Quantitative Portfolio Optimisation, Asset Allocation and Risk Management - 1403904588, Notas de estudo de Economia

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QUANTITATIVE PORTFOLIO

OPTIMISATION, ASSET

ALLOCATION AND

RISK MANAGEMENT

Mikkel Rasmussen

Quantitative Portfolio Optimisation,

Asset Allocation and Risk Management

QUANTITATIVE PORTFOLIO

OPTIMISATION, ASSET

ALLOCATION AND

RISK MANAGEMENT

M i k k e l R a s m u s s e n

© Mikkel Rasmussen 2003

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 1–4039–0458–8 hardback

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 4 3 2 1 12 11 10 09 08 07 06 05 04 03

Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham and Eastbourne

C O N T E N T S

vii

ix

L I S T O F F I G U R E S

L I S T O F F I G U R E S

xiii

xv

• AND ANALYSIS PART I A BASIS FOR QUANTITATIVE MANAGEMENT
• Chapter 1 Asset Management Basics
  • Introduction
  • Asset Management Objectives
  • The Case for Quantitative Management
  • Structure of this Book
• Chapter 2 Asset Returns
  • Defining Investment Returns
  • Examples from the Real World
  • Excess Returns and Risk-free Rates
  • Residual/Abnormal Returns
  • Time-weighted Returns (TWR)
  • Summary
  • Appendix
• Chapter 3 Asset Risk
  • Risk is Not Just a Four-letter Word
  • Defining Risk
  • A Brief Note on Normality
  • Summary
• Chapter 4 Asset Pricing
  • Pricing and Valuation
  • Determining the Discount Rate
  • The Dividend Discount Model (DDM)
  • The Discounted Cash Flow Model (DCF)
  • Old vs. New Economy – A Valuation Example
  • Implied Growth Rates
  • The Capital Asset Pricing Model (CAPM)
  • The Security Market Line (SML)
  • The Characteristic Line (CL)
  • The Arbitrage Pricing Theory (APT)
  • Summary
• PART II MODERN PORTFOLIO THEORY
• Chapter 5 Portfolio Characterisation
  • Introduction
  • Portfolio Return – The Sum of its Parts
  • Portfolio Risk – Less Than the Sum of its Parts
  • The Nature of Diversification
  • Summary
  • Appendix
  • Efficient Portfolios Chapter 6 Quantitative Portfolio Optimisation and
  • Portfolio Efficiency
  • Quantitative Portfolio Optimisation
  • The Efficient Frontier
  • Benefits from International Diversification
  • Optimisation and Diversification
  • Summary
  • Appendix
• Chapter 7 Estimating Model Parameters
  • Expected Return and Risk
  • The CAPM Revisited
  • Factor Models – The APT Revisited
  • Volatility and Correlation
  • Return Distributions (Risk Characterisation)
  • The Correlation Structure
  • Summary
• PART III ASSET ALLOCATION
• Chapter 8 Investment Objectives and Benchmark Selection
  • The Investment Policy Statement
  • Choosing the Benchmark
  • Summary
  • Asset Allocation Chapter 9 Quantitative Portfolio Construction and
  • The Asset Allocation Decision
  • Traditional Portfolio Construction Techniques
  • Quantitative Portfolio Optimisation for Asset Allocation
  • Introducing an MSCI Global Sector Model
  • Summary
  • Allocation (QRMCSAA) Chapter 10 Quasi-Random Monte Carlo Simulated Asset
  • Quantitative Optimisation and Monte Carlo Simulations
  • The Efficient Ridge
    • Asset Allocation The Quasi-Random Monte Carlo Simulated
  • Summary
  • Appendix
• Chapter 11 Refining the QRMCSAA Model
  • Bayesian Priors and Stein Estimators
  • Optimal Return Shrinkage
  • Optimal Covariance Matrix Shrinkage
  • Summary
• Chapter 12 Strategic and Tactical Asset Allocation
  • Introduction
  • SAA vs. TAA – Theory
  • SAA vs. TAA – Practice
  • Summary
• Chapter 13 Sector Rotation
  • The Sector Rotation Framework
  • Conceptual Framework
  • A Note on Determining Appropriate Model Inputs
  • Asset Allocation Through the Business Cycle
  • Summary
• PART IV QUANTITATIVE RISK MANAGEMENT
• Chapter 14 Tracking Error and Information Ratio
  • Definitions of Tracking Error
  • Risk Geometry
  • Information Ratio
  • Active Management Value Added
  • Summary
• Chapter 15 Sector Risk Model
  • The Global Perspective
  • Risk Characterisation
  • Constructing the Model
  • Portfolio Risk-Management Implications
  • MSCTR and MSCAR for the Global Sector Model
  • The Efficient Ridge Revisited
  • General Thoughts on Active Risk Management
  • Summary
  • Appendix 15A: Sector Indices and Volatilities
  • Appendix 15B: Sector Returns
  • Appendix 15C: Sector Return Distributions
  • Appendix 15D: Portfolio Volatility and Tracking Error
  • Appendix 15E: Portfolio Beta
• Chapter 16 Value-at-Risk (VaR) and Extreme Value Theory (EVT)
  • The Basics
  • Variance–Covariance VaR
  • Historical Simulation of VaR
  • Multivariate Normal Distributions
  • Monte Carlo Simulated VaR
  • VaR Along the Efficient Frontier
  • Marginal Contributions to VaR
  • Extreme Value Theory (EVT)
  • Summary
  • Appendix 16A: Sector Tail Return Frequencies
  • Appendix 16B: Sector Multivariate Normal Distribution
  • Appendix 16C: Sector Extreme Value Charts
• Appendix Notation
• Glossary
• Index
• 2.1 The compounding effect LIST OF FIGURES
  • Index, 1995/1–2002/4 (daily observations) 2.2 Performance of the DJIA, the S&P500 and the NASDAQ Composite
• 2.3 Daily returns on the Dow Jones Industrial Average, 1995/1–2002/4
• 2.4 Daily returns on the S&P500, 1995/1–2002/4
• 2.5 Daily returns on the NASDAQ Composite, 1995/1–2002/4
• 2.6 Annualised returns on the Dow Jones Industrial Average, S&P
  • Index and NASDAQ Composite Index, 1995–2001
  • S&P500 and the NASDAQ Composite Index, 1995–2001 2.7 Average annual returns on the Dow Jones Industrial Average,
  • Hang Seng Indices, 1995/1–2002/5 2.8 Average annual returns on the FTSE100, DAX30, Tokyo SE and
• 2.9 Monthly excess returns on IBM vs. the S&P500, 1995/1–2002/4
• 3.1 Hypothetical monthly performance of two global equity funds
• 3.2 Variance (average of the sum of squared deviations from the mean)
  • on daily observations 1995/1–2001/5 3.3 Three-month moving average of three-month volatilities, based
• 3.4 Normal, skewed and kurtotic return distributions
• 4.1 Discounting $1,000,000 at different discount rates over 30 years
• 4.2 Company A : FCFs, present value of FCFs and terminal value
• 4.3 Sensitivity – Company A : short-term growth and discount rate
• 4.4 Sensitivity – Company A : perpetual growth and discount rate
• 4.5 Sensitivity – Company A : short-term growth and perpetual growth
• 4.6 Company B : FCFs, present value of FCFs and terminal value
• 4.7 Sensitivity – Company B : short-term growth and discount rate
• 4.8 Sensitivity – Company B : discount rate and perpetual growth
• 4.9 Sensitivity – Company B : short-term growth and perpetual growth
• 4.10 Framework for calculating implied growth rates
• 4.11 Implied 10-year growth rate, Sony Corp.
• 4.12 Discounted earnings per share, Sony Corp.
• 4.13 Sensitivity analysis – Sony Corp.
• 4.14 The Security Market Line
• 4.15 The revised Security Market Line
• 4.16 The Characteristic Line – Sony vs. TOPIX, 1995/1–2002/5
• 5.1 Correlation coefficient of +1
• 5.2 Correlation coefficient of
• 5.3 Correlation coefficient of −
• 5.4 Diversification at work – 2-asset portfolio
• 5.5 Portfolio volatility as 70 MSCI world stocks are successively added
• 6.1 Asset and minimum-variance portfolio volatilities
• 6.2 Asset and minimum-variance portfolio Sharpe Ratios
• 6.3 Return/risk combinations with correlation coefficient of − 0.5
• 6.4 The efficient frontier for a five-asset portfolio
• 6.5 Correlation matrix – four US equity indices and cash
• 6.6 Efficient frontier – four US equity indices and cash
• 6.7 Correlation matrix – four US equity indices, MSCI-W ex US and cash
• 6.8 Efficient frontier – five US equity indices, MSCI-W ex US and cash
• 6.9 Sharpe Ratios for the two efficient frontiers
• 6.10 The efficient surface – varying minimum cash position
  • are successively added to the portfolio 6.11 Volatility of optimised portfolio as 70 randomly chosen stocks
  • as the 70 randomly chosen stocks become available 6.12 Number of stocks included in the minimum-variance portfolio
• 7.1 Asset Allocation Line – one risky asset A and the market portfolio M
• 7.2 60-day moving average volatility of the S&P500,1996/1–2002/5
• 7.3 Exponentially weighted volatility of the S&P500,1996/1–2002/5
• 7.4 Weighting schemes (per cent) of different forecast methods
• 7.5 Return frequency distribution for the S&P500,1995/1–2002/5
• 7.6 Return frequency distribution for the NASDAQ, 1995/1–2002/5
• 7.7 Return frequency distribution for the TOPIX, 1995/1–2002/5
• 7.8 3-month correlation coefficients for the Dow Jones,1995/4–2002/4
