www.factset.com
Multi-Asset Class (MAC) III
Risk Model:
Axioma AXWW4 Equity
Ivan Mitov,
Director of Risk Research
Tom Davis,
Director of Fixed Income and Derivatives Research
Viviana Vieli,
Director of Model Validation
Chris Desmarais,
Associate Director of Risk Product Management
Jevgenij Kusakovskij,
Associate Director of Risk Research
Artis Svilans,
Senior Quantitative Researcher
Jason Levine,
Senior Quantitative Researcher
White Paper
Multi-Asset Class (MAC) III Risk Model:
Axioma AXWW4 Equity
Ivan Mitov
Director of Risk Research
ivan.mitov@factset.com
Tom Davis
Director of Fixed Income and Derivatives Research
todavis@factset.com
Viviana Vieli
Director of Model Validation
vvieli@factset.com
Chris Desmarais
Associate Director of Risk Product Management
cdesmarais@factset.com
Jevgenij Kusakovskij
Associate Director of Risk Research
jevgenij.kusakovskij@factset.com
Artis Svilans
Senior Quantitative Researcher
artis.svilans@factset.com
Jason Levine
Senior Quantitative Researcher
jason.levine@factset.com
February 21, 2024
Copyright © 2024 FactSet Research Systems Inc. All rights reserved. FactSet Business Use Only
www.factset.com | 1
White Paper
Contents
Introduction
5
Model Overview
7
Risk Models
1 Equity Model . . . . . . . . . . . . . .
1.1 Estimation Process . . . . . . . . .
1.2 Coverage and Estimation Universe .
1.3 Factors . . . . . . . . . . . . . . .
Market-Based Factors . . . . . . . . .
Fundamental Factors . . . . . . . . .
Industry Factors . . . . . . . . . . .
Global Market Factor . . . . . . . . .
Country Factors . . . . . . . . . . .
Currency Factors . . . . . . . . . . .
Local Factors . . . . . . . . . . . . .
1.4 Specific Risks . . . . . . . . . . . .
2 Fixed Income Model . . . . . . . . . .
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Euro-Sovereign Bonds . . . . . . . . . . . . . .
U.S. Municipal Bonds . . . . . . . . . . . . . .
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2.1 FI Pricing Methodology and Risk Factor Definitions
2.2 Interest Rate Risk . . . . . . . . . . . . . . . . .
2.3 Spread Risk . . . . . . . . . . . . . . . . . . . . .
Spread Model Methodology Overview
Sovereign, Municipal, and Quasi-Government Bonds
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Developed Market Supranational, Sub-Sovereign, and Agency Bonds
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Developed Market Corporate Bonds . . . . . . . . .
Local Currency Market Corporate Bonds . . . . . . .
Emerging Market Bonds Denominated in USD and EUR .
Structured Credit Products . . . . . . . . . . . . . . .
Corporate Bonds
U.S. Mortgage-Backed and Mortgage-Related Securities
U.S. Asset-Backed Securities
Non-U.S. ABS and CMBS
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2.4 Inflation-Linked Bonds .
2.5 Convertible Bonds . . .
3 Commodity Model . . . . .
4 Private Equity Model . . .
4.1 Fund Universe . . . . .
Japanese MBS
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Copyright © 2024 FactSet Research Systems Inc. All rights reserved. FactSet Business Use Only
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www.factset.com | 2
White Paper
4.2 Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
Necessary Complications .
4.3 Results . . . . . . . . .
5 Derivatives . . . . . . . . .
5.1 Equity Derivatives . . .
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3 Parametric Risk Measures . . . . . . . . . . .
3.1 Portfolio Volatility (TEV) . . . . . . . . .
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4 Scenario Analysis . . . . . . . . .
4.1 Factor Stress Testing . . . . .
4.2 Extreme Event Stress Testing .
4.3 Extreme Event Simulation . .
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Structure Equation
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The Impact of Options on Return Distributions
. . . . . .
Equity and Equity Index Futures .
The Equity Volatility Surface . . .
5.2 Fixed Income Derivatives . . . .
Eurodollar Futures (EDFs) . . . .
Government Bond Futures . . . .
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Evaluation of Risk
1 Covariance Matrix Computation . . . .
1.1 EWMA Model . . . . . . . . . . . .
1.2 Covariance Matrix Regularization . .
2 Monte Carlo Risk Measures . . . . . . .
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Equity Option Pricing
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Option on Government Bond Futures
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Interest Rate Swap . . . . . . .
Swaption . . . . . . . . . . . .
Interest Rate Cap and Floor . .
Callable Bond . . . . . . . . .
Option on Bond
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Credit Default Swaps, Indices, and Swaptions
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Credit Default Index Option .
Currency Forward . . . . . . .
Currency Futures . . . . . . . .
Currency Options . . . . . . .
6 Return-Based Model . . . . . . .
Credit Default Index
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2.1 Simulating Distribution of Portfolio Returns
2.2 Estimating Portfolio Risk Measures . . . .
2.3 Marginal Contributions to Risk . . . . . . .
3.2 Marginal TEV and Factor Contributions to TEV
3.3 Value-at-Risk and Expected Tail Loss . . . . . .
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Example of Risk Report
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Conclusion
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Bibliography
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Appendix
73
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1 Principal Component Analysis of Yield Curves
2 Spread Model Types . . . . . . . . . . . . . . .
2.1 Relative Spread Change Models . . . . . . .
2.2 Specialized Linear Spread Models . . . . . .
2.3 Contingent Claim Analysis Model . . . . . .
3 Fixed Income Spread Model Factors . . . . . .
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3.1 Sovereign, Municipal, and Quasi-Government Bonds .
Euro-Sovereign Bonds . . . . . . . . . . . . . . . . .
U.S. Municipal Bonds . . . . . . . . . . . . . . . . . .
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U.S. Mortgage-Backed Securities and Mortgage-Related Securities .
ABS, Non-U.S. CMBS, and Japanese MBS . . . . . . . . . . . .
4 Relative Spread Change Model Factor Groups . . . . . . . . . .
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5 Return-Based Model Factors
5.1 Alternative Investments . .
5.2 Equity . . . . . . . . . . .
5.3 Real Estate . . . . . . . .
5.4 Fixed Income . . . . . . .
5.5 Commodities . . . . . . .
5.6 Currencies . . . . . . . . .
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Developed Market Supranational, Sub-Sovereign, and Agency Bonds
3.2 Corporate Bonds . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
Europe . . . . . . . . . . .
Japan and Australia . . . .
Local Currency Markets . .
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3.3 Emerging Market Bonds Denominated in USD and EUR .
3.4 Structured Credit Products . . . . . . . . . . . . . . . . .
U.S. and Canada
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4.1 Revere Business and Industry Classification System Sector and Industry Factor Definitions
4.2 Emerging Market Regional Factor Definitions . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
A growing number of investment managers today regard the function of the enterprise risk management
system as moving above and beyond simple compliance into the heart of the investment process. Risk models
are no longer simply used to monitor risk, but to actually drive portfolio performance. They are used for
risk forecasting (as forward-looking tools to predict future volatility), risk analysis (as tools to understand
current risk exposures), and risk attribution (as guides to past performance, evaluating actual returns in
relation to risk exposures of the portfolio).
A typical portfolio contains hundreds or thousands of securities, and quantifying the risks of each portfolio
holding and their complex interplay can be quite a daunting task. Factor risk models provide portfolio
managers with a convenient way of reducing this complexity to a manageable set of precise market risk
exposures. They supply a detailed understanding of the sources of risk in each portfolio, help portfolio
managers decompose risk into contributions from well-defined market factors, and quantify correlations
between these contributions. Factor risk models are indispensable for understanding the contributions of
different portfolio components to total risk and for measuring and controlling portfolio risk exposures.
A typical factor risk model represents the return of every security in the market as a function of a limited,
predefined set of factors. It could be a simple linear function, in which case we have a linear model that is
simple to implement but that does not necessarily capture the full behavior of complex financial instruments,
especially in the tails of the return distribution. Alternatively, the model may compute the return of each
instrument as a nonlinear function of the factors; for example, when the model is based on full and exact
repricing of the instruments or a nonlinear interaction between factors and sources of valuation, as in the case
of structural spread models. Such nonlinear models will be more difficult to implement and will require more
computational power to run, but they will provide more precise estimates of tail and complex derivatives
risk.
In both cases, the risk factor model estimates the portfolio’s risk characteristics using a covariance matrix of
factor returns. For most portfolios, the number of factors in the model is much smaller than the number of
securities in the portfolio. Thus, the factor covariance matrix requires fewer parameters to estimate, leading
to a much more stable and robust estimation of portfolio risks. Moreover, the factors can be designed in such
a way as to increase the stability of the covariance matrix and the quality of the risk estimators.
The model’s factors are designed, on the one hand, to capture systematic market fluctuation and, on the
other hand, to reveal a detailed structure of the market exposures of each portfolio. For example, a single
interest rate factor would capture the general movement of the interest rate curve but not the details of
the portfolio exposures to the short and long ends of the curve or the slope of the curve. In this case, it is
evident that more interest rate risk factors are required to quantify the portfolio exposure to these risks. At
the same time, the total number of interest rate factors should remain limited to keep the dimensionality of
the problem under control. Thus, factor risk model construction is, among other things, an exercise in the
optimal choice of the set of risk factors.
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Multi-asset class risk factor models are especially difficult because of the complexity of calculating risks across
different asset types. However, they are becoming absolutely necessary now as more investment managers
operate on global, multi-asset type markets. Historically, the risks of such portfolios have been estimated by
independently modeling risks of holdings of each asset type, like equities or fixed income, and then aggregating
those risks into a multi-asset class portfolio risk. This can result in significant errors in estimated risk because
this methodology does not take into account correlations between different asset classes. A multi-asset class
risk factor model that is based on a set of factors that capture systematic shifts in all markets (equities, fixed
income, commodities, etc.) remedies this deficiency and allows the portfolio manager to accurately estimate
portfolio risk and its components stemming from different asset type markets.
In this paper, we discuss the structure of the FactSet multi-asset class (MAC) factor risk framework and
the types of factors used in its models, and describe estimation techniques used to compute risk factors and
the corresponding security returns. We also describe the methodology behind the computation of various
portfolio risk measures and their components, and illustrate the application of these measures to portfolio
risk analysis.
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Model Overview
The FactSet multi-asset class (MAC) risk framework is a set of tools that investors can utilize to estimate,
monitor, and control the exposure of their portfolios to market risk (either on an absolute basis or relative to
a benchmark). The framework is based on fitting and forecasting the joint distribution of the future portfolio
returns and allows for the calculation of various risk statistics such as Tracking Error Volatility (TEV), Valueat-Risk (VaR), and Expected Tail Loss (ETL), as well as other characteristics of the return distribution (e.g.,
kurtosis, skewness). The framework covers two distinct approaches for risk estimation—Monte Carlo based
and parametric. The Monte Carlo MAC model (MC MAC) employs Monte Carlo simulations from the fitted
risk factors’ multivariate probability distribution and precise pricing methods for highly nonlinear securities
such as derivatives. The Linear MAC model (LMAC) employs a parametric (closed-form) solution for the
portfolio risk based on the parametric form and the properties of this distribution. As such, LMAC is more
suitable for portfolios that are mostly linear with respect to the risk factors.1
The MAC framework consists of the following components:
• A set of factor models that capture different common systematic components of returns across different
asset classes.
• A set of models that forecast the volatilities and correlations of risk factors. The models are used to
construct the forecast for the joint distribution of the factors.
• Frameworks for estimating portfolio risk metrics (e.g., total VaR with its factor and security contributions) from Monte Carlo simulations and pricing algorithms (MC MAC) or closed-form solutions
(LMAC).
• FactSet’s flexible reporting platform, which allows for detailed reporting of the results of risk computations as well as tight integration of portfolios and data to additional FactSet analytic systems.
• A stress testing module, which lets users perform both historical stress tests and hypothetical factor
shocks using any of FactSet’s extensive data libraries.
Traded securities are exposed to various types of risk, and factor risk models are estimated by identifying
common systematic sources of these risks. There are multiple ways of defining the systematic risk factors.
For example, one can use existing economic or financial time series as common risk drivers. In this case, the
sensitivities of each security to these factors are estimated using time series regression of the security returns
on factor returns. Alternatively, one can first define the factor sensitivities; for example, as dummy variables
having values of either 0 or 1 or as in fixed income models, using the analytics calculated from pricing
models such as duration or convexity. In this case, the factor returns are estimated using a cross-sectional
regression of security returns on corresponding sensitivities. FactSet employs both methods to construct its
set of multi-asset class factor models.
1 That
is why there is a coverage difference between the two models.
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The FactSet MAC framework uses separate factor risk models for equities, fixed income, commodities, and
alternative assets and combines them into one multi-asset class model by modeling and forecasting the joint
distribution of all the risk factors. Each risk model is described in detail in the Risk Models chapter of the
paper. The public equity model is a currency-invariant linear risk model that decomposes stock return risk
into effects of style, industry, region, market, and possibly other factor types. This component is modular
in the framework, as different models of equal explanatory power may offer different perspectives on the
risk of the asset class. Different perspectives suit different investment objectives, so this flexibility is highly
desirable. Particular details of this component can be found in Section 1, Risk Models chapter.
The fixed income (FI) model, described in Section 2 of the Risk Models chapter, is also a fundamental factor
model that uses pricing models for different types of FI securities available in FactSet as its foundation.
The pricing models precompute a set of analytics that are used as sensitivities to the FI risk factors. The
model is based on a general decomposition of the return of each FI security into the part driven by the yield
curve movements and the part defined by the changes of the option adjusted spread (OAS) of the security.
(For particular asset classes, additional terms are considered in the expansion.) Thus the FI risk model can
usually be represented as a combination of several submodels—the yield curve (interest rate risk) model that
applies to all FI securities and a number of spread models, each specific to a particular asset class.
The interest rate risk component is calculated using a number of key rates along sovereign yield curves (every
day FactSet computes key rate durations of several million FI securities on more than 60 sovereign yield
curves). Meanwhile, the spread risk computation approach varies depending on the type of fixed income
security. The biggest asset types in this market use specialized linear spread models. The models are built
using model selection techniques to accurately capture return dynamics with a small number of intuitive
factors. In the MC MAC model, some of the securities exhibiting spread risk and not being covered by
the aforementioned approach are modeled using a structural credit model (Merton model) that relates the
spread to return and volatility of the underlying equity. These models are described in detail in subsections
of Section 2 of the Risk Models chapter.
The FactSet commodity model (Section 3, Risk Models chapter) is a hybrid linear model that consists of
five cross-sectional factors predicated on short-term momentum measures, along with 10 time-series factors
chosen to be most common across assets. The model factors are estimated using a universe of highly liquid
commodity indexes, continuous front-month futures contracts, ETFs, and mutual funds. First, the style
factors are computed using cross-sectional regression. The residual returns from that regression are used
to estimate sensitivities to the time-series factors that include several FX rates, a set of major commodity
indexes, a volatility index, and the three-month Treasury Bill rate. The model is constructed to be simple
and has a fully transparent methodology that complements the multi-asset class model, and it works well
with balanced portfolios of equities, fixed income, currencies, and commodities.
The FactSet Private Equity model (Section 4, Risk Models chapter) is a linear Bayesian estimation that is
meant to capture well-known empirical features of the private fund market. These include but are not limited
to the J-curve effect, the information lag due to infrequent valuations, and the dependence on the public
equity markets. The model is estimated on fund cash flow data, rather than on aggregate market indexes,
so it is very granular and utilizes the largest set of data points available. The structure equation describing
the components of the model mimics Merton’s Capital Asset Pricing Model, while the estimation process
includes additional peculiarities to account for artifacts in the data set driven by how PE fund accounting
incorporates information more slowly than public markets.
FactSet calculates factor returns and exposures for most models overnight and archives the results, creating a
continually growing database of historical data that is used in risk model parameters estimation and scenario
analysis.
The ultimate goal of every risk model is the ability to accurately forecast the distribution of portfolio returns
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at a given horizon. A factor risk model links the portfolio return distribution to the joint distribution of
factor returns through security pricing algorithms, making the task of forecasting the return distribution
equivalent to the task of forecasting the joint factor distribution. Under the most widely used modeling
assumptions, the factor covariance matrix is sufficient to fully describe the risk factors distribution. The
quality and accuracy of the model covariance matrix define the accuracy of the risk measures produced by
the model, making the factor covariance matrix a cornerstone of every multifactor risk model. The FactSet
MAC risk framework employs the covariance matrix constructed from the historical time series of risk factor
returns of all the framework’s risk models.
The basic problem of forecasting volatility and correlations is that these variables are unobservable and
known to evolve stochastically over time. Any volatility model is always an attempt to forecast a set of
latent variables based on a series of historic observations of other variables, in the case of a risk model—of
observed or computed risk factor returns. MAC framework employs the classical, exponentially weighted
moving average (EWMA) method of covariance matrix estimation that produces stable, slowly varying
portfolio volatility and VaR forecasts. The method is designed as a short-term model, meaning a one-day
to two-month horizon. The details of the implementation of the method are described in Section 1 of the
Evaluation of Risk chapter. The covariance matrix of the MAC model is calculated daily, resulting in a rich
database of covariance matrices that can be used for historical simulations and scenario analysis.
Since the covariance matrix of the model is computed from a limited data set of observed factor time series,
it is almost always noisy and ill-conditioned. The high amount of noise in the matrix will result in spurious
correlations between factors and ultimately will translate into inaccurate risk predictions. In addition, if
the number of time observations is less than the number of factors in the model (which is possible for
large multi-asset class models), the matrix will be “rank deficient,” meaning that it is possible to use it to
construct apparently riskless portfolios. In practice, such a matrix cannot be used to construct Monte Carlo
simulations of the factor distribution and will result in model failure. As a result, applying a regularization
procedure to every covariance matrix estimator designed to remedy these deficiencies is always necessary. The
methodology employed in the FactSet MAC framework to construct a robust, invertible, and positive definite
sample covariance matrix is described in Section 1.2 of the Evaluation of Risk chapter. The methodology
is based on random matrix theory and employs spectral decomposition to regularize the distribution of the
matrix eigenvalues and produce a well-conditioned matrix.
Immediate calculations of risk measures for multi-asset class portfolios always present a trade-off between
speed and accuracy. The fastest methods rely on simplifying assumptions that a portfolio’s value changes
linearly with changes in market risk factors. Greater realism in measuring changes in portfolio value generally
comes at the price of increased complexity and nonlinearity of the pricing functions. For linear models, the
risk metrics can be computed by simple algebraic manipulations of the factor covariance matrix and vectors
of portfolio weights and factor sensitivities. More realistic, nonlinear pricing functions make it impossible
to use this simple computational method. In contrast, Monte Carlo simulation is applicable with virtually
any model of changes in risk factors and mechanism for determining a portfolio’s value in each market
scenario. In our framework, both representations can be used to compute a set of risk measures (TEV, VaR,
ETL) reflecting contributions of each security and/or factor to the total portfolio risk, as well as the risk
of the portfolio as a whole. The methodologies are described in Sections 2 and 3 of the Evaluation of Risk
chapter.
We should note that while historical data, in general, allows risk models to create reasonable forecasts of
the volatility and correlation of factors (and thus provide the estimates of future portfolio returns), these
are only the estimates of risk. Major market events such as a financial crisis are usually followed by a
period of extremely high volatility and significantly increased and sometimes reversed correlations between
different asset classes. Because correlations and volatilities during these periods are so different from historical
patterns, no risk model can ever predict when such events will occur. Thus, portfolio managers should not
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rely exclusively on risk models to manage the risk of the portfolios. It is always desirable to complement the
risk model analysis with what-if analysis, e.g., scenario analysis that estimates future portfolio returns under
stressed conditions, with extreme values of market volatilities and correlations. Methodologies of various
scenario analysis capabilities of the FactSet risk framework are described in Section 4 of the Evaluation of
Risk chapter.
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Risk Models
1 Equity Model
The public equity components of the framework rely on the cross-sectional Axioma’s World-Wide Fundamental Equity Risk Model SH 4 for risk measurement of equity assets. Factor returns from the equity model
are used in constructing the full covariance matrix used in the MAC framework, using FactSet methodology
to compute covariance terms for both the equity and cross-asset-type block of the matrix. Asset sensitivities
to each factor in the equity model are also extracted to compute risk contributions. The original covariance
matrix from the Axioma equity model is not used.
The equity risk model captures local asset excess returns using five sets of factors: a global market factor,
countries, industries, styles, and local factors. Currency sensitivities are added as a secondary step in the
process of risk estimation. Additional details of the model and the full list of factors can be found in [1].
Hereunder, we only summarize the estimation procedure and provide more details on the last three factor
blocks in respective sections.
1.1 Estimation Process
Constrained robust linear regression using Huber weight function and square-root USD capitalization weights.
Style, industry and country factors are included in the regression. Local factors are estimated via an auxiliary regression on the residuals. The capitalization-weighted industry and country factor returns are each
constrained to sum to zero. Currency factor returns are computed directly from exchange rates.
1.2 Coverage and Estimation Universe
As of 2017, the model covers roughly 42,700 securities (over 77,700 historically). The estimation universe
includes assets with sufficient size and liquidity, using selection criteria similar to those employed by major index providers. More granular, localized rules are also applied on a per-market basis to filter certain exchanges,
asset types, etc. In early 2017, the estimation universe contained 12,700 securities on average.
1.3 Factors
Market-Based Factors
• Market Sensitivity: 1-year weekly beta
• Volatility: 3-month average of absolute returns over cross-sectional standard deviation, orthogonalized
to the Market Sensitivity
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• Short-Term Momentum: Cumulative return over past month
• Liquidity: Natural logarithm of the ratio of 1-month average daily volume and 1-month average market
capitalization, inverse of 3-month Amihud illiquidity ratio, and proportion of returns-traded over last
calendar year
• Exchange Rate Sensitivity: 1-year weekly beta to returns of currency basket containing USD, EUR,
GBP, JPY, CNY
• Medium-Term Momentum: Cumulative return over past year excluding the most recent month
• Size: Natural logarithm of market capitalization
Fundamental Factors
• Value: Book-to-price
• Earnings Yield: Earnings-to-price and estimated earnings-to-price
• Leverage: Total debt (current and long-term liabilities) to total assets and total debt to equity
• Growth: Sales growth, estimated sales growth, earnings growth, estimated earnings growth
• Profitability: Return-on-equity, return-on-assets, cash flow to assets, cash flow to income, gross margin,
and sales-to-assets
• Dividend Yield: Ratio of sum of the dividends paid (excluding non-recurring, special dividends) over
the most recent year to average market capitalization
Industry Factors
The industry effects are captured by 68 factors based on the GICS® 2016 Industries. Exposures to these factors
are binary, i.e. having values of either 0 or 1. Assets with no official GICS® are given industry membership
based on internal research and are explicitly labeled as such in product deliverables.
Global Market Factor
Regression intercept term that affects all equity assets with unit exposure. Allows the model to better
distinguish between country and industry risk contribution effects.
Country Factors
Binary exposures based on an asset’s country of quotation, business activities or domicile. In most cases
this is equivalent to the market where an asset trades. The issuer’s home country is used for foreign listings,
depository receipts, and similar instruments.
Currency Factors
Binary assignments to the primary currency of an asset’s country.
Local Factors
Meant to capture strong residual structure in certain markets not captured by others factors. The model
currently has one such factor: Domestic China.
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1.4 Specific Risks
Idiosyncratic risk is estimated as the exponentially-weighted standard deviation of daily specific returns,
with a half-life parameter of 60 days, using 500 days of total history. Newey-West adjustment accounting for
1 day of autocorrelation is also incorporated.
