Portfolio risk analysis gregory connor 9780691128283

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Portfolio Risk Analysis

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Portfolio Risk Analysis

Gregory Connor

Lisa R. Goldberg

Robert A. Korajczyk

Princeton University Press
Princeton and Oxford

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Copyright © 2010 by Princeton University Press
Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Connor, Gregory.
Portfolio risk analysis / Gregory Connor, Lisa R. Goldberg, Robert A.
Korajczyk.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-12828-3 (alk. paper)
1. Portfolio management. 2. Risk management. I. Goldberg, Lisa R.
II. Korajczyk, Robert A., 1954– III. Title.
HG4529.5.C657 2010
332.6–dc22

2009050913

British Library Cataloging-in-Publication Data is available
This book has been composed in LucidaBright using TEX
Typeset and copyedited by T&T Productions Ltd, London
Printed on acid-free paper.

∞

press.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1

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To our families

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Contents

Acknowledgments

xi

Introduction

xiii

Key Notation

xix

1

Measures of Risk and Return
1.1
Measuring Return
1.2
The Key Portfolio Risk Measures
1.3
Risk–Return Preferences and Portfolio Optimization
1.4
The Capital Asset Pricing Model and Its Applications to
Risk Analysis
1.5
The Objectives and Limitations of Portfolio Risk Analysis

1
1
6
12

2

Unstructured Covariance Matrices
2.1
Estimating Return Covariance Matrices
2.2
The Error-Maximization Problem
2.3
Portfolio Choice as Decision Making under Uncertainty

36
36
47
54

3

Industry and Country Risk
3.1
Industry–Country Component Models
3.2
Empirical Evidence on the Relative Magnitudes of Country
and Industry Risks
3.3
Sector–Currency Models of Corporate Bond Returns

61
61

4

Statistical Factor Analysis

4.1
Types of Factor Models
4.2
Approximate Factor Models
4.3
The Arbitrage Pricing Theory
4.4
Small-n Estimation Methods
4.5
Large-n Estimation Methods
4.6
Number of Factors

79
79
82
86
88
93
98

5

The Macroeconomy and Portfolio Risk
5.1
Estimating Macroeconomic Factor Models
5.2
Event Studies of Macroeconomic Announcements

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23
31

73
77

101
101
110


viii

Contents
5.3
5.4
5.5

Macroeconomic Policy Endogeneity
Business Cycle Betas
Empirical Fit and the Relative Value of Macroeconomic
Factor Models

112
115

Security Characteristics and Pervasive Risk Factors

6.1
Equity and Fixed-Income Characteristics
6.2
Characteristic-Based Factor Models of Equities
6.3
The Fama–French Model and Extensions
6.4
The Semiparametric Approach to Characteristic-Based
Factor Models

117
117
122
130

7

Measuring and Hedging Foreign Exchange Risk
7.1
Definitions of Foreign Exchange Risk
7.2
Optimal Currency Hedging
7.3
Currency Covariances with Stock and Bond Returns
7.4
Macroeconomic Influences on Currency Returns

134
134
142

149
151

8

Integrated Risk Models
8.1
Global and Regional Integration Trends
8.2
Risk Integration across Asset Classes
8.3
Segmented Asset Allocation and Security Selection
8.4
Integrated Risk Models

155
155
158
159
162

9

Dynamic Volatilities and Correlations
9.1
GARCH Models
9.2
Stochastic Volatility Models
9.3
Time Aggregation

9.4
Downside Correlation
9.5
Option-Implied Volatility
9.6
The Volatility Term Structure at Long Horizons
9.7
Time-Varying Cross-Sectional Dispersion

167
167
178
180
181
184
187
188

6

116

132

10 Portfolio Return Distributions
10.1 Characterizing Return Distributions
10.2 Estimating Return Distributions
10.3 Tail Risk
10.4 Nonlinear Dependence between Asset Returns

191
191
196
203
207

11 Credit Risk
11.1 Agency Ratings and Factor Models of Spread Risk
11.2 Rating Transitions and Default
11.3 Credit Instruments
11.4 Conceptual Approaches to Credit Risk
11.5 Recovery at Default
11.6 Portfolio Credit Models
11.7 The 2007–8 Credit-Liquidity Crisis