• 7.9 Distribution of correlation coefficients – Dow Jones and TOPIX
• 9.1 Efficient frontier – five US equity indices, MSCI-W ex US and cash
• 9.2 Asset allocation along the efficient frontier, 3D
• 9.3 Asset allocation along the efficient frontier, cumulative percentages
• 9.4 Correlation matrix – MSCI Global Sector Model
• 9.5 The efficient frontier – MSCI Global Sector Model
• 9.6 Asset allocation topography along the efficient frontier
• 9.7 Expected Sharpe Ratio along the efficient frontier
• 10.1 Monte Carlo Simulation of portfolio returns
• 10.2 Return distributions for 10 MSCI global sectors, 1995/1–2002/5
• 10.3 Return distribution, minimum-variance portfolio
• 10.4 Return distribution, middle-variance portfolio
• 10.5 Return distribution, maximum-variance portfolio
• 10.6 The efficient frontier – MSCI Global Sector Model
• 10.7 The efficient ridge: 3D
• 10.8 The efficient ridge: 2D
• 10.9 Return distributions for 10 MSCI Global Sectors, 1995/1–2002/5
• 10.10 QRMCSAA – MSCI Global Sector Model
• 10.11 Efficient frontier – MSCI Global Sector Model, health care: 12%
• 10.12 Asset allocation – MSCI Global Sector Model, health care: 12%
• 10.13 QRMCSAA – MSCI Global Sector Model, health care: 12%
• 10.14 The efficient frontier – MSCI Global Industry Group Model
• 10.15 The efficient ridge: 3D
• 10.16 The efficient ridge: 2D
• 10.17 Asset allocation – MSCI Global Industry Sub-group Model
• 10.18 QRMCSAA – MSCI Global Industry Sub-group Model
• 11.1 Optimal shrinkage factor – Stein-I
• 11.2 Optimally shrunk historical returns – Stein-I
• 11.3 Efficient frontier – Stein-I return shrinkage
• 11.4 Asset allocation topography – Stein-I
• 11.5 QRMCSAA – Stein-I return shrinkage
• 11.6 Optimal shrinkage factor – Stein-II
• 11.7 Optimally shrunk historical returns – Stein-II
• 11.8 The efficient frontier – Stein-II return shrinkage
• 11.9 Asset allocation topography – Stein-II
• 11.10 QRMCSAA – Stein-II
• 11.11 The efficient frontier – equal correlations
• 11.12 Asset allocation – equal correlations
• 11.13 QRMCSAA topography – equal correlations
• 11.14 Optimal covariance shrinkage
• 11.15 The efficient frontier – optimal covariance shrinkage
• 11.16 Asset allocation topography – optimal covariance shrinkage
• 11.17 QRMCSAA topography – optimal covariance shrinkage
• 12.1 Long-run consensus efficient frontier
• 12.2 Asset allocation – long-run consensus efficient frontier
• 12.3 QRMCSAA – long-run consensus efficient frontier
• 12.4 Efficient frontier – manager with superior information
• 12.5 Asset allocation – manager with superior information
• 12.6 QRMCSAA – manager with superior information
• 12.7 Active return efficient frontier
• 12.8 Active return asset allocation
• 12.9 Active bets along the active return efficient frontier
• 12.10 QRMCSAA – tactical asset allocation
• 12.11 QRMCSAA – active bets along the active return
  • Sector Model, Ledoit 12.12 QRMCSAA – tactical asset allocation topography: MSCI Global
  • MSCI Global Sector Model, Ledoit 12.13 QRMCSAA – active bets along the active return efficient frontier:
• 13.1 A stylised economic cycle for macro, earnings and equities
• 13.2 The equity market cycle
• 13.3 Equity market characteristics
• 13.4 Average sector correlation with the MSCI World Index
• 13.5 Correlation coefficient frequency chart – 10 MSCI sectors
• 13.6 Selected volatility levels over time
• 13.7 Average MSCI sector return frequency chart
• 13.8 Asset allocation – Phase 1: MSCI Global Sector Model
• 13.9 QRMCSAA – Phase 1: MSCI Global Sector Model
• 13.10 Asset allocation – Phase 2: MSCI Global Sector Model
• 13.11 QRMCSAA – Phase 2: MSCI Global Sector Model
• 13.12 Asset allocation – Phase 3: MSCI Global Sector Model
• 13.13 QRMCSAA – Phase 3: MSCI Global Sector Model
• 13.14 Asset allocation – Phase 4: MSCI Global Sector Model
• 13.15 QRMCSAA – Phase 4: MSCI Global Sector Model
• 14.1 Geometric relations between portfolio and benchmark risk
• 14.2 Expected information ratio along the efficient frontier
• 14.3 Value added as a function of residual risk
• 15.1 Information technology index and moving volatility
• 15.2 Information technology daily returns
• 15.3 Information technology return frequency
• 15.4 Sector correlations with other sectors
• 15.5 Sector correlation matrix
