Quantitative Portfolio Optimisation, Asset Allocation and ...978-0-230-51285-6/1.pdf · PART I A BASIS FOR QUANTITATIVE MANAGEMENT AND ANALYSIS 1 Chapter 1 Asset Management Basics

Quantitative Portfolio Optimisation, Asset Allocation and Risk Management

QUANTITATIVE PORTFOLIO OPTIMISATION, ASSET

ALLOCATION AND

RISK MANAGEMENT

Mikkel Rasmussen

* © Mikkel Rasmussen 2003 Softcover reprint of the hardcover 1st edition 2003 978-1-4039-0458-4

All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission.

No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP.

Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages.

The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988.

First published 2003 by PALGRAVE MACMILLAN Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y. 10010 Companies and representatives throughout the world

PALGRAVE MACMILLAN is the global academic imprint of the Palgrave Macmillan division of St. Martin's Press, LLC and of Palgrave Macmillan Ltd. Macmillan• is a registered trademark in the United States, United Kingdom and other countries. Palgrave is a registered trademark in the European Union and other countries.

ISBN 978-1-349-50944-7 DOI 10.1057/9780230512856

ISBN 978-0-230-51285-6 (eBook)

This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources.

A catalogue record for this book is available from the British Library.

A catalog record for this book is available from the Library of Congress.

Editing and origination by Aardvark Editorial, Mendham, Suffolk

10 9 8 7 6 5 12 11 10 09 08 07 06 OS

Transferred to Digital Printing 2008

CONTENTS

List of Figures ix List of Tables xiv

PART I A BASIS FOR QUANTITATIVE MANAGEMENT AND ANALYSIS 1

Chapter 1 Asset Management Basics 3

Introduction 3 Asset Management Objectives 4 The Case for Quantitative Management 4 Structure of this Book 6

Chapter 2 Asset Returns 9

Defining Investment Returns 9 Examples from the Real World 12 Excess Returns and Risk-free Rates 18 Residual/ Abnormal Returns 19 Time-weighted Returns (TWR) 20 Summary 20 Appendix 20

Chapter 3 Asset Risk 23

Risk is Not Just a Four-letter Word 23 Defining Risk 26 A Brief Note on Normality 35 Summary 37

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CONTENTS

Chapter 4 Asset Pricing 38

Pricing and Valuation 38 Determining the Discount Rate 39 The Dividend Discount Model (DDM) 41 The Discounted Cash Flow Model (DCF) 43 Old vs. New Economy -A Valuation Example 50 Implied Growth Rates 59 The Capital Asset Pricing Model (CAPM) 63 The Security Market Line (SML) 63 The Characteristic Line (CL) 66 The Arbitrage Pricing Theory (APT) 68 Summary 70

PART II MODERN PORTFOLIO THEORY 71

Chapter 5 Portfolio Characterisation 73

Introduction 73 Portfolio Return - The Sum of its Parts 74 Portfolio Risk - Less Than the Sum of its Parts 75 The Nature of Diversification 87 Summary 91 Appendix 92

Chapter 6 Quantitative Portfolio Optimisation and Efficient Portfolios 97

Portfolio Efficiency 97 Quantitative Portfolio Optimisation 99 The Efficient Frontier 111 Benefits from International Diversification 117 Optimisation and Diversification 125 Summary 127 Appendix 128

Chapter 7 Estimating Model Parameters 138

Expected Return and Risk 138 The CAPM Revisited 139 Factor Models - The APT Revisited 143 Volatility and Correlation 146 Return Distributions (Risk Characterisation) 153 The Correlation Structure 158 Summary 162

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CONTENTS

PART Ill ASSET ALLOCATION 165

Chapter 8 Investment Objectives and Benchmark Selection 167

The Investment Policy Statement 167 Choosing the Benchmark 171 Summary 175

Chapter 9 Quantitative Portfolio Construction and Asset Allocation 177

The Asset Allocation Decision 177 Traditional Portfolio Construction Techniques 178 Quantitative Portfolio Optimisation for Asset Allocation 186 Introducing an MSCI Global Sector Model 194 Summary 199

