\A Random Matrix Approach to Portfolio Management and Financial Networks" Author: Supervisor: Nicolas A. Eterovic Professor Sheri Markose A Thesis submitted for the degree of Doctor of Philosophy Department of Economics University of Essex 2016 Contents 1 Introduction 1 1.1 Motivation . 1 1.2 Objectives . 2 1.3 Thesis Outline . 3 2 Literature Review 6 2.1 Introduction . 6 2.2 Applications of Random Matrix Theory to Portfolio Management . 8 2.2.1 Laloux et al. (1999, 2000) . 8 2.2.2 Plerou et al. (1999, 2000a,b, 2002); Gopikrishnan et al. (2001) . 9 2.2.3 Pafka and Kondor (2003, 2004); Pafka et al. (2004) . 11 2.2.4 Other similar studies . 13 2.3 Applications of Random Matrix Theory to Network Stability . 15 2.4 Financial Contagion in a Cross-Border Setting and the Role of Financial Networks 17 3 Background and Methodologies 20 3.1 Introduction . 20 3.2 Random Matrix Theory (RMT) . 21 3.2.1 The Mar˘cenko-Pastur Distribution (MP) . 24 3.3 The RMT-Filtering . 25 3.3.1 Laloux et al. (2000)'s Filter . 26 3.3.2 Plerou et al. (2002)'s Filter . 27 i 3.4 Methodology for Portfolio Analysis . 27 3.4.1 In-sample Methodology . 28 3.4.2 Out-of-sample Methodology . 29 3.4.3 Summary of Performance Measures . 33 3.5 The Eigen-Pair Method for Network Stability . 33 4 Eigenvalue Analysis for the FTSE 100 40 4.1 Introduction . 40 4.2 A Spectral Analysis: The RMT Approach in Practice . 41 4.3 The Number of Significant Components in the Deviating Eigenvectors . 45 4.4 The Evolution of the Largest Eigenvalues and Eigenvectors Over Time . 47 4.5 Asymmetries and Company Participation in Financial Correlations with RMT Tools 49 4.6 Summary of main findings . 50 5 Portfolio Analysis for the FTSE 100: RMT-Filtering 52 5.1 Introduction . 52 5.2 Illustration of Noise Filtering to Improve Portfolio Risk Estimates . 54 5.3 Sensitivity of the RMT Approach to the Data Size Q = T=N . 58 5.4 In-Sample Analysis . 60 5.5 Out-of-Sample Analysis . 62 5.5.1 Realised Risk Estimates . 63 5.5.2 Risk-Adjusted Returns Estimates . 65 5.5.3 Are RMT-Filtered Portfolios More Diversified? . 68 5.5.4 Robustness Test to the Recent Economic Crisis: Sub-sample Estimates . 69 5.6 Concluding Remarks . 74 6 A Comparative Study: Regime Switching Models and RMT-Filtering 77 6.1 Introduction . 77 6.2 Regime-Switching Models . 80 6.2.1 A Parsimonious RS-Model: The Beta-CAPM . 80 6.2.2 Data Analysis: The Existence of Different Regimes in the FTSE 100 Index 84 ii 6.3 Spectral Properties of Regime-Dependent Correlation Matrices: Combining RMT with MRS . 88 6.4 RMT-Based Correlation Estimators . 90 6.5 In-Sample Analysis . 91 6.6 Out-of-Sample Analysis . 92 6.6.1 Realised Risk Estimates . 93 6.6.2 Risk-Adjusted Returns Estimates . 94 6.7 Implications for Asset Allocation . 96 6.8 Concluding Remarks . 98 7 An Application of RMT to an Emerging Market Published as: \Separating the Wheat from the Chaff: Understanding Portfolio Returns in an Emerging Mar- ket"(2013) (With Dalibor S. Eterovic). Emerging Markets Review, vol. 16, issue C, pages 145-169. 100 7.1 Introduction . 100 7.2 Eigenvalue Analysis for the Chilean Stock Market . 102 7.3 Portfolio Analysis for the Chilean Stock Market . 107 7.3.1 Noise Filtering Using the Sample Correlation Matrix . 107 7.3.2 Improving our Results by Getting Rid of the Non-diversifiable Risk . 111 7.3.3 Improving Standard Correlation Models with RMT Tools . 113 7.4 Macroeconomic Determinants of the Chilean Stock Market Returns . 116 7.5 Concluding Remarks . 122 8 Early Warning and Systemic Risk in Core Global Banking: Financial Network and Market Price-based Methods 124 8.1 Introduction . 124 8.2 Systemic Risk Indices (SRIs) in a Cross-Border Setting . 128 8.2.1 Eigen-Pair Method based on Network Analysis . 129 8.2.2 Marginal Expected Shortfall (MES) . 130 8.2.3 SRISK . 132 8.2.4 ∆CoVaR . 133 iii 8.3 Data Description . 135 8.3.1 BIS Consolidated Banking Statistics and Bankscope Data . 135 8.3.2 MSCI Financials Data . 136 8.4 Network Analysis of the Core Global Banking System . 136 8.5 Spectral and Market Price-Based SRIs . 139 8.6 Systemic Risk Importance and Vulnerability in the Global Banking System . 142 8.6.1 Systemic Importance and Vulnerability of Core Eurozone Countries . 147 8.6.2 Systemic Importance and Vulnerability of Eurozone Periphery Countries . 149 8.6.3 What Explains Market Price-Based SRIs? . 