Problem Formulation Master Thesis Robin The Problem With Markowitz - - PowerPoint PPT Presentation

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

OPTIMAL LINEAR COMBINATIONS OF PORTFOLIOS SUBJECT TO ESTIMATION RISK

Robin Jonsson1

1Division of Applied Mathematics

Mälardalen University

Master Presentation for Financial Engineering, 2015 MMA 891, Project in Mathematics With Specialization in Finance, 30 ECTS Credits

DIVISION OF APPLIED MATHEMATICS

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

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

TABLE OF CONTENTS

1

INTRODUCTION Problem Formulation Solutions Main Literature

2

Estimation Risk Parameter Uncertainty Estimation Risk in Portfolios

3

Combined Portfolio Rules Formulation of the Rule Other Combined Rules

4

Performance Evaluation Experimental Design Combination Coefficients Out-of-Sample Results

5

Conclusion

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Problem Formulation

The Problem With Markowitz Portfolio Selection Maximum likelihood estimators Weighs toward outliers Magnitude of estimation error for realized returns Limits usefulness of mean-variance (MV) optimization Simpler Portfolio rules are at least as good How can the problems be controlled such that mean-variance optimization becomes useful to portfolio selection?

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Solutions

Solutions that Improve Performance Better estimators

• Shrinkage of the covariance matrix

Linear portfolio combinations

• Convexity of Two- and Three-fund rules
• Combine an equally weighted portfolio with a MV rule

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Main Literature I

Multivariate Statistics Muirhead, R. J.(1982) Aspects of Multivariate Statistical Theory Wiley Series In Probability and Mathematical Statistics, John Wiley & Sons. Covariance Estimation Ledoit, O., & Wolf, M. (2004a) A well-conditioned estimator for large-dimensional covariance matrices Journal of multivariate analysis, 88(2), 365-411. Ledoit, O., & Wolf, M. (2004b) Honey, I shrunk the sample covariance matrix The Journal of Portfolio Management, 30(4), 110-119.

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Main Literature II

Portfolio Selection Under Parameter Uncertainty Kan, R., & Zhou, G. (2007) Optimal portfolio choice with parameter uncertainty Journal of Financial and Quantitative Analysis, 42(03), 621-656. Kan, R., & Smith, D. R. (2008) The distribution of the sample minimum-variance frontier Management Science, 54(7), 1364-1380. Tu, J., & Zhou, G. (2011) Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies Journal of Financial Economics, 99(1), 204-215.

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Parameter Uncertainty

We are accounting for estimation risk by not knowing the true parameters.

Parametric Risk Premiums Parameter Proxy Expectation Distribution µ ˜ µ µ N(µ, Σ) Σ ˜ Σ

T−1 T Σ

WN(T − 1, Σ)/T Σ−1 ˜ Σ−1

T T−N−2Σ−1

W−1

N (T − 1, Σ)T

µg ˜ µg µg N(µg, σ2

g(1 + ˆ

ψ2)/T) µ2

g

˜ µ2

g

[T(1+ψ2)−2]σ2

g

T(T−N−1)

+ µ2

g

Identities for W = Σ− 1

2 ˜

ΣΣ− 1

2 ∼ WN(T − 1, I)/T

E[W−1] =

T T−N−2I

E[W−2] =

• T2(T−2)

(T−N−1)(T−N−2)(T−N−4)

• I

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Utility and Risk function

The utility of a mean–variance portfolio is measured by U(w) = w′µ − γ 2w′Σw (1) The risk function is a linear certainty equivalent return based on quadratic utility. R(w∗, ˜ w) = E[L(w∗, ˜ w)] = U(w∗) − E[U(˜ w)], (2) measures the expected out-of-sample loss between portfolio rules due to estimation risk.