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2 Fixed Income Model
2.1 FI Pricing Methodology and Risk Factor Definitions
FactSet provides terms and conditions on over three million fixed income (FI) securities. Given the size of
the fixed income universe and the large number of FI portfolios, the computational burden posed by the full
pricing of FI securities is a challenge faced by any integrated risk solution. To deal with this complication,
FactSet uses an efficient estimation method based on a multivariate Taylor series approximation of fixed
income returns. The idea of this approach is to reduce the computational burden by using a robust set of
risk factor sensitivities for each fixed income security, pre-computed daily in a separate process. Below we
describe the details of this methodology.
The market value M V (t) of an FI security at time t can be expressed through the security price P f (t) and
its face value F (t) as:
M V (t) = P f (t)F (t).
Here P f (t) is the full or “dirty” price of the security. If the security pays coupons, the full price will
jump down by the value of the coupon at each cash flow date. In other words, the security price will have
large fluctuations that are not actually related to any risk but are there because of deterministic coupon
payment. So, for risk purposes, it is better to distinguish between the random (risky) portion of the price
and deterministic coupon payment—accrued interest AI(t). This is done through the definition of clean price
P c (t) as the portion of the security’s price that does not include accrued interest:
P f (t) = P c (t) + AI(t).
If the face value of the security changes in time due to paydown (for example, because of prepayments as
in the case of a mortgage-backed security), the return of the security (relative change of its market value)
between the time t and t + ∆t can be expressed as:
r=
∆M V
(P c (t + ∆t) + AI(t + ∆t))F (t + ∆t) − (P c (t) + AI(t))F (t)
=
.
M V (t)
P f (t)F (t)
Introducing a paydown ratio
γ=
F (t + ∆t)
,
F (t)
we obtain the following equation for the FI security return:
r=
∆AI
∆P c
γ + γ − 1 + f γ,
f
P (t)
P (t)
rai
(1)
rpdwn
where we defined the accrued interest return rai and a paydown return rpdwn .
The price of any fixed income security can be computed as the risk-neutral expectation of all its discounted
cash flows. Thus, the main factor that underlies all FI prices (and returns) is the yield curve that is used to
compute the discount factors. When the price of other, more complex FI securities is computed using those
discount factors, the result does not coincide with their market price because those securities are riskier than
the sovereign bonds, and their holders demand higher yields as compensation for the extra risk. This risk
premium is quantified by a spread (OAS in the case of FI securities with optionality) that is added to the
yield curve to compute discount factors used to price these securities. Thus, the price of every FI security
can be represented as a function of interest rates (yield curve), spread, and time. The dependence on time
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reflects the fact that even when the curve and spread remain unchanged, the price of the FI security will
change when its maturity shortens (rolls down the yield curve).
We can use a Taylor expansion to write the change of the market price of the security as:
∆P c =
! ∂P c
∂P c
1 ! ∂2P c 2
∂P c
1 ∂2P c
∆t +
∆ri +
∆ri + ... +
∆OAS +
∆OAS 2 + ...
2
2
∂t
∂r
2
∂r
∂OAS
2
∂OAS
i
i
i
i
(2)
The first term in this expression represents the time dependence of the security price and thus defines the rolldown return. The next two terms (as well as the omitted higher-order terms) include partial sensitivities to
key rates ri along the security-specific yield curve. When computing the security return, these sensitivities
c
1 ∂2P c
turn into familiar key rate durations KRDi = − P1f ∂P
∂ri and key rate convexities KRCi = P f ∂r 2 . The
i
∂P
next set of terms accounts for spread risk with spread duration defined as DOAS = − P1f ∂OAS
and spread
2 c
1 ∂ P
convexity as COAS = P f ∂OAS 2 . Second- and higher-order terms are not used in the linear parametric
framework, so these are going to be omitted for consistency of notation.1 Finally, additional components
need to be considered in the expansion of certain complex instruments for accurate risk evaluation. These
can be dependencies of convertible bonds on underlying equities, interest rate volatility effects on callable
municipal bonds and mortgage-backed securities, inflation dependence of inflation-linked bonds, etc. These
are introduced and described in their respective sections.
c
The expansion equation (2) can be combined with the full return equation (1) to obtain the expression which
determines the risk measures of fixed income securities:
r = rai + rpdwn +
!
γ ∂P c
∆t − γ
KRDi ∆ri + ... − γDOAS ∆OAS +...
f
P ∂t
i
rroll
(3)
rOAS
ryc
Here, in addition to the accrued interest and paydown returns, we have introduced the roll return rroll as
well as the yield curve return ryc and the spread return rOAS .
The pricing function in equation (3) naturally defines the FI risk factors. For securities without principal
paydown (γ = 1) or in models with deterministic paydown, rai and rroll are deterministic. The projected
values of these components are known with certainty and do not need to be included in the set of random
variables modeled with a joint multivariate distribution. Thus, FI risk factors can be divided into three
main groups: the yield curve (interest rate) factors that affect returns across all asset classes, spread factors
that model systematic returns specific to a particular asset class, and additional factors that are related to
particular instrument properties. These factors are covered in the sections that follow.
2.2 Interest Rate Risk
It follows from equation (3) that calculation of interest rate risk components of VaR (or any other risk
measure) is based on pre-computed exposures to 17 key rates or functions of these key rates along the
security-specific discount curve. FactSet’s analytic platform is computing yield curves daily for more than
50 sovereign bond markets. Using 17 key rates from all of these curves would amount to more than 850
yield curve factors. Adding more FI factors and other asset class factors on top would result in a very
high-dimensional model covariance matrix. This is something we would like to avoid as the quality of the
matrix diminishes rapidly with its increasing size (see Section 1 of the Evaluation of Risk chapter for details).
Besides, interest rate movements at similar maturities tend to be highly correlated, and this high correlation
1 It
is important to note that second-order terms are employed for interest rate risk in the MC MAC framework.
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could lead to unwarranted model instability. To increase the stability of the results and to keep the size of the
model covariance matrix limited, we employ a principal component analysis (PCA) algorithm to determine
the most important aspects of yield curve behavior. (A good review of the PCA approach to modeling yield
curve dynamics is given in [2].)
The PCA approach allows us to parsimoniously summarize yield curve dynamics with a reduced number of
factors that explain the greatest percentage of the curve movement. FactSet uses four principal components of
each curve as risk factors. The covariance matrix of the MAC model is computed using the PCA factors, which
can then be converted back into the key rates that determine the future distribution of return components.
In the MC MAC framework, the simulated PCA factor returns are transformed into the key rate returns,
and the fixed income repricing formula (3) is applied. In the LMAC framework, on the other hand, the same
transformations are applied to the covariance matrix. As a result, all reporting tools operate in terms of the
17 partial points of the curve, even though principal components are used to forecast the curve dynamics.
Details of the PCA algorithm are outlined in Appendix 1.
The FactSet MAC model uses the yield curves from the following sovereign bond markets: AUD, BRL, CAD,
CHF, CLP, CNY, CNH, COP, CZK, DKK, EUR, GBP, HKD, HUF, IDR, ILS, INR, ISK, JPY, KRW, MXN,
MYR, NGN, NOK, NZD, PEN, PHP, PLN, RON, RUB, SEK, SGD, THB, TRY, TWD, USD, and ZAR. In
addition, the following alternative risk-free rate curves are available: AONIA (Australia), CORRA (Canada),
EONIA and ESTR (Europe), NZIONA (New Zealand), SARON (Switzerland), SOFR (U.S.), SONIA (UK),
and TONAR (Japan).
2.3 Spread Risk
Spread Model Methodology Overview
FactSet employs different spread models for different asset classes. The majority of securities with persistent
spreads are modeled using linear cross-sectional models with carefully selected factors. The main advantage
of this approach is that it ensures consistent factor decomposition across different risk estimation frameworks.
In addition, our robust model selection methodologies ensure that the employed linear models are sufficiently
accurate and intuitive.
Linear spread models employed in the MAC model have a simple common structure of factors, which can
be augmented or complemented with factors that are tailored for the specific asset type. The largest group
of our models exploits a known empirical observation that the spread volatility is proportional to the level
of security spread regardless of the credit rating ([3] and [4]). This model methodology is commonly referred
to as Duration Times Spread (DTS) because of the characteristic risk exposure that it implies. We provide
more details on this approach in Section 2.1 of the Appendix, while details of other types of linear models
are described in Section 2.2 of the Appendix.
Models are defined for spread levels OAS(t), spread level changes ∆OAS(t), or excess returns rexcess (t), which
can depend on the asset class and are explicitly stated as such. It should also be noted that idiosyncratic
risk components are parameterized for some security types as a function of the security’s attributes, e.g.,
ε(OAS, DOAS ). For simplicity of notation, models employing this feature contain the ε term, while the
independent parameters are suppressed. The list of factors for all linear spread models employed by FactSet’s
MAC model can be found in Section 3 of the Appendix.
Finally, the MC MAC framework employs the nonlinear contingent claim analysis (CCA) framework ([5])
for an additional set of securities that are not covered by the aforementioned models. Given the specificity
of this model and its limited coverage universe, the detailed description can be found in dedicated Section
2.3 of the Appendix.
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Sovereign, Municipal, and Quasi-Government Bonds
Euro-Sovereign Bonds
Most of the bonds issued by European sovereigns are denominated in a common currency—the euro. However,
each country within the Eurozone has different country-specific risks. In our approach, German government
bonds are modeled as government bonds with zero spread relative to the euro yield curve, and bonds from
other governments are priced relative to this curve. As a result, a broad range of spreads is observed, and
this universe exhibits the DTS behavior described in [3] for corporate bonds and in [4] for USD-denominated
sovereign bonds. Our research for the European sovereigns has been presented in [6].
For our estimation universe, we consider a union of IG and HY indexes that track EUR-denominated sovereign
debt: the Merrill Lynch All Euro Government Index (MLEZAS), the Merrill Lynch Euro Government Index
(MLEG00), and the Merrill Lynch Slovak Republic Government Index (MLG0SL). The resulting universe
includes all relevant countries and covers a broad range of credit ratings as is shown in Table 1.
After choosing the above set of indexes, we apply filters to guarantee that the bonds we use for factor return
estimation are highly liquid and have reliable analytics. In particular, for most countries, we consider only
non-callable fixed coupon bonds while allowing both fixed and variable coupon bonds for both Greece and
Cyprus. The latter is due to the sparsity of bonds and the nature of the debt issuance following the Greek
debt crisis and the subsequent debt restructuring. Details of analytics filters can be found in [6]. Details on
regression methodology and model selection can be found in the aforementioned reference as well. Further,
we only summarize the final results.
The model is estimated with Weighted Least Squares (WLS) regression and is defined by
∆OAS(t) = ∆fLS (t) + OAS∆fc/r (t) + OASβ SD ∆fRSD (t) + β SD ∆fASD (t) + ε,
(4)
where β SD = max{DOAS −DOAS , 0} is the short duration factor sensitivity that will be described shortly.
Different terms in model (4) capture different dynamics of a given OAS distribution. Factor fLS , referred to
as the low spread factor, captures parallel shifts of the entire distribution when spreads rise or decrease all
together, and it mainly drives the systematic spread volatility of bonds with the lowest spreads. Spread factor
fc/r is specific to every country or region listed in Table 1, and these factors capture how the distributions
tighten (or widen) during rallies (or corrections).
Country Group
Austria, Luxembourg
Finland
France
Italy
Netherlands
Ireland
Spain
Portugal
Belgium
Latvia, Lithuania,
Slovenia, Slovakia
Greece, Cyprus
S&P Rating
AA+, AAA
AA+
AA
BBB
AAA
A+
BBB+
BBBAA
A-, A-, A, A+
B-, BB
Table 1: Country groups for regional factors in the Euro-Sovereign Bond model.
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Factors fASD and fRSD are referred to as absolute short duration and relative short duration factors. They
capture the volatility component due to changes in curvature of the spread curve of the asset class because
of their nonlinear sensitivities β SD and OASβ SD , respectively. These two factors can also be thought of as
respective corrections to fLS and fc/r .
U.S. Municipal Bonds
The U.S. municipal bond model explains the market influences on the credit risk, including the effects from
different sectors, regions, and interest rate volatility. The latter factor proved to be highly significant as this
municipal bond market consists predominantly of callable bonds. Our research on this asset class is described
in [7], while only the main results of this DTS model are summarized further.
The model sectors break down into General Obligations and Revenue bonds from 12 broad industrial groups.
The latter are Authority, Building, Certificate of Participation, Economic Development Revenue, Education,
Health, Housing, School District, Tax Revenue, Transportation, Utility, and Other Small Industries.
The model introduces the following regional factors: the states of New York, California, New Jersey, Illinois,
Pennsylvania, Florida, Connecticut, Texas, Massachusetts, Washington, territory of Puerto Rico, and a group
that consists of Guam and the Virgin Islands. These regions were selected because the factors related to them
were shown to add explanatory power to the model.
The municipal bond class is distinguished from the European sovereigns introduced above by a much narrower distribution of spreads and a greater presence of callable bonds. To consider the interplay of these
peculiarities, the Taylor series approximation in equation (3) is augmented as follows:
r = rai + rpdwn + rroll + ryc + rOAS + rvega .
The accrued interest, paydown, roll, curve, and spread return components were introduced previously. The
component rvega arises from changes of interest rate volatility and is proportional to the vega duration
Dvega . The vega duration of a noncallable bond is zero. The spread and vega returns together form the
excess return
rexcess = r − (rai + rpdwn + rroll + ryc ) = rOAS + rvega ,
which captures the effects of callable bonds. To be compatible with the current model framework, our
municipal bond factor model was built to explain the DOAS -scaled excess return, i.e., −rexcess /DOAS . In the
case of a bond with no embedded options, this variable is simply equal to the change of spread ∆OAS.
The modeling universe of all available U.S. municipal bonds is filtered to remove securities that are likely to
exhibit large idiosyncratic movements and data outliers. Then, zero-coupon bonds and pre-refunded bonds
are removed from the universe. All bonds are removed from the universe six months before their maturity to
exclude prices that are converging to the par value. Callable bonds and sinking-fund bonds are removed from
the universe six months before the first call or sinking-fund date. Additional filters are applied to every daily
cross-section of observations to ensure that most liquid bonds are selected and analytics are of the highest
possible quality. Parameters of the filters were carefully tuned to ensure that all regions are represented when
estimating the factor returns. For finer details of the filtering procedure, the reader is referred to [7].
The model is defined by
−
1
DOAS
rexcess (t) =∆fsector−LS (t) + OAS∆fsector (t) + ∆fregion−LS (t) + OAS∆fregion (t)
+ β SD ∆fASD (t) + β LD ∆fALD (t) + Icall
Dvega
∆fvega (t) + ε
DOAS
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where β LD = max{DOAS − DOAS , 0} is the long duration sensitivity and Icall is an indicator function that
takes the value of 1 for callable bonds and 0 otherwise.
The ∆fsector−LS and ∆fregion−LS are the low spread factor returns, specific for model sectors and regions.
The ∆fsector and ∆fregion are the returns of relative spread factors that are also sector- and region-specific.
Returns of factors ∆fASD and ∆fALD are the absolute short duration and the absolute long duration factors
that are common for all bonds. The former factor has been introduced while describing the Euro-Sovereign
Bond model and the latter is its counterpart driving the long end of the spread curve. Finally, ∆fvega denotes
the vega factor return, which explains the changes in the interest rate volatility.
Developed Market Supranational, Sub-Sovereign, and Agency Bonds
Developed Market Supranational, Sub-Sovereign, and Agency (SSA) bond models are primarily built to cover
the corresponding securities issued in 13 of the major currencies — USD, EUR, JPY, GBP, CAD, AUD,
CHF, HKD, SEK, NZD, KRW, SGD, and NOK.
Agency bonds are issued by non-sovereign entities with close ties to, or the backing of, their sovereign
governments while sub-sovereign bonds are issued by the local authorities like counties, states, provinces,
or municipalities rather than national or federal governments. Bonds issued by supranational entities are
factored out as their credit risk is not tied to a specific country. Typically, such entities are set up as
investment or development banks, assistance funds, monetary unions, intergovernmental institutions, or
others.
The estimation universes for models are selected based on the available data on supranational, sub-sovereign,
and agency bonds at FactSet with fixed coupon type, additionally filtering for outliers. In most models, subsovereign and agency factors are estimated based on universes with bonds from developed market (DM)
countries; however, HKD and KRW models permit emerging market (EM) issuer bonds into estimation
universes to increase their sizes. U.S. municipal bonds are not permitted in the USD SSA model estimation
universe because they are covered by the U.S. municipal bond model. Similarly, issuers from EM are excluded
from the estimation universes of USD SSA and EUR SSA models, as the issues are respectively covered by
the EM USD and EM EUR spread models described further. The ∆OAS of bonds is regressed on the spread
factors using OLS regression methodology in each currency universe separately.
With the exception of the CAD model, agency and sub-sovereign bonds are exposed to the same SSA model
factors due to their similar spread change dynamics. The CAD model differentiates between agency and
sub-sovereign bonds by dedicated province and municipality spread factors.
Supranational bonds are treated distinctly from agency and sub-sovereign bonds in USD, EUR, GBP, CAD,
AUD, and CHF where models have dedicated supranational spread factors. Conversely, supranational bonds
in JPY, HKD, SEK, NZD, KRW, SGD, and NOK models expose supranational bonds to the same respective
factors as agency and sub-sovereign bonds in these currency universes.
The following aims to provide additional details on the SSA models and their functional form. The full list
of all SSA factors can be found in the Appendix Section 3.
USD
The spread model for USD-denominated SSA bonds has six group-specific spread factors ∆fgroup and an
absolute-short duration factor ∆fASD . The groups distinguish between U.S. agency bonds, Canadian, European, and Asia-Pacific agency/sub-sovereign bonds, U.S. supranational bonds, and European supranational
bonds.
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∆OAS(t) = OAS∆fgroup (t) + β SD ∆fASD (t) + ε.
EUR
The spread model for EUR-denominated SSA bonds has seven group-specific spread factors ∆fgroup and an
absolute-short duration factor ∆fASD . The groups distinguish between supranational bonds and agency/subsovereign bonds from German, French, Italian, Spanish, other DM European, and non-European DM issuers.
∆OAS(t) = OAS∆fgroup (t) + β SD ∆fASD (t) + ε.
CAD
The spread model for CAD-denominated SSA bonds has eight group-specific spread factors ∆fgroup . The
groups distinguish between supranational bonds, Canadian agency bonds, and provincial bonds from Ontario,
Quebec, Alberta, and other provinces, as well as Ontario municipal bonds and municipal bonds from other
Canadian provinces.
∆OAS(t) = OAS∆fgroup (t) + ε.
Note that Canadian agency/sub-sovereign entities issuing bonds in other DM currencies are expected to be
covered by the SSA model of the respective currency.
JPY
The spread model for JPY-denominated SSA bonds has four spread factors: low spread ∆fLS , spread ∆f ,
absolute-short duration ∆fASD , and absolute-long duration ∆fALD factors. These are selected to cover all
agency/sub-sovereign bonds issued in JPY.
∆OAS(t) = ∆fLS (t) + OAS∆f (t) + β SD ∆fASD (t) + β LD ∆fALD (t) + ε.
Note that Japanese agency/sub-sovereign entities issuing bonds in other DM currencies are expected to be
covered by the SSA model of the respective currency.
AUD, GBP, and CHF
The spread models for SSA bonds in AUD, GBP, and CHF share the same functional form of bond
type-specific spread factor for the two types of bonds – agency/sub-sovereign bonds and supranational
bonds.
∆OAS(t) = OAS∆ftype (t) + ε.
Factors are estimated in separate regressions for each currency universe.
HKD, SEK, NZD, KRW, SGD, and NOK
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Spread models for SSA bonds in HKD, SEK, NZD, KRW, SGD, and NOK share the same functional form
by using a spread factor in each universe.
∆OAS(t) = OAS∆f (t) + ε.
Factors are estimated in separate regressions for universes in each currency, and both agency/sub-sovereign
bonds and supranational bonds in these currencies are expected to be covered by the same spread factor.
Corporate Bonds
As can all major market actors, corporations can raise capital by issuing debt instruments to finance their
operations and for growing their business. For investors, these securities offer higher-yield opportunities in
exchange for taking on higher risks. Securities in this market segment are known to be effectively modeled
with the DTS methodology ([3] and [4]), and we have adopted this approach as well.
Developed Market Corporate Bonds
The DM corporate bond spread model decomposes the systematic risk into effects of currency denomination
(USD, EUR, GBP, JPY, AUD, CAD, and CHF), sector assignment, and industry classification in this order
of priority. Considered currency universes are distinguished by different sizes in terms of market value and
numbers of issues. Consequently, the particular choice of factors on the sector and industry levels was guided
by data availability and our statistical testing methodology ([8]). As a result, the USD component has the
highest factor granularity, and it was made coarser for other currency universes. For reference, final definitions
of sectors and industries for all seven currencies are presented in Section 4.1 of the Appendix.
General principles of our regression universe definition, factor selection methodology, and testing procedures
were described in [8]. DM corporate bond model components leverage factor types that were already introduced in the section dedicated to the research of the Euro-Sovereign Bond model, so it can be consulted for
interpretation of factor types. Further, we only summarize the details distinguishing different DM corporate bond model components, grouping them by currency and similar functional forms of ∆OAS modeling
expressions.
USD
Spread changes of bonds in the Financials and Industrials sectors of the USD universe are modeled as
"
#
∆OAS(t) =∆fsector−LS (t) + OAS ∆findustry (t) + IHY ∆fsector−HY (t)
+ OASβ
SD
∆fsector−RSD (t) + β
SD
(5)
∆fsector−ASD (t) + ε,
while for bonds in the Utilities sector we have
∆OAS(t) = OAS∆findustry (t) + OASβ SD ∆fsector−RSD (t) + β SD ∆fsector−ASD (t) + ε.
Here, ∆findustry denotes the components of industry spread factors. Sector-wide factors are denoted as
∆fsector−LS (low spread), ∆fsector−HY (high-yield), ∆fsector−RSD (relative short duration), and ∆fsector−ASD
(absolute short duration). In model (5), IHY denotes an indicator function that is equal to 1 if the composite
rating2 of the bond is lower than BBB-. By this definition, ∆fsector−HY factor acts as a correction to the
component explaining the industry effects.
2 The
composite ratings are defined as the maximum of the numerical ratings by S&P, Moody’s, and Fitch.
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Factor returns for this component set are estimated using WLS. The estimation universe consists of most
liquid USD-denominated fixed coupon bonds in the union of Merrill Lynch U.S. Corporate (MLC0A0) and
U.S. High Yield (MLH0A0) indexes. Only non-callable bonds are chosen from MLC0A0, while both noncallable and callable bonds are picked from MLH0A0 to ensure that the sample accurately represents the
entire currency universe.
EUR and GBP
Bonds in the EUR and GBP universes are modeled with separate models that share the same functional
form:
"
#
∆OAS(t) =∆fsector−LS (t) + OAS ∆findustry (t) + IHY ∆fHY (t)
+ OASβ SD ∆fsector−RSD (t) + β SD ∆fsector−ASD (t) + ε.
The expression is similar to (5) with a difference in that the high-yield factors ∆fHY in the EUR and
GBP universes are common between all sectors. Factor returns for both currency universes are estimated in
separate ordinary least squares (OLS) regressions. Estimation universes are defined to keep the most liquid
fixed coupon bonds in unions of representative indexes with the same callable/non-callable bond rules as in
the USD case. The starting point of the EUR universe is the union of Merrill Lynch Euro Corporate Index
(MLER00) and Yield Index (MLHE00), while for the GBP universe—Merrill Lynch Sterling Corporate Index
(MLUR00) and Merrill Lynch Sterling High Yield Index (MLHL00).
JPY
Spread changes of bonds in the JPY universe are modeled as follows:
∆OAS(t) = OAS∆findustry (t) + β SD ∆fsector−ASD (t) + ε.
