Portfolio Rebalancing

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.

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

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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

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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

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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

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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