Market risk analysis ii practical financial econometrics carol alexander 0470998016

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Market Risk Analysis
Volume II
Practical Financial Econometrics

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Market Risk Analysis
Volume II
Practical Financial Econometrics
Carol Alexander

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Published in 2008 by

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,
West Sussex PO19 8SQ, England
Telephone +44 1243 779777

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Copyright © 2008 Carol Alexander
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To Rick van der Ploeg

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Contents
List of Figures

xiii

List of Tables

xvii

List of Examples

xx

Foreword

xxii

Preface to Volume II

xxvi

II.1 Factor Models
II.1.1 Introduction
II.1.2 Single Factor Models
II.1.2.1 Single Index Model
II.1.2.2 Estimating Portfolio Characteristics using OLS
II.1.2.3 Estimating Portfolio Risk using EWMA
II.1.2.4 Relationship between Beta, Correlation and Relative
Volatility
II.1.2.5 Risk Decomposition in a Single Factor Model
II.1.3 Multi-Factor Models
II.1.3.1 Multi-factor Models of Asset or Portfolio Returns
II.1.3.2 Style Attribution Analysis
II.1.3.3 General Formulation of Multi-factor Model
II.1.3.4 Multi-factor Models of International Portfolios
II.1.4 Case Study: Estimation of Fundamental Factor Models
II.1.4.1 Estimating Systematic Risk for a Portfolio of US Stocks
II.1.4.2 Multicollinearity: A Problem with Fundamental Factor
Models
II.1.4.3 Estimating Fundamental Factor Models by Orthogonal
Regression
II.1.5 Analysis of Barra Model
II.1.5.1 Risk Indices, Descriptors and Fundamental Betas
II.1.5.2 Model Specification and Risk Decomposition
II.1.6 Tracking Error and Active Risk
II.1.6.1 Ex Post versus Ex Ante Measurement of Risk and Return
II.1.6.2 Definition of Active Returns
II.1.6.3 Definition of Active Weights
II.1.6.4 Ex Post Tracking Error

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Contents

II.1.6.5 Ex Post Mean-Adjusted Tracking Error
II.1.6.6 Ex Ante Tracking Error
II.1.6.7 Ex Ante Mean-Adjusted Tracking Error
II.1.6.8 Clarification of the Definition of Active Risk
II.1.7 Summary and Conclusions
II.2 Principal Component Analysis
II.2.1 Introduction
II.2.2 Review of Principal Component Analysis
II.2.2.1 Definition of Principal Components
II.2.2.2 Principal Component Representation
II.2.2.3 Frequently Asked Questions
II.2.3 Case Study: PCA of UK Government Yield Curves
II.2.3.1 Properties of UK Interest Rates
II.2.3.2 Volatility and Correlation of UK Spot Rates
II.2.3.3 PCA on UK Spot Rates Correlation Matrix
II.2.3.4 Principal Component Representation
II.2.3.5 PCA on UK Short Spot Rates Covariance Matrix
II.2.4 Term Structure Factor Models
II.2.4.1 Interest Rate Sensitive Portfolios
II.2.4.2 Factor Models for Currency Forward Positions
II.2.4.3 Factor Models for Commodity Futures Portfolios
II.2.4.4 Application to Portfolio Immunization
II.2.4.5 Application to Asset–Liability Management
II.2.4.6 Application to Portfolio Risk Measurement
II.2.4.7 Multiple Curve Factor Models
II.2.5 Equity PCA Factor Models
II.2.5.1 Model Structure
II.2.5.2 Specific Risks and Dimension Reduction
II.2.5.3 Case Study: PCA Factor Model for DJIA

Portfolios
II.2.6 Summary and Conclusions
II.3 Classical Models of Volatility and Correlation
II.3.1 Introduction
II.3.2 Variance and Volatility
II.3.2.1 Volatility and the Square-Root-of-Time Rule
II.3.2.2 Constant Volatility Assumption
II.3.2.3 Volatility when Returns are Autocorrelated
II.3.2.4 Remarks about Volatility
II.3.3 Covariance and Correlation
II.3.3.1 Definition of Covariance and Correlation
II.3.3.2 Correlation Pitfalls
II.3.3.3 Covariance Matrices
II.3.3.4 Scaling Covariance Matrices
II.3.4 Equally Weighted Averages
II.3.4.1 Unconditional Variance and Volatility
II.3.4.2 Unconditional Covariance and Correlation
II.3.4.3 Forecasting with Equally Weighted Averages