Factor returns are estimated using OLS regressions in the estimation universe defined to retain the most
liquid fixed coupon non-callable bonds in the Merrill Lynch Japan Corporate Index (MLJC00).
AUD and CAD
For AUD and CAD universes, we employ the following:
∆OAS(t) = OAS∆findustry (t) + IFinancials β SD ∆fFinancials−ASD (t) + ε.
It needs to be pointed out that, due to the relatively smaller sizes of these universes, nonlinear absolute short
duration factors are defined only for the Financials sector. It was not possible to reliably define separate
factors for the Industrials and Utilities sectors.
Factor returns for both currency universes are estimated in separate OLS regressions, using universes of fixed
coupon non-callable bonds selected by our filtering methodology from broad market indexes. Merrill Lynch
Australia Corporate Index (MLAUC0) was employed for the AUD universe, while Merrill Lynch Canada
Corporate Index (MLF0C0) and Merrill Lynch Canada High Yield Index (MLHC00) were employed for the
CAD universe.
CHF
The universe of CHF-denominated bonds is the smallest in the set that this section is concerned with. Only
two coarse sector spread factors were defined, so the spread changes of bonds in this universe are modeled
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as
∆OAS(t) = OAS∆findustry (t) + ε.
Factor returns are estimated using OLS regressions in the universe of fixed coupon non-callable corporate
bonds in the Swiss Bond Index (CH0027441887).
Local Currency Market Corporate Bonds
This group of models covers corporate bonds issued in CNY, HKD, INR, KRW, MYR, NOK, NZD, SEK,
SGD, THB, and ZAR. The granularity of spread models varies considerably depending on universe specifics.
All market segments were found to benefit from OAS sensitive factor(s) in line with DTS methodology,
and most models benefit from short- (or long-) duration sensitive factors. Models in CNY, KRW, and SGD
capture additional industry effects.
Readily available and reliable market indexes are uncommon in this market segment. To select the most liquid
and representative set of bonds for the estimation universe of any model, we adopted similar methodology
to that already deployed for the DM corporate models. As in the previous case, fixed coupon non-callable
corporate bond data available at FactSet and a number of liquidity and outlier filters were applied. All
local currency market corporate spread models use OLS regression methodology separately on each currency
universe. The following specifies spread models explicitly for all currencies and comments on the important
specifics.
CNY
The spread model for corporate CNY-denominated bonds also includes bonds that are issued by the three
government-backed banks (policy banks, or PBs in short)—Chinese Development Bank, Agricultural Development Bank of China, and Export-Import Bank of China. The spread model accounts for it using separate
industry spread factors ∆findustry for PB bonds and other corporate bonds.
∆OAS(t) = ∆fLS (t) + OAS∆findustry (t) + β SD ∆fASD (t) + β LD ∆fALD (t) + ε.
Other factors—low spread ∆fLS , absolute-short duration ∆fASD , and absolute-long duration ∆fALD factors—
are currency-wide and are, therefore, shared between PB and other corporate bonds.
HKD
The spread model for HKD-denominated corporate bonds has a low spread factor ∆fLS as well as a spread
factor ∆f and absolute-short duration factor ∆fASD
∆OAS(t) = ∆fLS (t) + OAS∆f (t) + β SD ∆fASD (t) + ε.
INR, SEK, NOK, and MYR
Spread models for corporate bonds in this group of currencies share the same functional form, and each has
a currency-wide spread factor ∆f and an absolute-short duration factor ∆fASD .
∆OAS(t) = OAS∆f (t) + β SD ∆fASD (t) + ε.
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Regressions are performed on a by-currency basis so that calculations of median duration and therefore
exposure β SD are also universe specific.
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KRW
The spread model for KRW-denominated corporate bonds has industry-specific spread factors ∆findustry and
an absolute-short duration factor ∆fASD .
∆OAS(t) = OAS∆findustry (t) + β SD ∆fASD (t) + ε.
NZD and ZAR
The spread models for corporate bonds in NZD and ZAR share the same functional form and have one
currency-wide spread factor each, fit on each currency universe separately.
∆OAS(t) = OAS∆f (t) + ε.
SGD
The spread model for SGD-denominated corporate bonds has a low spread factor ∆fLS and industry-specific
spread factors ∆findustry .
∆OAS(t) = ∆fLS (t) + OAS∆findustry (t) + ε.
THB
The spread model for THB-denominated corporate bonds has a low spread factor ∆fLS as well as a spread
factor ∆f and an absolute-long duration factor ∆fALD
∆OAS(t) = ∆fLS (t) + OAS∆f (t) + β LD ∆fALD (t) + ε.
Emerging Market Bonds Denominated in USD and EUR
The EM bond DTS model decomposes the systematic risk of corporates, agencies, and sovereigns into effects
of currency denomination (USD and EUR) and country of risk on the highest level. Where possible, the
effects of issuer class were considered by defining two separate factors: one for corporates and another for
agencies and sovereigns. However, some country groups were not large enough throughout the intended
history of the model to reliably estimate factor returns, and coarser regional groups were defined as a
result. Final definitions of regional groups for both model components are presented in Section 4.2 of the
Appendix.
This model was constructed by following the same methodology as for DM corporate bonds ([8]), while
particular details and empirical study for this asset class have been summarized in [9]. Only spread factors
were used when defining the model, so the final expression takes a very simple form:
∆OAS(t) = OAS∆fregion (t) + ε.
Factor returns are estimated using a robust linear regression and OLS for USD and EUR components, respectively. Regressions are performed in universes derived from broad market indexes. For the USD-denominated
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bonds, we chose the EM segment in the union of Merrill Lynch Global Broad Market Index (MLGBMI) and
Global High Yield Country External Corporate & Government Index (MLIM00). For the EUR bonds, we
chose bonds from the Merrill Lynch Emerging Markets Corporate Plus Index (MLEMCB). The universes are
further constrained to issuers from factor group definitions (Section 4.2 of the Appendix) that are classified
as Corporate and Quasi & Foreign Government in the Merrill Lynch classification scheme.
Structured Credit Products
U.S. Mortgage-Backed and Mortgage-Related Securities
Mortgage-backed securities (MBS) are securities that pass the principal and interest payments on a pool
of mortgages to the investor. Most MBS in the U.S. are issued by the government agencies and quasigovernmental agencies—Government National Mortgage Association (GNMA), Federal National Mortgage
Association (FNMA), and Federal Home Loan Mortgage Corporation (FHLMC). Other mortgage-related
securities also include collateralized mortgage obligations (CMOs), which are structured securities backed by
MBS, and residential ABS, typically structured securities backed by non-agency issued mortgages.
MBS
The estimation universe for MBS spread factors is based on ICE BofA US Mortgage Backed Securities
index (MLM0A0) constituency. Fixed rate agency MBS spread changes are modeled by mortgage issuer- and
term-specific factor ∆fg and a market-wide factor ∆fW AC :
∆OAS(t) = ∆fg (t) + (WAC − RP M M S ) ∆fWAC (t) + ε.
With the unit exposure, factors ∆fg represent spread level changes of current coupon securities characteristic
of the collateral group g. Factor ∆fWAC , on the other hand, represents the portion of the spread change that
scales with the difference between the gross weighted average coupon (WAC) of the security and the primary
mortgage market survey (PMMS) rate RP M M S of fixed rate mortgages. This exposure can be interpreted as
the moneyness of the embedded prepayment option of the underlying mortgages or equally as a refinancing
incentive. The relevant rate RP M M S is estimated given securities term and using the published PMMS rates
one week prior to the forecast date.
The groups g themselves are found comparing the behavior of spread changes between finer groups based on
mortgage term and issuer agency categories. The resulting groups read as follows:
• 15-year MBS issued by FNMA, FHLMC, and GNMA
• 20-year MBS issued by FNMA and FHLMC
• 30-year MBS issued by FNMA and FHLMC
• 30-year MBS issued by GNMA
Similar to the case of U.S. municipal bonds, the model for MBS also captures volatility (or vega) effects,
but these are addressed differently. The vega return rvega is modeled proportional to the vega duration
Dvega :
rvega = Dvega ∆fvega (t) + ε.
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In this context, factor ∆fvega represents a driver or volatility return in the MBS market.
Both the spread model and volatility model employ the OLS regression methodology for calculating corresponding factors ∆fg , ∆fWAC , and ∆fvega . More information on the model definition, selection and backtesting details can be found in [10].
CMO
Agency CMO spread changes are modeled by a separate model. The estimation universe for CMO spread
factors is based on ICE BofA Agency CMO Index (MLCMOS) constituency containing various CMO tranches
with the underlying collateral of U.S. Agency MBS. Similar practices to those employed in grouping securities
of other spread models were employed to group the index securities in two tranche groups. Each of these
groups is modeling the spread changes by a refinancing duration factor ∆fRτ that captures the level changes
of the corresponding spread curve specific to that tranche group τ .
As a result, the systematic spread changes for CMOs are modeled as proportional to the refinancing duration
DR according to the following expression:
∆OAS(t) = DR ∆fRτ (t) + ε.
The grouping τ distinguishes the Interest-Only (IO) and Principal-Only (PO) tranches from other tranches
found on the index:
• IO and PO tranches
• PAC, SEQUENTIAL, TAC, VDAM, Z, and other tranches
Both the selection of the factor type and grouping were based on empirical research on the available history
of CMO index constituency at FactSet. The final form of the model was determined using a combination
of analysis from cross-sectional OLS regression statistics on numerous model forms and the corresponding
research stage back-tests on the same models.
CMBS
Commercial Mortgage-Backed Securities (CMBS) have a dedicated spread model based on the constituency of
the ICE BofA US Fixed Rate CMBS Index (MLCMBS). Spread factors take into consideration the collateral
type, seniority, and long and short duration effects.
∆OAS(t) = ∆fcollat (t) + OAS∆fsen (t) + X SD ∆fSD (t) + X LD ∆fLD (t) + ε
The spread model distinguishes between agency and non-agency collateral types using a collateral spread
factor ∆fcollat and between senior and non-senior tranches using a seniority spread factor ∆fsen . The short
and long duration effects are modeled using factors ∆fSD and ∆fLD , respectively.
The corresponding short and long duration exposures X SD and X LD range from 0 to 1 depending on the
security’s spread duration DOAS . Exposure X SD = 1 when DOAS < 0.5, and linearly goes to 0 from 0.5 to
3 years. Similarly, X LD = 1 when DOAS > 6, and linearly goes down to 0 from 6 to 3 years.
The model derives factors using Robust Linear Model regression methodology [11] with Huber loss function
(t = 2.0) to diminish the effect of extreme observations on the spread factors.
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Other U.S. Mortgage-Related Securities
The spread risk of manufactured housing ABS and home equity ABS is approximated using the MBS spread
model by exposing the corresponding securities to the MBS spread factors in the same way as the 30-year
conventional MBS. In contrast, the spread risk of adjustable rate and hybrid adjustable rate mortgages is
approximated by exposure to the CMO refinancing duration spread factor that covers non-IO and non-PO
tranches.
U.S. Asset-Backed Securities
The U.S. asset-backed securities (ABS) are covered by a spread model calibrated to a universe of U.S.
ABS/CMBS, excluding residential mortgage-related securities. Note that CMBS remains in the estimation
universe while a dedicated U.S. ABS model is in research. At present, the linear spread level regression model
captures the spread sensitivity to rating levels and seasoning.
OAS(t) = flevel (t) + W ALAfseason (t) + I(AA)fAA (t) + I(A)fA (t) + I(BBB)fBBB (t)
Boolean ratings sensitivities I(rating) are used to capture rating-specific spread dynamics. When rating
information is not available, a spread rating is assigned based on a spread level. The seasoning sensitivity is
captured by the weighted-average loan age WALA exposure.
Non-U.S. ABS and CMBS
The non-U.S. ABS/CMBS securities are covered by a spread model based on a universe of UK and Eurozone ABS and CMBS (excluding residential mortgage-related securities). The model uses a linear spread
level regression model that captures the spread sensitivity to rating levels and seasoning. The non-U.S.
ABS/CMBS distinguishes between above-AA and below-AA ratings:
OAS(t) = flevel (t) + W ALAfseason (t) + I(A − BBB)fA−BBB (t).
Japanese MBS
The Japanese MBS securities are issued by the Japan Housing Finance Agency and its predecessor entities.
The overall spread of JPY MBS securities is known to be low and not volatile. Our research suggested that
the spread movements of individual Japanese MBS are highly correlated with each other as well as the index.
In fact, the average correlation between the changes in OAS of each individual security versus the index is
0.90. For these reasons, we opted for a simple model where the systematic change of each JPY MBS security
is estimated as the change in the mean OAS of a representative universe. The OAS of the securities in the
calibration universe is calculated using FactSet’s Japanese MBS prepayment model and FactSet’s Monte
Carlo MBS pricing model.
2.4 Inflation-Linked Bonds
Inflation-linked (IL) bonds are financial instruments that have their principal balance tied to an inflation
index. These securities are typically analyzed in terms of real yields when considered in isolation. However,
because the MAC model uses nominal returns, it is more insightful to analyze them in terms of factors
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driving (nominal) yield curves and inflation curves. A more detailed study of IL bond risk is presented in
[12], while here we only summarize the final conclusions.
In the present context, a change in the nominal curve affects the present value of expected future cash
flows, while a change in the inflation curve affects expected cash flows themselves. IL bonds are given
an inflation assumption I constructed from an inflation curve rI . If available, it is derived from inflation
swap rates, as these are considered to capture market participants’ inflation expectations more accurately.
In less mature markets, the breakeven rates are employed. Regardless of the chosen inflation expectation
source in a particular market, inflation factors are modeled the same way as the nominal curve factors (see
Section 2.2).
In our model, inflation swap market data are available for the following inflation indexes: CPURNSA
(U.S.), CHAW (UK), CPIHARM00XTOBAC (Germany), FRPR0403231 (France), ITPR0027835 (Italy),
and AUSCPI0001 (Australia). Breakeven inflation data are used for V41690973 (Canada) and JPPR0382789
(Japan).
For IL securities, equation (3) can be extended as
r = rai + rpdwn + rroll + ryc −
!
P IDi ∆riI +
i
!
Xi ∆fi + . . .
(6)
i
rinf l
Here, P IDi is the partial inflation duration of key rate i and ∆riI is the corresponding inflation rate change.
Inflation return is now denoted as rinf l . For higher forecast accuracy, additional factors fi with corresponding
exposures Xi were incorporated for some subclasses of IL securities, and these are introduced next.
For segments of sovereign issuers where inflation swaps are not available, the breakeven data are used to
construct the inflation curve rI and additional factors are not required.
r = rai + rpdwn + rroll + ryc + rinf l + . . .
For sovereign markets where inflation swaps are available and are used to construct rI , an additional set of
factors needs to be incorporated to capture the differences between breakeven and inflation swap rates. We
will refer to the term structure of these differences as the inflation asset-swap spread and denote it as riasw .3
For bonds in this segment, equation (6) becomes
!
r = rai + rpdwn + rroll + ryc + rinf l −
P IDi ∆riiasw + . . .
i
Finally, inflation swaps are widely available in the Euro-Sovereign segment and are thus employed for constructing rI . However, the German yield curve is used for discounting, which leads to persistent spreads
that need to be accounted for. The model of Euro-Sovereign spreads has been introduced in equation (4) of
Section 2.3, and we will denote it as ∆OAS EuroSov here to avoid confusion. With this notation, equation
(6) takes the following form:
r = rai + rpdwn + rroll + ryc + rinf l − DOAS ∆OAS EuroSov + . . .
2.5 Convertible Bonds
Convertible bonds are securities where the bond holder has the option to surrender the bonds to the issuer
and receive a pre-specified number of shares of an underlying equity. Like corporate bonds, these securities
3 Peculiarities
of this spread were explored and discussed in [13].
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are exposed to government yield curve risk and corporate spread risk. However, unlike standard corporate
bonds, convertible bonds are exposed to the additional direct risk of price movements of the underlying
equity. These securities are known to be nonlinear and are, thus, only accurately modeled in the MC MAC
framework. Their risk is captured by two additional terms based on the change from the base case equity
price versus the Monte Carlo simulated equity price—δE. The risk is reported in the convertible bond factor,
and the return due to that factor rCB is computed as:
1
rCB = ∆CB δE + ΓCB δE 2 .
2
The delta ∆CB and gamma ΓCB are computed nightly by the FactSet fixed income calculation engine and
represent the first and second-order numerical derivatives of convertible bond price versus small changes
in the stock price, holding OAS constant. FactSet’s pricing model for convertibles is based on the work of
Andersen and Buffum [14] that was further explored by Zeitsch et al. [15]. The model incorporates credit risk
on both cash and equity components through a jump-diffusion process. The default rate is linked to the level
of equity prices, which makes the model a quasi-two-factor model. It is implemented by solving the resulting
partial differential equation using a finite difference approach and a carefully calibrated equity-dependent
default intensity. Technical details of the implementation can be found in [16].
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3 Commodity Model
The FactSet commodity model is a factor-based hybrid model, employing multiple stages of regression to
capture risk across a wide variety of commodity sectors. The model is designed to capture systematic risk
from exposures to factors common across all commodities, as well as idiosyncratic risk specific to each
commodity. The FactSet commodity model covers roughly 100 indexes, ETFs, and mutual funds as well as
approximately 500 unique commodities. Each specific contract for a future is mapped to its front month
contract, meaning the entire curve of any given commodity is represented within the model.
The model’s estimation universe is comprised of highly liquid indexes and front-month contracts. Certain
commodities, such as those with an open interest below a minimum threshold or those with exorbitant or
missing returns for periods which we use to model common factors, are removed from the estimation universe
so that their returns do not impact systematic factor risk but are still exposed to these factors.
A two-stage linear regression model is employed to decompose the returns of each commodity. First, the style
factors are computed in a cross-sectional regression of the forward daily return of all assets in the calibration
universe on factor sensitivities β:
(7)
r = β10m f10m + β30m f30m + β5β f5β + β20β f20β + βvol fvol + ε.
The model uses five style factors: 10-day and 30-day momentum factors f10m and f30m , five-day and 20-day
beta factors f5β and f20β , and a volatility factor fvol .
The sensitivities of each security to the momentum factors are represented by a measure of the correlation
of the security price with time. Specifically, the short period (T = 10 days) and the long period (T = 30
days) correlation coefficients of security price with time are computed as:
ρ=
T
$
i=1
(pi − p̄)(ti − t̄)
(T − 1)σp σt
and the t-statistics of these coefficients are then used as a measure of the security momentum and a sensitivity
to the momentum factors:
%
T −2
βT m = ρ
.
1 − ρ2
Sensitivities to the beta factors are computed using a slope of the regression line of prices on time with the
regression computed on a short (T = 5 days) or long (T = 20 days) window as:
Pti = βT ti + α where ti ∈ {1, . . . , T }.
Each sensitivity is computed as the regression slope normalized by the last price in the series:
βT β =
βT
.
PT
And finally, the sensitivity to the idiosyncratic volatility factor is represented by the standard deviation of
the asset return measured over 22 trading days:
&
'
22
' 1 !
βvol = (
(rt − r)2 .
N − 1 i=1 i
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Before performing the factor regression (7), each of these sensitivities is checked for outliers greater than
three standard deviations from the mean and then normalized to a z-score.
Residuals from the first stage of the regression are then used as dependent variables in a second stage. The
second stage employs a time series regression of these residuals against the returns of several market indexes
for 250 days:
ε = β6 f6 + β7 f7 + β8 f8 + β9 f9 + β10 f10 + ϵ,
where
• β6 = S&P GSCI Index return
• β7 = Dow Jones-UBS Composite Index return
• β8 = Reuters-CRB Total Return Commodity Index return
• β9 = CBOE S&P 500 Volatility Index (VIX) return
• β10 = 3M U.S. T-Bill return (computed as a first difference)
The first three indexes each have very different returns and volatilities, so measuring a commodity portfolio’s
exposure to these separate composite indexes offers differing perspectives. This is a result of the S&P GSCI
Index and the Dow Jones-UBS Composite Index being long-only commodity indexes, while Reuters-CRB
Total Return Commodity is a long/short index. Also, the asset allocations across the commodities that make
up the indexes are very different. Additionally, VIX was selected to represent equity volatility, and the 3M
U.S. T-Bill was used to offer a risk-free asset return to the model.
Currency is accounted for last in the regression process by adjusting the residual of each asset by daily
currency return. The exposure for each asset is set as a binary value to each currency: 1 for the local currency
and 0 for all others. When a security is priced in a currency different from the model’s base numeraire, the
security’s residual is then adjusted by the factor return for that currency:
ϵnumeraire (t) = ϵlocal (t) − rF X (t).
The factor returns created from this regression process are then used as input into the factor return matrix
and, eventually, the overall covariance matrix for the FactSet multi-asset class risk model.
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4 Private Equity Model
The FactSet Private Equity Risk Model combines public market factors, private market drivers, and idiosyncratic risk to explain the returns of Private Equity (PE) Funds. It currently covers four PE fund
classes—Buyout, Growth Equity, Real Estate, and Venture Capital; and includes Secondaries as a subclass
of Buyout.
PE Funds are different from public equities and open-end mutual funds in some important ways. PE funds’
net asset values (NAVs) are not based on competitive market pricing, but come from subjective accounting
by the General Partners (GPs). PE funds are planned to be finite-lived and the GPs decide when to call in
committed capital and make distributions; the Limited Partners (LPs) do not have precise control over the
timing of cash flows.
4.1 Fund Universe
The FactSet Cobalt Database provides fund-by-fund qualitative and quantitative data for private equity funds
of various fund classes—Buyout, Growth Equity, Real Estate, Venture Capital, Private Debt, Infrastructure,
and Natural Resources. This model is currently restricted to Buyout (BO), Growth Equity (GR), Real Estate
(RE), and Venture Capital (VC) PE funds. Each of these broad categories is divided into subcategories
(Style/Focus values), but for this model, we aggregate the subcategories into these coarser categories so that
there is sufficient data to calibrate the model.
There are two tables we use from the PE source data—descriptive data and cash flow (time-series) data.
The FactSet Cobalt Database contains both descriptive data and time-series data for many funds, but only
descriptive data for many others. Time-series data is necessary to estimate the model. This distinction
separates our estimation universe from the coverage universe. The estimation universe only includes the
funds that have both descriptive data and time-series data, while the coverage universe includes both the
estimation universe and the funds that have only descriptive data. This means that we can find fund-specific
parameters for funds in the estimation universe, while assigning category-average values for funds available
only in the coverage universe. Table 2 shows the sizes of the historical coverage and estimation universe by
fund class as of December 2022, including funds that have already been liquidated.
BO
GR
RE
VC
Coverage Universe
2355
991
877
3583
Estimation Universe
843
136
191
350
Table 2: Size of Estimation and Coverage Universes
4.2 Model Equation
In this section, we detail a bird’s-eye view of the model. We keep the notation as simple as possible. Full
model details are provided in [17].
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Structure Equation
We have defined the following model structure to model returns over the life of a private equity fund:
rti = β i λν i (t) Ft +
) *+ ,
public market
pt
)*+,
systematic private
+ αi + σν i (t) ϵit
)
*+
,
fund-specific
(8)
Equation (8) captures the major effects of the PE investment dynamics explicitly. It explains the return
of the fund rti for period t as a linear combination of the public market return Ft connected with a β i of
the fund to the public market, the systematic private component pt , and fund idiosyncratic αi and residual
component ϵit ∼ N (0, 1). The model echoes CAPM for publicly traded investments, as well as the more
detailed Fama-French extensions, but the styles are replaced with the systematic private component as the
common driver for PE fund returns distinct from the public market returns.