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Estimation Risk in the GMV Portfolio

The loss between the unconstrained MV portfolio (w∗), and the

• ut-of-sample GMV portfolio (ˆ

w) is considered. The rules are given by w∗ = 1 γ Σ−1µ, and ˆ w = 1 γ ˜ Σ−1˜ µg1N. (3) We have, E [U(ˆ w)] = E 1 γ ˜ µg1′

N ˜

Σ−1µ − 1 2γ ˜ µ2

g1′ N ˜

Σ−1Σ ˜ Σ−11N

• = 1

γ E

• ˜

µg1′

N ˜

Σ−1µ

• − 1

2γ E

• ˜

µ2

g1′ N ˜

Σ−1Σ ˜ Σ−11N

• = 1

γ E

• ˜

µg1′

NΣ− 1

2 W−1Σ− 1 2 µ

• − 1

2γ E

• ˜

µ2

g1′ NΣ− 1

2 W−2Σ− 1 2 1N

• ,

= c1µg1′

NΣ−1µ

γ − c2 2γ

• T(1 + ψ2) − 2
• σ2

g

T(T − N − 1) + µ2

g

• ,

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Proposition 1

Proposition 1 Given the quadratic utility function in (1), the expected loss

• f holding the out-of-sample GMV portfolio instead of the

MV portfolio under parameter uncertainty is R(ˆ w, w∗) = θ2 2γ − c1µ2

gσ2 g

γ + c2 2γ

• T(1 + ψ2) − 2
• σ2

g

T(T − N − 1) + µ2

g

• ,

by the risk function in (2). c1 = T/(T − N − 2), c2 = T2(T − 2)/ [(T − N − 1)(T − N − 2)(T − N − 4)]

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Estimation Risk for R(ˆ w, w∗)

Estimation Risk for R(ˆ w, w∗) Panel A: θ2 = 0.2 Annual µ = 5 % Annual µ = 10 % N T γ = 1 γ = 3 γ = 1 γ = 3 60 10.26 3.42 11.00 3.67 10 120 10.17 3.39 10.66 3.55 240 10.15 3.38 10.54 3.51 60 10.85 3.62 13.34 4.45 25 120 10.28 3.43 11.06 3.69 240 10.18 3.39 10.67 3.56 60 102.84 34.28 381.08 127.03 50 120 10.73 3.58 12.85 4.28 240 10.26 3.42 11.00 3.67 100 120 58.84 19.61 205.11 68.37 240 10.67 3.56 12.63 4.21

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Combined Portfolio Rules

Linear portfolio combinations are theoretically convex If they are also empirically convex, we can formulate linear out-of-sample portfolios that outperform their components I propose to combine the equally weighted portfolio with a particular mean-variance portfolio Rules and Estimator wEW = 1 N , wLW = 1 γ ˜ Σ−1

LW ˜

µ (4) ˜ Σ−1

LW =

• αF + (1 − α) ˜

Σ −1 (5)

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Formulation of the Rule

Portfolio Rule and Estimated Delta wC = (1 − δ)wEW + δwLW, δ∗ = ˜ η1 − ˜ η13 ˜ η1 − 2˜ η13 + ˜ η3 ˜ η1 = wEW′ ˜ ΣwEW − 2 γ wEW′ ˜ µ + ˜ θ2 2γ . (6) ˜ η13 = 1 γ

• c2

1 − δ

• wEW′ ˜

µ − wEW′ ˜ ΣP˜ µ − ˜ θ2 γ

• + 1

γ ˜ µP˜ µ − wEW′ ˜ µ + ˜ θ2 γ

• .

(7) ˜ η3 = c2 γ2 N + T ˜ θ2 T(1 − δ)2 − c1 N + T ˜ µ′P˜ µ T(1 − δ) + N + T ˜ µ′P ˜ ΣP˜ µ T − 2c1˜ θ2 γ2(1 − δ) + ˜ µ′P˜ µ + ˜ θ2 γ2 . (8)

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Proposition 2

Proposition 2 Assume that T > N + 4 such that the second moment of the inverse Wishart distribution exists. Then there exist a combination of wEW and wLW, given by wC = (1 − δ)wEW + δwLW, such that the estimated optimum is wC = (1 − ˜ δ∗)wEW + ˜ δ∗wLW, where ˜ δ∗ = (˜ η1 − ˜ η13) / (˜ η1 − 2˜ η13 + ˜ η3) and ˜ η1, ˜ η13 and ˜ η3 are given by (6),(7) and (8) respectively.