212
213
217
218
220
232
232
238

12 Transaction Costs and Liquidity Risk
12.1 Some Basic Terminology
12.2 Measuring Transactions Cost

241
241

246

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ix

12.3 Statistical Properties of Liquidity
12.4 Optimal Trading Strategies and Transaction Costs

261
266

13 Alternative Asset Classes
13.1 Nonsynchronous Pricing and Smoothed Returns
13.2 Time-Varying Risk, Nonlinear Payoff, and Style Drift
13.3 Selection and Survivorship Biases
13.4 Collectibles: Measuring Return and Risk with Infrequent
and Error-Prone Observations
13.5 Summary

271
271
284
291

14 Performance Measurement
14.1 Return-Based Performance Measurement
14.2 Holdings-Based Performance Measurement and Attribution
14.3 Volatility Forecast Evaluation
14.4 Value-at-Risk Hit Rates
14.5 Forecast and Realized Return Densities

299
299
303
309
316
317

15 Conclusion
15.1 Some Key Messages
15.2 Questions for Future Research

319
319
320

References

323

Index

345

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295
298


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Acknowledgments

Our thanks to Patrick Braun, Aaron Brown, David Buckle, Scott Cogswell,
Christopher Culp, Dan Dibartolomeow, Stuart Doole, Janice Eberly,
Ed Fishwick, Kay Giesecke, Rupert Goodwin, Michael Hayes, Ely Klepfish, Jason MacQueen, Charles Cederfeldt-Malinas, David Miles, Guy
Miller, Brian O’Kelly, Andrew Patton, Riccardo Rebonato, Ronnie Sadka,
Bernd Scherer, Alan Scowcroft, James Sefton, Peter Shepard, David
Tang, Michela Verardo, and Tim Wilding for helpful comments on the
manuscript, and to Jonathan Brogaard, Pooja Kesavan, Anu Kulkarni,
Sharad Prakash, Zhigang Qiu, Terence Teo, Yi Yu, and Alminas Zaldokas
for excellent research assistance and for their infectious enthusiasm for
financial research. Lisa Goldberg is grateful to her colleagues at MSCI
Barra for their insights and invaluable contributions to this book.
We thank Richard Baggaley of Princeton University Press, our editor,
for constant support and feedback on this project. We also thank Sam

Clark, at T&T Productions Ltd, for converting our disparate TEX code into
a finished product. We also thank Laden Gehring and Stephanie Winters,
of MSCI Barra, for assistance with graphics.
We would like to thank MSCI Barra for its generous support of this
project through a donation to the Financial Markets Group at London
School of Economics. We also thank Northfield Information Services and
UBS Inc., who sponsored and helped organize a Portfolio Risk Forecasting Workshop at the Financial Markets Group (in spring 2006) on the
topic of this book. Gregory Connor wishes to acknowledge support from
the Science Foundation of Ireland under grant 08/SRC/FM1389. Robert
Korajczyk wishes to acknowledge the research support of the Zell Center
for Risk Research and the Jerome Kenney Fund.

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Introduction

This book provides a quantitative, technical treatment of portfolio risk
analysis with a focus on real-world applications. It is intended for
both academic and practitioner audiences, and it draws its inspiration and ideas from both the academic and practitioner research literature. Quantitative modeling for portfolio risk management is an
active research field. Virtually all institutional investment management

firms use quantitative models as an integral part of their portfolio riskmanagement procedures. Research and development on these models
takes place at investment management firms, brokerage houses, investment consultants, and risk-management software providers. Academic
researchers have explored the econometric foundations of portfolio risk
analysis models, the relative performance of the various types of models, the implications of the various models for the understanding of
capital market behavior, and the models’ implications for asset pricing equilibrium. This book attempts to synthesize the academic and
practitioner research in this field. We argue that portfolio risk analysis requires a balanced, multidisciplinary perspective combining statistical modeling, finance theory, microeconomics, macroeconomics, and a
behavioral–institutional understanding of modern capital markets.

Who Should Read This Book?
Among practitioners, an ideal reader of this book would be someone
with a good background in finance and statistics working in the riskmanagement office of an institutional fund manager. He or she1 may be
relying entirely on vendor software for portfolio risk analysis, or entirely
on routines developed in-house, or on a combination of in-house and
vendor products. The book is not a how-to manual for building a portfolio risk analysis model, but it gives the reader a solid understanding
about the many difficult issues and choices in model design and estimation and about current research frontiers in estimating and evaluating these models. Practitioners working in related functions, including
1 To avoid unnecessary verbal clutter, in the remainder of the book we use the male
pronoun for gender-neutral third-person singular.