β i is specific to each fund and is assumed to be the “true” beta of the fund, a single parameter capturing the
fund’s connection to the public market in a single constant parameter. The natural expectation is for VC
funds to have beta materially larger than 1, due to the riskiness of VC investments being more sensitive to the
market than average, while BO funds have betas close to 1 as their portfolios consist of mature companies,
and GR in the middle. RE funds should also have betas greater than 1. Note that β i also implicitly captures
the effect of leveraged returns—funds that use more leverage to purchase their investments will have higher
beta because that cash is not coming from committed capital and consequently increases the returns of the
fund rti . Additionally, if capital is called and sits in the fund before being invested, that would lower the
effective leverage and, hence, the estimated β i .
ν i (t) denotes the age of fund i in period t and therefore λν i (t) accounts for the age effects on β i . The
idiosyncratic component σν i (t) ϵit has volatility that also depends on the age of the fund at time t.
Necessary Complications
Additional complexity in the model arises from several directions:
• Funds in our model have investment type either BO, GR, RE, or VC. Most parameters in the model
vary by fund type.
• As the public market component is dependent on the PE fund strategy, each PE fund can have an
individual public market index based on investment regions, industry, or style tilts. That complexity
is captured in our model by decomposing the public market return Ft into several regions—North
America, Europe, and Asia-Pacific—and into GICS sectors. As a result, many of the risk components
and parameters are effectively a weighted sum of the parameters corresponding to funds’ specific regions
and sectors.
• The systematic private component pt also varies by the region of investment, as well as by fund type.
In practice, this means that the actual systematic private component is again a weighted sum of the
systematic private components that correspond to the fund’s class-region.
• Fund αi is hard to measure simultaneously with β i , so in this iteration of the model, we take αi to be
shared between all funds of a given class.
• The idiosyncratic volatility σν i (t) depends on the age of the private equity fund and is again effectively
a weighted sum of the per-region estimated idiosyncratic volatilities.
For brevity, we are omitting in this section some of the indexes for each variable.
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Further complications also arise because the data that is used to estimate the model parameters is subject
to biases, the most significant of which is a lagged dependence on the public market. That complication,
however, is not an inherent property of the fund returns, but is rather an artifact of the data collection
mechanism, so it is not a part of the structure equation (8).
4.3 Results
The model estimation is performed in two steps: 1. fitting the model using only public market explanatory
factors and 2. taking the public market parameters as given, including our systematic private component as
an additional factor and recomputing αi and σν .
There is a complication in step 1 of the estimation that does not appear in the equation (8): due to the slow
update of private fund valuations to public market information, we include a lag-structure with exponential
decay to explain the fund returns. For a fund of various ages, this lag-structure tells the effect of current and
past public market returns on the current rolling annual fund return.
Figures 1a and 1b show the lag-age structure for funds in each fund class. The estimated lag-age weights
are displayed in two ways. The difference between the two images is similar to transposing a matrix. Each
figure has the weights on the y-axis. Figure 1a has lags by number of quarters on the x-axis, and each line
C
corresponds to one of the four ages, giving us λC
ν uk (ν). Note that the zero-age curve is zero for lags beyond
one quarter, as fund returns should not be based on any returns before the fund was created.
Because of the complexity of the model, each of the two estimation steps uses a Bayesian fitting process,
specifically a Markov Chain Monte Carlo (MCMC) sampling.
This permits us to find model parameters and confirm the results visually. The main trace plots for the
public market parameter estimation (step 1) are shown in Figure 2. On the left are the distributions of each
parameter and on the right are the values of the parameters through the sequence of 1,000 iterations. In
Figure 2, each of the plots shows the posterior distribution for each of the 12 categories parameter values.
The trace plot is color-coded, each color representing one of the 12 class-region categories. The solid lines
are for the first iteration path and the dotted lines are for the second iteration path. In the plots on the
left, we can see that each of the distributions appears similar to a bell curve and that each solid curve
is similar to the corresponding dotted curve, showing that the model converges regardless of the choice of
starting points in the chain parameters. On the right, we can see that the iteration paths are relatively stable,
lack systematic behavior, and appear independent of the previous values in the chain, as is expected in a
well-behaved MCMC run.
Figure 3 shows scatter plots of β i versus vintage year for funds in each category. These plots reveal many
things. First, there are more funds invested in North America than in Europe and Asia-Pacific and the most
BO and second-most VC funds. Second, before 2000 there were mostly BO funds, with VC accelerating in
the late 1990s; GR and RE are more recent fund classes. Third, fund creation slowed for a few years starting
in 2008, during the global financial crisis. Fourth, the β i values for each category rise and fall in waves; for
example, BO funds created in the early 2000s have higher β i values than funds created around 2010. The
other fund classes also have their own waves.
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C
(a) by Lag—λC
ν uk (ν) for each k
(b) by Fund Age (stacked)—sum is λC
ν for each ν
Figure 1: Age-decay weights for beta
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Figure 2: Trace plot from public estimation step
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rti
βi
ν i (t)
λν
Ft
pt
αi
σν
ϵit
Definition
fund return
fund long-term beta
fund age at time t
beta age dependence
public market return
systematic private component
fund alpha
fund-specific volatility
fund residual
Dimensionality
time-series for each fund
one for each fund
one for each fund for each period
for each class
time-series for each region-sector pair
time-series for each class and region
for each class and region
for each class and region
time-series for each fund
Table 3: Structure equation variables
Figure 3: β i vs. vintage year
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5 Derivatives
Most financial derivatives have nonlinear payoff functions and are better served with simulation-based frameworks. As such, this section focuses on methodologies relevant for the MC MAC model, and it is written
from this perspective. It should be noted, however, that linear derivatives are covered by both MC MAC
and LMAC frameworks.
5.1 Equity Derivatives
There are two aspects of equity derivatives to consider when calculating risk. First, the return distributions
of portfolios that include equity derivatives will not be normal, necessitating full repricing for each future
scenario. Second, the dynamics of the volatility surface will need to be modeled to determine the implied
volatility in each future scenario.
The Impact of Options on Return Distributions
To gain some intuition around the resulting non-normal
distributions and their impact on VaR, we will examine
two standard option strategies.
We start with a portfolio containing one share of a fund
that tracks the S&P500, initially at $2, 085.51 with an
assumed volatility of 17.8%. The return distribution of
this portfolio is assumed to be normal as seen in Figure
4.
The VaR can be√analytically calculated in this case to
be V aR5% = zσ Thoriz = 10.727%.
A protective put is the combination of a long stock and
a long put option. It is a strategy intended to protect
the investor from downturns in the stock; the put acts
as insurance, paying precisely when the portfolio is otherwise adversely affected by the stock movement. The
return distribution is then capped on the downside, re- Figure 4: Simulated return distribution of a portfolio of
sulting in a highly non-normal distribution (Figure 5a). a single equity
The portfolio VaR is significantly reduced by this option strategy. In the present case with simplifying assumptions,4 the VaR is determined by simulation to be
V aR5% = 8.65%, demonstrating the ability of this strategy to reduce the downside risk of the portfolio.
Another indication that a normal distribution is inadequate is that in this case, the 99% VaR is identical to
the 99.5% VaR, a scaling that cannot be explained by a normal distribution.
A covered call is the combination of a long stock and a short call option. The resulting option premium
will generate return at the expense of giving up possible future appreciation of the underlying stock. This
arises in an opposite-looking distribution as compared to the protective put. As one can see in Figure 5b,
the upside gains are eliminated and the median return is above zero.
4 The
future volatility is held at the 17.8% level and the option expiry is less than the horizon time with a strike value of
$1, 900.
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In the case of the covered call, there is no true risk reduction of downside measures like VaR, since the
strategy sells upside potential for increased revenue—in the present case, the simulated V aR5% = 10.25%.5
The slight decrease is due to the increased return of the portfolio by selling the option and receiving the
option premium.
These two simple examples demonstrate that the VaR of portfolios containing even simple equity option
strategies can lead to drastically non-normal returns distributions, and therefore full repricing of these
options is required to adequately measure risk.
(a) Protective put option strategy
(b) Covered call option strategy
Figure 5: Simulated portfolio distributions
Equity Option Pricing
The pricing for equity options is based on the Black-Scholes model and the assumption of a normal distribution of equity returns.
For European-style options (those that can only be exercised at the maturity date), the price of a call option
with strike K can be computed with the Black-Scholes equation:
"
#
C(t) = P (t, T ) F (t, T )N (d1 ) − KN (d1 − v) ,
where P (t, T ) is the zero-coupon bond as seen today maturing on the same date as the option expires
d1 =
v=
and N (x) is the cumulative normal distribution.6
)
ln F (t,T
+v
K
√
,
v
-
T
σ 2 (s)ds,
t
5 The
option strike in this case was $2, 300.
that the volatility does not have to be constant in time; the Black-Scholes formula uses the total realized variance of
the underlying equity.
6 Note
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The situation is more difficult when the option can be exercised at any time up until maturity (Americanstyle options). These options have no fixed payoff condition; exactly where the owner exercises the option is
a dynamic function of the stock price and cannot be determined a priori.
One possible solution is to use a binomial tree to numerically solve for the price of an American-style option.
The binomial tree is a discretized approach where the final distribution is sampled on a finite set of points,
and the risk-neutral probabilities are used to determine the option price at an earlier time. This is then
iterated until the starting level of the equity is reached at time t.7
The benefit of this approach is that the continuation value can be calculated at every “node” and therefore,
the optimal exercise time for American options is handled naturally. The main drawback of this approach
as it relates to Monte Carlo VaR is that on each forward simulation, the number of calculations required for
each option increases the complexity of the calculation by an order of magnitude.
At FactSet, we have consequently used an approximate closed-form approach to value American options
based on an early paper by Barone-Adesi and Whaley [18]. In this paper, they examine commodity options
and determine an approximate closed-form solution based on the difference between the cost of carry for the
underlying and the risk-free rate.
The starting point is noting that the same partial differential equation (PDE) applies to both American and
European options and therefore to the American option premium as well. To see this, simply note that the
PDE can be written as a linear operator LC(S, T ) = 0, where:
L=
∂
1
∂2
∂
+ S 2 σ 2 (t) 2 + rS
− r.
∂t 2
∂S
∂S
Since CA (S, T ) and CE (S, T ) both solve the PDE, so does their difference representing the American option
premium, which we will denote ϵ(S, T ) = CA (S, T ) − CE (S, T ). The next step is to use the parameterization8 :
ϵ(S, T ) = g(T )f (S, g),
where g(T ) = 1 − e−rT . Barone-Adesi and Whaley then go on to show that an approximate solution to the
PDE is given by:
CA (S, T ) = CE (S, T ) + g(T )α1 S β ,
(9)
where β depends on the parameters of the stock (volatility, risk-free rate, and dividend yield) and α will be
determined by examining when the early exercise of an American option is optimal.
There are a number of bounds on the price of an option. First, the value of an American option is greater
than that of a European option since the owner has more optionality. The value of a call option cannot be
more than the value of the underlying stock. The value of an American call option can never be less than the
intrinsic value (S − K). This is subtly, but importantly, different than European options that have a lower
bound of (P (t, T )F (t, T ) − P (t, T )K). In the case of a non-dividend-paying stock, this lower bound is higher
than the American lower bound, implying that the owner of an American option on a non-dividend-paying
stock will never choose to exercise the stock earlier than maturity.
However, when the stock pays a dividend (or is a commodity with a cost of carry different than the riskfree rate), then this lower bound can be less than the European lower bound. In this case, early exercise is
possible. If we denote the S ∗ as the stock price at which early exercise first becomes optimal, then α1 can
be solved by setting the left-hand side of equation (9) equal to the exercise value (S ∗ − K) and equating the
7 This
8 This
is the approach taken in the Portfolio Analysis product.
parameterization seems to be ubiquitous in American option pricing; see [19].
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first derivative. This leads to the Barone-Adesi and Whaley approximation for the value of an American call
option:
. /β
S
CA (S, T ) = CE (S, T ) + α2
,
S∗
where:
α2 =
.
S∗
β
/0
"
#1
1 − e−qT N d1 (S ∗ ) ,
and q is the constant dividend yield of the stock. Barone-Adesi and Whaley demonstrate that S ∗ can be
solved in an efficient manner using an iterative method.
Equity and Equity Index Futures
Equity and Equity Index Futures (EIF) are contracts where one party has the obligation to pay the difference
between the reference security/index price on a specific future date and a price struck at the initiation
of the contract. EIFs are marked-to-market daily, which results in no net exposure to default of either
counterparty.
In the FactSet risk model, the underlying equity or equity index is exposed to the equity risk model factors.
In the Monte Carlo simulation, the price change of the underlying equity is simulated in the local currency
using the joint probability distribution of these equity risk model factors. The new price is then subtracted
from the contract price to determine the return of the index future in each scenario.
The Equity Volatility Surface
The second important consideration is how the volatility surface changes as the factors change on the forward
scenarios.
The implied volatility of options is seen empirically to strongly depend on the strike, a phenomenon known as
the volatility smile. For equities and equity indexes, the longer maturity option volatilities are monotonically
decreasing functions of strike (sometimes referred to as the volatility skew) whereas short-dated maturities
do not exhibit monotonicity, exhibiting decreasing volatility for strikes lower than the at-the-money (ATM)
strikes and increasing for strikes higher.
FactSet uses a statistical volatility surface model inspired by the work of R. G. Tompkins [20]. The volatility
surface is first fit with a five-factor regression in the variance and time variables:
σrel ≡
σ
= β1 ν 2 + β2 ν + β3 t + β4 νt + β5 ,
σATM
(10)
√
where ν ≡ µ/ t and µ = ln(K/S). Options prices are notoriously difficult to work with. Therefore an
extensive set of filters are used to ensure the quality of the inputs used for the factor estimation using
equation (10). The coefficients are fit nightly on the number of options.
5.2 Fixed Income Derivatives
For all fixed income derivative types, in contrast to the equity derivatives, FactSet employs the analytical
approximation based on Taylor expansion (see Section 2.1 of the Risk Models chapter) with appropriate
adjustments. The first- and second-order KRDs used in the Taylor expansion formula (3) are computed
using stochastic models that take into account the optionality embedded in various FI instruments.
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Eurodollar Futures (EDFs)
A Eurodollar Future is a contract where one party has the obligation to pay the difference between the
London Interbank Offered Rate (LIBOR) on a specific future date and a rate struck at the initiation of the
contract. EDFs are marked-to-market daily, meaning that the counterparty that has lost will put money into
a margin account, resulting in no net exposure to a default of either counterparty.
This daily margining also results in an extra step in the expected value calculation, known as the convexity
adjustment. This term arises9 from the benefit that the long receives from the positive correlation between
the value of the contract and rising interest rates. As interest rates rise and the long gains value, the short
must add additional money to the margin account precisely when the interest owed will be higher. Conversely,
as rates fall, the long must add money to the margin account precisely when rates are lower and the interest
required to be paid is smaller. This benefit to the long has monetary value.
The calculation of the convexity adjustment requires an interest rate model, and FactSet employs the Black
model of lognormal LIBOR rates.
Government Bond Futures
Government bond futures are run using a LIBOR Market Model (LMM) in Monte Carlo. Two hundred
paths are run and the cheapest to deliver (CTD) is determined on each path. The value of the CTD is then
discounted using the stochastic discount factor on each path, and the resulting present value from each path
is averaged.
Option on Government Bond Futures
The CTD bond is chosen as the CTD as of the valuation date. A closed-form solution of the Black model is
then used to price the option on this bond, which assumes that the bond price is a lognormal process.
Option on Bond
For over the counter (OTC) options on bonds, FactSet employs the Hull-White model, a mean-reverting
normal model of the short rate with a mean-reverting parameter of a = 0.05. The short rate is technically
the instantaneous rate of lending for an infinitesimal period of time; however, it can be proxied by the
overnight rate or more correctly just thought of as a mathematical construction representing the single
stochastic factor driving movements of the entire yield curve term structure.
The method is closed form; however, it is based on a trick by Jamshidian [21]. The reason special treatment
is required is that a bond option is not simply an option on a single cash flow; rather, the buyer of the option
is entitled to many cash flows, a fact that needs to be accounted for in the determination of the price. It
is not acceptable to simply treat the collection of cash flows as a collection of individual options, since the
buyer of the option only has one decision to make.
What Jamshidian shows is that there is a cash flow specific strike, related to the actual strike, such that all
cash flow options (or optionlet, to borrow from the term caplet) are exercised at the same time.
9 Technically,
this term arises from the fact that the expected value is not taken in the natural numeraire of the contract,
implying the contract is not a martingale.
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Interest Rate Swap
The interest rate swap (IRS) market is the largest and most important of all fixed income derivative markets.
According to the latest Bank of International Settlements survey,10 the IRS market has a gross market value
of $8, 915 billion.
The bulk of the IRS market is in “vanilla IRS,” which are instruments that swap a fixed stream of cash
flows for a stream of cash flows whose magnitude is set according to a prevailing interest rate, commonly
LIBOR; however, this rate could be an overnight rate such as the U.S. Fed Funds rate. The magnitude of
the cash flows is cτ N , where c is the relevant coupon, τ is the accrual fraction between the coupon start and
end dates, and N is the notional principal value (referred to simply as the notional). Even though the cash
flows are based on the notional, this amount is not exchanged for the simple fact that this would have no
economic impact.
Other types of swaps include:
• Cross-currency swaps. These can be either fixed-float, float-float, or fixed-fixed (sometimes called FX
swaps). For traditional cross-currency swaps, the notional is exchanged at swap initiation and maturity,
since the economic impact of the final exchange is non-zero based on foreign exchange rate movements.
Since large fluctuations in the foreign exchange rate can lead to a large economic impact, the market
has started trading “mark-to-market” (or resetting notional) cross-currency swaps. On each payment
date, one counterparty makes a notional payment whose magnitude is set so that the ratio of the
notionals is equal to the spot foreign exchange rate.
• Zero-coupon swaps. Interim cash flows are not paid but start to accrue interest, and all cash flows are
paid at maturity.
• Inflation swaps. The inflation-linked leg follows equation (6) to compute its return, with a small difference from an inflation-linked security issued on the same inflation index. The inflation asset-swap
spread component of the return is not necessary and is removed from the equation. In the EuroSovereign segment, the spread term is removed as well.
• Amortizing swaps. The notional changes over time according to an agreed-upon schedule.
Interest rate swaps are linear instruments, meaning their price can be determined solely based on the current
market forecast of interest rates, without the need for a stochastic model. For explicit information on how
FactSet prices IRS, please refer to the white paper [22].
Swaption
Swaption is an option to enter a fixed-to-float interest rate swap at a specified time in the future. The
nomenclature for an interest rate swaption is determined by the direction of the fixed leg of the underlying
swap. A payer swaption is the option to enter into a swap where the buyer pays the fixed leg of the swap,
which is beneficial when interest rates rise. A receiver swaption is an option to enter into a swap where the
long receives the fixed leg, which is beneficial when interest rates fall.
Swaptions have three important parameters:
1. Strike—the fixed rate associated with the underlying swap
2. Option expiry—the length of the option; the amount of time until the long must make the exercise
decision
3. Swap tenor—the length of the underlying swap
10 As
of H2 2016.
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For instance, a 2x10 payers swaption struck at ATM+100 is a contract where, in 2y time, the long has the
option to enter into a 10y swap with a fixed leg that is today’s at-the-money swap rate plus 1%.
A closed-form solution in the Black model of the swap rate, where the swap rate follows a lognormal process,
is used to determine the price of swaptions at FactSet. Note that even though a closed-form solution exists,
swaptions in the future horizons in the MAC model are priced using the KRD/KRC formalism.
Interest Rate Cap and Floor
Interest rate caps and floors are priced using a closed-form solution in the Black model of individual LIBOR
rates. Each cap can be deconstructed into a single period cap, known as a caplet. By virtue of this decomposition, each LIBOR rate can be modeled in isolation, and therefore, the correlated dynamics of neighboring
LIBOR rates do not need to be taken into account.
Callable Bond
Again the Hull-White model, a mean-reverting normal model of the short rate with mean reversion speed
a = 0.05, is used as the model to value callable bonds. The calculation method chosen is the trinomial tree,
which is a discretized backward induction technique where the short rate is determined on a set of nodes that
expand around the expected value, which can be set to exactly reproduce the term structure as seen on the
valuation date. The payoff at maturity is determined on each final timestep of the tree, and the continuation
value can then be determined backward until the valuation node is reached.
Credit Default Swaps, Indices, and Swaptions
Ever since the credit “big bang” in 2011, there has been a standardized method to price credit default
swaps (CDSs). FactSet uses this pricing methodology. Credit default swaps name a “reference entity” in the
contract, in which one counterparty would like protection from the adverse effects of a credit event on this
entity. This reference entity can be a corporation, like Radio Shack, or it could be a specific bond. CDS
contracts have two legs, similar to Interest Rate Swaps. The protection buyer makes fixed-rate payments to
the other counterparty, called the protection seller. The protection seller only makes a payment after the
occurrence of a “credit event,” which is precisely defined and adjudicated by the International Swaps and
Derivatives Association (ISDA). The amount of the payment depends on the amount that was able to be
recovered by the protection buyer as a result of the credit event (known as the recovery rate R). In the
pre-standardized CDS contracts, the fixed coupon that the protection buyer would pay was determined by
a breakeven analysis based on the market expectations of the probability of default and the recovery rate of
the reference entity. However, today’s current standard contract specifies that the coupon is either 100bps
or 500bps. This implies that contracts no longer have zero value at initiation and therefore, current CDS
contracts are quoted in “upfront fee” or “premium”; that is the present value of the two legs, again taking
into account the market expectations of the probability of default and the recovery rate of the reference
entity.
To compute the risk of CDS contracts, FactSet employs the DTS methodology similar to corporate bonds.
With this method there is an implicit assumption that the credit default swap spread equals the spread
of the reference entity and CDS basis is negligible. For single name CDS, the financial information of the
reference entity is used.
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Credit Default Index
A Credit Default Index (CDX) is a basket of CDSs, typically comprised of between 40 and 125 entries.
Originating in 2002, the products have grown in volume and in popularity to be amongst the most liquid instruments in the credit markets. The CDX market benefits from contract standardization and other
operational efficiencies that make them suitable for low cost exposure to the credit market.
CDX contracts are priced in the MAC model with the assumption that index basis is negligible. In this
manner the index can be treated as a weighted basket of individual CDSs when a look-through map is
available. When one is not, the CDX will be repriced using the Merton Contingent Claim Analysis (CCA)
framework ([47]), a model which assesses credit risk as a function of the firm’s assets and liabilities using an
options pricing framework. Where available, CDXs are proxied to a representative basket of like securities
using the framework’s proxy approach. Further details of this process can be found in Section 2.3 of the
Appendix.
Credit Default Index Option
CDS Index Options (Credit Default Swaptions, or CDXO) are European-style options that can be exercised
at a forward start date that gives the buyer the right to enter into a CDS index contract. CDXOs are
generally written with shorter time frames (one to 12 months) in mind. Options are generally written on
five-year on-the-run indices; however, other series are gaining in popularity.
There are two types of CDX options: payers and receivers. A payer option holder has the right but not the
obligation to buy protection on the underlying index at the strike spread level at option expiry. They are
entering the contract to pay the fixed coupon of the CDX contract. A receiver option holder has the right to
sell protection at the strike spread level. Investors typically buy out of the money payer options to protect a
long credit portfolio in the event of dramatic credit spread widening; in other words, a holder of the payer
option is short credit risk. Another strategy often employed is writing a payer option to reduce the cost of
holding a short position in the underlying index. If a reference entity underlying an individual CDS defaults
prior to the maturity of the contract, the swaption is knocked out and the contract nullified; however, options
written on a CDS index do not have any knock-out feature. Because of the lack of a no-knock feature, the
price of the option needs to take into account the expected value from index spread payoff as well as the
number of defaults at expiry.
