldórsson,et al.  An Interior-Point Method for a Class of Saddle-Point Problems , 2003 .

[33]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[34]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[35]  Olivier Ledoit,et al.  A well-conditioned estimator for large-dimensional covariance matrices , 2004 .