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Other Combined Rules

The Traditional Markowitz Rule w1 = (1 − δ)wEW + δ w,

• w = 1

γ

• Σ−1 ˜

µ. The Kan & Zhou (2007) Two-Fund Rule w2 = (1 − δ)wEW + δ wKZ,

• w = 1

γ

• αKZ

Σ−1 ˜ µ + (1 − αKZ)µg Σ−11N

• .

The Jorion (1986) Rule w3 = (1 − δ)wEW + δ wPJ,

• wPJ = 1

γ

• Σ−1

PJ µPJ.

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Experimental Design

Normalized Weights wNorm = wi |1′

Nwi|

Performance Measures SRi,t = µi,t σi,t , CERi,t = µi,t − γ 2σ2

i,t.

The reported measures are averages; SRi and CERi. Data S & P 500 Constituent List 414 assets, 2000-01-01 to 2015-01-31 monthly data T = [60, 90, 120, 150], N = [10, 25, 50, 75] “Rolling sample” approach

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Ratio of Extreme Coefficients

Figure: Ratio of Extreme Combination Coefficients

ˆ δi = max

• 0, min{ˆ

δi, 1}

• .

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Convergence Ratio of Sophisticated Rule

Ratio of Sophisticated Weights Parameters T 60 120 240 480 960 3000 6000 Panel 1: w∗

LW

˜ δC 0.227 0.287 0.191 0.215 0.239 0.262 0.268 (0.354) (0.182) (0.065) (0.023) (0.013) (0.006) (0.005) Panel 2: wPJ ˜ δPJ 0.142 0.399 0.121 0.139 0.173 0.190 0.193 (0.058) (0.278) (0.206) (0.039) (0.018) (0.013) (0.012) Panel 3: wML ˜ δML 0.142 0.275 0.481 0.668 0.807 0.930 0.964 (0.155) (0.061) (0.043) (0.025) (0.014) (0.005) (0.003) Panel 4: wKZ ˜ δKZ 0.932 0.923 0.915 0.933 0.958 0.984 0.992 (0.158) (0.080) (0.043) (0.025) (0.014) (0.005) (0.003)

Table: Ratio of sophisticated weights

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Sharpe Ratios

Out-of-Sample Sharpe Ratios Portfolio Size (N) Rules 10 25 50 75 10 25 50 75 Panel 1: T = 60 Panel 2: T = 90 EW 0.75 0.77 0.77 − 0.60 0.65 0.67 0.65 LW (ccm) 0.04 0.23 −0.08 − 0.18 0.16 0.02 0.03 LW (sid) 0.03 0.23 −0.08 − 0.17 0.17 0.02 0.03 ML 0.19 −0.09 −0.08 − 0.16 0.22 −0.08 −0.36 KZ 0.62 0.51 0.11 − 0.51 0.51 0.42 0.08 PJ −0.17 0.12 0.14 − 0.12 0.22 0.24 0.03 CLW (ccm) 0.32 0.31 0.17 − 0.33 0.53 0.67 0.35 CLW (sid) 0.43 0.44 0.45 − 0.45 0.65 0.65 0.48 CML 0.30 0.30 0.38 − 0.52 0.60 0.46 −0.34 CKZ 0.69 0.63 0.16 − 0.56 0.62 0.56 0.18 CPJ −0.06 0.42 0.56 − 0.40 0.62 0.57 0.58 Panel 3: T = 120 Panel 4: T = 150 EW 1.19 1.18 1.18 1.21 2.01 2.09 2.24 2.25 LW (ccm) 0.86 0.97 0.85 0.78 1.04 0.69 0.61 0.41 LW (sid) 0.88 0.98 0.85 0.78 1.04 0.69 0.61 0.43 ML 0.87 0.92 0.72 0.50 1.03 0.67 0.65 0.48 KZ 1.32 1.43 1.29 1.17 1.66 1.53 1.53 1.47 PJ 0.77 0.90 0.72 0.51 0.91 0.64 0.64 0.48 CLW (ccm) 1.13 1.27 1.28 1.29 1.78 1.70 1.88 2.05 CLW (sid) 1.17 1.29 1.27 1.36 1.83 1.83 1.81 1.84 CML 1.19 1.24 1.17 1.10 1.89 1.93 2.18 2.14 CKZ 1.28 1.42 1.30 1.27 1.84 1.78 1.96 2.03 CPJ 1.12 1.29 1.29 1.27 1.68 1.55 2.06 2.15

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Certainty Equivalent Returns

Certainty Equivalent Returns, γ = 3 Portfolio Size Rules 10 25 50 75 10 25 50 75 Panel 1: T = 60 Panel 2: T = 90 EW 8.55 9.26 8.97

• 6.67

6.68 6.95 7.19 LW (ccm) < -5000

• 258.06
• 662.55
• < -5000
• 186.71
• 467.82
• 24.27

LW (sid)

• 419.60
• 3385.47
• 493.55
• 58.22
• 7.60
• 8.14
• 8.04

ML

• 509.85
• 166.62
• 220.86
• 272.55
• 17.54
• 80.90

< -5000 KZ 5.28

• 61.10

6.49

• 5.56

4.12 0.61

• 14.11

PJ

• 2033.73
• 50.79

< -5000

• 1086.61
• 17.57
• 98.12

< -5000 CLW (ccm)

• 758.89
• 106.84
• 68.71
• < -5000
• 12.90

6.75 0.67 CLW (sid)

• 21.33

6.36

• 28.56
• 2.88

4.58 5.52 4.07 CML

• 78.87
• 17.06
• 18.41
• 41.10

2.48

• 22.68
• 677.74

CKZ 6.71

• 27.20

8.08

• 6.86

5.28 3.25

• 8.14

CPJ

• 252.93

7.24

• 1009.60
• 393.30

5.46 5.05

• 644.81

Panel 3: T = 120 Panel 4: T = 150 EW 13.95 14.29 14.55 14.21 18.59 19.38 19.17 19.17 LW (ccm) 11.60 11.40 10.03 9.29 12.05 11.71 6.14 3.22 LW (sid) 12.51 12.94 12.24 12.50 13.30 13.72 9.90 7.96 ML 11.80 10.50 0.13

• 7.12

11.75 11.71 6.23 2.60 KZ 13.11 13.31 13.18 12.55 16.51 15.93 14.19 14.53 PJ 10.65 10.16 4.24

• 4.07

10.38 11.40 6.20 2.76 CLW (ccm) 13.65 14.61 15.05 14.70 17.32 17.45 17.45 17.25 CLW (sid) 14.14 14.71 13.95 14.19 17.78 17.78 16.01 13.98 CML 13.55 14.03 14.03 14.01 17.18 18.56 18.29 18.43 CKZ 13.31 14.01 13.96 13.94 17.37 17.17 16.48 16.69 CPJ 13.92 14.42 14.59 14.56 16.80 17.28 17.22 17.83

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Plots of Convexity I

Plots of combined rules in (σ, µ)–space

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Plots of Convexity II

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Conclusion

Estimation risk penalizes the mean-variance investor Linear combinations are less prone to estimation error due to diversification There exists empirical convexity Current analytical solutions rely to much on statistics Estimators are sensitive to sample dimensions (N, T) Estimation error is much more severe than bias

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Further Research

N-fund rules Estimators that are more stable Numerical methods for optimal combinations

Master Thesis Robin Jonsson INTRODUCTION

Problem Formulation Solutions Main Literature

Estimation Risk

Parameter Uncertainty Estimation Risk in Portfolios

Combined Portfolio Rules

Formulation of the Rule Other Combined Rules

Performance Evaluation

Experimental Design Combination Coefficients Out-of-Sample Results

Conclusion

Time for questions and discussion!