The MC MAC model uses a modified form of the Black model to price CDXO, with the spread being scaled
by a forward starting intrinsic annuity. Assumptions are made that the strike of the CDX option is fixed at
today’s level, that coupons are paid continuously, that CDX index basis is negligible, and finally that the
hazard rate is constant for each CDS (though subject to change per simulation). The model takes the form
of:
Voption (t, Te , Tm ) = AI (t, Te , Tm )Black(SID (t, Te , Tm ), K̃, Te − t, σ̂, χ)
where Te is time to expiration of the option, TM is time to maturity of the option, σ̂ is volatility implied
from the options price, χ is the sign switch variable for the option type (+1 for payer and -1 for receiver),
AI (t, Te , Tm ) is the forward index annuity used to represent the present value of the default risk, calculated
as:
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AI (t, Te , Tm ) =
J
!
ωj Aj (t, Te , Tm )
j=1
=
J
!
ω j Et [
P (Te , tes )
Aj (Te , Tm )Iτ >Te ]
βt (Te ))
ω j Et [
P (Te , tes )
Iτ >Te
βt (Te ))
j=1
=
J
!
j=1
=
J
!
ω j Et [
j=1
-
Tm
P (Te , u)e−λj u du]
Te
2 −erTe 32
3
P (Te , tes )
Iτ >Te
e−(rTe +λj )Tm − e−(rTe +λj )Te ]
βt (Te ))
rT e + λ j
where wj is the weight of CDSj in the index; rTe is the forward rate as of the option expiration, λj is the
Sj
bondOASi,j
hazard rate as inferred by the credit triangle λj = 1−R
=
and R is the recovery rate modeled
1−R
by the user. With the assumption that the CDX index basis is negligible, the forward starting spread of the
option is then modified to be:
SID
=
$J
1
j=1 βt (Te )) Iτ >Te ωj Aj (Te , Tm )Sj (Te , Tm )
$J
j=1 Iτ >Te ωj Aj (Te , Tm )
Because the hazard curve is considered flat, modifying the strike to account for the annuity scale is not
necessary. Instead, the model only needs to adjust for the strike when the CDXO is quoted in price terms
and not spread terms, as seen in indices such as the CDX.NA.HY and CDX.EM.
(Kspread − C)A(Te , tM )P (T0 , Tes )
A(Te , tM ))
(K − C)AI (t, Te , Tm )
=C+
AI (t, Te , Tm )
= C + (K − C)
Spread Strike : K̃spread = C +
=K
P rice Strike : K̃spread = C +
1 − Pindex
100
AI (t, Te , Tm )
where K is the option strike and C is the CDX coupon rate.
Currency Forward
Currency Forwards require no modeling and are priced as two zero coupon bonds in different currencies
where the ratio of the two principals is set equal to the spot FX rate at the contract initiation date.
Currency Futures
Currency Futures (CF) are contracts where one party has the obligation to exchange one currency for another
on a specified future date at a rate specified at the initiation of the contract. In FactSet, currency futures are
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loaded as “physical” security in the contract’s base currency. In the universe options of Portfolio Analysis, the
user can choose to then “show currency exposure” that splits the position into two, with one leg representing
the base currency and the other the term currency. This option should be used for the purpose of risk
analysis.
Currency Options
Currency Options are securities that give the purchaser the right, but not the obligation, to exchange money
denominated in one currency into another currency at a rate agreed at the initiation of the contract. They
are typically traded OTC and used to hedge currency exposures. To value currency options, FactSet uses a
modified version of the same pricing algorithms as is used for equity or index options that are detailed in
Section 5.1 of this chapter, which takes into account the risk-free rate of each currency. FactSet does not
currently have a currency volatility surface model and computes implied volatility based on the contract price
supplied by the client. FactSet then reprices the option in the Monte Carlo simulation process by holding
the implied volatility constant. If no price is supplied by the clients, the historical volatility is used.
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6 Return-Based Model
At a time when markets are trending in ways to incorporate diversification among portfolios via nontraditional asset types, regulatory oversight has mandated such portfolios be viewed not from an esoteric
perspective but based on processes designed to measure variance on all assets. The return based risk (RBR)
model is designed to fill the niche in the asset management space where a portfolio manager may have access
to a return stream of an asset, but the underlying holdings may not be known.
In other MAC risk models, the exposures to model factors are known for a predefined universe of assets. In
the context of the RBR model, a dynamic approach is taken to modeling the risk of securities—exposures
are calculated at runtime from a regression against the predefined factors within the risk model. The model
employs observed market returns as time series of risk factors, alongside a proprietary algorithm to model
variance of almost any asset type. The RBR risk factors encompass a wide range of asset classes, including
equity, fixed income, hedge funds, commodities, and other alternatives, all based on market indices. The
range of these asset types can be quite diverse, so a balance between an all-encompassing range of factors
spanning each asset class and limitations on data availability, frequency, methodology, and integration must
be chosen. As a result, more than 100 time-series factors have been selected as a representation of client
holdings, including 17 hedge funds from the SPAR HFR database, six real estate benchmarks, 11 equity
indices, 18 fixed income indices, 15 indices representing global interest rates, seven commodity indices,
including timber, and 26 currencies. (The full universe of the RBR factors is listed in Appendix 5.)
The model makes very little assumption of the source or distribution of these returns. The only requirement
is that the known return series must be daily and have a minimum of 180 days to calculate risk, though
in future iterations the model will accept monthly returns series. The model is designed to work based on
limited information and should only be used when more detailed alternatives do not exist.
Regressing the returns of an asset against all of the model factors surely would introduce multicollinearity in
the model, leading to unstable results that are difficult to interpret. Since we are not making assumptions on
the underlying returns series and cannot limit ourselves to a small subset of factors given the variety of assets
employed by portfolio managers, a programmatic approach is necessary to choose the correct predictors for
each asset. FactSet has developed a clustering algorithm based on the Hierarchical Agglomerative Clustering
approach (see, for example, [23], Chapter 17) to allow this process to occur at runtime. Its goal is to select
a set of statistically significant factors with a high correlation to the asset, while at the same time avoiding
inter-factor correlation. The result is a parsimonious selection of independent factors for a wide range of
assets that are generally uncovered by other models.
The factor selection algorithm is based on Spearman’s rank correlation coefficient between factors and between factors and asset return time series. It chooses a subset of factors that have the highest correlation
with the asset, then defines clusters of highly correlated factors within the subset and uses a representative
factor from each cluster in the final model to ensure minimal cross-correlation among factors and thus avoid
multicollinearity in model regression. The final step of the algorithm is the stepwise regression of the asset
return time series on all chosen factors. The Akaike information criterion corrected for sample size is used to
rank the regressions, and the subset of factors with the highest AIC and significant values of the T-statistics
is selected for the asset. In the end, the asset can load on a maximum of eight factors and a minimum of
one. Typically, the asset ends with sensitivities to two to four statistically significant factors with a fairly
large adjusted R2 .
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Evaluation of Risk
The FactSet MAC risk framework employs various risk measures such as Tracking Error Volatility (TEV),
Value-at-Risk (VaR), or Expected Tail Loss (ETL) to analyze the risk of a portfolio on an absolute basis
or relative to a benchmark. TEV—the volatility of the excess return of the portfolio vs. benchmark index—
provides the measure of the variability of the excess return. The V aRα associated with probability α (e.g.,
α = 95%) is the lower bound for the loss incurred by a portfolio with a probability of 1 − α, and provides a
measure for tail risk of a portfolio. For example, if a portfolio’s V aR95 (at a 95% confidence level) is reported
as 10 bp, then there is a 95% probability that the portfolio loss will be smaller than 10 bp (the portfolio
return will be greater than −10 bp). Alternatively, one could say that the portfolio’s return is expected to be
worse than −10 bp 5% of the time. VaR measures only a threshold value and does not provide information
about the extent of the losses beyond that. ETL overcomes this shortcoming by measuring the average loss of
all the worst-case scenarios beyond the threshold, which gives a better representation of the potential losses
of the portfolio at a given probability. All these risk measures are derived from the probability distribution
of the portfolio returns estimated by the MAC risk model.
The FactSet risk model represents the return of each security through changes in systematic risk factors
and the security’s exposures to each of them. Each systematic factor is treated as a random variable, and a
portfolio’s return distribution is ultimately modeled through the joint distribution of these variables. Using
the wealth of data available at FactSet, we employ statistical techniques to generate time series of historical
observations for each of these factors. These time series, in conjunction with volatility and correlation models,
serve to compute the covariance matrix of the factors’ joint distribution. The factor covariance matrix is
used to predict the volatilities and correlations of the factors, and thus represents a key component of the
risk model.
Having specified the behavior of risk factors and their correlations, the risk model can estimate the distribution of security and portfolio returns in two different ways. In the parametric approach of LMAC, risk
measures, e.g., TEV, VaR, ETL, etc., can be computed using closed-form expressions. In the Monte Carlo
approach of MC MAC, the process is more involved. Multiple samples from the factors’ joint distribution
are generated. For each simulated scenario, the factor returns are used to compute the individual security
returns. For linear models such as Equity and Commodity models, the factor returns are multiplied by the
corresponding sensitivities of each security. For nonlinear FI models, the pricing function is applied to the
simulated factors to obtain the FI security returns. The simulations are repeated several thousands of times
to generate an entire distribution of possible returns for each security in a given portfolio. Then the results
of all scenarios are aggregated to get the entire return distribution for the portfolio. Once we have generated
the entire distributions for returns, we can easily calculate the risk measures at any chosen confidence interval. Whatever approach is used, the risk model also provides the breakdown of portfolio risk into additive
contributions of various risk factors, allowing portfolio managers to quickly identify sources of risk in the
portfolio and see if it matches their desired risk profile.
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1 Covariance Matrix Computation
The purpose of any risk model is to provide a prediction, a forecast, of the future volatility of portfolio returns,
rather than just to model the variability of the past returns. Factor risk models rely on the forecasted joint
distribution of factors to estimate the risk metrics of a portfolio. This distribution is defined through the factor
covariance matrix that combines forecasts for the volatilities and correlations of the model factors, and thus
represents a critical component for constructing a high-quality risk model. The need for forecasting volatility
and correlations, not just estimating them based on past return data, determines the main components of
the covariance matrix model employed in the FactSet MAC risk framework.
The difficulty of creating a good volatility (and correlations) forecasting model becomes apparent once
one realizes that volatility is inherently unobserved and evolves more or less randomly through time. (In
mathematical terms, the volatility is a latent stochastic process.) The recognition of the time-varying nature
of volatility leads to the volatility models that take conditional flavor. In particular, having a return process
with observations available at equally spaced discrete points in time as rt ≡ r(t), t = 1, 2, ..., we define its
conditional variance as:
0
1
2
2
σt+1|t
= V ar rt+1
|Ft ,
(11)
where Ft is the information set that reflects all relevant information
4 2 5 through time t. This conditional variance
does not necessarily equal the unconditional variance V ar rt+1
, but it effectively incorporates the most
recent information available up to the last observation time t into the one-step-ahead forecasts for time t + 1.
It is also very important to note that some information in the set Ft is not actually available—since the
volatility is a latent process and we do not have observations of the volatility at previous time points—and so
the “true” volatility, conditional or not, cannot be determined exactly, but only extracted with some degree
of error. Effectively, the latent nature of the volatility process turns the volatility estimation problem into a
filtering problem on the set of past returns. But under the simplifying assumption that we are dealing with
full information sets, so that the conditional volatility is directly observable, the equation (11) provides an
unbiased estimate of the future volatility.
The simplest way to provide a low biased estimate of time-varying volatility based on the actual return data
is to use the exponentially weighted moving average (EWMA) filter that effectively became the market-wide
standard for risk models since it was first employed by JPMorgan RiskMetrics [24]. This method is also used
in the FactSet MAC.
Volatility and correlation forecasting methodology is an important component of any factor risk model.
At the same time, for extensive multi-asset class models such as FactSet MAC, another key challenge in
estimating the factor covariance matrix lies in the dimensionality of the problem. A multi-asset model can
easily contain more than 1,000 factors, requiring an estimation of more than a million independent elements
of the matrix. An estimator of such a matrix is susceptible to noise and spurious relationships that are
unlikely to persist out-of-sample. If the covariance matrix is computed naively, by simply computing the
pairwise correlations of the time series of all the factors taken over a short period of time, then it is likely
to be extremely ill-conditioned. For instance, if the number of time observations is less than the number of
factors in the model, the matrix will be “rank deficient,” meaning in practice that it can be used to construct
apparently riskless portfolios. Thus, an estimator of a covariance matrix obtained from the observed time
series of risk factors has to be regularized—a special procedure has to be applied to filter the excess noise in
the matrix and ensure that the matrix is symmetric positive definite (i.e., well-conditioned).
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1.1 EWMA Model
This widely used method to forecast the conditional volatility of a factor return process rt employs the
moving average estimator for the conditional volatility while assigning more weight to recent observations
than to the observations farther in the past. In the EWMA model, the importance of past observations
decreases smoothly as time passes with the speed of this decrease defined by the model decay constant
δ:
M
(1 − δ) ! i−1
2
σt+1|t =
δ (rt−i+1 − r̄t )2 .
1 − δ M i=1
Here r̄t is the weighted mean of the factor return time series.1
In the EWMA model, the value of δ determines the trade-off between the responsiveness and persistence
of the estimator. Short half-life results in a very responsive estimator that can incorporate large volatility
shocks. On the downside, they tend to decrease too much in low-volatility periods, thus being surprised when
that period comes to an end. Estimators with longer half-lives are not quite as responsive, but they do not
decrease as much in low-volatility periods. In the standard MAC daily model, the value of δ = 0.9944 is
used, corresponding to approximately 125 day half-life of the decaying sequence δ i−1 .
The effect of decaying weights can be interpreted from the perspective of memory we impose on past data.
For the EWMA estimator, the memory slowly fades as time passes. The estimator cutoff value M is chosen
such that the memory of the farthest-away sample of the return time series was practically nonexistent. In
practice, we chose the value of M = 800, which provides a long enough time series of data from the EWMA
memory point of view and, at the same time, supplies enough time points in the time series to construct a
robust covariance matrix (see Section 1.2 below).
To construct a covariance matrix, the time series of factors from all MAC models (e.g., equity and commodity
factor returns, spread, and interest rate factors) are combined in a matrix F, where the factors form columns
and the rows run along the time dimension. In theory, the covariance matrix estimator Σ can be computed
directly from the factor matrix F as:
2
3T
Σ = W1/2 F W1/2 F,
(12)
where W is the diagonal matrix with exponentially decayed weights at the diagonal. However, sometimes
historical factor time series do not have the same length, making it impossible to use equation (12) directly.
In particular, this happens when new factors are introduced into the model (for example, a yield curve from
a country that was not available before a certain date). To be able to incorporate new factors into the model
as soon as possible, the factor can be included in the model when it has 250 days of history (this length
still provides enough data from the EWMA memory point of view). This makes it necessary to compute the
factor variances and covariances pairwise, column by column. Using factor time series of variable lengths does
result in undesirable properties of the covariance matrix as a whole, but these negative effects are mitigated
to a large degree by the regularization procedure applied to the resulting matrix.
The EWMA covariance matrix is computed daily and stored in the FactSet database. The database can be
used to run historical computations of portfolio risks, as well as scenario analysis based on specific market
correlation events. The MAC model can also be run with custom historical look-back and decay parameters
if required.
1 For
daily time series of certain factors, such as interest rate changes, the mean can be assumed to be equal to zero.
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1.2 Covariance Matrix Regularization
The methodology used by the FactSet MAC model to obtain the robust estimator of the model factor
covariance matrix is based on the random matrix theory. (The detailed description of the methodology can
be found in [25]). The advantage of the proposed method is that it not only guarantees that the resulting
estimator will be positive definite, but also reduces the amount of noise in the estimator and minimizes the
differences between the estimator and the (generally unknown) true covariance matrix of the model.
Random matrix theory (a good review of the theory can be found, for example, in [26]) allows one to derive
the formula for the distribution of eigenvalues of a sample correlation matrix obtained from independent,
identically distributed random time series. It shows that all eigenvalues of the matrix are concentrated
in a well-defined region between certain values of λmin and λmax —the support region of the eigenvalue
probability distribution. Since the random processes we are correlating are independent, the true correlation
matrix should be diagonal and only have one eigenvalue. Thus, the width of the support region effectively
identifies the range of errors in the matrix estimator due to the limited sample size.
The sample correlation matrix of the risk model factor time series will have the distribution of eigenvalues
that is wider than in the case of a pure random matrix as there will be a number of large eigenvalues lying
outside of the support region of a random matrix. A simple and intuitive assumption to make is that the
components of the correlation matrix that are defined by the small eigenvalues within the random matrix
support region (i.e., are orthogonal to the space of the large eigenvalues) are dominated by noise. In other
words, only the eigenvalues of the sample matrix that lie outside of the support region [λmin , λmax ] contain
information relevant to the actual correlation matrix.
The underlying assumption we use to construct the optimal estimator is that the eigenvectors of the optimal
estimator are the same as the eigenvectors of the sample
estimator (for detailed discussion see [27]). This assumption allows us to construct the optimal matrix by doing
spectral decomposition of the original estimator, adjusting the eigenvalues, and reconstructing the optimal estimator using the same eigenvectors. This adjustment is
similar to principal component analysis, when only the
principal components that carry information are used,
and the ones that carry only the noise are discarded. In
the same spirit, we reconstruct the optimal estimator of
the correlation matrix using only the eigenvalues outside
of the RMT support region λ > λmax . After the matrix
is reconstructed, we replace the diagonal elements of the
new estimator with 1, which ensures that the estimator
has the properties of a correlation matrix.
500
400
300
200
100
0
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
600
400
200
0
-0.15
Figure 6: Distributions of correlation values in the
sample matrix before (top) and after (bottom) regularization
To illustrate the technique, we constructed an example (100 × 100) matrix with all correlations equal to 0.1.
We use this matrix to construct the sample matrix estimator using a (100 × 1000) set of random samples
drawn from standard normal distribution. The resulting correlation coefficient estimates vary between −0.1
and 0.27. The top panel of Figure 6 shows the distribution of the elements of the sample matrix (excluding
diagonal elements) before regularization is applied. While the distribution of the true correlation matrix
should be a single vertical line at 0.1, the estimation noise results in a distribution approximately centered
around the true value with a standard deviation of about 0.05. The bottom panel of that figure shows the
same distribution after we applied the regularization procedure described above. Clearly, the width of the
distribution is significantly reduced.
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2 Monte Carlo Risk Measures
2.1 Simulating Distribution of Portfolio Returns
The Monte Carlo process of estimating the portfolio risk measures starts by simulating the future distribution
of the risk model factor returns. The FactSet MAC risk model assumes that all the model factors are jointly
distributed with a multidimensional normal distribution described by the factor covariance matrix. The
Monte Carlo process generates a number of samples from the factor joint distribution that are later converted
into a set of samples from the return distribution of the portfolio. A standard Monte Carlo run generates
K = 5, 000 samples for each factor of the model. Then the appropriate pricing function is applied to each
factor scenario, resulting in a number of samples from the joint distribution of contributions from each factor
to the return of a given security.
S Securities
For a portfolio of S securities and a risk model of F
factors, the result of the Monte Carlo simulation can
r1,1,1 . . . . . . . . . r1,K,1
rs
. . . . . . . . . r1,K,1
r1,1,1
be visualized as a 3D matrix that has F vertical slices;
to
. . . . . . . . . r1,K,1
r
c
1,1,1
Fa r r1,1,1. .... ..... . .. . .. . .. . .r r1,K,1 . ..
each slice is a matrix of S rows and K columns (FigF
1,1,1
1,K,1
. . .
. ... . . .. . . . . r1,K,1
r1,1,1
ure 7). Each row of the slice matrix is the set of returns
.. .. . .
. .... .. . .. . . . . . r1,K,1
r1,1,1
r1,1,1 . .... . . . . . . .. . r1,K,1 .. . . .
for one security/one factor, obtained on K Monte Carlo
. ... . . . . . . r1,K,1 .. . . . .
r1,1,1
.
. ... . . .. . . .. . . r.1,K,1
r
.
1,1,1
samples. This matrix can be aggregated along different
.. .. . . .. .. r. S,1,1
. .... .. . .. . . . ... . .. rS,K,1
.
.
.
.
.
.
.
.
.
.
.
r
r
.
.
.
.
S,1,1
S,K,1
.
.
directions to obtain the distribution of returns of indi.. .. . .. . rS,1,1 . .... .. . . . . . ... . . rS,K,1
. . .. r. rS,1,1. . .. . .. .. .. . .. .. .. ... .r. rS,K,1
vidual securities rsk (a set of k = 1, ..., K samples from
S,1,1
S,K,1
. . rS,1,1
. . . . . . . .. . .. rS,K,1
th
.. .. r. S,1,1
return distribution of the s security) or the distribu. . . . . . . ... . .. rS,K,1
. rS,1,1 . . . . . . . . . . rS,K,1
. . . . . . . . . rS,K,1
tion of returns from individual risk factors rf k (a set
rS,1,1
. . . . . . . . . rS,K,1
rS,1,1
of k = 1, ..., K samples from return distribution of the
f th risk model factor) or aggregated along both security
K Samples
and factor dimensions to obtain the distribution of total
portfolio returns rkp .
Figure 7: Results of Monte Carlo sampling with K
samples for a portfolio of S securities, using the model
The generated portfolio return distribution is used to of F factors
estimate the total portfolio risk measures, while the distributions of security and factor returns allow the model
to produce marginal contributions to risk. Below we consider the process of estimating VaR and component
VaR as an example.
2.2 Estimating Portfolio Risk Measures
The VaR associated with probability 1 − α (e.g., α = 1%) is the lower bound for the loss incurred by a
portfolio with probability α:
α = P (rp ≤ −V aRα ) =
−V
-aRα
fr (x)dx,
−∞
where rp is the portfolio return and fr (x) is the return distribution function. This equation can be rewritten
as an expectation of the indicator function defined as
6
1 if x ≤ a
I(x ≤ a) =
.
0 if x > a
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With the definition above, α is expressed as
α=
-∞
−∞
I(x ≤ −V aRα )fr (x)dx = Er [I(x ≤ −V aRα )].
The expectation representation can be used to compute VaR when we generate a sample of independent and
identically distributed portfolio returns rkp , k = 1, ..N when using the Monte Carlo method. In this case, the
estimator for the expectation is:
α=
N
1 !
I(rkp ≤ −V aRα )
N
k=1
and in this form, it can be used to find VaR from a sorted list of sample returns given a value of α. If the
value of VaR equals the nth return in the sample V aRα = −rnp , then for every sample with k ≤ n in the
sorted list the indicator function is equal to one, and for every sample with k > n the indicator function is
zero. Thus, the confidence level for that value of VaR is:
αV aR =
So we can just find the value n in the list such that
−rn as an estimate of VaR.
n
.
N
is closest to the given value of α and use the sample
n
N
2.3 Marginal Contributions to Risk
The top-level risk analysis relies on aggregate portfolio risk measures such as Value-at-Risk (VaR) or Expected
Shortfall (ES). Further decomposition of the risk measures allows the portfolio manager to understand the
sources of risk in terms of specific market factors. For capital allocation, measurement of risk-adjusted
performance, developing hedging strategies, and in general, understanding the impact of different risk factors
and components on portfolio risk, it is useful to allocate the risk to elements of the portfolio based on their
marginal contributions to total risk. If we represent the return of a portfolio as a weighted sum of return
contribution from different components (be it individual securities or risk model factors), the marginal
contribution to portfolio VaR from component i, M V aRα,i , can be defined as the change in portfolio VaR
resulting from a marginal change in the ith component position:
M V aRα,i =
∂V aRα
.
∂wi
(13)
This metric allows portfolio managers to find the components that can be used to significantly revise the
overall risk of the portfolio with minimal change to capital allocation.
