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BarraRiskModel.pdf
About Barra
A Pioneer in Risk Management
Contacting Barra
Other Barra Resources
Introduction
Further references
1. Forecasting Risk with Multiple- Factor Models
What Are Multiple-Factor Models?
How Do Multiple-Factor Models Work?
Advantages of Multiple-Factor Models
An Illustration of Multiple-Factor Models
Model Mathematics
Risk Prediction with MFMs
Summary
2. Forecasting Equity Risk
A Historical Perspective
Barra’s Equity Multiple-Factor Model
Common Factors
Specific Risk
3. Barra Equity Risk Modeling
Model Estimation Overview
Data Acquisition
Descriptor Selection and Testing
Descriptor Standardization
Risk Index Formulation
Industry Allocation
Factor Return Estimation
Covariance Matrix Calculation
Specific Risk Modeling
Updating the Model
4. Forecasting Fixed-Income Risk
A Historical Perspective
Barra’s Multiple-Factor Model
Common Factors
Specific Risk
Summary
5. Interest Rate Risk Modeling
Estimation Process Overview
Term Structure Specification
Factor Shape Determination
Factor Exposure Calculation
Factor Return Estimation
Term Structure Covariance Matrix Construction
Covariance Matrix Rotation
Updating the Model
6. Spread Risk Modeling
Swap Spread Risk Model
Detailed Credit Spread Risk Model
Emerging-Market Risk Modeling
Updating the Model
7. Specific Risk Modeling
Heuristic Models
Transition-Matrix-Based Model
Updating the Model
8. Currency Risk Modeling
Model Structure
Data Acquisition and Return Calculation
Estimation of the Covariance Matrix
Time-Scaling Currency Risk Forecasts
Updating the Model
9. Integrated Risk Modeling
Model Integration Overview
Building Global Asset Class Models
Global Equities
Global Bonds
The Currency Model
Putting It All Together-A Multi-Asset Class Risk Model
Summary
Glossary
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
U
V
W
Y
Z
Contributors
Index
Barra Risk Model
Barra Risk Model
Handbook
Copyright © 2004 Barra, Inc and/or its subsidiaries and affiliates (“Barra”). All rights reserved. This publication shall be used only in accordance with the terms of
your license agreement with Barra, but in any event, only for your internal business use.This publication is the confidential and proprietary property of Barra. No
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erenced in this publication are used for identification purposes only and may be trademarks of their respective owners.
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HEREIN (AND ANY RESULTS TO BE OBTAINED FROM THE USE THEREOF), INCLUDING BUT NOT LIMITED TO ALL WARRANTIES OF MERCHANTABILITY, FITNESS
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sory services.
RV 03-2004
Contents
About Barra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
A Pioneer in Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Contacting Barra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vi
Other Barra Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vi
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1. Forecasting Risk with Multiple-Factor Models . . . . . . . . . . . .1
What Are Multiple-Factor Models? . . . . . . . . . . . . . . . . . . . . . . . . 1
How Do Multiple-Factor Models Work? . . . . . . . . . . . . . . . . . . . . 2
Advantages of Multiple-Factor Models. . . . . . . . . . . . . . . . . . . . . . 2
An Illustration of Multiple-Factor Models . . . . . . . . . . . . . . . . . . . 3
Model Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Single-Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Multiple-Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Multiple-Asset Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Risk Prediction with MFMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
The Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Deriving the Variance-Covariance Matrix of Asset Returns. . . 10
Final Risk Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2. Forecasting Equity Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
A Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Barra’s Equity Multiple-Factor Model . . . . . . . . . . . . . . . . . . . . . 19
Common Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Risk Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Industries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Specific Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Contents
i
3. Barra Equity Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . 21
Model Estimation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Descriptor Selection and Testing . . . . . . . . . . . . . . . . . . . . . . . . . 24
Descriptor Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Risk Index Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Industry Allocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Factor Return Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Covariance Matrix Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Exponential Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Covariance Matrix Scaling: Computing Market Volatility . . . 29
Specific Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Updating the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4. Forecasting Fixed-Income Risk . . . . . . . . . . . . . . . . . . . . . . 39
A Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Barra’s Multiple-Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Common Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Interest Rate Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Spread Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Specific Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.