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Contents

II.3.5 Precision of Equally Weighted Estimates
II.3.5.1 Confidence Intervals for Variance and Volatility
II.3.5.2 Standard Error of Variance Estimator
II.3.5.3 Standard Error of Volatility Estimator
II.3.5.4 Standard Error of Correlation Estimator
II.3.6 Case Study: Volatility and Correlation of US Treasuries
II.3.6.1 Choosing the Data
II.3.6.2 Our Data
II.3.6.3 Effect of Sample Period
II.3.6.4 How to Calculate Changes in Interest Rates
II.3.7 Equally Weighted Moving Averages
II.3.7.1 Effect of Volatility Clusters
II.3.7.2 Pitfalls of the Equally Weighted Moving Average Method
II.3.7.3 Three Ways to Forecast Long Term Volatility
II.3.8 Exponentially Weighted Moving Averages
II.3.8.1 Statistical Methodology
II.3.8.2 Interpretation of Lambda
II.3.8.3 Properties of EWMA Estimators
II.3.8.4 Forecasting with EWMA
II.3.8.5 Standard Errors for EWMA Forecasts

II.3.8.6 RiskMetricsTM Methodology
II.3.8.7 Orthogonal EWMA versus RiskMetrics EWMA
II.3.9 Summary and Conclusions
II.4 Introduction to GARCH Models
II.4.1 Introduction
II.4.2 The Symmetric Normal GARCH Model
II.4.2.1 Model Specification
II.4.2.2 Parameter Estimation
II.4.2.3 Volatility Estimates
II.4.2.4 GARCH Volatility Forecasts
II.4.2.5 Imposing Long Term Volatility
II.4.2.6 Comparison of GARCH and EWMA Volatility Models
II.4.3 Asymmetric GARCH Models
II.4.3.1 A-GARCH
II.4.3.2 GJR-GARCH
II.4.3.3 Exponential GARCH
II.4.3.4 Analytic E-GARCH Volatility Term Structure Forecasts
II.4.3.5 Volatility Feedback
II.4.4 Non-Normal GARCH Models
II.4.4.1 Student t GARCH Models
II.4.4.2 Case Study: Comparison of GARCH Models for the
FTSE 100
II.4.4.3 Normal Mixture GARCH Models
II.4.4.4 Markov Switching GARCH
II.4.5 GARCH Covariance Matrices
II.4.5.1 Estimation of Multivariate GARCH Models
II.4.5.2 Constant and Dynamic Conditional Correlation GARCH
II.4.5.3 Factor GARCH

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104
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107
109
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115
115
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x

Contents

II.4.6 Orthogonal GARCH
II.4.6.1 Model Specification
II.4.6.2 Case Study: A Comparison of RiskMetrics and O-GARCH
II.4.6.3 Splicing Methods for Constructing Large Covariance
Matrices

II.4.7 Monte Carlo Simulation with GARCH Models
II.4.7.1 Simulation with Volatility Clustering
II.4.7.2 Simulation with Volatility Clustering Regimes
II.4.7.3 Simulation with Correlation Clustering
II.4.8 Applications of GARCH Models
II.4.8.1 Option Pricing with GARCH Diffusions
II.4.8.2 Pricing Path-Dependent European Options
II.4.8.3 Value-at-Risk Measurement
II.4.8.4 Estimation of Time Varying Sensitivities
II.4.8.5 Portfolio Optimization
II.4.9 Summary and Conclusions
II.5 Time Series Models and Cointegration
II.5.1 Introduction
II.5.2 Stationary Processes
II.5.2.1 Time Series Models
II.5.2.2 Inversion and the Lag Operator
II.5.2.3 Response to Shocks
II.5.2.4 Estimation
II.5.2.5 Prediction
II.5.2.6 Multivariate Models for Stationary Processes
II.5.3 Stochastic Trends
II.5.3.1 Random Walks and Efficient Markets
II.5.3.2 Integrated Processes and Stochastic Trends
II.5.3.3 Deterministic Trends
II.5.3.4 Unit Root Tests
II.5.3.5 Unit Roots in Asset Prices
II.5.3.6 Unit Roots in Interest Rates, Credit Spreads and Implied
Volatility
II.5.3.7 Reconciliation of Time Series and Continuous Time Models
II.5.3.8 Unit Roots in Commodity Prices

II.5.4 Long Term Equilibrium
II.5.4.1 Cointegration and Correlation Compared
II.5.4.2 Common Stochastic Trends
II.5.4.3 Formal Definition of Cointegration
II.5.4.4 Evidence of Cointegration in Financial Markets
II.5.4.5 Estimation and Testing in Cointegrated Systems
II.5.4.6 Application to Benchmark Tracking
II.5.4.7 Case Study: Cointegration Index Tracking in the Dow
Jones Index
II.5.5 Modelling Short Term Dynamics
II.5.5.1 Error Correction Models

