Understanding and Controlling High Factor Exposures of Robust Portfolios

Transcription

1 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

2 Contents Section 1. Introduction Section 2. Robustness Measure Section 3. New Robust Portfolio Formulation Section 4. Empirical Analysis Section 5. Conclusions 2

3 Section 1 INTRODUCTION 3

4 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

5 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

6 Section 2 ROBUSTNESS MEASURE 6

7 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

8 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

9 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

10 Section 3 NEW ROBUST PORTFOLIO FORMULATION 10

11 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

12 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

13 Constraints on Factor Exposure We will find additional constraints to the robust formulation with ellipsoidal uncertainty sets. [ Robust formulation ] [ Additional constraints] 13

14 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

15 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

16 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

17 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

18 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

19 Summary of the New Robust Formulations 19

20 Section 4 EMPIRICAL ANALYSIS 20

21 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

22 Empirical Results (1/3) F1: Yearly value of factor loading (top: 0.01, bottom: 0.05) Norm of factor loading MV R R1 R4' Norm of factor loading MV R R1 R4'

23 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 MV R R1 R4' Proportion of variance from factors MV R R1 R4'

24 Empirical Results (3/3) RM: Robustness measure (top: = 0.01, bottom: = 0.05) 3.0E E 05 Robustness measure 2.0E E E 05 MV R R1 R4' 5.0E E E E 04 Robustness measure 2.5E E E E 04 MV R R1 R4' 5.0E E

25 Section 5 CONCLUSIONS 25

26 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

27 THANK YOU! Q&A 27

28 REFERENCE 28

29 Alexander,C.,Dimitriu,A.,2004.Sources of outperformance in equity markets.journal of Portfolio Management 30 (4), Ben Tal,A.,Nemirovski,A.,1998.Robust convex optimization.mathematics of Operations Research 23 (4), Best,M.J.,Grauer,R.R.,1991.On the sensitivity of mean variance efficientportfolios to changes in asset means:some analytical and computational results.review of Financial Studies 4(2), Broadie,M.,1993.Computing efficientfrontiers using estimated parameters.annals of Operations Research 45 (1), Chopra,V.K.,Ziemba,W.T.,1993.The effect of errors in means,variances,and covariances on optimal portfolio choice.journal of Portfolio Management 19 (2),6 11. Clark,R.G.,de Silva,H.,Wander,B.H.,2002.Risk allocation versus asset allocation.journal of Portfolio Management 29 (1),9 30. Costa,O.L.V.,Paiva,A.C.,2002.Robust portfolio selection using linear matrix inequalities.journal of Economic Dynamics &Control 26, Goldfarb,D.,Iyengar,G.,2003.Robust portfolio selection problems.mathematics of Operations Research 28 (1),1 38. El Ghaoui,L.,Oks,M.,Oustry,F.,2003.Worst case value at risk and robust portfolio optimization:aconic programming approach.operations Research 51 (4), Fabozzi,F.J.,Huang,D.,Zhou,G.,2010.Robust portfolios:contributions from operations research and finance.annals of Operations Research 176, Fabozzi,F.J.,Kolm,P.N.,Pachamanova,D.A.,Focardi,S.M.,2007a.Robust portfolio optimization.journal of Portfolio Management 33, Fabozzi,F.J.,Kolm,P.N.,Pachamanova,D.A.,Focardi,S.M.,2007b.Robust Portfolio Optimization and Management.Wiley,Hoboken,NJ.

30 Fama,E.F.,French,K.R.,1993.Common risk factors in the returns on stocks and bonds.journal of Financial Economics 33(1),3 56. Fama,E.F.,French,K.R.,1995.Size and book to market factors in earnings and returns.journal of Finance 50 (1), Fama,E.F.,French,K.R.,1997.Industry costs of equity.journal of Financial Economics 43 (2), Fung,W.,Hsieh,D.A.,1997.Empirical characteristics of dynamic trading strategies:the case of hedge funds.review of Financial Studies 10 (2), Kanniappan,P.,Sastry,S.M.A.,1983.Uniform convergence of convex optimization problems.journal of Mathematical Analysis and Applications 96,1 12. Kim,W.C.,Kim,J.H.,Ahn,S.H.,Fabozzi,F.J.,accepted for publication a.what do robust equity portfolio models really do? Annals of Operations Research. Kim,J.H.,Kim,W.C.,Fabozzi,F.J.,accepted for publication b.recent developments in robust portfolios with aworst case approach.journal of Optimization Theory and Applications. Kim,W.C.,Kim,J.H.,Fabozzi,F.J.,2012.Deciphering robust portfolios.working paper. Kim,W.C.,Kim,J.H.,Mulvey J.M.,Fabozzi,F.J.,2013.Focusing on the worst state for robust investing.working paper. Markowitz,H.,1952.Portfolio selection.journal of Finance 7(1), Strang,G.,1988.Linear Algebra and its Applications,3rd ed.thomson Learning,Inc (Chapter6). Stubbs,R.A.,Vance,P.,2005.Computing Return Estimation Error Matrices for Robust Optimization.Axioma,Inc. Tütüncü,R.H.,Koenig,M.,2004.Robust asset allocation.annals of Operations Research 132, Zhu,S.,Cui,X.,Sun,X.,Li,D.,2011.Factor risk constrained mean variance portfolio selection:formulation and global optimization solution approach.journal of Risk 14 (2),51 89.