Interest Rate Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . 51
Estimation Process Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Term Structure Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Estimation Algorithm Implementation . . . . . . . . . . . . . . . . . 58
Factor Shape Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Factor Exposure Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Factor Return Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Term Structure Covariance Matrix Construction . . . . . . . . . . . . . 66
Covariance Matrix Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Updating the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
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6. Spread Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71
Swap Spread Risk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Data Acquisition and Factor Return Estimation. . . . . . . . . . . 72
Factor Exposure Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 72
Detailed Credit Spread Risk Model . . . . . . . . . . . . . . . . . . . . . . . 74
Currency Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Factor Return Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Covariance Matrix Estimation. . . . . . . . . . . . . . . . . . . . . . . . 79
Factor Exposure Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 80
Emerging-Market Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . 80
Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Data Acquisition and Factor Return Estimation. . . . . . . . . . . 82
Covariance Matrix Estimation. . . . . . . . . . . . . . . . . . . . . . . . 82
Factor Exposure Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 83
Updating the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7. Specific Risk Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
Heuristic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Sovereign, U.S. Agency, and MBS Specific Risk Estimation . . 86
Corporate Bond Specific Risk Estimation . . . . . . . . . . . . . . . 87
Transition-Matrix-Based Model. . . . . . . . . . . . . . . . . . . . . . . . . . 88
Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Transition Matrix Generation . . . . . . . . . . . . . . . . . . . . . . . . 89
Rating Spread Level Calculation . . . . . . . . . . . . . . . . . . . . . . 90
Credit Migration Forecasting . . . . . . . . . . . . . . . . . . . . . . . . 93
Updating the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
8. Currency Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .99
Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Data Acquisition and Return Calculation . . . . . . . . . . . . . . . . . 100
Estimation of the Covariance Matrix . . . . . . . . . . . . . . . . . . . . . 100
Currency Correlation Model . . . . . . . . . . . . . . . . . . . . . . . . 101
Currency Volatility Model. . . . . . . . . . . . . . . . . . . . . . . . . . 102
Contents
iii
Volatility Across Markets . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Time-Scaling Currency Risk Forecasts . . . . . . . . . . . . . . . . . . . . 107
Updating the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.
Integrated Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 111
Model Integration Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Building Global Asset Class Models . . . . . . . . . . . . . . . . . . . . . 112
The Structure of Local Models . . . . . . . . . . . . . . . . . . . . . . 113
Aggregating Local Models . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Implementing Global Factor Models . . . . . . . . . . . . . . . . . . 115
Consistency Between Local Models and Global Model . . . . 116
Global Equities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Global Equity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Exposures of Local Equity Factors to Global Equity Factors . 117
Estimating Returns to Global Equity Factors . . . . . . . . . . . . 117
Computing Covariances of Global Equity Factors . . . . . . . . 119
Scaling to Local Markets. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Global Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Global Bond Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Exposures of Local Bond Factors to Global Bond Factors. . . 124
Computing Covariances Of Global Bond Factors . . . . . . . . 124
The Currency Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Putting It All Together—A Multi-Asset Class Risk Model . . . . . 127
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
iv
Barra Risk Model
Handbook
About Barra
In recent years the investment management industry has adjusted
to continuing changes—theoretical advances, technological devel-
opments, and volatility. To address these, investment managers
and financial institutions require the most advanced and powerful
analytical tools available.
A Pioneer in Risk Management
As the leading provider of global investment decision support
tools and innovative risk management technology, Barra has
responded to these industry changes by providing quantitative
products and services that are both flexible and efficient.
Barra products are a combination of advanced technology and
superior analytics, research, models, and data that provide clients
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Barra uses the best data available to develop econometric financial
models. In turn, these models are the basis of software products
designed to enhance portfolio performance through returns fore-
casting, risk analysis, portfolio construction, transaction cost anal-
ysis, and historical performance attribution.
With more than 80 researchers in offices around the world and
products that cover most of the world’s traded securities, Barra
maintains one of the strongest risk management research practices
in the world today.
About Barra
v
Contacting Barra
Barra support analysts are in Berkeley, London, and Hong Kong.
Together they form the Global Support Desk, which is available
24 hours a day, Monday through Friday. You can reach the Global
Support Desk as follows:
Email:
supportdesk@barra.com
Phone: North America:
888.588.4567
Europe:
Asia:
+44 (0)20.7618.2222
+81.3.5402.4151 (Japanese)
+81.3.5402.4151 (English)
For local access numbers, visit the Client Support web site,
http://support.barra.com.
The Global Support Desk is your first point of contact concern-
ing your Barra product or service. Support desk analysts are avail-
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model questions.
For suggestions or comments regarding this documentation, send
your e-mails to publications@barra.com.
Other Barra Resources
You can visit the Library at http://support.barra.com for more
information on the topics discussed in this handbook.
In addition to handbooks and reference guides, Barra offers
numerous workshops and seminars throughout the year. For more
information, visit the Events Calendar at http://www.barra.com.
vi
Barra Risk Model
Handbook
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