Robust Portfolio Optimization SOCP Formulations

Transcription

1 1 Robust Portfolio Optimization SOCP Formulations There has been a wealth of literature published in the last 1 years explaining and elaborating on what has become known as Robust portfolio optimization. I will not elaborate on the derivation of the objective function; rather I have listed a set of references (Page 6) where this material can be accessed. The book Robust Portfolio Optimization and Management by Fabozzi, Kolm, Pachamanova, and Focardi, (27), gives a comprehensive coverage of this work. In the practioner world, this work is most closely associated with Ceria and Stubbs (CS), who work at the software vendor Axioma Inc. The essential innovation of this work is to consider asset means as being derived from an (unknown) probability distribution. The worst case mean vector is computed by minimizing portfolio expected return for a given variance target. The mean vector used for the sample optimization lies somewhere between this worst case and the observed sample vector. There is however no definitive answer as to what the weightings should be, using a heuristic of some sort is suggested for parameterization. The classic MV objective function and Robust objective functions are: Classic mean-variance maximand: Robust mean-variance maximand: w R s w Ω s w w R s κ Σ 1/2 w w Ω s w where κ = confidence interval specific scaling of asset mean returns standard deviations. Σ = covariance matrix of asset mean returns. The robust formulation is not a quadratic program, because of the presence of the square root term this maximization is referred to as a second order cone program (SOCP). For the purpose of simulated comparisons, we have: κ = Σ = Ω s ( 1 ). 1 N where 1 = identity matrix, and N is the sample size. The Delta portfolio and the Robust solution were compared in simulations where 2 populations were sampled once each. Population portfolio Sharpe ratios are lie in the interval [,8] and sample sizes vary greatly with some being as low as K+1, so the impact of very high estimation error can be observed. As alluded to in the non-technical summary document, there are two portfolio optimization problems, the case where the mean vector is independent of the (residual) covariance matrix, and the case where there is dependence (equilibrium returns). Both cases were investigated and are illustrated in the graphs below. In Figures (2.1) and (2.2) we compare the ex post Sharpe ratios of the Delta portfolio and the Robust (CS) portfolio, for the independent and dependent case respectively. The data is sorted in ascending order by the CS ratios. The Delta portfolio has consistently higher ratios than the CS portfolio. In Figure (3.1) we compare the ex post mean-variance utility of the Delta portfolio and the Robust (CS) portfolio, for the independent case. The data is sorted in ascending order by the CS utility values. The Delta portfolio has consistently higher ex post expected utility than the CS portfolio.

2 2 In Figure (3.2) we compare the ex post mean-variance utility of the MVO portfolio and the Robust (CS) portfolio, for the independent case. The data is sorted in ascending order by the CS utility values. The CS portfolio has consistently higher ex post expected utility than the MVO portfolio. In Figure (3.3) we compare the ex post mean-variance utility of the Delta portfolio and the Robust (CS) portfolio, for the Dependent case. The results differ little from Figure (3.1). In Figure (3.4) we compare the ex post mean-variance utility of the MVO portfolio and the Robust (CS) portfolio, for the Dependent case. The results differ little from Figure (3.2). 1 Critique of the Robust Formulation Although the math is elegant, the reason for the failure of this SOCP to fully address the problem of optimization in the presence of sample error lies in the formulation of the problem. The formulation s first set up is to look at the distribution in the mean vectors (holding the weights constant) and then revert to optimizing over the weights. The key thing is to consider the variation together, there is a covariance in the estimation error in the mean vectors and the estimation error in the weights, so these comparative, statics will be inadequate. Furthermore this distribution of population solutions (or sample optimals), needs to be compared with the distribution of returns where all possible population weight vectors combined with the most likely population mean vector (ex post), to compute a bias in portfolio return (the bias always being equal to the mean ex ante expected return from the sampling of the Null population). Furthermore, this presentation of the problem entirely overlooks the typically significant contribution of estimation error in the higher moments. The diagonalization of the asset mean return covariance also has a deleterious effect on the solution: if the mean vector and the covariance matrix are independent, then any dilution of the correlations will lead to lower Sharpe ratios (if the mean return covariance matrix is the return covariance matrix scaled by 1/N then it won t change the solution from the classic formulation).

3 3 2 Sharpe or Information Ratios Figure (2.1) Ex Post Sharpe/Information Ratio: and CS Robust Portfolio Mean Return Vector Independent of Covariance Matrix Figure (2.2) Ex Post Sharpe/Information Ratio: and CS Robust Portfolio Mean Return Vector Dependent of Covariance Matrix

4 4 3 Expected Utility/Certainty Equivalent Figure (3.1) Ex Post Expected Utility: and CS Robust Portfolio Mean Return Vector Independent of Covariance Matrix Figure (3.2) Ex Post Expected Utility: MV Portfolio and CS Robust Portfolio Mean Return Vector Independent of Covariance Matrix 5 Mean-Variance -5-1

5 5 Figure (3.3) Ex Post Expected Utility: and CS Robust Portfolio Mean Return Vector Dependent of Covariance Matrix Figure (3.4) Ex Post Expected Utility: MV Portfolio and CS Robust Portfolio Mean Return Vector Dependent of Covariance Matrix 3 2 Mean-Variance

6 6 References Best, M. J. & Grauer, R. R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: Some analytical and computational results. Review of Financial Studies, 4: Ceria, S. & Stubbs, R. A. (26). Incorporating estimation errors into portfolio selection: Robust portfolio construction. Journal of Asset Management, 7: DeMiguel, V. & Nogales, F. J. (29). Portfolio selection with robust estimation. Operations Research, forthcoming. Fabozzi, F. J., Kolm, P. N., Pachamanova, D. A., Focardi, S. M., 27a. Robust portfolio optimization. Journal of Portfolio Management, 33, Fabozzi, F. J., Kolm, P. N., Pachamanova, D. A., Focardi, S. M., 27b. Robust Portfolio Optimization and Management. Hoboken, NJ: Wiley. Garlappi, L., Uppal, R., & Wang, T. (27). Portfolio selection with parameter and model uncertainty: A multi-prior approach. Review of Financial Studies, 2:41. Goldfarb, D. & Iyengar, G. (23). Robust Portfolio Selection Problems. Mathematics of Operations Research, 28:1 38. Lobo, M. S., Boyd, S., 2. The worst-case risk of a portfolio. Technical report, Stanford Scherer, B. (27). Can robust portfolio optimisation help to build better portfolios? Journal of Asset Management, 7: