Master Thesis Robin Jonsson O PTIMAL L INEAR C OMBINATIONS OF I NTRODUCTION P ORTFOLIOS S UBJECT TO E STIMATION Problem Formulation Solutions R ISK Main Literature Estimation Risk Parameter Uncertainty Robin Jonsson 1 Estimation Risk in Portfolios Combined Portfolio Rules 1 Division of Applied Mathematics Formulation of the Mälardalen University Rule Other Combined Rules Performance Master Presentation for Financial Engineering, 2015 Evaluation MMA 891, Project in Mathematics With Specialization in Experimental Design Combination Finance, 30 ECTS Credits Coefficients Out-of-Sample Results Conclusion
D IVISION OF A PPLIED M ATHEMATICS Master Thesis in Applied Mathematics, With Specialization in Finance Date: June 5, 2015 Project Name: Optimal Linear Combinations of Portfolios Subject to Estimation Risk Author : Robin Jonsson Supervisors : Lars Petterson, Senior Lecturer Linus Carlsson, Senior Lecturer Examiner : Anatoliy Malyarenko, Professor Comprising : 30 ECTS credits
T ABLE OF C ONTENTS I NTRODUCTION Master Thesis 1 Problem Formulation Robin Jonsson Solutions I NTRODUCTION Main Literature Problem Formulation Solutions Estimation Risk 2 Main Literature Parameter Uncertainty Estimation Risk Estimation Risk in Portfolios Parameter Uncertainty Estimation Risk in 3 Combined Portfolio Rules Portfolios Formulation of the Rule Combined Portfolio Rules Other Combined Rules Formulation of the Rule Other Combined Performance Evaluation 4 Rules Experimental Design Performance Evaluation Combination Coefficients Experimental Design Combination Out-of-Sample Results Coefficients Out-of-Sample Results Conclusion 5 Conclusion
Problem Formulation Master Thesis Robin The Problem With Markowitz Portfolio Selection Jonsson I NTRODUCTION Maximum likelihood estimators Problem Formulation Solutions Weighs toward outliers Main Literature Estimation Magnitude of estimation error for realized returns Risk Parameter Limits usefulness of mean-variance (MV) optimization Uncertainty Estimation Risk in Portfolios Simpler Portfolio rules are at least as good Combined Portfolio Rules Formulation of the Rule Other Combined Rules How can the problems be controlled such that Performance Evaluation mean-variance optimization becomes useful to portfolio Experimental Design Combination selection? Coefficients Out-of-Sample Results Conclusion
Solutions Master Thesis Robin Jonsson I NTRODUCTION Solutions that Improve Performance Problem Formulation Solutions Main Literature Better estimators Estimation Risk - Shrinkage of the covariance matrix Parameter Uncertainty Estimation Risk in Linear portfolio combinations Portfolios Combined - Convexity of Two- and Three-fund rules Portfolio Rules - Combine an equally weighted portfolio with a MV rule Formulation of the Rule Other Combined Rules Performance Evaluation Experimental Design Combination Coefficients Out-of-Sample Results Conclusion
Main Literature I Master Thesis Multivariate Statistics Robin Jonsson Muirhead, R. J.(1982) I NTRODUCTION Aspects of Multivariate Statistical Theory Problem Formulation Solutions Wiley Series In Probability and Mathematical Statistics , Main Literature John Wiley & Sons. Estimation Risk Covariance Estimation Parameter Uncertainty Estimation Risk in Ledoit, O., & Wolf, M. (2004a) Portfolios A well-conditioned estimator for large-dimensional Combined Portfolio Rules covariance matrices Formulation of the Rule Journal of multivariate analysis , 88(2), 365-411. Other Combined Rules Performance Ledoit, O., & Wolf, M. (2004b) Evaluation Experimental Design Honey, I shrunk the sample covariance matrix Combination Coefficients The Journal of Portfolio Management , 30(4), 110-119. Out-of-Sample Results Conclusion
Main Literature II Master Thesis Portfolio Selection Under Parameter Uncertainty Robin Jonsson Kan, R., & Zhou, G. (2007) I NTRODUCTION Optimal portfolio choice with parameter uncertainty Problem Formulation Journal of Financial and Quantitative Analysis , 42(03), Solutions Main Literature 621-656. Estimation Risk Kan, R., & Smith, D. R. (2008) Parameter Uncertainty The distribution of the sample minimum-variance Estimation Risk in Portfolios frontier Combined Portfolio Rules Management Science , 54(7), 1364-1380. Formulation of the Rule Other Combined Tu, J., & Zhou, G. (2011) Rules Performance Markowitz meets Talmud: A combination of Evaluation sophisticated and naive diversification strategies Experimental Design Combination Coefficients Journal of Financial Economics , 99(1), 204-215. Out-of-Sample Results Conclusion