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Contents

II.5.5.2 Granger Causality
II.5.5.3 Case Study: Pairs Trading Volatility Index Futures
II.5.6 Summary and Conclusions

xi

246
247
250

II.6 Introduction to Copulas
II.6.1 Introduction
II.6.2 Concordance Metrics
II.6.2.1 Concordance
II.6.2.2 Rank Correlations
II.6.3 Copulas and Associated Theoretical Concepts
II.6.3.1 Simulation of a Single Random Variable
II.6.3.2 Definition of a Copula
II.6.3.3 Conditional Copula Distributions and their Quantile Curves
II.6.3.4 Tail Dependence
II.6.3.5 Bounds for Dependence
II.6.4 Examples of Copulas
II.6.4.1 Normal or Gaussian Copulas
II.6.4.2 Student t Copulas
II.6.4.3 Normal Mixture Copulas
II.6.4.4 Archimedean Copulas
II.6.5 Conditional Copula Distributions and Quantile Curves
II.6.5.1 Normal or Gaussian Copulas
II.6.5.2 Student t Copulas
II.6.5.3 Normal Mixture Copulas

II.6.5.4 Archimedean Copulas
II.6.5.5 Examples
II.6.6 Calibrating Copulas
II.6.6.1 Correspondence between Copulas and Rank Correlations
II.6.6.2 Maximum Likelihood Estimation
II.6.6.3 How to Choose the Best Copula
II.6.7 Simulation with Copulas
II.6.7.1 Using Conditional Copulas for Simulation
II.6.7.2 Simulation from Elliptical Copulas
II.6.7.3 Simulation with Normal and Student t Copulas
II.6.7.4 Simulation from Archimedean Copulas
II.6.8 Market Risk Applications
II.6.8.1 Value-at-Risk Estimation
II.6.8.2 Aggregation and Portfolio Diversification
II.6.8.3 Using Copulas for Portfolio Optimization
II.6.9 Summary and Conclusions

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283
285
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287
290
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295
298

II.7 Advanced Econometric Models
II.7.1 Introduction
II.7.2 Quantile Regression
II.7.2.1 Review of Standard Regression
II.7.2.2 What is Quantile Regression?
II.7.2.3 Parameter Estimation in Quantile Regression

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xii

Contents

II.7.3

II.7.4

II.7.5

II.7.6

II.7.7

II.7.2.4 Inference on Linear Quantile Regressions
II.7.2.5 Using Copulas for Non-linear Quantile Regression
Case Studies on Quantile Regression
II.7.3.1 Case Study 1: Quantile Regression of Vftse on FTSE 100
Index
II.7.3.2 Case Study 2: Hedging with Copula Quantile Regression

Other Non-Linear Regression Models
II.7.4.1 Non-linear Least Squares
II.7.4.2 Discrete Choice Models
Markov Switching Models
II.7.5.1 Testing for Structural Breaks
II.7.5.2 Model Specification
II.7.5.3 Financial Applications and Software
Modelling Ultra High Frequency Data
II.7.6.1 Data Sources and Filtering
II.7.6.2 Modelling the Time between Trades
II.7.6.3 Forecasting Volatility
Summary and Conclusions

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II.8 Forecasting and Model Evaluation
II.8.1 Introduction
II.8.2 Returns Models
II.8.2.1 Goodness of Fit
II.8.2.2 Forecasting
II.8.2.3 Simulating Critical Values for Test Statistics
II.8.2.4 Specification Tests for Regime Switching Models
II.8.3 Volatility Models
II.8.3.1 Goodness of Fit of GARCH Models
II.8.3.2 Forecasting with GARCH Volatility Models
II.8.3.3 Moving Average Models
II.8.4 Forecasting the Tails of a Distribution
II.8.4.1 Confidence Intervals for Quantiles
II.8.4.2 Coverage Tests
II.8.4.3 Application of Coverage Tests to GARCH Models
II.8.4.4 Forecasting Conditional Correlations
II.8.5 Operational Evaluation
II.8.5.1 General Backtesting Algorithm
II.8.5.2 Alpha Models
II.8.5.3 Portfolio Optimization
II.8.5.4 Hedging with Futures
II.8.5.5 Value-at-Risk Measurement
II.8.5.6 Trading Implied Volatility
II.8.5.7 Trading Realized Volatility
II.8.5.8 Pricing and Hedging Options
II.8.6 Summary and Conclusions

