Portfolio Rebalancing: (Chapman and Hall/CRC Financial Mathematics Series)

The goal of Portfolio Rebalancing is to provide mathematical and empirical analysis of the effects of portfolio rebalancing on portfolio returns and risks. The mathematical analysis answers the question of when and why fixed-weight portfolios might outperform buy-and-hold portfolios based on volatilities and returns. The empirical analysis, aided by mathematical insights, will examine the effects of portfolio rebalancing in capital markets for asset allocation portfolios and portfolios of stocks, bonds, and commodities.

Table of Contents:
Contents Preface, vii Chapter 1 ◾ Introduction 1 1.1 RISK MANAGEMENT 1 1.2 REBALANCING ALPHA 2 1.3 DIVERSIFICATION RETURN, VOLATILITY EFFECT 3 1.4 SERIAL CORRELATION AND REBALANCING ALPHA 5 1.5 NEW TOPICS IN PORTFOLIO REBALANCING 6 1.6 OUTLINE OF THE BOOK 7 Chapter 2 ◾ A Brief Review of Portfolio Theory 9 2.1 ARITHMETIC AND GEOMETRIC MEANS 9 2.2 RETURN VOLATILITIES 11 2.3 RELATIONSHIPS BETWEEN ARITHMETIC AND GEOMETRIC MEANS 13 2.3.1 Analytic Approximation 13 2.3.2 Empirical Examination 15 2.4 PORTFOLIO RETURN AND VOLATILITY 19 2.5 SERIAL CORRELATION AND VOLATILITY OF MULTIPERIOD RETURNS 23 2.5.1 Single Asset Multi-Period Volatility 24 2.5.2 Portfolio Multi-Period Volatility 26 PROBLEMS 27 K26247.indb 5 14-09-2018 17:22:11 Chapter 3 ◾ Portfolio Rebalancing 29 3.1 SIMPLE EXAMPLES 29 3.2 REBALANCING LONG-ONLY PORTFOLIOS 32 3.3 REBALANCING LONG-SHORT PORTFOLIOS 36 3.4 REBALANCING ALPHA 41 3.4.1 Rebalancing Alpha of Asset Allocation Portfolios 42 3.4.2 Periodic Rebalancing versus Threshold Rebalancing 46 PROBLEMS 47 Chapter 4 ◾ Volatility Effect and Return Effect 49 4.1 DEFINITIONS OF TWO EFFECTS 50 4.2 POSITIVE RETURN EFFECT OF LONG-ONLY PORTFOLIOS 52 4.2.1 Jensen’s Inequality 52 4.2.2 Return Effect of Long-Only Portfolios 53 4.3 POSITIVE VOLATILITY EFFECT OF LONG-ONLY PORTFOLIOS 53 4.3.1 Cauchy’s Inequality 54 4.3.2 A Two-Asset Two-Period Case 54 4.3.3 An M-Asset Two-Period Case 57 4.3.4 The General Case 58 4.4 CASES OF POSITIVE AND NEGATIVE REBALANCING ALPHAS 61 4.4.1 The Case of Positive Rebalancing Alpha 61 4.4.2 The Case of Negative Rebalancing Alpha 62 4.5 TWO-ASSET LONG-SHORT PORTFOLIOS 63 4.5.1 Negative Return Effect of Two-Asset Long-Short Portfolios 63 4.5.2 Negative Volatility Effect of Two-Asset Long-Short Portfolios 65 PROBLEMS 66 K26247.indb 6 14-09-2018 17:22:11 Chapter 5 ◾ Analysis of Volatility Effect 69 5.1 "DIVERSIFICATION RETURN" 69 5.1.1 Two-Asset "Diversification Return" 71 5.1.2 Pairwise Decomposition of "Diversification Return" 72 5.1.3 Another Decomposition of "Diversification Return" 74 5.2 MAXIMIZING "DIVERSIFICATION RETURN" 75 5.3 DIVERSIFICATION RETURNS OF LONG-SHORT PORTFOLIOS 78 5.3.1 Two-Asset Long-Short Portfolios 78 5.3.2 Inverse and Leveraged Exchange-traded Funds 79 