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xiv

Introduction

portfolio managers, investment consultants, and risk analysis software
providers, are also part of our target audience. As an important caveat,
those working in the central risk-management offices of investment

banks, with responsibility for a full range of trading desks, will find that
this book is not comprehensive, since we do not cover risk analysis for
financial engineering.
Among academics, an ideal reader of this book would be a graduate
student or faculty member interested in portfolio-risk-related research.
Many of the best new ideas in academic finance come from the interface with business practice. The book might be suitable as a main
or secondary text in an advanced master’s level course on portfolio
management or financial risk management.

Topics Covered
In this section we discuss some broad topics that are included or
excluded from the book and then briefly describe the content of each
chapter.
Managerial, Accounting, and Regulatory Issues
The main objective of the modeling techniques described in this book is
to provide accurate, reliable portfolio risk analysis and forecasts. Portfolio risk analysis serves as a crucial input into a range of managerial
and regulatory decisions, but we do not address, except peripherally, the
broader policy problems of financial risk management and regulation.
We also do not address accounting and disclosure issues.
Portfolio Risk Analysis versus Portfolio Management
The book is addressed to the problems of portfolio risk analysis rather
than the broader problems of portfolio management. There are two
main reasons for this. First, the methods and techniques for portfolio
risk analysis are distinct from those for portfolio management. Second,
almost all institutional managers separate the portfolio management
function from the risk analysis function; there are very strong business
and regulatory reasons for this separation. Many of the worst scandals
in the investment management industry occurred when this functional
separation was breached or inadequately monitored. We touch upon
portfolio management issues outside the risk analysis domain, such as

asset valuation, security selection, and trading strategies, whenever it is
illuminating for our main focus.
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Introduction

xv

Portfolio versus Asset Risk Analysis
Consider a set of n asset returns r and a set of portfolio weights w.
Let rw denote the portfolio return associated with this particular set
of portfolio weights. If we know that the portfolio weights are permanently fixed, then rw can be treated as if it were a single asset return.
We can analyze the risk of rw without analyzing the numerous risk relationships among all the constituent asset returns. The focus of the book
is on portfolio risk analysis, by which we mean either that we do not
know the portfolio weights w when we build the risk analysis model or
we are allowing w to vary. We need to create a risk analysis model for
potential portfolios rw that will be accurate across a wide range of portfolio weight vectors w. This requires a risk model of all n asset returns
r and their interdependencies, rather than just a model of a particular
portfolio return rw .
Financial Engineering
Financial engineering and derivatives-trading strategies pose especially
difficult problems for risk analysis. Financial engineers are willing and
able to trade, at an acceptable price, virtually any nonlinear function
of any future asset price or price path. This transforms the risk analysis problem into a subdiscipline of financial engineering; the portfolio
aspect diminishes in importance. There are many high-quality texts on
risk analysis for financial engineering and derivatives trading;2 we do not
attempt to cover this material. Risk managers oriented toward derivatives will find that this book covers only one part of their problem: risk

analysis for portfolios of primary assets not including any financially
engineered securities.3 This is an important component of risk analysis
for financial engineering, but it is not the whole story.
Limits to Coverage of Research Literature
Although the general perspective in this book is strongly empirical, we
generally do not attempt to replicate the vast array of empirical findings on portfolio risk analysis. Instead, we provide references to empirical findings in the research literature and incorporate them into our
analysis without attempting to re-derive them from raw data. We provide empirical illustrations when it seems particularly illuminating to do
so, to underscore major conceptual points in the analysis or to highlight
counterintuitive empirical findings.
2 Including, for example, Christoffersen (2003), Crouhy et al. (2001), Dowd (2005), and
Jorion (2007).
3 We allow for the simplest types of derivatives overlays such as equity index futures.

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xvi

Introduction

Chapter Summaries
Chapter 1 sets out basic measures of return and risk. It compares arithmetic and logarithmic returns and discusses the relative advantages of
each. It introduces the key measures of portfolio risk including variance, value-at-risk, and expected shortfall. It discusses the objectives of
portfolio risk management and its limitations.
Chapter 2 examines the estimation and use of unstructured return
covariance matrices. It discusses the problem of estimation error in
covariance matrices and the implications for their use in portfolio
management. Chapter 3 examines industry–country component models. In these simple models the cross section of returns is divided into