The marginal contributions to VaR can
$in turn be used to define the additive VaR components that will sum
to the total portfolio risk as V aRα = i CV aRα,i . It can be shown that VaR can be decomposed as:
V aRα =
n
!
i
wi
∂V aRα
,
∂wi
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and thus the additive component of VaR in terms of the marginal contribution to VaR can be defined
as:
CV aRα,i = wi M V aRα,i .
(14)
It can be shown that the marginal contribution to portfolio VaR, defined by equation (13), is the conditional
expectation of the component return, conditioned on the value of V aRα :
M V aRα,i =
∂V aRα
= −E[ri |rp = −V aRα ].
∂wi
(15)
Here the index i can represent either the security index s (when the marginal contribution from individual
security is computed) or the factor index f (when contribution from individual risk factor is computed). It
is not possible to use this formula directly to estimate marginal VaR values in the Monte Carlo process—the
sampling variability of the estimate is large and will not go down as we increase the number of samples for
the simulation. The problem is that the contribution to VaR from a given component depends on the single
return sample that happens to be the αth return observation for the portfolio (the simulated V aRα ). The
contribution to VaR depends on that single return observation in such a way that the sampling variability
does not change with the number of trials in the simulation.
For example, an estimator for a conditional expectation (15) of the sth security return in this case will
be:
E[rs |rp = −V aRα ] =
$K
p
k=1 rsk I(rk = −V aRα )
.
p
k=1 I(rk = −V aRα )
$N
(16)
Unfortunately, if we just generate a single sample of K values of rsk , the sums in the estimator for conditional
expectation will only have a single non-zero term. In other words, we will have a single MC sample in the
region of interest.
We can remedy this situation by generating a number of samples of K portfolio returns such that we will
have multiple realizations of rkp = V aRα . This, however,
inefficient. Instead, we can relax the
7 is extremely
7
condition in the expectation (16) from rp = −V aRα to 7rp + V aRα 7 < ε so that the estimator of the marginal
contribution to VaR becomes:
M V aRα,s
7 p
7
$K
7r + V aRα 7 < ε)
7
7
r
I(
sk
k=1
k
= −E[rs |7rp + V aRα 7 < ε] = − $N
.
7 p
7
7
7
k=1 I( rk + V aRα < ε)
(17)
7
7
Because the averaging region of formula (17) 7rp + V aRα 7 < ε is located in the tail of the portfolio returns
distribution, the weighted average of the conditional mean returns will be less negative than the portfolio
quantile return. In other words, the estimator (17) is biased and the weighted sum of marginal contributions
is expected to be less than the estimated portfolio VaR:
n
!
wi M V aRα,s < V aRα .
i
To correct that, we introduce the normalization factor ω defined as:
ω = $n
s
V aRα
,
ws M V aRα,s
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and define the adjusted estimator for marginal contribution to VaR as
M V aRα,s = ωM V aRα,s .
The corresponding estimate for the CVaR follows from equation (14). Due to the adjustment factor ω,
the sum of the CVaRs exactly equals the initially estimated overall portfolio VaR, as required by CVaR
definition.
The introduction of the normalization factor ω is very similar to the method of control variates in Monte
Carlo estimations. The method relies on knowing the expectation of an auxiliary simulated random variable
called a control. The known expectation is compared with the estimated expectation obtained by simulation.
The observed discrepancy between the two is then used to adjust estimates of other (unknown) quantities
that are the primary focus of the simulation. In our case, the portfolio VaR is used as a control variate
for component VaR estimators. A more detailed analysis of the adjusted estimator and justification of the
normalization procedure can be found in [28].
The size of the averaging region ε in formula
σES
aR
(17) has to be chosen to have a sufficient number
α V aRα σV aR VσVaR
% ESα σES ES
%
α
α
of points in the neighborhood of VaR to bring
90
4.79
0.13
2.66
6.55 0.11
1.60
the variance of the Monte Carlo estimator down.
95
6.16
0.14
2.23
7.69 0.10
1.26
At the same time, the width of the region has
97
7.06
0.12
1.63
8.44 0.10
1.16
to be restricted to limit the variability of the
99
8.61
0.13
1.55
9.92 0.13
1.29
portfolio return within the neighborhood. The
Monte Carlo process can provide a measure of Table 4: Estimators and errors for VaR and ES of Barclays
the variance of the estimator. From the same EUR Aggregate Index for different confidence levels α
computation, one can obtain both an estimated
result and an objective measure of the statistical
uncertainty in the result. In our case, we use multiple Monte Carlo simulations of a portfolio VaR and its
components at different confidence levels with K = 5000 samples as described above. We run each simulation
multiple times and record the means and standard deviations of estimated VaR and MVaR values. We use
this data to evaluate the adequacy of the number of samples for our purposes and to establish an acceptable
averaging region ε for computing marginal contributions to VaR that provide reasonable balance between
bias and variance of the estimators.
Table 4 shows the estimated Value-at-Risk (V aRα ) and Expected Shortfall (ESα ) values for different confidence levels α computed for Barclays
EUR Aggregate Index. Also shown are Monte Carlo standard deviations
of the estimators σV aR and σES and the corresponding relative error for
each estimator, expressed as a percentage of the estimator itself. It is clear
that with the employed number of samples (K = 5, 000), the error of both
risk measures never exceeds 3%.
The index we use for testing purposes contains around 4, 000 securities.
We analyze the performance of the Monte Carlo algorithm by comparing the distributions of the errors of the components corresponding to
each security obtained at different values of the averaging region width
parameter ε (eq. (17)).
α
90
95
97
99
Mean
1%
5%
0.15 0.07
0.12 0.06
0.1 0.04
0.08 0.05
1%
0.11
0.08
0.08
0.06
σ
5%
0.05
0.04
0.03
0.04
Table 5: Parameters of error distributions
Table 5 shows the mean and standard deviation of the distribution of errors obtained with different values of
averaging region width parameter ε (1% and 5%) for different VaR threshold probabilities α. With the value
of ε equal to 5% of the total number of samples, the center of the error distribution is located at around
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6%, while its width never exceeds 5%. Thus, the value of ε = 5% is sufficient to keep the errors of the VaR
components within a 10% range.
To estimate the portfolio’s excess return (tracking error) distribution, we also perform the same simulation for
the portfolio’s benchmark and take the difference in returns (portfolio minus benchmark) for each simulation
run. The simulated distributions for total return and tracking error are used to calculate other risk measures
(TEV, ES) and their marginal contribution and components in a similar manner. A more detailed description
of the methodology and test results is provided in [29].
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3 Parametric Risk Measures
3.1 Portfolio Volatility (TEV)
A linear factor model expresses return rj of a security Sj as a weighted sum of factor returns fi , i = 1, ...M ,
where M is the number of factors in the model.
rj =
M
!
(18)
lij fi + εj .
i
Here lij is the loading (sensitivity) of the security j to the factor fi , and εj is the idiosyncratic component
of the security return (the portion of the return not explained by systematic factors).
The portfolio return is, in turn, a function of individual security returns and the market weights of these
securities in the portfolio. For a portfolio of K securities with weights wj , we can write the portfolio return
as
K
!
rp =
w j rj .
j
Taking into account the factor model decomposition of the security return rj (equation (18)), the portfolio
return can be written as
⎡
⎤⎡ ⎤
l11 . . . l1M
f1
⎥⎢ . ⎥
4
5⎢ . .
.
T
T
T
T
⎢
⎥
⎢
..
.. ⎦ ⎣ .. ⎥
rp = w1 . . . wN ⎣ ..
⎦ + w E = w × L × f + w E.
1
lN
...
M
lN
fM
N ×M
1×N
M ×1
where L is the portfolio loadings matrix with ith column representing the vector of factor loadings for the
ith security, and E is a vector of idiosyncratic components εj . This formula illustrates the main feature
of the linear factor model—its ability to describe the return of a portfolio using a small set of factors
f that are designed to capture systematic market movements and relate them to the specific features of
individual portfolios through the portfolio loading matrix L. The idiosyncratic return wT E is the residual
component that cannot be explained by the systematic factors. In a factor risk model, idiosyncratic returns
are uncorrelated among themselves and also are not correlated with the factor returns. Therefore, correlations
across different components of the portfolio are defined by the correlation between factors of the model.
If the portfolio manager is interested in benchmark-relative return and risk of the portfolio, the above formula
can easily be rewritten for the excess return (tracking error) by replacing the vector of portfolio weights with
the difference between the portfolio and benchmark weights
rpexcess = (w − wb ) × LT × f + (w − wb ) E.
T
T
(If a security is not included in the portfolio or the benchmark, its weight in the portfolio or benchmark
vector of weights is set to zero.) If the portfolio is evaluated on an absolute basis, it is equivalent to the case
of a cash benchmark, and thus a concept of tracking error and tracking error volatility (TEV) is applicable
in both cases.
The volatility of the portfolio return, or tracking error volatility, under this model can then be computed
as
>
T
T
T EV = (w − wb ) LT ΣL (w − wb ) + (w − wb ) ΣE (w − wb )
(19)
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where Σ is the risk factor covariance matrix, and ΣE is the covariance matrix of idiosyncratic returns. Since
the idiosyncratic components of the model are uncorrelated, the matrix ΣE is diagonal. In a large portfolio,
contribution of the idiosyncratic components to the portfolio volatility is decreasing proportionally to the
number of securities in the portfolio (diversification effect), and the systematic part of the return volatility
dominates equation (19).
Thus, the TEV can be understood as consisting of two components—the systematic component that stems
from the systematic movements of the market and is captured by the factor model:
>
T
T EVsys = (w − wb ) LT ΣL (w − wb )
and the idiosyncratic component that represents the aggregated effect of specific risks of individual securities,
T EVid .
>
T EVid =
(w − wb ) ΣE (w − wb ).
T
Note that while idiosyncratic and systematic variances are additive, the two TEV components are not. The
additive contributions to risk can be defined as the ratio of the corresponding variances to the total TEV of
the portfolio.
CT EVsys
=
CT EVid
=
2
T EVsys
T EV
T EVid2
T EV
(20)
(21)
where CT EVsys is the systematic contribution to TEV and CT EVid is the idiosyncratic contribution. It is
easy to see that they add up to the total risk of the portfolio:
T EV = CT EVsys + CT EVid .
(22)
3.2 Marginal TEV and Factor Contributions to TEV
Apart from giving a portfolio manager a fast and simple way of estimating total portfolio risk, linear risk
models can provide detailed insight into the structure of portfolios. By estimating the contributions of each
factor to the total risk measure, the models help analyze a portfolio’s exposure to major risk sources and
guide portfolio managers in constructing, optimizing, and hedging their portfolios.
This analysis starts from the computation of portfolio exposure to individual factors. The exposure λi to the
risk factor fi can be expressed as
K
!
λi =
(wj − wjb )lij
j
Each of the portfolio exposures to an individual factor multiplied by the volatility of the factor can serve as
a measure of the risk of this exposure in isolation (as if all other risks of the portfolio were fully hedged).
Together, these standalone contributions represent the risk of the portfolio if all sources of risk were independent:
>!
>!
ST EV =
λ2i σi2 =
ST EVi2 .
Here σi2 are the variances of individual factors, i.e., the diagonal elements of the factor covariance matrix Σ,
and ST EV and ST EVi designate standalone TEV of the portfolio and the standalone factor contributions
respectively.
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Obviously, the standalone contributions do not allow the portfolio manager to understand interactions between different sources of risk. This can be done only if correlations are also taken into consideration, i.e., if
we consider contributions to the total portfolio volatility that take into account the full covariance matrix
Σ, not just its diagonal elements. In this case, the total variance of the portfolio is not the sum of the
individual variances (as is evident from equation (19)), but can be expressed through a weighted sum of
marginal contributions to variances, each depending not only on the variance of individual factors but also
on covariances between all factors to which the portfolio is exposed.
A marginal contribution is defined as a change in TEV resulting from a marginal change in the ith factor
exposure. Mathematically, it can be expressed as a partial derivative of the TEV with respect to the portfolio
exposure to the ith factor:
∂T EV
M T EVi =
.
∂λi
This metric is very useful from the risk and portfolio management point of view. For example, it allows
portfolio managers to find the components that can be used to significantly revise the overall risk of the
portfolio with minimal change to capital allocation. Under a linear factor model, the marginal contribution
to risk can be easily computed as
Σi λ i
M T EVi =
T EV
where Σi is the ith row of the factor covariance matrix and λ is the vector of portfolio factor exposures λi
that is defined as
λ = L (w − wb ) .
The marginal contribution is defined in terms of the full TEV of the portfolio, not just the systematic portion
T EVsys . However, it is easy to see that the weighted sum of all marginal contributions, weighted with the
portfolio factor sensitivities λi , is equal to the systematic contribution to tracking error volatility CT EVsys
defined by (20):
!
CT EVsys =
λi M T EVi .
i
Thus, the total portfolio risk (TEV) can be decomposed into a sum of systematic and idiosyncratic contributions as in (22), or, on a more detailed level, into a sum of factor contributions and the idiosyncratic
term:
!
T EV = CT EVsys + CT EVid =
CT EVi + CT EVid
i
where the individual factor contribution CT EVi is defined as
CT EVi = λi M T EVi .
3.3 Value-at-Risk and Expected Tail Loss
Under the assumption that the factors are distributed normally, the distribution of the portfolio returns
in the linear model is also normal. Therefore, the linear risk model is best suited for the evaluation of the
TEV as a main risk measure of a portfolio. The normality assumption and linearity of the security prices
as functions of risk factors do not allow the model to estimate tail risk with a high degree of accuracy.
Nevertheless, the model can be used to produce useful estimates of VaR and ETL, provided that the user
of the model remembers that these estimates are made under the assumption of normality of the return
distribution.
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In the linear risk model, the tail risk measures, VaR and ETL, can be computed directly from the value of
TEV. The VaR associated with probability 1 − α (e.g., α = 1%) is the lower bound for the loss incurred by
a portfolio with probability α:
α = P (rp ≤ −V aRα ) =
−V
- aRα
fr (x)dx
−∞
where rp is the portfolio return and fr (x) is the return distribution function. Thus, the value of V aRα can
be computed from the portfolio return volatility as
V aRα = Φ−1 (1 − α)T EV
(23)
where Φ−1 (x) is the inverse cumulative standard normal distribution function.
The Expected Tail Loss at the level α can also be computed as
"
#
φ Φ−1 (α)
ET Lα =
T EV
1−α
(24)
where φ(x) is the standard normal distribution function.
The various contributions to VaR and ETL can also be computed directly using the corresponding contributions to TEV in formulas (23) and (24).
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4 Scenario Analysis
The risk measures discussed above—VaR, ETL, TEV—are effectively the measures of the parameters of
the probability distribution of the portfolio returns. As such, they serve as good statistical, probabilistic
measures of risk but provide little information on possible losses or gains of the portfolio under specific
market conditions. Value-at-Risk informs portfolio managers of the probability of certain losses but gives
little indication under what specific market scenario these losses might be realized. Thus it is very useful
to complement the risk management framework by scenario analysis capabilities designed to quantify the
potential losses under specific, user-defined market events.
In the FactSet risk management framework, three types of scenario analysis or stress tests are available:
• Factor Stress Testing
• Extreme Event Stress Testing
• Extreme Event Simulation
4.1 Factor Stress Testing
In factor stress testing, it is possible to apply a shock to individual factors to see what impact the shock could
have on the portfolio as a whole. Both endogenous factors (factors within the risk model) and exogenous
factors (factors that are not in the risk model) can be stressed.
The FactSet framework can perform two types of factor stress tests—time weighted and event weighted. In
the time-weighted algorithm, the model computes the covariance matrix of the risk model factors together
with the exogenous factors in essentially the same way as described above. In the event-weighted algorithm,
the model assigns higher weights to the periods that have factor returns similar to the predefined factor
shock amount. First, the absolute difference between the shock amount and the historical returns of the
stressed factor is calculated and sorted in ascending order, and then the exponential weights are assigned to
each period so that periods with a factor return similar to the shock will have higher weights. For example,
if the scenario starts with shocking the oil factor by 20%, historical periods that had changes of 18% will
have more weight than a period where oil changed by 5%. Exactly how much more weight they are given
is determined by the decay rate. In both time-weighted and event-weighted modes, the user can define the
amount of history used and the decay factor.
The covariance matrix of all the factors is used to compute the sensitivity β of the stressed factor fs to each
risk model factor fi as:
Cov(fs , fi )
βsi =
.
V ar(fs )
These sensitivities are then used to estimate the amount of each factor return ∆fi that is implied by the
defined shock of the stressed factor ∆fs :
∆fi = βsi ∆fs .
For example, let’s assume that the beta between the stress factor (oil) and a risk model factor, e.g., the
size factor, is 0.5. This means that for each unit of exposure to the size risk factor, every 1% change in the
oil factor will have a positive 0.5% impact. If you shock oil by 20% and assume that the portfolio has an
exposure of 1 to size risk factor, the contribution of the size risk factor will be 10%.
The total return of the portfolio implied by the given factor shock is computed by applying the appropriate
pricing function to each factor return and aggregating the resulting security returns (see Chapter “Risk
Models” for discussions of the pricing functions of different asset class risk models).
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4.2 Extreme Event Stress Testing
Extreme event stress testing takes today’s portfolio and hypothesizes what its return would be if an extreme
event were to happen again. Current factor exposures are used with actual factor returns from historical
events to derive the impact of each event on portfolio returns.
To create an extreme event stress test, the user needs to define the start date and end date of the event.
For example, Figure 8 illustrates the definition of the COVID-19 selloff (02/2020) test for today’s portfolio.
To compute the implied return of the portfolio under this test, the factor returns for the period between
02/19/2020 and 03/23/2020 are used in the security pricing functions, together with the factor portfolio
exposures as of today.
4.3 Extreme Event Simulation
Extreme event simulation is similar to VaR calculations described in Sections 2 and 3 of this chapter, except
the covariances from the event date are used in combination with the current portfolio exposures. To create
an extreme event stress test (simulation approach), the user needs to define the date of the event (see Figure
9). The calculation will be based on the covariance matrix as of this date. Figure 10 shows an example of
the extreme event simulation report.
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Figure 8: Extreme event stress test definition
Figure 9: Extreme event simulation definition
Figure 10: Extreme event simulation example using the Credit Crisis of 2008
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Example of Risk Report
FactSet’s multi-asset class risk model is fully integrated
across FactSet’s portfolio analytics suite. The suite includes a flexible, highly customizable risk reporting platform that allows for detailed reporting of the results of
the risk computations and scenario analysis tightly integrated with data from other FactSet systems such as
fixed income analytics, performance attribution, and others. The framework can be used to generate consolidated
analytics and risk reports in a client-ready PDF presentation or to view dynamic asset and factor-level data with a
customizable level of detail. The users can define custom
risk reports that allow them to analyze risk exposures of
the portfolio from different points of view (e.g., portfolio
holdings, analytics, risk factor exposures) and study the
decomposition of risk measures of the portfolio in both
factor and asset space to a desired degree of granularity.
This section will walk the reader through several layers
of a sample risk report to illustrate the process of risk
analysis that the portfolio manager can carry out using
the FactSet platform.
We will consider a sample risk report for a global balanced
fixed income portfolio managed against a global aggregate
bond index. The FactSet risk framework allows the user
to analyze the portfolio both on an absolute basis and
relative to a benchmark. The following example assumes
that a portfolio is actively tracking a benchmark, i.e., it
is allowed to deviate from the benchmark to a certain
extent to obtain superior returns. The degree of the deFigure 11: Summary section of risk report
viation from the benchmark is usually specified in terms
of the Tracking Error Volatility that provides an estimate
of the magnitude of the expected difference between the
portfolio and benchmark returns. At the same time, the portfolio manager might want to monitor the tail
risk of the portfolio, either on an absolute basis as a minimum possible absolute loss at certain probability
(absolute VaR), or on the relative basis as the Value-at-Risk of the tracking error. The risk report helps the
manager to judge whether the TEV of the portfolio is within the manager’s mandate and the tail risk is
within the acceptable range to understand how the factor and asset composition of the portfolio relates to
the benchmark and affects the total risk, and to make decisions regarding portfolio compositions that are
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consistent with the manager’s views on risk budget and expected portfolio return.
The analysis starts with the summary report (Figure 11) that compares portfolio and benchmark exposures
to major sources of market risk and looks at their risk on absolute and relative levels. The report shows
the higher-level decomposition of risk, displaying aggregate risk components corresponding to the main risk
drivers of the portfolio.
The first section of the summary report (Portfolio Characteristics) shows the extent to which both portfolio
and benchmark securities are covered by the analytic models employed by the risk framework. In particular,
one can see that the system provides full coverage for the portfolio (all 170 securities are covered), and only
0.2% of benchmark securities (46 out of more than 19,000) have problems. The user can generate a detailed
report that will outline specific reasons for lack of coverage of every problematic security.
The Fixed Income Characteristics section shows the main characteristics of both the portfolio and benchmark
and their exposures to major sources of risk—their respective duration, convexity, spread duration, and
spread. One can see that this specific portfolio has a slightly shorter duration than benchmark, somewhat
larger convexity, and about 7 bp lower yield. In other words, even on an aggregated level, one can see that
the portfolio tracks the benchmark closely but not perfectly, and one should expect nontrivial contributions
to portfolio risk from both yield curve and spread components.
The next section of the summary report shows absolute risks of both portfolio and benchmark and their
decomposition into major factor contributions that
can be described in terms of categories or groups of
risk factors, e.g., Yield Curve, Muni, MBS, Credit
Spread. Each asset class is exposed to a specific set of
risk factors in addition to factors common to all assets in that market (for example, Yield Curve factors
are common to all fixed income assets). The analysis
of the factor contributions to risk allows the portfolio manager to understand the relationship between
common and specific sources of risk in the portfolio
and benchmark. In our example, one can see that
in both portfolio and benchmark, the yield curve is
by far the largest contributor to total risk, as is expected for any unhedged fixed income portfolio. (We
use 95% VaR as a measure of risk in this report,
but these fields, as well as all others, are easily customizable by the user and can show different threshold VaR as well as portfolio and benchmark return
volatilities.) Since both portfolio and benchmark are
global, i.e., contain securities denominated in different currencies, the second largest contributor to risk
is the currency exposure. Its sources will be analyzed
further when we examine the factor level report that
Figure 12: Risk decomposition by factor contributions
shows, in particular, the exposures of both portfolio
and benchmark to different risk factors. The thirdand fourth-largest risk contributions on an absolute level are credit spreads and mortgage-backed security
spreads.
As indicated by the duration exposures, the benchmark effectively hedges the majority of the portfolio yield
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curve risk. As a result, seen in the last section of the summary (Relative Risk), the yield curve component of
the tracking error volatility of the portfolio/benchmark pair becomes much smaller (only about 2 bp) than
contributions from other risk factors. The total monthly TEV of the portfolio is 75 bp (annualized) and is
mainly composed of the currency, credit spread, and MBS spread components with FX exposure giving the
largest contribution to risk and credit spread the smallest.
The factor risk model can be used for detailed analysis of the possible imbalances in the portfolio vs.
benchmark risk, and the decomposition shown in Figure 11 is just the first step of this analysis. Each major
factor group described above can be presented as a set of detailed risk factors, and the risk report can
be configured to look at all levels of granularity of risk exposures down to contributions from individual
factors or each security in the portfolio. In the report, any set of factors can be grouped to show aggregate
higher-level contributions to risk, or ungrouped to the desired level of granularity. For example, Figure 12
shows the details of exposure and risk contribution from Yield Curve and Currency factors and the aggregate
contribution from other factor groups. (The portfolio is exposed to more than one sovereign curve, but in
the figure we have shown only the USD curve to save space.)