152 8.7 Concluding Remarks . 154 9 Conclusions and Future Work 157 9.1 Goals of the Thesis . 157 9.2 Concluding Remarks . 158 9.2.1 Sample Correlation Matrix . 158 9.2.2 Other Correlation Estimators . 159 9.2.3 Combining RMT with VAR Analysis . 160 9.2.4 Systemic Risk in a Cross-Border Setting and the Role of Financial Networks161 9.3 Summary . 161 9.4 Future Work . 162 A Tables & Figures from Chapter 5 164 B Tables & Figures from Chapter 7 166 C Additional Analyses from Chapter 8 168 C.1 Proof Equation 4.1 . 168 C.2 Simulation Algorithm for LRMES . 169 C.3 Breakdown of the Core Global Banking System: SRISK# . 170 C.4 Estimation Methodology for the Capital Threshold for Losses . 171 C.5 Cross-Border Flows Breakdown used to Rescale λmax . 173 C.6 Alternative Measures of Systemic Instability from Network Theory . 176 iv List of Figures 3.1 The Mar˘cenko-Pastur Distribution for different values of Q. When T , ! 1 Q , the noise band shrinks to zero. 24 ! 1 4.1 The Mar˘cenko-Pastur Distribution vs. The Sample Eigenvalue Density of C2007. We can see two clear deviations from the upper edge of the spectrum. 42 4.2 Eigenvector Components of the Two Largest Eigenvalues: Year 2007. 43 4.3 Eigenvector Components of the Two Smallest Eigenvalues: Year 2007. 44 4.4 Eigenvector Components Associated with Two Noisy Eigenvalues: Year 2007. Their components do not exhibit a clear preference for some stocks over others. 45 4.5 Time Evolution of the Eigenvector Components. 48 4.6 Time evolution of the Largest Eigenvalue (Left y-axis) vs. Average Correlation Coefficient and the FTSE Returns (Right y-axis). 49 4.7 Time evolution of the Participation Ratio (PR) Associated to Largest Eigenvector of C vs. the PR for Random Returns (Right y-axis) and the Average Correlation Coefficient (Left y-axis). 50 5.1 Efficient Frontier during 2007. Top panel shows the difference between predicted and realised risk, for the same family of optimal portfolios using the filtered cor- relation matrix, Cf . Bottom panel shows the difference between predicted and realised risk, but using the unfiltered correlation matrix, C. As it can be seen, the predicted risk is closer to the realised one when we use the filtered correlation matrix. Short-sale permitted. 55 v 5.2 Percentage of Deviating Eigenvalues as a Function of T . The larger T gets, the smaller the noise band of the MP spectrum and the more eigenvalues deviate from the upper edge λ+.................................... 60 5.3 Mean Bootstrapped (in-sample) for the ratio γ of realised to predicted risk, for Ω the filtered approach,γ ¯f = fr , compared to the unfiltered approach,γ ¯u = Ωur , Ωf Ωu as a function of Q = T=N. Notice that filtering is most effective for low values of Q. Short-sale allowed. 61 5.4 Cumulated Wealth at the end of the out-of-sample period for different estimation windows, with and without short-sale......................... 66 5.5 Panel (a): Eigenvalue Distribution for the Pre-crisis Period (2003-2006). Panel (b): Eigenvalue Distribution for the Crisis period (2007-2010). For a given value of Q = 13:54. ...................................... 70 5.6 Number of Deviating Eigenvalues included in the RMT-filtering over the out-of- sample period. For a given value of Q, there are eigenvalues entering and exiting the Mar˘cenko-Pastur distribution. 75 6.1 Smoothed and Filtered Probabilities for Regime 1 - Whole Sample Estimates (2000-2012). 85 6.2 FTSE 100 Returns and Index - Whole Sample Estimates. 86 6.3 Distribution of the Sample Correlation matrix estimated on a daily rolling window of length 250 days. Note that the distribution shifts to the right post year 2008. 88 6.4 Eigenvalue Distributions by Regime vs. the Mar˘cenko-Pastur Distribution. 89 6.5 Mean Bootstrapped (in-sample) for the ratio of realised to predicted risk, for the filtered regime-dependent correlation matrices,γ ¯f = Ωfr , compared to the unfil- Ωf tered approach,γ ¯u = Ωur , as a function of Q = T=N. Notice that filtering is most Ωu effective for low values of Q............................... 92 6.6 Cumulated Wealth at the end of the out-of-sample period for different estimation windows, with and without short-sale......................... 97 7.1 Eigenvalue Distribution of the Sample Correlation Matrix vs. The Mar˘cenko- Pastur Distribution. Year 2005 . ..