industry-related returns, country-related returns, and asset-specific (neither country- nor industry-related) returns. This simple decomposition
is surprisingly powerful in explaining the common components in the
cross section of equity returns, and also has growing relevance for
corporate bond markets as they broaden and deepen internationally.
Factor models of security returns are typically categorized as statistical, economic, or characteristic-based. Chapter 4 describes factor models of security returns and discusses statistical factor analysis. Chapter 5
deals with macroeconomic factor models in which the pervasive factors
in returns are observable economic time series, such as output, inflation,
and interest rate changes. Chapter 6 treats characteristic-based factor
models, in which the factor sensitivities of assets are tied to the corporate characteristics and/or cash flow characteristics of assets. Not all
factor models fit neatly into one of these three categories. We also discuss “hybrid” factor models, which have features from more than one of
these pure types. Chapter 7 analyzes foreign exchange risk.
An important design choice in risk model construction is the approach
to integration of risk analysis across asset classes and across national
borders. This problem of risk model integration is considered in chapter 8. This chapter also surveys research on the level and trend of crossborder capital market integration, which has obvious relevance for the
choice of integrated versus segregated risk modeling.
Due to its analytical convenience, the first eight chapters of the book
have an emphasis on unconditional portfolio return variance as a risk
measure. Chapters 9–14 broaden the perspective, using many other risk
metrics besides variance, and explicitly account for risk dynamics. Chapter 9 explores models of dynamic volatility and dynamic correlations,
and the choice of forecast horizon. Chapter 10 considers density estimation and the related problem of tail estimation and value-at-risk measures. Chapter 11 discusses credit risk, and chapter 12 liquidity risk.
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Introduction

xvii

Chapter 13 looks at risk analysis for alternative asset classes such as

hedge funds, venture capital, and commodities. Chapter 14 deals with
the performance evaluation of portfolio risk–return realizations and
also the performance evaluation of portfolio risk-forecasting models.
Chapter 15 provides a brief conclusion.

Useful Background
Understanding the material in the book requires at least an intermediatelevel background in statistics, linear algebra, and finance theory. Some
sections require more advanced knowledge in statistics or finance
theory. For those who wish to refresh their knowledge or study these topics independently, we suggest some appropriate finance and statistics
texts.
Greene (2008) provides a solid foundation in statistics and econometrics appropriate for understanding the material covered in this book.
Readers wanting a finance-focused treatment at a more advanced level
may benefit from reading Campbell et al. (1997). For general finance
background, Bodie et al. (2009) and Elton et al. (2010) are two possible
references.

Approximations Used in the Book
Statistical approximations are very important in portfolio risk analysis. Diversification is a key principle of portfolio management, and by
its nature it relies on statistical approximations. There are also useful
approximations as the chosen return interval becomes short or as the
sample used for risk model estimation becomes large.
We use a simple common notation for the different types of approximations used in the book. Most of the approximations rely on one of
three limiting variables: either the number of assets n, the number of
time periods T , or the return measurement interval ∆ (monthly, weekly,
daily, hourly, etc.). We take the limiting approximation for large n, or
large T , or for small ∆, holding all other variables constant. Which of
these three limiting variables is being used in the approximation is
indicated by a superscript on the approximately equals symbol:
n

≈,

T

≈,

∆

≈.

If f and g go to zero with ∆, but the difference goes to zero more quickly,
in the sense that
f −g ∆
≈ 0,
∆
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xviii

Introduction

we write
o(∆)

f ≈ g,
meaning that f − g goes to zero relative to the magnitude, or “order,”
of ∆. This is useful if we are looking at two returns over a short time

interval and want to say that the returns are approximately the same,
even though both are approximately zero. The symbol “≈” has the standard definition from introductory calculus; those who received at least
a “B” in their introductory calculus course do not need to read this long
footnote, while those who received a “C” or worse will not want to, but
we include it anyway for completeness.4
We also rely on the two basic statistical approximations: limit in probability and limit in distribution. We add the superscript “pr” for limit
ˆ
in probability and “di” for limit in distribution. So, for example, let m
denote the sample mean from T independent observations of a random
variable with a true mean of zero, a variance of one, and finite higher
moments. Then
pr,T
ˆ ≈ 0
m
is our notation for the law of large numbers (the sample mean approaches zero in probability) and
di,T

ˆ ≈ N(0, 1)
(T 1/2 )m
is our notation for the central limit theorem (the sample mean scaled by
the square root of the number of observations is approximately normal
in its distribution). Readers not familiar with these standard statistical
approximations are referred to Greene (2008, appendix D) or any standard statistics textbook. In a few isolated places we use approximations
∗
that do not fit neatly into these simple categories; we use the notation ≈
in these cases, and give references outside the text.

4 Recall

from introductory calculus that

T

f ≈a
means that for any
> 0 there exists a T ∗ such that |f (T ) − a| <
for all T > T ∗ ;
n

f ≈a
has the analogous definition with n replacing T as the limiting variable. Approximations
based on the return interval differ in that the limiting approximation relies on a suitably
small (rather than suitably large) value of the limiting variable:
∆

f ≈a
means that for any
> 0 there exists a δ such that |f (∆) − a| <
for all ∆ < δ.

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Key Notation

This section gives some of the key notation that we use throughout the
text. It is not a comprehensive list; it covers only the most commonly
used symbols in the book. We use bold font for vectors and matrices
and regular font for real values; so for example r is an individual asset
or portfolio return and r is a vector of asset returns.
0n×k
1

n

an n × k matrix of zeros
an n-vector of ones

arg max

in an optimization problem this denotes the
value of the choice variable that gives a
maximum of the objective function

B

the n × k matrix of the assets’ exposures to
the factors

C

the n × n matrix of return covariances

Cf

the k × k matrix of factor return covariances

Cε

the n × n matrix of asset-specific return
covariances

cum(·)

the cumulative distribution function of an asset
or portfolio return; hence cum(a) = Pr(r
a)

cumloss (·)

the cumulative distribution function of an asset
or portfolio loss; cumloss (a) = Pr(−r
a)

den(·)

the density function of an asset or portfolio return

Diag[a1 , . . . , an ]

an n × n diagonal matrix with elements a1 , . . . , an
along the diagonal

E[·]

the expectation operator for a random variable

E ∗ [·]

the expectation operator for a random variable
under the risk-neutral probability measure (see
chapter 1 for a discussion)

ES(1 − α)

the expected shortfall of an asset or portfolio,
with confidence level 1 − α (see chapter 1 for
a description)
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xx

Key Notation

f

the k vector of factor returns

loss

minus the return on an asset or portfolio

L(·)

the likelihood function of a sample of data for a set
of estimated parameters (see chapter 1 for details)

n

the number of assets in the investment universe

pi

the price of asset i

Pr(·)

the probability function of a set of events;
so for example Pr(ri > r0 ) is the probability
that the return to asset i, ri , is larger than
the riskless return, r0

Q(α)

the αth quantile of the cumulative probability
distribution of a random return

r

the return on an individual asset or portfolio

r

the n-vector of asset returns

R

the n × T matrix of returns on n assets for a
sample period of T periods

r0

the riskless return

rl

the logarithmic return on an asset or portfolio

rw

the return on a portfolio w

t

any particular time period

T

the total number of time periods; also
the last time period, as in t = 1, T , or
in continuous time t ∈ [0, T ]

VaR(1 − α)

the value-at-risk of an asset or portfolio with
confidence level 1 − α (see chapter 1 for a
detailed description)

w

the n-vector of portfolio weights

x

the excess return on an individual asset or
portfolio (return minus the riskless return)

x

the n-vector of asset excess returns

X

the n × T matrix of excess returns on n assets
for a sample period of T periods

∆

a small discrete unit of time
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Key Notation

xxi

ε

the n-vector of asset-specific or nonmarket returns

µ

the n-vector of expected returns

σ2

the variance of a random variable; correspondingly
σ is the standard deviation of a random variable

n

≈
T

≈
pr,n

≈

pr,T

≈

di,n

≈

di,T

≈

o(∆)

≈

∗

approximately equal for large n
(see the introduction for a discussion)
approximately equal for large T
(see the introduction for a discussion)
the probability limit for large n
(see the introduction for a discussion)
the probability limit for large T
(see the introduction for a discussion)
the distribution limit for large n
(see the introduction for a discussion)
the distribution limit for large T
(see the introduction for a discussion)
o(∆)

of smaller order, so that a ≈ b means that a − b
goes to zero relative to the magnitude of ∆ for
small ∆ (see the introduction for a discussion)

≈

approximately equal using some approximation
measure other than the five shown above
(see the introduction for a discussion)

∼ N(µ, σ 2 )

denotes that a random variable is normally
distributed with mean µ and variance σ 2



the transpose of a vector or matrix;
so for example if a and b are n-vectors,
then a b is the inner product of the vectors

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Portfolio Risk Analysis

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