Chapter 10 Quasi-Random Monte Carlo Simulated Asset Allocation (QRMCSAA) 201

Quantitative Optimisation and Monte Carlo Simulations 201 The Efficient Ridge 205 The Quasi-Random Monte Carlo Simulated Asset Allocation 215

Summary 223 Appendix 225

Chapter 11 Refining the QRMCSAA Model 239

Bayesian Priors and Stein Estimators 239 Optimal Return Shrinkage 242 Optimal Covariance Matrix Shrinkage 255 Summary 270

Chapter 12 Strategic and Tactical Asset Allocation 273

Introduction 273 SAA vs. TAA- Theory 274 SAA vs. TAA- Practice 281 Summary 290

Chapter 13 Sector Rotation 291

The Sector Rotation Framework 291 Conceptual Framework 293 A Note on Determining Appropriate Model Inputs 299 Asset Allocation Through the Business Cycle 303 Summary 313

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CONTENTS

PART IV QUANTITATIVE RISK MANAGEMENT 315

Chapter 14 Tracking Error and Information Ratio 317

Definitions of Tracking Error 317 Risk Geometry 320 Information Ratio 324 Active Management Value Added 327 Summary 330

Chapter 1 S Sector Risk Model 332

The Global Perspective 332 Risk Characterisation 333 Constructing the Model 340 Portfolio Risk-Management Implications 345 MSCTR and MSCAR for the Global Sector Model 347 The Efficient Ridge Revisited 359 General Thoughts on Active Risk Management 364 Summary 375 Appendix 15A: Sector Indices and Volatilities 377 Appendix 15B: Sector Returns 380 Appendix 15C: Sector Return Distributions 383 Appendix 15D: Portfolio Volatility and Tracking Error 386 Appendix lSE: Portfolio Beta 389

Chapter 16 Value-at-Risk (VaR) and Extreme Value Theory (EVT) 392

The Basics 392 Variance-Covariance VaR 395 Historical Simulation of VaR 396 Multivariate Normal Distributions 404 Monte Carlo Simulated VaR 414 VaR Along the Efficient Frontier 415 Marginal Contributions to VaR 416 Extreme Value Theory (EVT) 419 Summary ~1

Appendix 16A: Sector Tail Return Frequencies ~3

Appendix 16B: Sector Multivariate Normal Distribution ~6 Appendix 16C: Sector Extreme Value Charts ~9

Appendix Notation

Glossary

432

434 441 Index

LIST OF FIGURES

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2.1 The compounding effect 11 2.2 Performance of the DJIA, the S&P500 and the NASDAQ Composite

Index, 1995/1-2002/4 (daily observations) 12 2.3 Daily returns on the Dow Jones Industrial Average, 1995/1-2002/4 13 2.4 Daily returns on the S&P500, 1995/1-2002/4 14 2.5 Daily returns on the NASDAQ Composite, 1995/1-2002/4 14 2.6 Annualised returns on the Dow Jones Industrial Average, S&P500

Index and NASDAQ Composite Index, 1995-2001 15 2.7 Average annual returns on the Dow Jones Industrial Average,

S&P500 and the NASDAQ Composite Index, 1995-2001 16 2.8 Average annual returns on the FTSE1 00, DAX30, Tokyo SE and

Hang Seng Indices, 1995/1-2002/5 17 2.9 Monthly excess returns on IBM vs. the S&P500, 1995/1-2002/4 19

3.1 Hypothetical monthly performance of two global equity funds 25 3.2 Variance (average of the sum of squared deviations from the mean) 27 3.3 Three-month moving average of three-month volatilities, based

on daily observations 1995/1-2001/5 31 3.4 Normal, skewed and kurtotic return distributions 35

4.1 Discounting $1 ,000,000 at different discount rates over 30 years 49 4.2 Company A: FCFs, present value of FCFs and terminal value 52 4.3 Sensitivity- Company A: short-term growth and discount rate 52 4.4 Sensitivity - Company A: perpetual growth and discount rate 53 4.5 Sensitivity - Company A: short-term growth and perpetual growth 54 4.6 Company 8: FCFs, present value of FCFs and terminal value 56 4.7 Sensitivity - Company 8: short-term growth and discount rate 57 4.8 Sensitivity - Company 8: discount rate and perpetual growth 58 4.9 Sensitivity- Company 8: short-term growth and perpetual growth 58

f ix)