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References

377

Index

387

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List of Figures

II.1.1

II.1.2

II.1.3

II.1.4

II.1.5
II.1.6

II.1.7
II.1.8
II.1.9
II.2.1
II.2.2
II.2.3

II.2.4

II.2.5

EWMA beta and

systematic risk of the
two-stock portfolio
EWMA beta, relative
volatility and correlation of
Amex ( = 095)
EWMA beta, relative
volatility and correlation of
Cisco ( = 095)
Two communications
stocks and four possible
risk factors
A fund with ex post
tracking error of only 1%
Irrelevance of the
benchmark for tracking
error
Which fund has an ex post
tracking error of zero?
Forecast and target active
returns
Returns distributions for
two funds
UK government zero
coupon yields, 2000–2007
Volatilities of UK spot
rates, 2005–2007
Eigenvectors of the UK
daily spot rate correlation
matrix
Eigenvectors of the UK

daily short spot rate
covariance matrix
UK government interest
rates, monthly, 1995–2007

II.2.6
8
II.2.7
9

II.2.8

9

II.2.9

21
35

II.2.10
II.2.11
II.2.12

36
38
40
42
54

II.2.13

II.3.1
II.3.2
II.3.3
II.3.4

55
II.3.5
58
II.3.6
61
II.3.7
64

Eigenvectors of the
UK monthly spot rate
covariance matrix
First principal component
for UK interest rates
Constant maturity futures
on West Texas Intermediate
crude oil
Eigenvectors of crude oil
futures correlation matrix
Credit spreads in the euro
zone
First two eigenvectors on
two-curve PCA
Three short spot curves,
December 2001 to August
2007

Eigenvectors for multiple
curve PCA factor models
Confidence interval for
variance forecasts
US Treasury rates
Volatilities of US interest
rates (in basis points)
MIB 30 and S&P 500 daily
closing prices
Equally weighted moving
average volatility estimates
of the MIB 30 index
EWMA volatility estimates
for S&P 500 with different
lambdas
EWMA versus equally
weighted volatility

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xiv

List of Figures

II.3.8
II.3.9

II.4.1
II.4.2

II.4.3

II.4.4
II.4.5

II.4.6
II.4.7
II.4.8

II.4.9

II.4.10
II.4.11
II.4.12
II.4.13
II.4.14
II.4.15

II.4.16

Standard errors of EWMA
estimators
Comparison of the
RiskMetrics ‘forecasts’ for
FTSE 100 volatility
Solver settings for GARCH
estimation in Excel
Comparison of GARCH
and EWMA volatilities for
the FTSE 100
Term structure GARCH
volatility forecast for FTSE
100, 29 August 2007
The mean reversion effect
in GARCH volatility
Effect of imposing long
term volatility on GARCH
term structure
E-GARCH asymmetric
response function
E-GARCH volatility

estimates for the FTSE 100
Comparison of GARCH
and E-GARCH volatility
forecasts
GBP and EUR dollar rates
A-GARCH volatilities of
GBP/USD and EUR/USD
Covariances of GBP/USD
and EUR/USD
F-GARCH volatilities and
covariance
Constant maturity crude oil
futures prices
Constant maturity natural
gas futures prices
Correlation between
2-month and 6-month
crude oil futures forecasted
using RiskMetrics EWMA
and 250-day methods
Correlation between
2-month and 6-month
natural gas futures
forecasted using
RiskMetrics EWMA and
250-day methods

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139

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145

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154

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168
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174

175

175

II.4.17 O-GARCH 1-day volatility
forecasts for crude oil
II.4.18 O-GARCH 1-day
correlation forecasts for
crude oil
II.4.19 O-GARCH 1-day volatility
forecasts for natural gas

II.4.20 O-GARCH 1-day
correlation forecasts for
natural gas
II.4.21 Comparison of normal
i.i.d. and normal GARCH
simulations
II.4.22 Comparison of symmetric
normal GARCH and
asymmetric t GARCH
simulations
II.4.23 High and low volatility
components in normal
mixture GARCH
II.4.24 Simulations from a Markov
switching GARCH process
II.4.25 Correlated returns
simulated from a bivariate
GARCH process
II.4.26 GARCH correlation of
the returns shown in
Figure II.4.25
II.4.27 Comparison of EWMA
and GARCH time varying
betas
II.5.1 Mean reversion in
stationary processes
II.5.2 Impulse response for an
ARMA(2,1) process
II.5.3 A stationary series
II.5.4 Correlogram of the spread

in Figure II.5.3
II.5.5 Simulation of stationary
process with confidence
bounds
II.5.6 Two random walks with
drift
II.5.7 Stochastic trend versus
deterministic trend
processes

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209

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List of Figures

II.5.8
II.5.9
II.5.10
II.5.11
II.5.12
II.5.13
II.5.14

II.5.15
II.5.16

II.5.17

II.5.18
II.5.19

II.5.20

II.5.21
II.5.22

II.5.23

II.5.24

II.5.25

II.5.26

II.6.1

FTSE 100 and S&P 500
stock indices
$/£ exchange rate
UK 2-year interest rates
The iTraxx Europe index
Volatility index futures
Cointegrated prices, low
correlation in returns
Non-cointegrated prices
with highly correlated
returns
FTSE 100 and S&P 500
indices, 1996–2007
FTSE 100 and S&P 500
indices in common
currency, 1996–2007
Residuals from