• 15.6 Marginal sector contributions to total risk
• 15.7 Marginal sector contributions to active risk
• 15.8 Total and active risk for changing information technology sector
• 15.9 Relative marginal sector contributions to total risk
• 15.10 Relative marginal sector contributions to active risk
• 15.11 Sector component Betas
• 15.12 Beta for the information technology sector
• 15.13 MSCTR during the estimation period
• 15.14 MSCAR during the estimation period
• 15.15 QRMCSAA – Ledoit: MSCI Global Sector Model
• 15.16 MSCTR along the efficient ridge
• 15.17 Active weights along the efficient ridge
• 15.18 Tracking error along the efficient ridge
• 15.19 MSCAR along the efficient ridge
• 15.20 Beta along the efficient ridge
• 15.21 Size and direction of sector bets and Betas
• 15.22 Portfolio factor exposures relative to the benchmark
• 15.23 Size of the most ‘risky’ bets
• 15.24 Volatility of the most ‘risky’ bets
• 15.25 Hope and confidence: expected contribution to tracking error
• 15.26 Matching active risk with active return
• 16.1 Normal distribution and 95% confidence level
• 16.2 Normal distribution at varying time horizons
• 16.3 Return frequency on the MSCI World, 1995/1–2002/5
• 16.4 Tail return frequency on the MSCI World, 1995/1–2002/5
• 16.5 Two normal distributions and a bivariate (combined) distribution
• 16.6 Normal and bivariate distributions, MSCI World 1995/1–2002/5
• 16.7 Normal and bivariate distributions, MSCI World 1995/1–2002/5
• 16.8 VaR of equal-weighted portfolio for the MSCI Global Sector Model
• 16.9 Extreme value theory and the information technology sector
• 16.10 Extreme value theory and the telecommunications sector
• 2.1 Index returns, 1995–2001 LIST OF TABLES
• 2.2 Estimated risk-free rates of return
• 3.1 Variance, standard deviation and volatility using monthly returns
  • observations) 3.2 12-month averages of volatilities (3 months of daily
• 4.1 Annual income statement, IBM 1995–1999
• 4.2 Annual balance sheet, IBM 1995–1999
• 4.3 Annual cash flow statement, IBM 1995–1999
• 4.4 Valuation – Company A (‘Old Economy’)
• 4.5 Valuation – Company B (‘New Economy’)
• 6.1 Returns, volatilities and correlations: three-asset portfolio
• 6.2 Asset class characteristics – four US equity indices and cash
• 6.3 MVP and EWP – four US equity indices and cash
  • ex US and cash 6.4 Asset characteristics – four US equity indices, MSCI-W
• 6.5 MVP and EWP – five asset classes and cash
• 6.6 Correlations among major international equity indices
  • equity indices 7.1 Daily mean return, volatility, kurtosis and skewness of
• 7.2 Return distributions expressed by standard deviations
• 9.1 MSCI Global Sector Model data
• 10.1 Sector composition, minimum-variance portfolio
• 10.2 Sector composition, middle-variance portfolio
• 10.3 Sector composition, maximum-variance portfolio
  • Global Sector 11.1 Stein-I optimal shrinkage adjusted returns – MSCI
• 11.2 Stein-II optimal shrinkage-adjusted returns
• 11.3 Equal correlations – grand mean approach
• 11.4 No shrinkage variance–covariance matrix – ‘naive’ approach
• 11.5 Optimal covariance shrinkage
• 12.1A MSCI Global Sector Model
• 12.1B Weight limits – MSCI Global Sector Model
• 13.1 The economic cycle
• 13.2 Market dynamics during the economic cycle
• 13.3 The economic cycle
• 13.4 Phase 1 – model inputs
• 13.5 Phase 2 – model inputs
• 13.6 Phase 3 – model inputs
• 13.7 Phase 4 – model Inputs
• 14.1 Empirical distribution of information ratios
• 15.1 Portfolio and benchmark weights
• 15.2 Effect on MSCTR and MSCAR of sector rotation
• 15.3 Historical correlation coefficient levels (relative)
• 15.4 Historical correlation coefficient levels (relative)
• 16.1 Normal distribution vs. actually observed equity returns
• 16.2 Normal distribution vs. actually observed equity returns
  • 1995/1–2002/5 16.3 Distribution of returns for the 10 MSCI World Sectors:
• 16.4 Distribution of returns for major stock indices
  • 1995/1–2002/5 16.5 Bivariate and normal distributions, MSCI Sectors,
  • 1995/1–2002/5 16.6 Bivariate and normal distributions, country indices,
• 16.7 VaR estimates using three estimation techniques
• 16.8 VaR estimates using three estimation techniques
• 16.9 Sector contributions to total portfolio VaR
• 16.10 Marginal sector contributions to VaR

