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Information-Theoretic Approaches to Portfolio Selection

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Information-Theoretic Approaches to Portfolio Selection Information-theoretic approaches to portfolio selection Nathan LASSANCE Doctoral Thesis 2 | 2020 Université catholique de Louvain LOUVAIN INSTITUTE OF DATA ANALYSIS AND MODELING IN ECONOMICS AND STATISTICS (LIDAM) Universite´ catholique de Louvain Louvain School of Management LIDAM & Louvain Finance Doctoral Thesis Information-theoretic approaches to portfolio selection Nathan Lassance Thesis submitted in partial fulfillment of the requirements for the degree of Docteur en sciences ´economiques et de gestion Dissertation committee: Prof. Fr´ed´ericVrins (UCLouvain, BE), Advisor Prof. Kris Boudt (Ghent University, BE) Prof. Victor DeMiguel (London Business School, UK) Prof. Guofu Zhou (Washington University, USA) Prof. Marco Saerens (UCLouvain, BE), President Academic year 2019-2020 \Find a job you enjoy doing, and you will never have to work a day in your life." Mark Twain Contents Abstract vii Acknowledgments ix Research accomplishments xii List of Figures xii List of Tables xv List of Notation xvii Introduction1 1 Research background7 1.1 Mean-variance approaches..........................7 1.1.1 Definitions...............................8 1.1.2 Estimation risk............................ 10 1.1.3 Robust mean-variance portfolios................... 13 1.2 Higher-moment approaches.......................... 19 1.2.1 Efficient portfolios.......................... 21 1.2.2 Downside-risk criteria......................... 23 1.2.3 Indirect approaches.......................... 25 1.3 Risk-parity approaches............................ 25 1.3.1 Asset-risk parity........................... 26 1.3.2 Factor-risk parity........................... 28 1.3.3 Criticisms............................... 29 1.4 Information-theoretic approaches...................... 30 1.5 Thesis contributions............................. 32 2 Minimum R´enyi entropy portfolios 35 2.1 Introduction.................................. 35 2.2 The notion of entropy............................ 36 2.2.1 Shannon entropy........................... 37 2.2.2 R´enyi entropy............................. 37 2.3 R´enyi entropy and risk measurement.................... 39 2.3.1 Connection with deviation risk measures.............. 39 2.3.2 The subadditivity property...................... 40 2.3.3 Exponential R´enyi entropy as a flexible risk measure....... 42 2.3.4 Appeal of the case α 2 [0; 1]..................... 44 2.4 R´enyi entropy and portfolio selection.................... 45 iii Contents iv 2.4.1 Definition............................... 45 2.4.2 Entropy and higher moments: A Gram-Charlier expansion.... 45 2.5 Robust m-spacings estimation........................ 49 2.5.1 Motivation and expression for the m-spacings estimator...... 49 2.5.2 Properties of the m-spacings estimator............... 51 2.5.2.1 Asymptotic bias...................... 51 2.5.2.2 Robustness to outliers................... 52 2.6 Out-of-sample performance.......................... 53 2.6.1 Data and methodology........................ 53 2.6.2 Results................................. 56 2.7 Conclusion................................... 58 2.8 Appendix................................... 59 2.8.1 Proofs of results............................ 59 2.8.1.1 Proposition 2.1....................... 59 2.8.1.2 Theorem 2.1........................ 59 2.8.1.3 Proposition 2.2....................... 63 2.8.1.4 Theorem 2.2........................ 63 2.8.1.5 Proposition 2.3....................... 65 2.8.1.6 Proposition 2.4....................... 67 2.8.2 Additional empirical results..................... 68 3 Optimal portfolio diversification via independent component analysis 71 3.1 Introduction.................................. 71 3.2 PCA versus ICA: from decorrelation to independence........... 75 3.2.1 Principal component analysis.................... 76 3.2.2 Independent component analysis.................. 77 3.2.3 Numerical illustration........................ 79 3.3 Factor-variance parity via uncorrelated factors............... 81 3.3.1 Derivation of the factor-variance-parity portfolios......... 81 3.3.2 Arbitrariness of the decorrelation criterion............. 83 3.4 Higher-moment diversification via ICA................... 85 3.4.1 IC-variance-parity portfolios..................... 85 3.4.2 Kurtosis diversification........................ 86 3.4.3 Data-driven shrinkage portfolio................... 89 3.5 Factor-risk parity with higher-moment risk measures........... 90 3.5.1 Parsimonious estimation of higher moments with ICs....... 90 3.5.2 IC-risk parity with modified Value-at-Risk............. 91 3.6 Out-of-sample performance.......................... 94 3.6.1 Data and methodology........................ 94 3.6.2 Calibration of K and δ ........................ 96 3.6.3 Results................................. 96 3.7 Conclusion................................... 99 3.8 Appendix................................... 100 3.8.1 Proofs of results............................ 100 3.8.1.1 Proposition 3.1....................... 100 Contents v 3.8.1.2 Theorem 3.1........................ 101 3.8.1.3 Theorem 3.2........................ 102 3.8.1.4 Excess kurtosis of PCVP portfolios in Example 3.5... 104 3.8.1.5 Proposition 3.2....................... 105 3.8.1.6 Proposition 3.3....................... 105 3.8.2 FastICA algorithm.......................... 106 3.8.3 Theoretical properties of IC-kurtosis-parity portfolios....... 108 3.8.4 Long-only factor-variance-parity portfolios............. 109 4 Robust portfolio selection using sparse estimation of comoment tensors113 4.1 Introduction.................................. 113 4.2 Dimension reduction............................. 117 4.2.1 Curse of dimensionality........................ 117 4.2.2 Reducing dimensionality via principal components......... 118 4.2.3 Approximation of comoment tensors................ 120 4.3 Sparse higher-comoment tensors....................... 121 4.3.1 Independent factor model...................... 122 4.3.2 Approximation via independent component analysis........ 123 4.3.3 Sparse estimate of optimal portfolio................. 124 4.4 Empirical analysis............................... 126 4.4.1 Data and methodology........................ 126 4.4.2 Independence of PCs versus ICs................... 128 4.4.3 Results................................. 129 4.5 Conclusion................................... 133 4.6 Appendix................................... 134 4.6.1 Proofs of results............................ 134 4.6.1.1 Proposition 4.1....................... 134 4.6.1.2 Corollary 4.1........................ 134 4.6.1.3 Theorem 4.1........................ 135 4.6.1.4 Proposition 4.2....................... 135 4.6.2 Robustness test: daily returns.................... 135 5 Portfolio selection: A target-distribution approach 137 5.1 Introduction.................................. 137 5.2 Minimum-divergence portfolio........................ 142 5.2.1 General formulation......................... 142 5.2.2 The Kullback-Leibler divergence................... 143 5.3 Targeting a generalized-normal return distribution............. 144 5.3.1 Minimum-divergence portfolio under a generalized-normal target. 145 5.3.2 The case of Gaussian asset returns................. 146 5.4 Targeting a Gaussian return distribution.................. 149 5.4.1 Decomposition of the KL divergence................ 149 5.4.2 The Dirac-delta target-return distribution............. 150 5.5 A reference-portfolio approach........................ 153 5.6 Estimation of the minimum-divergence portfolio.............. 155 Contents vi 5.6.1 Estimation for the generalized-normal target return........ 156 5.6.2 Estimation of the portfolio-return entropy H(P ).......... 156 5.6.3 Estimation of the expectation E[jP − α^jγ]............. 156 5.6.4 Estimation for the Gaussian target return............. 158 5.7 Out-of-sample performance.......................... 158 5.7.1 Data and methodology........................ 158 5.7.2 Reported portfolio strategies..................... 162 5.7.3 Results................................. 163 5.7.3.1 Full sample......................... 163 5.7.3.2 Financial crisis....................... 166 5.8 Conclusion................................... 168 5.9 Appendix................................... 169 5.9.1 Proofs of results............................ 169 5.9.1.1 Proposition 5.1....................... 169 5.9.1.2 Proposition 5.2....................... 170 5.9.1.3 Theorem 5.1........................ 171 5.9.1.4 Theorem 5.2........................ 171 5.9.1.5 Proposition 5.3....................... 171 5.9.1.6 Theorem 5.3........................ 172 5.9.1.7 Proposition 5.4....................... 173 5.9.1.8 Corollary 5.1........................ 173 5.9.2 Choice of kernel bandwidth..................... 174 5.9.3 Out-of-sample performance of all considered portfolio strategies. 174 5.9.3.1 Comparison with reference portfolios........... 174 5.9.3.2 Comparison with higher-moment portfolios....... 178 5.9.3.3 Results for the generalized-normal target with γ = 4.. 179 5.9.4 Entropy and diversification...................... 180 5.9.4.1 The case of i.i.d. asset returns............... 180 5.9.4.2 Empirical point of view.................. 182 6 Conclusion 185 6.1 Summary of main results........................... 185 6.2 Open questions for future research...................... 188 6.2.1 Specific questions........................... 188 6.2.2 General questions........................... 195 6.2.2.1 Properties of independent components.......... 195 6.2.2.2 The notion of diversification...............
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