DynOpt: Incorporating dynamics into mean-variance portfolio optimization

Mean-variance (MV) portfolio theory leads to relatively simple and elegant numerical problems. Nonetheless, the approach has been criticized for treating the market parameters as if they were constant over time. We propose a novel convex optimization problem that extends an existing MV formulation with chance constraint(s) by accounting for the portfolio dynamics. The core idea is to consider a multiperiod scenario where portfolio weights are implicitly regarded as the output of a state-space dynamical system driven by external inputs. The approach leverages a result on realization theory and uses the nuclear norm to penalize complex dynamical behaviors. The proposed ideas are illustrated by two case studies.

[1]  Stephen P. Boyd,et al.  A rank minimization heuristic with application to minimum order system approximation , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[2]  J. DiLellio.  A Kalman filter control technique in mean-variance portfolio management , 2015 .

[3]  Marc Moonen,et al.  A geometrical approach for the identification of state space models with singular value decomposition , 1988, ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing.

[4]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[5]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[6]  Bart De Moor,et al.  Subspace Identification for Linear Systems: Theory ― Implementation ― Applications , 2011 .

[7]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[8]  Donald Goldfarb,et al.  Robust Portfolio Selection Problems , 2003, Math. Oper. Res..

[9]  E. Fama,et al.  Multifactor Explanations of Asset Pricing Anomalies , 1996 .

[10]  W. Sharpe.  The Sharpe Ratio , 1994 .

[11]  Patrick Dewilde,et al.  Subspace model identification Part 1. The output-error state-space model identification class of algorithms , 1992 .

[12]  Pablo A. Parrilo,et al.  Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , 2007, SIAM Rev..

[13]  W. Sharpe.  A Simplified Model for Portfolio Analysis , 1963 .

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

[15]  Carol Alexander.  Moving Average Models for Volatility and Correlation, and Covariance Matrices , 2008 .

[16]  Srikanth Sridharan,et al.  A receding horizon control approach to portfolio optimization using a risk-minimax objective for wealth tracking , 2011, 2011 IEEE International Conference on Control Applications (CCA).

[17]  Johan A. K. Suykens,et al.  Convex Estimation of Cointegrated VAR Models by a Nuclear Norm Penalty , 2012 .

[18]  Donald Goldfarb,et al.  Robust convex quadratically constrained programs , 2003, Math. Program..

[19]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[20]  S. Van Huffel,et al.  Exact and Approximate Modeling of Linear Systems: A Behavioral Approach , 2006 .

[21]  Zoubin Ghahramani,et al.  A Unifying Review of Linear Gaussian Models , 1999, Neural Computation.

[22]  R. Shumway,et al.  AN APPROACH TO TIME SERIES SMOOTHING AND FORECASTING USING THE EM ALGORITHM , 1982 .

[23]  Lieven Vandenberghe,et al.  Interior-Point Method for Nuclear Norm Approximation with Application to System Identification , 2009, SIAM J. Matrix Anal. Appl..

[24]  M. Best,et al.  On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results , 1991 .

[25]  Carol Alexander Chair and.  Moving Average Models for Volatility and Correlation, and Covariance Matrices , 2008 .

[26]  R. Haugen,et al.  Commonality in the Determinants of Expected Stock Returns , 1996 .