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Imperial College London Imperial College Business School Three Essays on Variance Risk and Correlation Risk Xianghe Kong Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Finance of Imperial College and the Diploma of Imperial College October 2010 1 Abstract This thesis focuses on variance risk and correlation risk in the equity market, and consists of three essays. The first essay demonstrates that the variance risk, mea- sured as the difference between the realized return variance and its risk-neutral expectation, is an important determinant of the cross-sectional variation of hedge fund returns. Empirical evidence shows that funds with significantly higher loadings on variance risk outperform lower-loading funds on average. However, they incur severe losses during market downturns. Failure to account for variance risk results in overestimation of funds' absolute returns and underestimation of risk. The results provide important implications for hedge fund risk management and performance evaluations. The second essay examines the empirical properties of a widely-used correlation risk proxy, namely the dispersion trade between the index and individual stock options. I find that discrete hedging errors in such trading strategy can result in incorrect inferences on the magnitude of correlation risk premium and render the proxy unreliable as a measure of pure exposure to correlation risk. I implement a dynamic hedging scheme for the dispersion trade, which significantly improves the estimation accuracy of correlation risk and enhances the risk-return profile of the trading strategy. Finally, the third essay aims to forecast the average pair-wise correlations between stocks in the market portfolio. I investigate a comprehensive list of forecasting mod- els and find that past average correlation and the option-implied correlation provide superior out-of-sample forecasting performance compared to other predictors. I pro- vide empirical evidence showing that the forecasts of average correlation can improve the optimal portfolio choices and substantially enhance the performance of active correlation trading strategies. 2 Acknowledgements First and foremost, I would like to thank my supervisor, Dr. Robert Kosowski, for his ongoing guidance, patient discussions, and continuous encouragement. Most of the work in this thesis is developed from long discussions with him. I am also grateful to my supervisor Professor Andrea Buraschi, who recommends me to study at London Business School and gives me inspiring research ideas and suggestions. Special thanks go to my third supervisor, Professor Karim Abadir, for the insightful and rigorous comments on my work. I would like to thank my examiners, Professor Nigel Meade and Professor Stephen Taylor, for their comments and suggestions on this thesis. My thanks extend to my colleagues at the Business School. The last four years were a cherishable experience with them: Susan Shi Wu, Filip Zikes, Adam S. Iqbal, Leo Evans, Ludmila Dimtcheva, Marina Theodosiou, Zheng Ruan, Adam Golin- ski, Akindynos-Nikolaos Baltas, Farouk Jivraj, Fotios Amaxopoulos, Saad Badaoui, Nikhil Shenai, Qi Cao, Yeyi Liu, Dr. Robert Metcalfe, Yijiang Wu, Evelina Klerides. Special thanks to Julie Paranics and Lisa Umenyiora, for their administrative sup- port and assistance. I gratefully thank all my friends, in particularly, Dr. Lin Luo and Dr. Feng Chang, for sharing their invaluable experiences and giving great helps. Many thanks go in particular to my beloved husband, Fan Haijian, who has so much understanding and patience during the hard times and so much fun and love every minute. Last but most importantly, I thank my father, Kong Fanxing , my mother, He Yamei, for being the best parents in the world. I gratefully acknowledge the studentship from the Hedge Fund Center of Imperial College Business School. 3 4 Dedication To my parents and my husband Contents Abstract 1 Acknowledgements 3 1 Introduction and Literature Review 12 1.1 Introduction . 12 1.2 A Literature Review on Variance Risk Estimation . 15 1.2.1 Variance Risk Premium and Variance Swap . 15 1.2.2 Synthesizing the Variance Swap Rate . 17 1.2.3 Sensitivity Analysis . 18 1.2.4 Approximation Errors . 20 1.3 Appendix . 23 1.3.1 Estimation of Variance Swap Rate . 23 1.3.2 A Review on Model-Free Implied Variance . 25 2 Variance Risk and the Cross-Section of Hedge Fund Returns 33 2.1 Introduction . 33 5 CONTENTS 6 2.2 Data and Variance Risk Proxy . 40 2.2.1 Hedge Fund Data . 40 2.2.2 Variance Risk Proxy . 41 2.3 Empirical Results . 43 2.3.1 Hedge Fund Index and Variance Risk . 43 2.3.2 Cross-Sectional Analysis . 45 2.3.3 Cross-Sectional Analysis Conditional on Downside Market . 48 2.3.4 Fama-Macbeth Regressions . 49 2.4 Robustness Checks . 50 2.4.1 Alternative Proxy of Variance Risk . 50 2.4.2 Put Option Returns and Liquidity Risk . 51 2.4.3 Equally-Weighted Hedge Fund Indices . 51 2.4.4 TASS Hedge Fund Database . 