Engle–Granger regression
of FTSE 100 on S&P 500
DAX 30 and CAC 40
indices, 1996–2007
Residuals from
Engle–Granger regression
of DAX 30 on CAC40
UK short spot rates,
2000–2007
Comparison of TEVM and
cointegration tracking error
Residuals from
Engle–Granger regression
of log DJIA on log stock
prices
Comparison of
cointegration and TEVM
tracking
Difference between log
spot price and log futures
price
Impulse response of
volatility futures and their
spread I
Impulse response of
volatility futures and their
spread II
Bivariate normal copula
density with correlation 0.5

II.6.2
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219
220
221
222

II.6.3

226

II.6.5

II.6.4

II.6.6
226
II.6.7
233
II.6.8
233

234

II.6.9

234
II.6.10
235
237

II.6.11
II.6.12

239
II.6.13
241

II.6.14

242

II.6.15
II.6.16

245

249

II.6.17

250

II.6.18

261

Bivariate Student t copula
density with correlation 0.5
and 5 degrees of freedom

A bivariate normal mixture
copula density
Bivariate Clayton copula
density for  = 05
Bivariate Gumbel copula
density for  = 125
Bivariate normal copula
density with  = 025
Bivariate Student t copula
density with  = −025 and
seven degrees of freedom
Bivariate normal mixture
copula density with
 = 025, 1 = 05 and
2 = −05
Bivariate normal mixture
copula density with
 = 075, 1 = 025 and
2 = −075
Bivariate Clayton copula
density with  = 075
Bivariate Gumbel copula
density with  = 15
Quantile curves of normal
and Student t copulas with
zero correlation
Quantile curves for
different copulas and
marginals
Scatter plot of FTSE 100

index and Vftse index
returns, 2004–2006
Uniform simulations from
three bivariate copulas
Simulations of returns
generated by different
marginals and different
copulas
Marginal densities of two
gamma distributed random
variables
Distribution of the sum for
different correlation
assumptions

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261
262
262
263
270

270

270

270
270

270

277

278

282
288

289

294

294


xvi

List of Figures

II.6.19 Density of the sum of the
random variables in
Figure II.6.17 under
different dependence
assumptions
II.6.20 Optimal weight on FTSE
and Sharpe ratio vs
FTSE–Vftse returns
correlation
II.7.1 Quantile regression lines

II.7.2 Loss function for q quantile
regression objective
II.7.3 Distribution of Vftse
conditional on FTSE
falling by 1% (linear
quantile regression)
II.7.4 Calibration of copula
quantile regressions of
Vftse on FTSE
II.7.5 Distribution of Vftse
conditional on FTSE
falling by 1%
II.7.6 Distribution of Vftse
conditional on FTSE
falling by 3%
II.7.7 Vodafone, HSBC and BP
stock prices (rebased)

II.7.8

295

II.7.9
II.7.10
II.7.11

II.7.12
297
306
306

II.7.13
II.7.14

311

II.7.15

312

II.8.1
II.8.2

313

II.8.3

313

II.8.4

315

Comparison of FTSE index
and portfolio price
EWMA hedge ratio
Quadratic regression curve
Default probabilities
estimated by discrete
choice models

Sensitivity of default
probabilities to debt–equity
ratio
Vftse and FTSE 100
indices
A simulation from the
exponential symmetric
ACD(1,1) model
Historical versus realized
volatility of S&P 500
Likelihood comparison
S&P 500 Index January
2000–September 2007
Distribution of 30-day
GARCH forecasts on
FTSE 100
Distribution of spread
between implied and
GARCH volatilities

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318
320

324

324
326

334
335
354
368

370

371


List of Tables

OLS alpha, beta and
specific risk for two stocks
and a 60:40 portfolio
II.1.2 Results of style analysis
for Vanguard and Fidelity
mutual funds
II.1.3 Risk factor correlations
and volatilities
II.1.4 Risk factor covariance
matrix
II.1.5 Factor betas from
regression model
II.1.6 Multicollinearity in time
series factor models
II.1.7 Factor correlation matrix
II.1.8 Eigenvalues and
eigenvectors of the risk
factor covariance matrix

II.1.9 Using orthogonal
regression to obtain risk
factor betas
II.1.10 Values of a fund and a
benchmark
II.1.11 Values of a fund and two
benchmarks
II.1.12 TE and MATE for the
funds in Figure II.1.7
II.2.1 Correlation matrix of
selected UK spot rates
II.2.2 Eigenvalues and
eigenvectors of the
correlation matrix of UK
spot rates
II.1.1

