Understanding and Controlling High Factor Exposures of Robust Portfolios July 8, 2013 Min Jeong Kim Investment Design Lab, Industrial and Systems Engineering Department, KAIST Co authors: Woo Chang Kim, Jang Ho Kim and Frank J. Fabozzi
Contents Section 1. Introduction Section 2. Robustness Measure Section 3. New Robust Portfolio Formulation Section 4. Empirical Analysis Section 5. Conclusions 2
Section 1 INTRODUCTION 3
Robust Portfolio Optimization Mean variance model (Markowitz, 1952) Mean variance portfolio weight are sensitive to changes in means, variances, and covariance. Robust portfolio optimization (worst case formulation) 4
Behavior of Robust Portfolios Much researches has been focused on defining uncertainty in input parameters and formulating the problems using worst case optimization methods. The relationship between robustness correlation with factors (Kim et al. 2012a) Worst case formulation with ellipsoidal and box uncertainty sets Robustness leads to higher correlation with factors. Factors explain the underlying movement of the market Market index, size and book to market ratio factors (Fama and French 1993) Therefore, investors can lose control of the factor dependency. Research Objective Introduce formulations which maintain a desired level of factor exposure while conserving the robust effect of robust optimization. 5
Section 2 ROBUSTNESS MEASURE 6
Robustness Measure The meaning of robustness Assume is a function of an uncertain parameter The robustness of the movement of with respect to. Suitable uncertain function for measuring portfolio robustness Portfolio weight? Portfolio performance? The optimal portfolio is robust if the performance of the portfolio is stable with respect to input parametets. Robustness measure 7
Properties of the Robustness Measure (1/2) Robust formulation with ellipsoidal uncertainty set for We base our model on the robust formulation with an ellipsoidal uncertainty set. Portfolio performance Among many possibilities for representing portfolio performance, we focus on the objective function of the robust formulation. An important property which the robustness measure should satisfy is the decreasing pattern with respect to the delta. 8
Properties of the Robustness Measure (2/2) Claim 1 Let be the optimal solution to the optimization problem. If converges to as goes to infinity, then is the portfolio that minimizes the robustness measure. Claim 2 The robustness measure is a decreasing function of. That is, the optimal portfolio becomes more robust as increases and the robustness measure max,, decreases. 9
Section 3 NEW ROBUST PORTFOLIO FORMULATION 10
New Robust Portfolio Formulation We derive new robust formulations that control the dependency on fundamental factors by having the same factor exposure relative to a target portfolios. We again base our model on the robust formulation with an ellipsoidal uncertainty set. Parameter assumptions n risky securities r : (random) return vector of risky securities E Var is strictly positive definite where is the returns of m (< n) factors, is stock betas and is the idiosyncratic risk : a portfolio that satisfies the target factor exposure (we set it to the Markowitz portfolio) 11
Measuring the Proportion Invested in Factors What value can measure the exposure to factors? The expected portfolio return and the variance of returns becomes where E and Var. Exposure to factors The amount of return invested on factors: The proportion of variance depended on factors: By controlling the factor exposure with constraints that incorporate above values, we can construct portfolios that possess the same level of factor dependency as the target portfolio. 12
Constraints on Factor Exposure We will find additional constraints to the robust formulation with ellipsoidal uncertainty sets. [ Robust formulation ] [ Additional constraints] 13
Formulation Based on the Expected Return A constraint that matches the factor loading with the target portfolio Constraint 1 Additional constraints is Linear constraint 14
Formulation Based on the Variance of Returns (1/4) If the portfolio return is heavily affected by the factor returns, it means that the value of is relatively greater than. The proportion of variance explained by the variance of factors, can be used to match the factor exposure of portfolios. Constraint 2 Additional constraint is Non linear constraint! 15
Formulation Based on the Variance of Returns (2/4) To relax the constraint 2, let and, Using the Rayleigh s principle, the two values, and, can be easily found by calculating the maximum and minimum eigenvalues. The alternative approach should at least find an optimal portfolio that is not too close to either or. 16
Formulation Based on the Variance of Returns (3/4) Constraint 3 The factor exposure of portfolio should not be maximized or minimized unless that is the case for portfolio. Additional constraint is Linear constraints! 17
Formulation Based on the Variance of Returns (4/4) Constraint 4 Because the robust portfolios tend to increase the dependency on factors, it is more important for the optimal portfolio to not move towards than. Additional constraints are Linear constraints! 18
Summary of the New Robust Formulations 19
Section 4 EMPIRICAL ANALYSIS 20
Empirical Analysis We analyze portfolios formed from,, 1 and 4 with 0.5. Test measures Data description The three factor model proposed by Fama and French (1993, 1995) Daily returns from January 1970 to December 2011 for both factor and industry level returns. 21
Empirical Results (1/3) F1: Yearly value of factor loading (top: 0.01, bottom: 0.05) 1.6 1.4 2 Norm of factor loading 1.2 1 0.8 0.6 0.4 0.2 MV R R1 R4' 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 6 2 Norm of factor loading 5 4 3 2 1 MV R R1 R4' 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 22
Empirical Results (2/3) F2: Yearly value of proportion of variance from factors (top: 0.01, bottom: 0.05) 1 Proportion of variance from factors 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 MV R R1 R4' 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 1.2 Proportion of variance from factors 1 0.8 0.6 0.4 0.2 MV R R1 R4' 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 23
Empirical Results (3/3) RM: Robustness measure (top: = 0.01, bottom: = 0.05) 3.0E 05 2.5E 05 Robustness measure 2.0E 05 1.5E 05 1.0E 05 MV R R1 R4' 5.0E 06 0.0E+00 1970 1975 1980 1985 1990 1995 2000 2005 2010 3.5E 04 3.0E 04 Robustness measure 2.5E 04 2.0E 04 1.5E 04 1.0E 04 MV R R1 R4' 5.0E 05 0.0E+00 1970 1975 1980 1985 1990 1995 2000 2005 2010 24
Section 5 CONCLUSIONS 25
Conclusions We derive new robust formulations which maintain a desired level of factor exposure while conserving the robust effect of robust optimization. To derive the formulations, we introduce several approaches by adding constraints into the well known robust formulation with an ellipsoidal uncertainty set. Our empirical results show that the improved robust portfolios resolve the tilting effect of current robust formulations while still making the portfolios more robust than mean variance portfolios. 26
THANK YOU! Q&A 27
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