5.3.3 Leveraged "Long-Only" Portfolios 81 PROBLEMS 84 Chapter 6 ◾ Analysis of Return Effect 87 6.1 RETURN EFFECT OF LONG-ONLY PORTFOLIOS 87 6.1.1 Two-Asset Return Effect 90 6.1.2 Pairwise Decomposition of Return Effect 92 6.2 THE IMPACT OF CROSS-SECTIONAL SERIAL CORRELATIONS ON RETURN EFFECT 93 6.3 APPROXIMATING RETURN EFFECTS OF LONGSHORT PORTFOLIOS 96 6.3.1 Two-Asset Long-Short Portfolios 97 6.3.2 General Long-Short Portfolios 99 PROBLEMS 102 Chapter 7 ◾ Analysis of Rebalancing Alpha 103 7.1 REBALANCING ALPHA OF TWO-ASSET PORTFOLIOS 103 7.1.1 Pairwise t-Statistics 103 7.1.2 Probability of Positive Rebalancing Alpha 106 7.1.3 Expected Value and Standard Deviation of Rebalancing Alpha 110 7.1.4 Distribution of Rebalancing Alpha 112 K26247.indb 7 14-09-2018 17:22:11 7.2 REBALANCING ALPHA OF GENERAL PORTFOLIOS 116 7.2.1 Pairwise Decomposition of Rebalancing Alpha 116 7.2.2 An Alternative Decomposition of Rebalancing Alpha 118 7.2.3 Expectation of Portfolio Rebalancing Alpha 118 7.2.4 Rebalancing Alpha—S&P 500 Sector Portfolios 119 7.2.5 Cross-Sectional Serial Correlations of Sector Portfolios 125 7.2.6 Rebalancing Alpha of Varying Horizons 127 PROBLEMS 130 Chapter 8 ◾ Asset Allocation Portfolios 131 8.1 TRADITIONAL 60/40 PORTFOLIOS 131 8.2 RISK PARITY PORTFOLIOS 138 8.2.1 Risk Parity Portfolio without Leverage 138 8.2.2 Risk Parity Portfolio with Leverage 142 Chapter 9 ◾ Asset Class Portfolios 147 9.1 STOCK PORTFOLIOS 147 9.2 BOND PORTFOLIOS 156 9.3 COMMODITY PORTFOLIOS 164 Chapter 10 ◾ Rebalancing Alpha and Mean Reversion 173 10.1 TWO-ASSET TWO-PERIOD CASE 173 10.2 MULTIPLE-ASSET TWO-PERIOD CASE 176 10.3 TWO-ASSET THREE-PERIOD CASE 177 10.4 MULTIPLE-ASSET THREE-PERIOD CASE 182 10.5 THE GENERAL CASE 183 10.6 INCOMPLETE REBALANCE 187 PROBLEMS 191 Chapter 11 ◾ Risk and Return of Rebalancing Effects 193 11.1 TERMINAL WEALTH 193 11.2 EXPECTED TERMINAL WEALTH 195 K26247.indb 8 14-09-2018 17:22:11 11.2.1 Equal Expected Returns 196 11.2.2 General Case 196 11.3 VARIANCE OF TERMINAL WEALTH 198 11.4 COMPARISON OF TWO VARIANCES 201 11.5 A GENERAL TWO-ASSET CASE 209 11.6 THE IMPACT OF SERIAL CORRELATIONS 212 11.7 TERMINAL WEALTH OF LONG-SHORT PORTFOLIOS 218 APPENDIX 11.A RISK-ADJUSTED WEALTH OF TWO-ASSET LONG-ONLY PORTFOLIOS 225 APPENDIX 11.B EXPECTED TERMINAL WEALTH AND VARIANCE WITH SERIAL CORRELATIONS 226 PROBLEMS 231 Chapter 12 ◾ Threshold Rebalancing 233 12.1 RETURN DISPERSION OR WEIGHT DISPERSION AS A THRESHOLD 234 12.2 NUMERICAL SIMULATION OF THRESHOLD REBALANCING 235 BIBLIOGRAPHY, 243 INDEX, 245

About the Author :
Edward Qian is a Chief Investment Officer with PanAgora Asset Management. He has research experience and expertise in quantitative investing, portfolio theory, and asset allocation. He is the co-author of the bestselling book, Quantitative Equity Portfolio Management: Modern Techniques and Applications.