Eigenvalues of the UK
short spot rate covariance
matrix
II.2.4 Cash flows and PV01
vector for a UK bond
portfolio
II.2.5 Eigenvalues of UK
yield curve covariance
matrix
II.2.6 Eigenvalues for UK short
spot rates
II.2.7 Stress test based on PCA
factor model

II.2.8 Eigenvectors and
eigenvalues of the
three-curve covariance
matrix
II.2.9 Ticker symbols for DJIA
stocks
II.2.10 Cumulative variation
explained by the principal
components
II.2.11 PCA factor models for
DJIA stocks
II.2.12 Portfolio betas for the
principal component
factors, and systematic,
total and specific risk
II.3.1 Volatilities and correlations
of three assets
II.3.2 Closing prices on the
FTSE 100 index
II.3.3 Closing prices on the S&P
500 index
II.2.3

6

15
19
20
22
24

25

26

27
34
35
38
56

57

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69
75

79
82

83
84

85
96
100

102


xviii

List of Tables

II.3.5
II.3.6

II.4.1

II.4.2

II.4.3

II.4.4

II.4.5

II.4.6

II.4.7

II.4.8
II.4.9

II.4.10

II.4.11

II.4.12

II.4.13

Correlations between US
Treasury rates
Volatilities and correlation
of US Treasuries,
2006–2007
EViews and Matlab
estimation of FTSE
100 symmetric normal
GARCH
Estimation of FTSE
100 symmetric normal
GARCH, 2003–2007
Comparison of
symmetric and
asymmetric GARCH
models for the
FTSE 100
Parameter estimates
and standard errors of
GJR-GARCH models
Excel estimates of
E-GARCH parameters for
the FTSE 100
Student t GARCH
parameter estimates from

Excel and Matlab
Estimation of symmetric
and asymmetric normal
GARCH models for the
FTSE 100
Student t GARCH models
for the FTSE 100
Parameter estimates and
standard errors of NM(2)
A-GARCH models
PCA of 2mth–12mth crude
oil futures and natural gas
futures
Parameter settings for
symmetric and symmetric
GARCH simulations
Parameter settings for
normal mixture GARCH
simulations
Diagonal vech parameters
for correlated GARCH
simulations

113

115

140

140

149

151

153

158

160
161

163

176

183

184

187

II.4.14 Multivariate A-GARCH
parameter estimates
II.4.15 Optimal allocations under
the two covariance
matrices
II.5.1 Critical values of the
Dickey–Fuller distribution
II.5.2 Critical values of the

augmented Dickey–Fuller
distribution
II.5.3 Results of ADF(1) tests
II.5.4 Johansen trace tests on UK
short rates
II.5.5 Optimal weights on 16
stocks tracking the
Dow Jones Industrial
Average
II.5.6 ECMs of volatility index
futures
II.5.7 ECMs of volatility index
futures (tested down)
II.6.1 Calculation of Spearman’s
rho
II.6.2 Calculation of Kendall’s
tau
II.6.3 Ninety per cent confidence
limits for X2 given that
X1 = 3
II.6.4 Calibrated parameters for
Student t marginals
II.6.5 Empirical copula density
and distribution
II.6.6 Daily VaR of 1% based on
different dependence
assumptions
II.7.1 Quantile regression
coefficient estimates of
Vftse–FTSE model

II.7.2 Conditional quantiles of
Vftse
II.7.3 Estimation of discrete
choice models
II.8.1 Analysis of variance for
two models
II.8.2 Comparison of goodness
of fit

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197
217

217
219
238

241
248
248
257
257

279
282
284

291

310
311
323
344
344


List of Tables

II.8.3

II.8.4

Maximum R2 from
regression of squared
return on GARCH variance
forecast
Confidence intervals for
empirical quantiles of S&P
500

II.8.5

353

357

II.8.6

Hypothetical Sharpe ratios
from alpha model backtest
results
Coverage tails for VaR
prediction on the S&P 500
index

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366

369


List of Examples

OLS estimates of alpha and
beta for two stocks
II.1.2 OLS estimates of portfolio
alpha and beta
II.1.3 Systematic and specific risk
II.1.4 Style attribution
II.1.5 Systematic risk at the
portfolio level
II.1.6 Decomposition of
systematic risk into equity
and forex factors
II.1.7 Total risk and systematic
risk

II.1.8 Tracking error of an
underperforming fund
II.1.9 Why tracking error only
applies to tracking funds
II.1.10 Irrelevance of the
benchmark for tracking
error
II.1.11 Interpretation of
Mean-Adjusted Tracking
Error
II.1.12 Comparison of TE and