The Yield Curve factors allow the portfolio manager to analyze the duration risk of the portfolio
to a much greater detail than the simple duration
over/underweight view in the summary report. The
factors show exposure of the portfolio to each of the
17 key maturity points along the yield curve, letting
the user examine the risk of the portfolio that comes
from various possible curve movements. One can see
from the report that the portfolio is effectively long a
position in the middle of the curve (six-, seven-, and
nine-year points have major exposures) and short a
position at the long end of the curve (20-, 25-, and
30-year points). Thus the portfolio effectively has a
curve flattener in the zero to 10-year region and a
curve steepener in the 10- to 30-year region. The fact
that the points of the short duration result in a positive contribution to risk, while the regions of long
Figure 13: Asset level risk decomposition
key rate durations have a negative contribution, is
the consequence of the correlation structure of the
fixed income market captured by the factor model. Since fixed income market spreads are in general negatively correlated with rates, the long duration positions effectively hedge the spread risk (provide negative
contribution to TEV), while the short duration positions add to the volatility of returns together with the
spread volatility.
The Currency factor group shows contributions to the risk from individual FX exposures. One can see that the
portfolio is short EUR, GBP, and JPY with the majority of risk coming from the short EUR position.
The analysis of factor risk contributions can be complemented by examining the contributions to portfolio
risk from different asset classes, asset groups, or individual assets in the portfolio. Figure 13 shows an example
of such an analysis. In this type of report, the contributions to risk can be grouped by asset type (e.g., ABS,
CMBS, Corporate) or on a more granular level, by sectors (Financial, Industrial, Utilities) and industries (not
shown in the figure), and the contributions can be analyzed on individual asset level. The report compares
the market value of portfolio holdings with the benchmark and shows contribution to risk from each security
group. This kind of information would be especially useful for an equity portfolio where the weights of assets
in various groups (like industries or sectors) effectively define portfolio risk exposure. Applied to a fixed
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income portfolio, this report illustrates the shortcomings of market weight decomposition for analysis of
risk. For example, the analyzed portfolio is overweight Industrial Corporate bonds by 6.18% and only very
slightly underweight the Financial sector by -0.32%. Yet the largest contribution to risk (40 bp) comes from
the Financial sector. Thus, the market weight information alone, while important, tells us very little about
the actual risk composition of a fixed income portfolio and has to be combined with the factor exposures
and factor risk decomposition to produce an adequate picture of portfolio risk. Taking the above example
of Corporate bond risk, one should look at the factor decomposition report (Figure 12) and the currency of
denomination of individual assets to realize that most of the risk attributed to the Financial sector actually
comes from the FX exposure of the bonds in the sector. A detailed analysis of this kind can be performed
using a report showing various factor exposures (e.g., key rate durations, spread durations) grouped by asset
class, sector, or industry. However, even that kind of report will give only a basic understanding of risk
contributions as it will not take into account correlations between different factors, which brings us back
to the factor decomposition in Figure 12 that is based on risk contribution computations that use all the
information available in the factor correlation matrix (see Estimating Portfolio Risk Measures section).
A factor-based risk model allows for deep
analysis of the risk imbalances in the
portfolio. Each of the risk categories is
modeled with a detailed set of risk factors
that are designed to capture the particular sources of risk the asset class is exposed to. The above examples are just
a cursory demonstration of how the factor model can offer useful insights from
risk management and portfolio construction perspectives.
A complementary view of the portfolio
risk is provided by stress tests (or scenario analysis) that can be performed in
several ways. One may want to reprice
the whole portfolio under a particular
scenario on risk factors, such as interest
rates or spreads, and look at the hypoFigure 14: Stress test results
thetical return under that scenario. Another way is to evaluate how the portfolio
would have performed under particular
historical scenarios. And finally, one might want to examine the risk (in terms of VaR or TEV) the portfolio
would have during a specific historical period. (All these methods are described in detail in the Scenario
Analysis section.) Figure 14 illustrates the results of four different factor scenarios—depreciation of EUR,
U.S. Treasury and U.S. stock market sell-off, and oil price rally. In all four cases, the absolute and relative
portfolio returns are computed by simulating the changes in risk factors that correspond to a specific perturbation of the tested market variable. Thus, the results include all information on market correlations and
volatilities that is stored in the model covariance matrix and represent the most likely actual outcome of the
event that implicitly includes all hedging and diversification strategies built into the portfolio.
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Conclusion
Factor risk models help portfolio managers to describe risk exposures and imbalances of different portfolios
using a relatively small set of common, intuitive, and easily understandable factors. The analysis is performed
with the help of risk reporting tools that can present a coherent view of portfolio risk across different
portfolios, asset classes, and asset management styles. These reports can be used to assess risk/return tradeoffs in a given portfolio, guide the portfolio risk balancing process, and assist in the portfolio construction
process through risk/return optimization.
The FactSet multi-asset class risk model provides detailed information about the risk exposures of portfolios
and can be very valuable for portfolio management. It is built to be reactive to the current market environment and reflect the true risks a portfolio faces. The availability of both parametric and Monte Carlo
approaches offers flexibility to support a broad range of investment objectives and styles. The combination of
factors from global equities, commodities, currencies, and fixed income creates a single framework to investigate portfolios and their risks. The resulting risk statistics and analytics can be consumed in many FactSet
applications, automated with particular formats, used to create a dashboard view for all portfolios, or carry
out any ad-hoc analysis, giving users multiple options to analyze the risk of their portfolios. These reporting
tools allow the manager to merge all views and constraints into a final portfolio simply and objectively. The
manager can then compare the risk with the expected return of each of the views and decide on the optimal
portfolio allocation.
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Bibliography
[1] “Axioma AXWW4 Model Supplement: Equity Factor Risk Models,” Axioma White Paper, 2017.
[2] B. Golub and L. Tilman, Risk Management. Approaches for Fixed Income Market. Hoboken, NJ: John
Wiley & Sons, Inc., 2000.
[3] A. B. Dor, L. Dynkin, J. Hyman, P. Houweling, E. v. Leeuwen, and O. Penninga, “DTSSM (Duration
Times Spread): A New Measure of Spread Exposure in Credit Portfolios,” The Journal of Portfolio
Management, vol. 33, no. 2, pp. 77–100, 2007.
[4] A. B. Dor, L. Dynkin, J. Hyman, and B. D. Phelps, Quantitative Credit Portfolio Management: Practical
Innovations for Measuring and Controlling Liquidity, Spread, and Issuer Concentration Risk. Wiley,
1st ed., 2011.
[5] R. Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” The Journal of
Finance, vol. 29, no. 2, pp. 449–470, 1974.
[6] D. Mossessian and C. Westenberger, “FactSet European Sovereign Spread Model: Foundations,” FactSet
White Paper, 2018.
[7] A. J. Harju and D. Mossessian, “FactSet MAC U.S. Municipal Bond Risk Model,” FactSet White Paper,
2019.
[8] J. Kusakovskij, A. Svilans, C. Westenberger, and I. Mitov, “Daily Corporate Spread Factor Models:
USD, EUR, GBP, JPY, CHF, AUD, and CAD,” FactSet White Paper, 2020.
[9] J. Kusakovskij, A. Svilans, C. Westenberger, and I. Mitov, “Daily Spread Factor Models: USD- and
EUR-Denominated Emerging Market Debt,” FactSet White Paper, 2020.
[10] A. Svilans, D. Chen, R. Shaikhutdinov, and T. Davis, “FactSet US Agency Mortgage-Backed Securities
(MBS) Risk Model,” FactSet White Papers, 2022.
[11] P. J. Huber, Robust Statistics. New York: John Wiley & Sons, Ltd, 1981.
[12] I. Mitov, J. Kusakovskij, T. Davis, and R. Shaikhutdinov, “Inflation-Linked Bonds in the FactSet MultiAsset Class Model: Major Developed Markets,” FactSet White Paper, 2022.
[13] M. Deacon, A. Derry, and D. Mirfendereski, “Inflation-Linked Derivatives: Pricing, Hedging and Other
Technical Aspects,” in Inflation-Indexed Securities, pp. 269–296, John Wiley & Sons, Ltd, 2004. Chapter
9.
[14] L. Andersen and D. Buffum, “Calibration and Implementation of Convertible Bond Models,” Journal
of Computational Finance, vol. 7, no. 2, pp. 1–34, 2004.
Copyright © 2024 FactSet Research Systems Inc. All rights reserved. FactSet Business Use Only
www.factset.com | 71
White Paper
[15] P. Zeitsch, M. Hyatt, T. P. Davis, and X. Liu, “Convertible Bond Arbitrage and the Term Structure of
Volatility,” SSRN Scholarly Paper, 2020.
[16] M. Hyatt, F. Liu, and P. Zeitsch, “FactSet Convertible Bonds,” FactSet White Paper, 2021.
[17] B. Nonchev, J. Levine, and I. Mitov, “FactSet Global Private Equity Model,” FactSet White Paper,
2022.
[18] G. Barone-Adesi and R. E. Whaley, “Efficient Analytic Approximation of American Option Values,”
The Journal of Finance, vol. 42, no. 2, pp. 301–320, 1987.
[19] N. Ju and R. Zhong, “An Approximate Formula for Pricing American Options,” The Journal of Derivatives, vol. 7, no. 2, pp. 31–40, 1999.
[20] R. G. Tompkins, “Implied Volatility Surfaces: Uncovering Regularities for Options on Financial Futures,”
The European Journal of Finance, vol. 7, no. 3, pp. 198–230, 2001.
[21] F. Jamshidian, “An Exact Bond Option Formula,” The Journal of Finance, vol. 44, no. 1, pp. 205–209,
1989.
[22] T. Davis, “Interest Rate Swap Valuation,” FactSet White Paper, 2015.
[23] C. D. Manning, P. Raghavan, and H. Schütze, Introduction to Information Retrieval. Cambridge University Press, 2008.
[24] “RiskMetrics: Technical Document,” JP Morgan, 1996.
[25] D. Mossessian and V. Vieli, “Robust Estimation of Risk Factor Model Covariance Matrix,” FactSet
White Paper, 2016.
[26] Z. D. Bai, “Methodologies in Spectral Analysis of Large Dimensional Random Matrices, a Review,”
Statistica Sinica, vol. 9, pp. 611–677, 1999.
[27] O. Ledoit and M. Wolf, “Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices,”
The Annals of Statistics, vol. 2, no. 40, p. 1024–1060, 2012.
[28] W. G. Hallerbach, “Decomposing Portfolio Value-at-Risk: A General Analysis,” Journal of Risk, no. 5,
p. 1–18, 2002.
[29] D. Mossessian and V. Vieli, “Decomposing Portfolio Risk Using Monte Carlo Estimators,” FactSet White
Paper, 2016.
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Appendix
1 Principal Component Analysis of Yield Curves
PCA is a powerful statistical technique for simplifying a set of data in which a linear transformation is
applied to the original variables such that the new, transformed variables become independent (have zero
correlation), while all the information about the volatility of the original variables is maintained. The new
variables, called principal components (PC), have different explanatory power in terms of the volatility of
the original system. The purpose of PCA is to select those PCs that explain the variability of the data
to a required accuracy. A typical threshold criterion to select principal components is for their cumulative
volatility to exceed a certain percentage of the original volatility of the data set. This is equivalent to selecting
the largest eigenvalues of the original data covariance matrix such that their sum exceeds a certain percentage
of the total sum of that matrix’s eigenvalues.
To perform PCA, we take the historical time series of rate changes for the 17 key rates (tenors) for each
spot rate curve for the most recent T trading days. The returns are z-scored by subtracting the mean of each
tenor and dividing by the standard deviation. The time series of the normalized returns are combined in a
(17 × T ) matrix R. The covariance matrix of the z-scored returns
C
17×17
=
1
R × RT
T 17×T T ×17
can be decomposed as
C
17×17
= QΛQT ,
where Λ is a diagonal matrix of the eigenvalues of matrix C. The principal component projection matrix Q
consists of eigenvectors of the curve covariance matrix. The matrix of all principal components of the curve
P (17 × T ) is computed as
P = QT × R .
17×T
17×17
17×T
Based on our research, we pick four components out of the 17 as sufficient to explain more than 99% of
the yield curve dynamics. These first four time series of principal components form a (4 × T ) matrix P̂.
The P̂ matrices for all curves are combined with the time series of all other model factors for the purpose
of creating the MAC covariance matrix. The first four columns of the PC projection matrix Q are stored
in a new (17 × 4) matrix Q̂. This matrix is then used to either convert the covariance matrix of principal
components into the covariance matrix of key rates in the LMAC framework or to convert the simulated
values of the PCs stored in a (4 × 1) vector P̂ into the projected rate changes of the 17 maturities in the MC
MAC framework.
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2 Spread Model Types
2.1 Relative Spread Change Models
Our primary approach to modeling spread risk of credit securities is based on research in [3] and [4],
where authors observed a linear relationship between spread change volatility σ∆s and spread level s of
USD-denominated corporate and sovereign bonds. This result had far-reaching implications for the design of
spread risk models as it suggested dominant factors in these market segments, as we will now demonstrate.
The main advantage of this approach is that it enables the construction of accurate linear models that allow
a meaningful decomposition of portfolio risks into the credit risk of different countries, regions, or sectors.
To simplify the notation of the following discussion, we will suppress the time dependence of variables that
do not require forecasting, such as spreads, durations, etc.
It was suggested in [3] that spread change volatility σ∆s (t) of a particular security at time t is proportional
to the current spread level s:
"
#
σ∆s (t) = s × β(t) + σε ,
(25)
where the proportionality factor β(t) is common for all bonds in a given market segment regardless of their
credit rating. This relationship can be obtained assuming that the relative change of spread ∆s(t)/s of a
credit security can be modeled as
∆s(t)
= ∆f (t) + ε(t),
(26)
s
where ∆f (t) is a return of a common factor that drives spread changes across a market segment and ε(t)
is the idiosyncratic spread change of a given security only. If processes of ∆f (t) and ε(t) are uncorrelated,
expressions (25) and (26) become equivalent and β(t) equates to the volatility of ∆f (t). The latter can be
denoted as σ∆f (t).
If we use σ∆f (t) as a measure of spread risk of a portfolio of securities exposed to this factor, the sensitivity to
that risk will be quantified by Ds × s, where Ds denotes the spread duration. More specifically, we can write
the volatility of the portfolio excess return as a sum of systematic and idiosyncratic volatilities as
2
2
σexcess
(t) = σsyst.
(t) + σε2 (t).
Volatility of the systematic part can then be written as:
σsyst. (t) = Ds × σ∆s (t) = [Ds × s]
σ∆s (t)
= [Ds × s] σ∆f (t),
s
which confirms the statement above.
Our research referenced throughout Section 2.3 of the Model Overview chapter supported the empirical
observations of the aforementioned work and observed the effect in other market segments. We also identified
additional factors that govern the spread curves’ dynamics and improve model accuracy.
All of our relative spread change models are based on OAS computed by the FactSet analytics engine and
factor returns ∆fj (t) for different credit market segments, estimated by regressing cross-sectional ∆OAS(t)
observations on instrument sensitivities βj . The general form of models can be expressed as
!
∆OAS(t) =
βj ∆fj (t) + ε(OAS, DOAS ),
where ε(OAS, DOAS ) denotes the idiosyncratic component, which is also parametrized with bond-specific
attributes for different asset classes. To simplify the notation, the dependence of the idiosyncratic component
on OAS and DOAS is usually suppressed. Particulars of βj are described in dedicated subsections of particular
asset classes.
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2.2 Specialized Linear Spread Models
Security classes whose spread risk is defined by processes that are not directly related to the ratio of the
underlying company assets and liabilities are modeled using linear regression spread models. All linear spread
models start with the same factors specific to the asset classes they are used for, but these factors can then
be augmented through model selection or each model may be supplemented with additional factors tailored
for the characteristics of the given asset type. Detailed descriptions of the models are provided in Section 2.3
of the Model Overview chapter, while general details of these models and their use modeling frameworks are
presented below.
The regression model for spread is designed to capture the level and slope of the common spread factor as
a function of the security’s duration (level and slope of the spread curve):
!
OAS(t) = flevel (t) + Dfslope (t) +
βj fj (t).
(27)
Here, OAS and D are spread and duration of the security, flevel (t) and fslope (t) are the estimated slope and
level factors, and βj and fj (t) are additional sensitivities and factors, if any. Returns of the estimated time
series of factors are then obtained as the first differences of these series. The latter are then used as inputs
into the factor return matrix and eventually the overall covariance matrix.
For purposes of risk, a model equivalent to equation (27) is produced if spread level changes are used; that
is
!
∆OAS(t + 1) = ∆flevel (t + 1) + D∆fslope (t + 1) +
βj ∆fj (t + 1).
(28)
In this approach, regressions immediately produce the spread factors returns, and these are used in the
covariance matrix forecasts without taking the differences of the series.
Models that follow from equations (27) and (28) are used in all of our frameworks and are explicitly identified
as such in model definitions. As with relative spread models described in the previous section of the appendix,
security’s idiosyncratic risk can also be parametrized as a function of security attributed, e.g., ε(OAS, DOAS ).
For simplicity of notation, models where this feature is available contain the ε term, but the independent
parameters were suppressed.
Finally, it is important to note that the starting level and slope factors can be modified in the process of
model selection. These factors can turn out to be insignificant and discarded, e.g., the slope factor in the
Japanese MBS model. On the other hand, these factors can be made more granular to capture the rich
heterogeneity of some asset class, e.g., the level factor in the U.S. MBS model was split into four factors
specific to mortgage terms and issuer agency groupings. Augmentations of this sort are presented in the
specific subsection of Section 2.3 in the Model Overview chapter.
2.3 Contingent Claim Analysis Model
This section describes the principles of the CCA approach as it is applied to valuing credit risk of FI securities.
Specifics of the application of the methodology are applicable to different classes of securities covered only
in the MC MAC framework.
Balance sheet risk is the key to understanding credit risk and default probabilities under the CCA approach,
whether the balance sheet is of a corporation, financial institution, or sovereign entity. From the balance
sheet point of view, default occurs when the assets of the entity become insufficient to meet the amount of
debt owed to creditors at maturity; that is, when assets fall below a distress barrier defined by the total
value of the company’s liabilities. Thus, the uncertainty in the changes in the future asset value relative to
the promised payment on debt is the driver of the credit and default risk. As total assets decline toward the
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distress barrier or if the asset volatility increases such that the value of the assets becomes more uncertain,
the value of the risky debt declines and credit spread of the risky debt rises. In other words, credit spreads
in this model are attributed entirely to the risk-neutral expected default loss, which is positively related to
leverage and volatility in the firm value.
The contingent claim approach ([5]) to analyzing the credit risk of FI securities is based on three basic
assumptions: (i) values of liabilities are derived from assets; (ii) liabilities have different priorities and, in the
simplest model, can be divided into senior claims (debt) and junior claims (equity); and (iii) assets follow a
stochastic process. In this framework, both risky debt and equity can be modeled as contingent claims on
the assets. Equity is modeled as a call option on the company’s assets, while risky debt is viewed as the
difference between the book value of debt and a put option on the firm’s assets. Basically, CCA assumes
that owners of corporate equity hold a call option on the firm value after outstanding liabilities have been
paid off. Moreover, they have the option to default if their firm’s asset value falls below the present value of
the notional amount of outstanding debt at maturity. Thus, bond holders receive a put option premium in
the form of a credit spread above the risk-free rate in return for holding risky corporate debt (and bearing
the potential loss) due to the limited liability of equity owners.
It should be noted here that the approach used to determine the distress barrier is a defining element of the
CCA and has a great impact on the model results. The definition of the distress barrier depends on the type
of asset being modeled.
The FactSet CCA spread modeling framework is based on the Merton model developed in [5]. The model
starts with the balance sheet equation stating that, at any time t, the total market value of assets A(t) is
equal to the market value of the claims on the assets—equity E(t)—and risky debt D(t) maturing at time
T:
A(t) = E(t) + D(t)
The equity is modeled as an implicit call option on the assets with an exercise price equal to the promised
payments, B, maturing in T − t periods. Thus, the value of the equity can be computed using the BlackScholes-Merton formula for the value of a call:
E(t) = A(t)N (d1 ) − Be−r(T −t) N (d2 ),
where N (x) is the cumulative probability of the standard normal density function below x,
2 3 2
3
σ2
A
ln B
+ r + 2A (T − t)
√
d1 =
,
σA T − t
√
d 2 = d 1 − σA T − t
(29)
(30)
(31)
and r is the risk-free rate, and σA is the asset return volatility.
If the asset value is modeled as a geometric Brownian motion with volatility σA , the values of equity and
equity volatility are connected to the values of asset and asset volatility through the following relation (see
[5] for details):
EσE
=
AσA N (d2 ).
(32)
Neither the asset’s value nor its volatility is directly observable. However, given the values of E(t) and σE ,
the equations (32) and (29) can be used to get an implied estimate of the asset value and volatility. The
collective view of many market participants is incorporated in the observable market prices of liabilities, and
the change in the market price of these liabilities will determine its volatility. The contingent claims approach
implicitly assumes that market participants’ views on prices incorporate forward-looking information about
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the future economic prospects of the company. This assumption does not imply that the market is always
right about its assessment of credit risk, but that it reflects the best available collective forecast of the
expectations of market participants. An important aspect of this type of model is that equity needs to be
expressed in market value, and the volatility of equity has to be computed by means of the price evolution of
the company’s shares. This feature of the CCA framework highlights the main advantage of this approach:
the forward-looking character of the model incorporated in capital market expectations through price and
volatility movements.
In practice, having determined the value of the equity E(t) and equity volatility σE at the time of calibration,
the model solves equations (32) and (29) to obtain the starting values of the company assets A(t) and asset
volatility σA . Monte Carlo simulation is then used to forecast the distribution of equity values at the time of
risk model horizon t + ∆t using the equity-factor model, and the starting value A(t) is used in conjunction
with the relation (32) to obtain the distribution of values of the company assets from the terminal values of
the equity E(t + ∆t) in Monte Carlo simulation as:
A(t + ∆t) = A(t) +
E(t + ∆t) − E(t) σE
.
N (d2 )
σA
The simulated distribution of asset values at the risk model horizon is then used to compute the expected
value of the risky debt and derive the corresponding distribution of the spread changes. Under the CCA
approach, the risky debt is equivalent in value to default-free debt minus a guarantee against default. This
guarantee can be calculated as the value of a put P (t) on the asset with an exercise price equal to B. At
the same time, the risky debt can be represented as a spread-discounted value of the total company debt B,
which leads to the following equation for the spread s:
Be−(r+s)(T −t) = Be−r(T −t) − P (t).
The values of the spread at the risk horizon can be computed from equation (33)
.
/
1
P (t + ∆t) r(T −(t+∆t))
s(t + ∆t) = −
ln 1 −
e
,
T − (t + ∆t)
B
(33)
(34)
where the value of the put option P (t + ∆t) is computed using the simulated terminal values of the asset
A(t + ∆t).
The CCA approach to modeling corporate defaults and bond spreads has several well-known deficiencies.
However, when this approach is used to build the spread risk model, these deficiencies do not play a significant
role. The first one is the well-known concern that arises when one applies the Merton model to study default
risk: the probabilities obtained are in the risk-neutral measure and, consequently, one needs a model to go
from risk neutral probabilities to risk natural probabilities. This is not a concern for the systematic spread
risk model since we focus on the valuation of the contingent claims and not on probabilities of default.
The positive characteristic of the approach—that it strives to use information in various assets and markets
to gather information about systemic risk—is at the same time a concern since the implementation of the
model requires measurements of a number of variables that are not directly observable. This leads to a fairly
complex calibration procedure that relies on a number of heuristic assumptions. Also, in practice, structural
models tend to underestimate credit spreads (this is known as the “spread puzzle”). Both these issues are
mitigated by the fact that we use the approach not to measure the exact relationship between various asset
characteristics, but as a framework for quantifying a range of uncertainty in those characteristics. We do not
need to accurately model the absolute spread levels and default probabilities but rather to produce a robust
estimation for the volatility of spread changes over a given horizon.