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3

C H A P T E R 1

ASSET MANAGEMENT BASICS

INTRODUCTION

The basis of any investment is the desire to obtain a return on that investment.

Since there is no such thing as a free lunch, the investor or asset manager must

accept some amount of risk in order to obtain the return. In other words, the

risk taken on by the investor is the price paid for the opportunity for a positive

return, and the desired level of return thus determines the exact amount of risk

taken on by the investor. This is a fundamental investment relationship, which

investors must consider when deciding whether to invest in either a single

asset or a portfolio of assets.

So what does it take to be a successful asset manager? It is really quite

simple. The manager must be knowledgeable , decisive , and right. This obviously

looks true on paper, but is hard to achieve in the real world, and the track

record for the active management industry speaks for itself. On average,

approximately only half of all active equity portfolio managers outperform

their benchmark, and the number of managers who outperform consistently is

so low that it is hardly statistically significant.

Investment return and risk can be modelled quantitatively on both the asset

and the portfolio levels. The modelling of risk in particular is inherently

complex, since in reality risk is not just a single number but rather an infinite

number of possible future states, each with its own implications for the

investors’ expected cash flow. Therefore we first need to establish what an

asset is , and what characterises its return and risk. The same must be done for

portfolios. Having done this, we can then move on to considering optimal

portfolios with respect to return and risk, and eventually to optimal allocations

over the entire risk spectrum.