MATE
II.1.13 Which fund is more risky
(1)?
II.1.14 Which fund is more risky
(2)?
II.2.1 PCA factor model for a UK
bond portfolio
II.2.2 PCA factor model for
forward sterling exposures
II.2.3 PCA on crude oil futures

Immunizing a bond
portfolio using PCA
II.2.5 Asset–liability management
using PCA
II.2.6 Stress testing a UK bond
portfolio
II.2.7 PCA on curves with

different credit rating
II.2.8 PCA on curves in different
currencies
II.2.9 Decomposition of total risk
using PCA factors
II.3.1 Calculating volatility from
standard deviation
II.3.2 Estimating volatility for
hedge funds
II.3.3 Portfolio variance
II.3.4 Scaling and decomposition
of covariance matrix
II.3.5 Equally weighted average
estimate of FTSE 100
volatility (I)
II.3.6 Equally weighted average
estimate of FTSE 100
volatility (II)
II.3.7 Equally weighted
correlation of the FTSE 100
and S&P 500
II.3.8 Confidence interval for a
variance estimate
II.3.9 Confidence intervals for a
volatility forecast
II.3.10 Standard Error for
Volatility
II.2.4

II.1.1

4
5
12
15
17

19
22
34
34

35

37
37
41
41
63
68
70

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74
77
78
83
91

93
96
97

100

101

102
105
106
108


List of Examples

II.3.11 Testing the significance of
historical correlation
II.3.12 Historical volatility of MIB
30
II.4.1 GARCH estimates of FTSE
100 volatility
II.4.2 Imposing a value for long
term volatility in GARCH
II.4.3 An asymmetric GARCH
model for the FTSE 100
II.4.4 An E-GARCH model for
the FTSE 100
II.4.5 Symmetric Student t
GARCH

II.4.6 CC and DCC GARCH
applied to FOREX rates
II.4.7 F-GARCH applied to
equity returns
II.4.8 Pricing an Asian option
with GARCH
II.4.9 Pricing a barrier option
with GARCH
II.4.10 Portfolio optimization with
GARCH
II.5.1 Testing an ARMA process
for stationarity
II.5.2 Impulse response
II.5.3 Estimation of AR(2) model
II.5.4 Confidence limits for
stationary processes
II.5.5 Unit roots in stock indices
and exchange rates
II.5.6 Unit root tests on interest
rates
II.5.7 Unit root tests on credit
spreads
II.5.8 Unit roots in implied
volatility futures
II.5.9 Are international stock
indices cointegrated?
II.5.10 Johansen tests for
cointegration in UK interest
rates

109
116
139
145
148
153
158
167
170
190
191
196
205
207
210
210
218
220
221
221
232

237

II.5.11 An ECM of spot and
futures on the Hang Seng
index
II.5.12 Price discovery in the Hang
Seng index
II.6.1 Spearman’s rho

II.6.2 Kendall’s tau
II.6.3 Calibrating copulas using
rank correlations
II.6.4 Calibration of copulas
II.6.5 VaR with symmetric and
asymmetric tail dependence
II.6.6 Aggregation under the
normal copula
II.6.7 Aggregation under the
normal mixture copula
II.6.8 Portfolio optimization with
copulas
II.7.1 Non-linear regressions for
the FTSE 100 and Vftse
II.7.2 Simple probit and logit
models for credit default
II.7.3 Estimating the default
probability and its
sensitivity
II.7.4 Chow test
II.8.1 Standard goodness-of-fit
tests for regression models
II.8.2 Generating unconditional
distributions
II.8.3 Bootstrap estimation of the
distribution of a test
statistic
II.8.4 Quantile confidence
intervals for the S&P 500
II.8.5 Unconditional coverage test

for volatility forecast
II.8.6 Conditional coverage test
II.8.7 Backtesting a simple VaR
model
II.8.8 Using volatility forecasts to
trade implied volatility

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247
256
257
280
282
291
293
294
296
320
322