Parameter Uncertainty Master Thesis We are accounting for estimation risk by not knowing the true parameters. Robin Jonsson Parametric Risk Premiums I NTRODUCTION Problem Formulation Parameter Proxy Expectation Distribution Solutions Main Literature ˜ N ( µ , Σ ) Estimation µ µ µ Risk ˜ T − 1 W N ( T − 1 , Σ ) / T Σ Σ T Σ Parameter ˜ Uncertainty W − 1 Σ − 1 Σ − 1 T − N − 2 Σ − 1 T N ( T − 1 , Σ ) T Estimation Risk in Portfolios g ( 1 + ˆ N ( µ g , σ 2 ψ 2 ) / T ) µ g µ g ˜ µ g Combined [ T ( 1 + ψ 2 ) − 2 ] σ 2 Portfolio Rules µ 2 µ 2 g + µ 2 ˜ g g g T ( T − N − 1 ) Formulation of the Rule Identities for W = Σ − 1 ΣΣ − 1 2 ˜ 2 ∼ W N ( T − 1 , I ) / T Other Combined Rules Performance E [ W − 1 ] = T T − N − 2 I Evaluation � � Experimental Design T 2 ( T − 2 ) E [ W − 2 ] = Combination I ( T − N − 1 )( T − N − 2 )( T − N − 4 ) Coefficients Out-of-Sample Results Conclusion
Utility and Risk function Master Thesis Robin Jonsson The utility of a mean–variance portfolio is measured by I NTRODUCTION Problem Formulation U ( w ) = w ′ µ − γ Solutions 2 w ′ Σ w (1) Main Literature Estimation Risk The risk function is a linear certainty equivalent return Parameter Uncertainty based on quadratic utility. Estimation Risk in Portfolios Combined R ( w ∗ , ˜ w ) = E [ L ( w ∗ , ˜ w )] = U ( w ∗ ) − E [ U (˜ w )] , (2) Portfolio Rules Formulation of the Rule Other Combined measures the expected out-of-sample loss between Rules Performance portfolio rules due to estimation risk. Evaluation Experimental Design Combination Coefficients Out-of-Sample Results Conclusion
Estimation Risk in the GMV Portfolio Master Thesis The loss between the unconstrained MV portfolio ( w ∗ ), and the Robin out-of-sample GMV portfolio ( ˆ w ) is considered. The rules are given by Jonsson w ∗ = 1 w = 1 I NTRODUCTION γ Σ − 1 µ , Σ − 1 ˜ ˜ and ˆ µ g 1 N . (3) Problem Formulation γ Solutions Main Literature We have, Estimation Risk � 1 � Parameter Σ − 1 µ − 1 N ˜ N ˜ Σ − 1 Σ ˜ µ 2 Σ − 1 1 N Uncertainty E [ U (ˆ w )] = E γ ˜ µ g 1 ′ 2 γ ˜ g 1 ′ Estimation Risk in Portfolios = 1 − 1 � � � � Combined N ˜ Σ − 1 µ µ 2 N ˜ Σ − 1 Σ ˜ Σ − 1 1 N µ g 1 ′ g 1 ′ γ E ˜ 2 γ E ˜ Portfolio Rules Formulation of the Rule = 1 − 1 � N Σ − 1 2 W − 1 Σ − 1 � � N Σ − 1 2 W − 2 Σ − 1 � µ 2 2 1 N µ g 1 ′ g 1 ′ γ E ˜ 2 µ 2 γ E ˜ , Other Combined Rules Performance �� � T ( 1 + ψ 2 ) − 2 � σ 2 N Σ − 1 µ = c 1 µ g 1 ′ − c 2 Evaluation g + µ 2 , g Experimental Design T ( T − N − 1 ) γ 2 γ Combination Coefficients Out-of-Sample Results Conclusion
Proposition 1 Master Thesis Proposition 1 Robin Jonsson Given the quadratic utility function in (1), the expected loss I NTRODUCTION of holding the out-of-sample GMV portfolio instead of the Problem Formulation MV portfolio under parameter uncertainty is Solutions Main Literature �� � � Estimation 2 γ − c 1 µ 2 g σ 2 T ( 1 + ψ 2 ) − 2 σ 2 w , w ∗ ) = θ 2 Risk + c 2 g g + µ 2 R (ˆ , Parameter g Uncertainty γ 2 γ T ( T − N − 1 ) Estimation Risk in Portfolios Combined by the risk function in (2). Portfolio Rules Formulation of the Rule Other Combined Rules Performance Evaluation c 1 = T / ( T − N − 2 ) , Experimental Design Combination c 2 = T 2 ( T − 2 ) / [( T − N − 1 )( T − N − 2 )( T − N − 4 )] Coefficients Out-of-Sample Results Conclusion
Estimation Risk for R (ˆ w , w ∗ ) Master Thesis Estimation Risk for R (ˆ w , w ∗ ) Panel A: θ 2 = 0 . 2 Robin Jonsson Annual µ = 5 % Annual µ = 10 % I NTRODUCTION N T γ = 1 γ = 3 γ = 1 γ = 3 Problem Formulation Solutions 60 10.26 3.42 11.00 3.67 Main Literature 10 120 10.17 3.39 10.66 3.55 Estimation 240 10.15 3.38 10.54 3.51 Risk Parameter Uncertainty 60 10.85 3.62 13.34 4.45 Estimation Risk in Portfolios 25 120 10.28 3.43 11.06 3.69 Combined 240 10.18 3.39 10.67 3.56 Portfolio Rules Formulation of the Rule 60 102.84 34.28 381.08 127.03 Other Combined Rules 50 120 10.73 3.58 12.85 4.28 Performance 240 10.26 3.42 11.00 3.67 Evaluation Experimental Design 100 120 58.84 19.61 205.11 68.37 Combination Coefficients 240 10.67 3.56 12.63 4.21 Out-of-Sample Results Conclusion
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