52 2.5 Conclusion . 52 3 The Price of Correlation Risk: An Empirical Examination 75 3.1 Introduction . 75 3.2 The Theory . 78 3.2.1 Review of the Strategy . 78 3.2.2 The Fair Value of Correlation Risk . 80 3.2.3 Option-Implied Correlation . 82 CONTENTS 7 3.3 Data and Variable Construction . 83 3.3.1 Data . 83 3.3.2 Realized Correlation . 84 3.3.3 Model-Free Implied Variance and Correlation . 85 3.4 The Impact of Hedging Errors . 86 3.4.1 The Estimation Bias . 86 3.4.2 The Risk Attributions of a Dispersion Portfolio . 89 3.5 Robust Check . 92 3.5.1 Transaction Cost . 92 3.5.2 Sensitivity Analysis of Implied Correlation . 93 3.6 Conclusions . 96 4 Forecasting Average Correlation in Equity Market 103 4.1 Introduction . 103 4.2 Data . 108 4.3 Variables Construction . 108 4.3.1 Average Correlation (AC) . 109 4.3.2 Conditional Correlation from Multivariate GARCH Model . 110 4.3.3 Implied Correlation (IC) . 111 4.3.4 Variance Risk Premium (VRP) . 111 4.3.5 Stock Characteristics and Interest-Rate Related Variables . 112 CONTENTS 8 4.3.6 Difference in Beliefs Index (DBIX) . 112 4.4 Econometric Methodology . 114 4.4.1 Predictive Regression Model . 114 4.4.2 Forecast Combination . 115 4.4.3 Forecast Evaluation . 116 4.5 Empirical Results . 118 4.5.1 In-Sample Forecast . 118 4.5.2 Out-of-Sample Forecast . 119 4.5.3 Forecast Encompassing Test Results . 120 4.5.4 Mean Variance Portfolio . 121 4.5.5 Active Correlation Swap Trading Strategy . 123 4.6 Forecasts with Boundary Conditions . 125 4.7 Conclusion . 128 4.8 Appendix . 129 4.8.1 Empirical Estimation of DECO . 129 4.8.2 Estimation of Implied Correlation . 131 4.8.3 Construction of the Difference in Belief Index . 133 5 Conclusion 145 Bibliography 148 List of Tables 2.1 Summary Statistics of Hedge Fund Indices Returns . 54 2.2 Summary Statistics of Variance Risk Factor . 55 2.3 Time-Series Regression of Hedge Fund Returns on Different Risk Factors .............................. 56 2.4 Portfolios Sorted on Fund Exposures to Variance Risk . 58 2.5 Portfolios Sorted on Variance Risk in Downside Market . 62 2.6 Fama-Macbeth Regressions .................... 66 2.7 Time-Series Regression with Put Option Returns and Liq- uidity Risk Factor ........................... 67 2.8 Time-Series Regression with Equally-Weighted Hedge Fund Indices .................................. 69 2.9 Time-Series Regression with TASS Hedge Fund Data . 71 3.1 Summary Statistics of the Static Option Trading Strategies 97 3.2 Summary Statistics of the Dispersion Trade with Dynamic Hedging ................................. 98 3.3 Time-Series Regressions of Dispersion Trading Returns . 98 9 LIST OF TABLES 10 3.4 Dispersion Trade With Transaction Costs . 99 3.5 Sensitivities Analysis of Option-Implied Correlation . 100 4.1 Summary Statistics of the Predictors . 135 4.2 In-Sample Forecasting Results . 136 4.3 Out-of-Sample Forecasting Results . 137 4.4 Forecast Encompassing Test Results . 138 4.5 Out-of-Sample Mean Variance Portfolio Comparison . 139 4.6 Out-of-Sample Performance of Correlation Swap Trading . 139 4.7 Out-of-Sample Forecasting Results Using Logistic Transfor- mation ..................................140 List of Figures 1.1 Truncation Error ........................... 31 1.2 Discretization Error ......................... 32 2.1 Variance Risk Premium ....................... 73 2.2 Synthetic Variance Swap versus Variance Swap Quotations . 74 3.1 Time Series Plot of Realized and Implied Correlation . 101 3.2 Delta and Vega Exposures of Dispersion Portfolios . 102 4.1 Plot of Average Correlation . 141 4.2 Out-of-Sample Cumulative Square Prediction Error . 142 4.3 Out of Sample Forecasts of Average Correlation . 143 4.4 Cumulative Profits of Active versus Passive Correlation Trad- ing Strategies ..............................144 11 Chapter 1 Introduction and Literature Review 1.1 Introduction This thesis focuses on two risk factors in the equity market, namely, variance risk and correlation risk. They both play important roles in financial studies. I start the discussion with variance risk. The current state of literature suggests that variance risk is a systematic risk factor in the economy and it carries a negative risk pre- mium. An increase in variance represents a deterioration in investment opportunity set (see Campbell (1993) and Campbell (1996)) and also coincides with downward market movement (see Campbell and Hentschel (1992) and French, Schwert, and Stambaugh (1987)). Therefore risk-averse investors want to hedge against shocks in market variance and are willing to pay a premium for downside protection. The hedging motive is indicative of an economically large negative variance risk premium. Consistent with the theory, many studies provide supportive empirical evidence 1. 1Ang, Hoddrick, Xing, and Zhang (2006) form stock portfolios on their sensitivity to volatility risk and find evidence of a negative volatility risk in stock returns.

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