323
326
344
346

349
357

358
359
369
370


Foreword
How many children dream of one day becoming risk managers? I very much doubt little
Carol Jenkins, as she was called then, did. She dreamt about being a wild white horse, or a
mermaid swimming with dolphins, as any normal little girl does. As I start crunching into
two kilos of Toblerone that Carol Alexander-Pézier gave me for Valentine’s day (perhaps to
coax me into writing this foreword), I see the distinctive silhouette of the Matterhorn on the
yellow package and I am reminded of my own dreams of climbing mountains and travelling
to distant planets. Yes, adventure and danger! That is the stuff of happiness, especially when
you daydream as a child with a warm cup of cocoa in your hands.
As we grow up, dreams lose their naivety but not necessarily their power. Knowledge
makes us discover new possibilities and raises new questions. We grow to understand better
the consequences of our actions, yet the world remains full of surprises. We taste the
sweetness of success and the bitterness of failure. We grow to be responsible members of
society and to care for the welfare of others. We discover purpose, confidence and a role to
fulfil; but we also find that we continuously have to deal with risks.
Leafing through the hundreds of pages of this four-volume series you will discover one
of the goals that Carol gave herself in life: to set the standards for a new profession, that of
market risk manager, and to provide the means of achieving those standards. Why is market
risk management so important? Because in our modern economies, market prices balance
the supply and demand of most goods and services that fulfil our needs and desires. We can
hardly take a decision, such as buying a house or saving for a later day, without taking some
market risks. Financial firms, be they in banking, insurance or asset management, manage
these risks on a grand scale. Capital markets and derivative products offer endless ways to
transfer these risks among economic agents.

But should market risk management be regarded as a professional activity? Sampling the
material in these four volumes will convince you, if need be, of the vast amount of knowledge
and skills required. A good market risk manager should master the basics of calculus,
linear algebra, probability – including stochastic calculus – statistics and econometrics. He
should be an astute student of the markets, familiar with the vast array of modern financial
instruments and market mechanisms, and of the econometric properties of prices and returns
in these markets. If he works in the financial industry, he should also be well versed in
regulations and understand how they affect his firm. That sets the academic syllabus for the
profession.
Carol takes the reader step by step through all these topics, from basic definitions and
principles to advanced problems and solution methods. She uses a clear language, realistic
illustrations with recent market data, consistent notation throughout all chapters, and provides
a huge range of worked-out exercises on Excel spreadsheets, some of which demonstrate

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Foreword

xxiii

analytical tools only available in the best commercial software packages. Many chapters on
advanced subjects such as GARCH models, copulas, quantile regressions, portfolio theory,
options and volatility surfaces are as informative as and easier to understand than entire
books devoted to these subjects. Indeed, this is the first series of books entirely dedicated to
the discipline of market risk analysis written by one person, and a very good teacher at that.
A profession, however, is more than an academic discipline; it is an activity that fulfils
some societal needs, that provides solutions in the face of evolving challenges, that calls for
a special code of conduct; it is something one can aspire to. Does market risk management
face such challenges? Can it achieve significant economic benefits?
As market economies grow, more ordinary people of all ages with different needs and

risk appetites have financial assets to manage and borrowings to control. What kind of
mortgages should they take? What provisions should they make for their pensions? The range
of investment products offered to them has widened far beyond the traditional cash, bond
and equity classes to include actively managed funds (traditional or hedge funds), private
equity, real estate investment trusts, structured products and derivative products facilitating
the trading of more exotic risks – commodities, credit risks, volatilities and correlations,
weather, carbon emissions, etc. – and offering markedly different return characteristics from
those of traditional asset classes. Managing personal finances is largely about managing
market risks. How well educated are we to do that?
Corporates have also become more exposed to market risks. Beyond the traditional exposure to interest rate fluctuations, most corporates are now exposed to foreign exchange risks
and commodity risks because of globalization. A company may produce and sell exclusively
in its domestic market and yet be exposed to currency fluctuations because of foreign competition. Risks that can be hedged effectively by shareholders, if they wish, do not have
to be hedged in-house. But hedging some risks in-house may bring benefits (e.g. reduction
of tax burden, smoothing of returns, easier planning) that are not directly attainable by the
shareholder.
Financial firms, of course, should be the experts at managing market risks; it is their
métier. Indeed, over the last generation, there has been a marked increase in the size of
market risks handled by banks in comparison to a reduction in the size of their credit risks.
Since the 1980s, banks have provided products (e.g. interest rate swaps, currency protection,
index linked loans, capital guaranteed investments) to facilitate the risk management of their
customers. They have also built up arbitrage and proprietary trading books to profit from
perceived market anomalies and take advantage of their market views. More recently, banks
have started to manage credit risks actively by transferring them to the capital markets
instead of warehousing them. Bonds are replacing loans, mortgages and other loans are
securitized, and many of the remaining credit risks can now be covered with credit default
swaps. Thus credit risks are being converted into market risks.
The rapid development of capital markets and, in particular, of derivative products bears
witness to these changes. At the time of writing this foreword, the total notional size of all
derivative products exceeds $500 trillion whereas, in rough figures, the bond and money
markets stand at about $80 trillion, the equity markets half that and loans half that again.

Credit derivatives by themselves are climbing through the $30 trillion mark. These derivative
markets are zero-sum games; they are all about market risk management – hedging, arbitrage
and speculation.
This does not mean, however, that all market risk management problems have been
resolved. We may have developed the means and the techniques, but we do not necessarily

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