LIST OF FIGURES

4.10 Framework for calculating implied growth rates 60 4.11 Implied 1 0-year growth rate, Sony Corp. 61 4.12 Discounted earnings per share, Sony Corp. 61 4.13 Sensitivity analysis - Sony Corp. 62 4.14 The Security Market Line 65 4.15 The revised Security Market Line 66 4.16 The Characteristic Line- Sony vs. TOPIX, 1995/1-2002/5 67

5.1 Correlation coefficient of + 1 77 5.2 Correlation coefficient of 0 78 5.3 Correlation coefficient of -1 79 5.4 Diversification at work- 2-asset portfolio 88 5.5 Portfolio volatility as 70 MSCI world stocks are successively added 90

6.1 Asset and minimum-variance portfolio volatilities 105 6.2 Asset and minimum-variance portfolio Sharpe Ratios 105 6.3 Return/risk combinations with correlation coefficient of -Q.5 112 6.4 The efficient frontier for a five-asset portfolio 113 6.5 Correlation matrix - four US equity indices and cash 115 6.6 Efficient frontier- four US equity indices and cash 115 6.7 Correlation matrix - four US equity indices, MSCI-W ex US and cash 119 6.8 Efficient frontier- five US equity indices, MSCI-W ex US and cash 120 6.9 Sharpe Ratios for the two efficient frontiers 121 6.10 The efficient surface - varying minimum cash position 124 6.11 Volatility of optimised portfolio as 70 randomly chosen stocks

are successively added to the portfolio 125 6.12 Number of stocks included in the minimum-variance portfolio

as the 70 randomly chosen stocks become available 127

7.1 Asset Allocation Line - one risky asset A and the market portfolio M 141 7.2 60-day moving average volatility of the S&P500, 1996/1-2002/5 147 7.3 Exponentially weighted volatility of the S&P500, 1996/1-2002/5 149 7.4 Weighting schemes (per cent) of different forecast methods 152 7.5 Return frequency distribution for the S&P500, 1995/1-2002/5 154 7.6 Return frequency distribution for the NASDAQ, 1995/1-2002/5 154 7.7 Return frequency distribution for the TOPIX, 1995/1-2002/5 155 7.8 3-month correlation coefficients for the Dow jones, 1995/4-2002/4 159 7.9 Distribution of correlation coefficients - Dow jones and TOPIX 160

9.1 Efficient frontier- five US equity indices, MSCI-W ex US and cash 187 9.2 Asset allocation along the efficient frontier, 3D 190 9.3 Asset allocation along the efficient frontier, cumulative percentages 191 9.4 Correlation matrix - MSCI Global Sector Model 197 9.5 The efficient frontier- MSCI Global Sector Model 197 9.6 Asset allocation topography along the efficient frontier 198 9.7 Expected Sharpe Ratio along the efficient frontier 199

10.1 Monte Carlo Simulation of portfolio returns 204 10.2 Return distributions for 1 0 MSCI global sectors, 1995/1-2002/5 207 10.3 Return distribution, minimum-variance portfolio 208

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LIST OF FIGURES

10.4 Return distribution, middle-variance portfolio 210 10.5 Return distribution, maximum-variance portfolio 211 10.6 The efficient frontier - MSCI Global Sector Model 213 10.7 The efficient ridge: 3D 214 10.8 The efficient ridge: 2D 214 10.9 Return distributions for 10 MSCI Global Sectors, 1995/1-2002/5 218 10.10 QRMCSAA- MSCI Global Sector Model 219 10.11 Efficient frontier- MSCI Global Sector Model, health care: 12% 220 10.12 Asset allocation - MSCI Global Sector Model, health care: 12% 221 10.13 QRMCSAA- MSCI Global Sector Model, health care: 12% 222 10.14 The efficient frontier- MSCI Global Industry Group Model 235 10.15 The efficient ridge: 3D 236 10.16 The efficient ridge: 2D 236 10.17 Asset allocation - MSCI Global Industry Sub-group Model 237 10.18 QRMCSAA- MSCI Global Industry Sub-group Model 238

11.1 Optimal shrinkage factor - Stein-I 244 11.2 Optimally shrunk historical returns - Stein-1 245 11.3 Efficient frontier - Stein-1 return shrinkage 247 11.4 Asset allocation topography - Stein-1 248 11.5 QRMCSAA - Stein-1 return shrinkage 249 11.6 Optimal shrinkage factor - Stein-11 251 11.7 Optimally shrunk historical returns - Stein-11 251 11.8 The efficient frontier - Stein-11 return shrinkage 253 11.9 Asset allocation topography - Stein-11 254 11.10 QRMCSAA - Stein-11 254 11.11 The efficient frontier - equal correlations 256 11.12 Asset allocation - equal correlations 258 11.13 QRMCSAA topography - equal correlations 259 11.14 Optimal covariance shrinkage 265 11.15 The efficient frontier - optimal covariance shrinkage 268 11.16 Asset allocation topography- optimal covariance shrinkage 268 11.17 QRMCSAA topography - optimal covariance shrinkage 269

12.1 Long-run consensus efficient frontier 275 12.2 Asset allocation - long-run consensus efficient frontier 276 12.3 QRMCSAA - long-run consensus efficient frontier 276 12.4 Efficient frontier - manager with superior information 278 12.5 Asset allocation - manager with superior information 278 12.6 QRMCSAA - manager with superior information 279 12.7 Active return efficient frontier 282 12.8 Active return asset allocation 285 12.9 Active bets along the active return efficient frontier 286 12.10 QRMCSAA - tactical asset allocation 287 12.11 QRMCSAA - active bets along the active return 288 12.12 QRMCSAA- tactical asset allocation topography: MSCI Global

Sector Model, Ledoit 289 12.13 QRMCSAA - active bets along the active return efficient frontier:

MSCI Global Sector Model, Ledoit 289

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LIST OF FIGURES

13.1 A stylised economic cycle for macro, earnings and equities 294 13.2 The equity market cycle 294 13.3 Equity market characteristics 296 13.4 Average sector correlation with the MSCI World Index 300 13.5 Correlation coefficient frequency chart- 1 0 MSCI sectors 300 13.6 Selected volatility levels over time 301 13.7 Average MSCI sector return frequency chart 302 13.8 Asset allocation - Phase 1: MSCI Global Sector Model 305 13.9 QRMCSAA- Phase 1: MSCI Global Sector Model 306 13.10 Asset allocation - Phase 2: MSCI Global Sector Model 308 13.11 QRMCSAA- Phase 2: MSCI Global Sector Model 308 13.12 Asset allocation - Phase 3: MSCI Global Sector Model 309 13.13 QRMCSAA- Phase 3: MSCI Global Sector Model 311 13.14 Asset allocation - Phase 4: MSCI Global Sector Model 313 13.15 QRMCSAA- Phase 4: MSCI Global Sector Model 313

14.1 Geometric relations between portfolio and benchmark risk 321 14.2 Expected information ratio along the efficient frontier 326 14.3 Value added as a function of residual risk 328

15.1 Information technology index and moving volatility 335 15.2 Information technology daily returns 336 15.3 Information technology return frequency 336 15.4 Sector correlations with other sectors 338 15.5 Sector correlation matrix 339 15.6 Marginal sector contributions to total risk 349 15.7 Marginal sector contributions to active risk 350 15.8 Total and active risk for changing information technology sector 353 15.9 Relative marginal sector contributions to total risk 354 15.10 Relative marginal sector contributions to active risk 355 15.11 Sector component Betas 356 15.12 Beta for the information technology sector 357 15.13 MSCTR during the estimation period 358 15.14 MSCAR during the estimation period 359 15.15 QRMCSAA- Ledoit: MSCI Global Sector Model 360 15.16 MSCTR along the efficient ridge 361 15.17 Active weights along the efficient ridge 362 15.18 Tracking error along the efficient ridge 362 15.19 MSCAR along the efficient ridge 363 15.20 Beta along the efficient ridge 364 15.21 Size and direction of sector bets and Betas 369 15.22 Portfolio factor exposures relative to the benchmark 370 15.23 Size of the most 'risky' bets 372 15.24 Volatility of the most 'risky' bets 373 15.25 Hope and confidence: expected contribution to tracking error 373 15.26 Matching active risk with active return 374

16.1 Normal distribution and 95% confidence level 394 16.2 Normal distribution at varying time horizons 396

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16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10

LIST OF FIGURES

Return frequency on the MSCI World, 1995/1-2002/5 Tail return frequency on the MSCI World, 1995/1-2002/5 Two normal distributions and a bivariate (combined) distribution Normal and bivariate distributions, MSCI World 1995/1-2002/5 Normal and bivariate distributions, MSCI World 1995/1-2002/5 VaR of equal-weighted portfolio for the MSCI Global Sector Model Extreme value theory and the information technology sector Extreme value theory and the telecommunications sector

402 403 405 406 407 415 420 421

LIST OF TABLES

2.1 Index returns, 1995-2001 17 2.2 Estimated risk-free rates of return 18

3.1 Variance, standard deviation and volatility using monthly returns 29 3.2 12-month averages of volatilities (3 months of daily

observations) 30

4.1 Annual income statement, IBM 1995-1999 44 4.2 Annual balance sheet, IBM 1995-1999 45 4.3 Annual cash flow statement, IBM 1995-1999 47 4.4 Valuation - Company A ('Old Economy') 51 4.5 Valuation - Company B ('New Economy') 55 6.1 Returns, volatilities and correlations: three-asset portfolio 102 6.2 Asset class characteristics - four US equity indices and cash 114 6.3 MVP and EWP -four US equity indices and cash 116 6.4 Asset characteristics - four US equity indices, MSCI-W

ex US and cash 118 6.5 MVP and EWP - five asset classes and cash 121 6.6 Correlations among major international equity indices 123

7.1 Daily mean return, volatility, kurtosis and skewness of equity indices 155

7.2 Return distributions expressed by standard deviations 156

9.1 MSCI Global Sector Model data 195

10.1 Sector composition, minimum-variance portfolio 209 10.2 Sector composition, middle-variance portfolio 210 10.3 Sector composition, maximum-variance portfolio 211

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LIST OF TABLES

11.1 Stein-1 optimal shrinkage adjusted returns - MSCI Global Sector 246

11.2 Stein-11 optimal shrinkage-adjusted returns 252 11.3 Equal correlations - grand mean approach 257 11.4 No shrinkage variance-covariance matrix - 'naive' approach 266 11.5 Optimal covariance shrinkage 267

12.1A MSCI Global Sector Model 283 12.1 B Weight limits - MSCI Global Sector Model 284

13.1 The economic cycle 295 13.2 Market dynamics during the economic cycle 295 13.3 The economic cycle 303 13.4 Phase 1 - model inputs 304 13.5 Phase 2 - model inputs 307 13.6 Phase 3 - model inputs 310 13.7 Phase 4 - model Inputs 312

14.1 Empirical distribution of information ratios 327

15.1 Portfolio and benchmark weights 348 15.2 Effect on MSCTR and MSCAR of sector rotation 352 15.3 Historical correlation coefficient levels (relative) 367 15.4 Historical correlation coefficient levels (relative) 368

16.1 Normal distribution vs. actually observed equity returns 397 16.2 Normal distribution vs. actually observed equity returns 398 16.3 Distribution of returns for the 10 MSCI World Sectors:

1995/1-2002/5 400 16.4 Distribution of returns for major stock indices 401 16.5 Bivariate and normal distributions, MSCI Sectors,

1995/1-2002/5 408 16.6 Bivariate and normal distributions, country indices,

1995/1-2002/5 409 16.7 VaR estimates using three estimation techniques 412 16.8 VaR estimates using three estimation techniques 413 16.9 Sector contributions to total portfolio VaR 417 16.10 Marginal sector contributions to VaR 418