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Specifically, the FactSet credit spread model computes the distribution of the changes in the credit spread
given the changes in the stock price of the issuer from the Monte Carlo simulation as described above and
then passes these changes to the pricing equation (3) to obtain simulated distribution of returns of the
modeled credit security.
One aspect of the CCA spread model worth noting is the assumption of a 100% correlation between a
company’s spreads with the stock price, less the impact of interest rates. As a result, there are no contributions
from corporate bonds to the cross-asset covariance matrix; the interaction between spreads and other factors
in the risk model is expressed through the equity factor covariances with all other factors.
To be able to use the CCA framework to value corporate bonds, additional peculiarities need to be accounted for. One first needs to identify the equity
Country Group
to which the credit relates. The FactSet corporate
bonds model can use either a FactSet formula or clientsupplied inputs to identify the underlying equity. If the
Sector Subgroup
issuing corporation is publicly traded or has a corporate parent that is publicly traded, the framework relies
Industry Subgroup
on public equity information. For private corporations
with no public parent, a corporate proxy is created to
determine suitable equity inputs. FactSet maintains a
A Rating Group
B Rating Group
number of global corporate fixed income indexes. These
indexes are used daily to create a hierarchy of equity
proxy groups, as shown in Figure 15. Each bond issued
AAA
A
A
BBB
BB
B
by a private corporate entity is assigned to one of the
groups based on the information available for the bond.
The hierarchy of sectors and industries is created using Figure 15: Hierarchy of equity for a specific country to
FactSet’s classification system. The rating of a bond determine corporate proxy
is determined as a minimum value from three rating
agencies—S&P, Moody, and Fitch. Each proxy group is represented by an index of the corporate bonds from
publicly traded entities. These indexes are used to calculate the average equity model exposures, as well as
Merton model inputs such as equity market capitalization, implied equity volatility, and debt level for each
proxy group. Risk statistics of a bond from the non-publicly traded entity are then modeled using FactSet’s
corporate credit modeling framework with proxy values as inputs.
Once the equity value and volatility are determined for a given corporate bond, the model runs Monte Carlo
simulations and computes the distribution of asset values at the risk model horizon as described above.
However, in the case of corporate bonds, we cannot use equation (34) directly because structural models are
known to underestimate the value of the corporate bond’s spread (the effect known as a “spread puzzle”).
There are multiple reasons for this discrepancy, e.g., the liquidity risk premium, taxes, and cash outflows
to service debts (coupons and dividends). Besides, the true value of the distress barrier is also a random
variable. The model assumes that the default point is described by the firm’s leverage—the ratio of its assets
and liabilities, where only assets are changing. But firms often adjust their liabilities as they near default.
It is common to observe the liabilities of commercial and industrial firms increase as they near default while
the liabilities of financial institutions often decrease as they approach default. The difference is usually just a
reflection of the liquidity in the firm’s assets and thus their ability to adjust their leverage as they encounter
difficulties. Unfortunately, the Merton model is unable to specify the behavior of the liabilities or include
other sources of risk in the spread’s computation, and thus these effects must be captured by the calibration
procedure. The FactSet corporate spread model takes these effects into account by calibrating an effective
risk-neutral probability of default.
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The model starts by assuming that the value of the spread is determined exclusively by the default probability.
In case of default, the holder of the debt will be paid the total amount of debt times a recovery rate R. In case
of no default, the payout equals just the total value of the debt. Thus, if p(t) is the risk-neutral probability
of default at time t, the present value of the risky debt at that time is the risk-neutral expectation of the
payout:
Be−(r+s)(T −t) = p(t)RBe−r(T −t) + (1 − p(t))Be−r(T −t) .
(35)
At the same time, if we assume that the firm’s leverage L is a geometric Brownian motion with volatility
σL , it can be shown that the risk-neutral probability of default is:
2
3
σ2
ln(L) + r − 2L (T − t)
√
p = N (−
),
(36)
σL T − t
where r is the risk-free rate and N (x) is the cumulative normal distribution function.
The model uses the market value of the bond’s spread at time t to calibrate the current value of default
probability p(t) using equation (35) and the starting effective value of leverage L(t) using equation (36).
It then computes Monte Carlo simulated distribution of leverage using the simulated distribution of asset
values obtained in the previous step, assuming that the relative change of leverage between current time and
risk model horizon is the same as the relative change of the asset value:
L(t + ∆t) − L(t)
A(t + ∆t) − A(t)
=
L(t)
A(t)
and uses the simulated values of leverage to compute probabilities of default at the risk model horizon
p(t + ∆t) using equation (36). The final step is the computation of the distribution of spreads at time t + ∆t
as:
s(t + ∆t) = −
"
#
1
ln 1 − p(t + ∆t)(1 − R) .
T −t
For securities not modeled specifically by the linear models covered in the main text and where a public
cannot be identified, FactSet assigns an equity proxy using the hierarchy of equity groups described above.
Using the same hierarchy and the terms and conditions information available, a corporate proxy is assigned.
In these cases, however, it is likely that most detailed information on sector, industry, and ratings may
not be available. Thus, these miscellaneous securities would more likely be assigned to the country proxy
group.
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3 Fixed Income Spread Model Factors
3.1 Sovereign, Municipal, and Quasi-Government Bonds
Euro-Sovereign Bonds
Factor ID
Factor Name
EURO_SOV_SPR_LS
EURO_SOV_SPR_AT_LU
EURO_SOV_SPR_FI
EURO_SOV_SPR_FR
EURO_SOV_SPR_IT
EURO_SOV_SPR_NL
EURO_SOV_SPR_IE
EURO_SOV_SPR_ES
EURO_SOV_SPR_PT
EURO_SOV_SPR_BE
EURO_SOV_SPR_CEE
EURO_SOV_SPR_GR_CY
EURO_SOV_SPR_REL_SHORTDUR
EURO_SOV_SPR_ABS_SHORTDUR
Euro
Euro
Euro
Euro
Euro
Euro
Euro
Euro
Euro
Euro
Euro
Euro
Euro
Euro
Sovereign: Low Spread
Sovereign Austria, Luxembourg: Spread
Sovereign Finland: Spread
Sovereign France: Spread
Sovereign Italy: Spread
Sovereign Netherlands: Spread
Sovereign Ireland: Spread
Sovereign Spain: Spread
Sovereign Portugal: Spread
Sovereign Belgium: Spread
Sovereign Central and Eastern Europe: Spread
Sovereign Greece, Cyprus: Spread
Sovereign: Relative Short Duration
Sovereign: Absolute Short Duration
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U.S. Municipal Bonds
Factor ID
Factor Name
US_MUNI_GENERAL_OBLIGATION_IV
US_MUNI_AUTHORITY_IV
US_MUNI_BUILDING_IV
US_MUNI_CERT_OF_PART_IV
US_MUNI_ECONOMIC_DEV_IV
US_MUNI_EDUCATION_IV
US_MUNI_HEALTH_IV
US_MUNI_HOUSING_IV
US_MUNI_SCHOOL_DISTRICT_IV
US_MUNI_TAX_IV
US_MUNI_TRANSPORTATION_IV
US_MUNI_UTILITY_IV
US_MUNI_OTHER_IV
US_MUNI_GENERAL_OBLIGATION_OAS
US_MUNI_AUTHORITY_OAS
US_MUNI_BUILDING_OAS
US_MUNI_CERT_OF_PART_OAS
US_MUNI_ECONOMIC_DEV_OAS
US_MUNI_EDUCATION_OAS
US_MUNI_HEALTH_OAS
US_MUNI_HOUSING_OAS
US_MUNI_SCHOOL_DISTRICT_OAS
US_MUNI_TAX_OAS
US_MUNI_TRANSPORTATION_OAS
US_MUNI_UTILITY_OAS
US_MUNI_OTHER_OAS
US_MUNI_PUERTO_RICO_IV
US_MUNI_ILLINOIS_IV
US_MUNI_NEW_JERSEY_IV
US_MUNI_CALIFORNIA_IV
US_MUNI_PENNSYLVANIA_IV
US_MUNI_FLORIDA_IV
US_MUNI_CONNECTICUT_IV
US_MUNI_TEXAS_IV
US_MUNI_MASSACHUSETTS_IV
US_MUNI_WASHINGTON_IV
US_MUNI_NEW_YORK_IV
US_MUNI_PUERTO_RICO_OAS
US_MUNI_ILLINOIS_OAS
US_MUNI_NEW_JERSEY_OAS
US_MUNI_CALIFORNIA_OAS
US_MUNI_PENNSYLVANIA_OAS
US_MUNI_FLORIDA_OAS
US_MUNI_CONNECTICUT_OAS
US_MUNI_TEXAS_OAS
US_MUNI_MASSACHUSETTS_OAS
US_MUNI_WASHINGTON_OAS
US_MUNI_NEW_YORK_OAS
US_MUNI_SHORT_SPREAD_DURATION
US_MUNI_LONG_SPREAD_DURATION
US_MUNI_VEGA_IF_CALLABLE
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
Muni
General Obligation
Authority
Building
Certificate of Participation
Economic Development
Education
Health
Housing
School District
Tax
Transportation
Utility
Other
General Obligation OAS
Authority OAS
Building OAS
Certificate of Participation OAS
Economic Development OAS
Education OAS
Health OAS
Housing OAS
School District OAS
Tax OAS
Transportation OAS
Utility OAS
Other OAS
Puerto Rico
Illinois
New Jersey
California
Pennsylvania
Florida
Connecticut
Texas
Massachusetts
Washington
New York
Puerto Rico OAS
Illinois OAS
New Jersey OAS
California OAS
Pennsylvania OAS
Florida OAS
Connecticut OAS
Texas OAS
Massachusetts OAS
Washington OAS
New York OAS
Short Spread Duration
Long Spread Duration
Vega if Callable
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Developed Market Supranational, Sub-Sovereign, and Agency Bonds
Factor ID
Factor Name
USD_SSA_SPR_US_AGEN_OAS
USD_SSA_SPR_CA_AGEN_OAS
USD_SSA_SPR_DM_EUROPE_AGEN_OAS
USD_SSA_SPR_DM_ASIA_PACIFIC_AGEN_OAS
USD_SSA_SPR_US_SUPRA_OAS
USD_SSA_SPR_DM_EUROPE_SUPRA_OAS
USD_SSA_SPR_ABS_SHORTDUR
EUR_SSA_SPR_DE_AGEN_OAS
EUR_SSA_SPR_FR_AGEN_OAS
EUR_SSA_SPR_IT_AGEN_OAS
EUR_SSA_SPR_ES_AGEN_OAS
EUR_SSA_SPR_ODM_EUROPE_AGEN_OAS
EUR_SSA_SPR_DM_NONEUROPE_AGEN_OAS
EUR_SSA_SPR_DM_SUPRA_OAS
EUR_SSA_SPR_ABS_SHORTDUR
JPY_SA_SPR_DM_AGEN_OAS
JPY_SA_SPR_ABS_SHORTDUR
JPY_SA_SPR_ABS_LONGDUR
JPY_SA_SPR_LS
GBP_SSA_SPR_DM_AGEN_OAS
GBP_SSA_SPR_SUPRA_OAS
CAD_SSA_SPR_AGENCY_CA_OAS
CAD_SSA_SPR_PROV_ONTARIO_OAS
CAD_SSA_SPR_PROV_QUEBEC_OAS
CAD_SSA_SPR_PROV_ALBERTA_OAS
CAD_SSA_SPR_PROV_OTHER_OAS
CAD_SSA_SPR_MUNI_ONTARIO_OAS
CAD_SSA_SPR_MUNI_OTHER_OAS
CAD_SSA_SPR_SUPRA_OAS
AUD_SSA_SPR_DM_AGEN_OAS
AUD_SSA_SPR_SUPRA_OAS
CHF_SSA_SPR_DM_AGEN_OAS
CHF_SSA_SPR_SUPRA_OAS
HKD_SSA_SPR_OAS
SEK_SSA_SPR_DM_OAS
NZD_SSA_SPR_DM_OAS
KRW_SSA_SPR_OAS
SGD_SA_SPR_DM_AGEN_OAS
NOK_SSA_SPR_DM_OAS
USD SSA: U.S. Agency Spread
USD SSA: Canada Agency Spread
USD SSA: DM Europe Agency Spread
USD SSA: DM Asia Pac Agency Spread
USD SSA: U.S. Supranational Spread
USD SSA: DM Europe Supranational Spread
USD SSA: Absolute Short Duration
EUR SSA: Germany Agency Spread
EUR SSA: France Agency Spread
EUR SSA: Italy Agency Spread
EUR SSA: Spain Agency Spread
EUR SSA: Other DM Europe Agency Spread
EUR SSA: DM X-Europe Agency Spread
EUR SSA: DM Supranational Spread
EUR SSA: Absolute Short Duration
JPY SA: DM Agency Spread
JPY SA: Absolute Short Duration
JPY SA: Absolute Long Duration
JPY SA: Low Spread
GBP SSA: DM Agency Spread
GPB SSA: Supranational Spread
CAD SSA: Canada Agency Spread
CAD Province: Ontario Spread
CAD Province: Quebec Spread
CAD Province: Alberta Spread
CAD Province: Other Spread
CAD Muni: Ontario Spread
CAD Muni: Other Spread
CAD SSA: Supranational Spread
AUD SSA: DM Agency Spread
AUD SSA: Supranational Spread
CHF SSA: DM Agency Spread
CHF SSA: Supranational Spread
HKD SSA: Spread
SEK SSA: Spread
NZD SSA: Spread
KRW SSA: Spread
SGD SA: Spread
NOK SSA: Spread
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3.2 Corporate Bonds
U.S. and Canada
Factor ID
Factor Name
USD_CORP_SPR_FIN_LS
USD_CORP_SPR_FIN_ABS_SHORTDUR
USD_CORP_SPR_FIN_REL_SHORTDUR
USD_CORP_SPR_FIN_HIGH_YIELD
USD_CORP_SPR_BANKING_OAS
USD_CORP_SPR_INSURANCE_OAS
USD_CORP_SPR_INV_SERV_OAS
USD_CORP_SPR_UTIL_OAS
USD_CORP_SPR_UTIL_ABS_SHORTDUR
USD_CORP_SPR_UTIL_REL_SHORTDUR
USD_CORP_SPR_IND_LS
USD_CORP_SPR_IND_ABS_SHORTDUR
USD_CORP_SPR_IND_REL_SHORTDUR
USD_CORP_SPR_IND_HIGH_YIELD
USD_CORP_SPR_CONS_SERV_OAS
USD_CORP_SPR_CONS_CYC_OAS
USD_CORP_SPR_CONS_NONCYC_OAS
USD_CORP_SPR_ENERGY_OAS
USD_CORP_SPR_HEALTHCARE_OAS
USD_CORP_SPR_IND_OAS
USD_CORP_SPR_NE_MATS_OAS
USD_CORP_SPR_TECH_OAS
CAD_CORP_SPR_IND_OAS
CAD_CORP_SPR_FIN_ABS_SHORTDUR
CAD_CORP_SPR_FIN_OAS
USD Corp: Financials Low Spread
USD Corp: Financials Absolute Short Duration
USD Corp: Financials Relative Short Duration
USD Corp: Financials High Yield
USD Corp: Financials/Banking Spread
USD Corp: Financials/Insurance Spread
USD Corp: Financials/Investment Services Spread
USD Corp: Utilities Spread
USD Corp: Utilities Absolute Short Duration
USD Corp: Utilities Relative Short Duration
USD Corp: Industrials Low Spread
USD Corp: Industrials Absolute Short Duration
USD Corp: Industrials Relative Short Duration
USD Corp: Industrials High Yield
USD Corp: Consumer Services Spread
USD Corp: Consumer Cyclicals Spread
USD Corp: Consumer Noncyclicals Spread
USD Corp: Energy Spread
USD Corp: Healthcare Spread
USD Corp: Industrials Spread
USD Corp: Non-Energy Materials Spread
USD Corp: Technology Spread
CAD Corp: Non-Financials Spread
CAD Corp: Financials Absolute Short Duration
CAD Corp: Financials Spread
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Europe
Factor ID
Factor Name
EUR_CORP_SPR_IND_LS
EUR_CORP_SPR_IND_ABS_SHORTDUR
EUR_CORP_SPR_IND_REL_SHORTDUR
EUR_CORP_SPR_CONS_CYC_OAS
EUR_CORP_SPR_CONS_NONCYC_OAS
EUR_CORP_SPR_IND_OAS
EUR_CORP_SPR_NE_MATS_OAS
EUR_CORP_SPR_TECH_OAS
EUR_CORP_SPR_FIN_LS
EUR_CORP_SPR_FIN_ABS_SHORTDUR
EUR_CORP_SPR_FIN_REL_SHORTDUR
EUR_CORP_SPR_BANKING_OAS
EUR_CORP_SPR_INV_SERV_OAS
EUR_CORP_SPR_UTIL_LS
EUR_CORP_SPR_UTIL_OAS
EUR_CORP_SPR_UTIL_ABS_SHORTDUR
EUR_CORP_SPR_UTIL_REL_SHORTDUR
EUR_CORP_SPR_HIGH_YIELD
GBP_CORP_SPR_IND_LS
GBP_CORP_SPR_IND_ABS_SHORTDUR
GBP_CORP_SPR_IND_REL_SHORTDUR
GBP_CORP_SPR_CONS_CYC_OAS
GBP_CORP_SPR_CONS_NONCYC_OAS
GBP_CORP_SPR_IND_OAS
GBP_CORP_SPR_TECH_OAS
GBP_CORP_SPR_FIN_LS
GBP_CORP_SPR_FIN_ABS_SHORTDUR
GBP_CORP_SPR_FIN_REL_SHORTDUR
GBP_CORP_SPR_BANKING_OAS
GBP_CORP_SPR_INV_SERV_OAS
GBP_CORP_SPR_UTIL_LS
GBP_CORP_SPR_UTIL_OAS
GBP_CORP_SPR_UTIL_ABS_SHORTDUR
GBP_CORP_SPR_UTIL_REL_SHORTDUR
GBP_CORP_SPR_HIGH_YIELD
CHF_CORP_SPR_IND_OAS
CHF_CORP_SPR_FIN_OAS
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
EUR
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
GBP
CHF
CHF
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Corp:
Industrials Low Spread
Industrials Absolute Short Duration
Industrials Relative Short Duration
Consumer Cyclicals Spread
Consumer Noncyclicals Spread
Industrials Spread
Non-Energy Materials Spread
Technology Spread
Financials Low Spread
Financials Absolute Short Duration
Financials Relative Short Duration
Financials/Banking Spread
Financials/Investment Services Spread
Utilities Low Spread
Utilities Spread
Utilities Absolute Short Duration
Utilities Relative Short Duration
High Yield
Industrials Low Spread
Industrials Absolute Short Duration
Industrials Relative Short Duration
Consumer Cyclicals Spread
Consumer Noncyclicals Spread
Industrials Spread
Technology Spread
Financials Low Spread
Financials Absolute Short Duration
Financials Relative Short Duration
Financials/Banking Spread
Financials/Investment Services Spread
Utilities Low Spread
Utilities Spread
Utilities Absolute Short Duration
Utilities Relative Short Duration
High Yield
Non-Financials Spread
Financials Spread
Japan and Australia
Factor ID
Factor Name
JPY_CORP_SPR_IND_ABS_SHORTDUR
JPY_CORP_SPR_CONS_OAS
JPY_CORP_SPR_IND_OAS
JPY_CORP_SPR_TECH_OAS
JPY_CORP_SPR_FIN_ABS_SHORTDUR
JPY_CORP_SPR_FIN_OAS
JPY_CORP_SPR_UTIL_ABS_SHORTDUR
JPY_CORP_SPR_UTIL_OAS
AUD_CORP_SPR_IND_OAS
AUD_CORP_SPR_FIN_ABS_SHORTDUR
AUD_CORP_SPR_FIN_OAS
JPY Corp: Industrials Absolute Short Duration
JPY Corp: Business/Consumer/Healthcare Spread
JPY Corp: Energy/Industrials/Materials Spread
JPY Corp: Technology/Telecommunications Spread
JPY Corp: Financials Absolute Short Duration
JPY Corp: Financials Spread
JPY Corp: Utilities Absolute Short Duration
JPY Corp: Utilities Spread
AUD Corp: Non-Financials Spread
AUD Corp: Financials Absolute Short Duration
AUD Corp: Financials Spread
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Local Currency Markets
Factor ID
Factor Name
CNY_CORP_SPR_LS
CNY_CORP_SPR_CORP_OAS
CNY_CORP_SPR_PB_OAS
CNY_CORP_SPR_ABS_SHORTDUR
CNY_CORP_SPR_ABS_LONGDUR
HKD_CORP_SPR_LS
HKD_CORP_SPR_OAS
HKD_CORP_SPR_ABS_SHORTDUR
INR_CORP_SPR_OAS
INR_CORP_SPR_ABS_SHORTDUR
KRW_CORP_SPR_ABS_SHORTDUR
KRW_CORP_SPR_FIN_OAS
KRW_CORP_SPR_IND_OAS
MYR_CORP_SPR_OAS
MYR_CORP_SPR_ABS_SHORTDUR
NOK_CORP_SPR_OAS
NOK_CORP_SPR_ABS_SHORTDUR
NZD_CORP_SPR_OAS
SEK_CORP_SPR_OAS
SEK_CORP_SPR_ABS_SHORTDUR
SGD_CORP_SPR_LS
SGD_CORP_SPR_FIN_OAS
SGD_CORP_SPR_IND_OAS
THB_CORP_SPR_LS
THB_CORP_SPR_OAS
THB_CORP_SPR_ABS_LONGDUR
ZAR_CORP_SPR_OAS
CNY Corp: Low Spread
CNY Corp: Spread
CNY Corp: Policy Bank Spread
CNY Corp: Absolute Short Duration
CNY Corp: Absolute Long Duration
HKD Corp: Low Spread
HKD Corp: Spread
HKD Corp: Absolute Short Duration
INR Corp: Spread
INR Corp: Absolute Short Duration
KRW Corp: Absolute Short Duration
KRW Corp Financials: Spread
KRW Corp Industrials: Spread
MYR Corp: Spread
MYR Corp: Absolute Short Duration
NOK Corp: Spread
NOK Corp: Absolute Short Duration
NZD Corp: Spread
SEK Corp: Spread
SEK Corp: Absolute Short Duration
SGD Corp: Low Spread
SGD Corp Financials: Spread
SGD Corp Industrials: Spread
THB Corp: Low Spread
THB Corp: Spread
THB Corp: Absolute Long Duration
ZAR Corp: Spread
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3.3 Emerging Market Bonds Denominated in USD and EUR
Factor ID
Factor Name
EM_USD_AFRICA_OAS
EM_USD_AMERICAS_OAS
EM_USD_ARGENTINA_OAS
EM_USD_ASIA_OAS
EM_USD_BRAZIL_CORP_OAS
EM_USD_BRAZIL_SOV_OAS
EM_USD_CHILE_OAS
EM_USD_CHINA_OAS
EM_USD_CIS_EX_RUSSIA_OAS
EM_USD_COLOMBIA_OAS
EM_USD_EAST_EUROPE_OAS
EM_USD_INDIA_CORP_OAS
EM_USD_INDONESIA_OAS
EM_USD_ISRAEL_OAS
EM_USD_MALAYSIA_OAS
EM_USD_MEXICO_OAS
EM_USD_MIDDLE_EAST_OAS
EM_USD_PERU_OAS
EM_USD_PHILIPPINES_OAS
EM_USD_RUSSIA_OAS
EM_USD_SOUTH_AFRICA_OAS
EM_USD_TURKEY_CORP_OAS
EM_USD_TURKEY_SOV_OAS
EM_USD_UAE_CORP_OAS
EM_USD_UAE_SOV_OAS
EM_USD_VENEZUELA_OAS
EM_EUR_AMERICAS_OAS
EM_EUR_CHINA_OAS
EM_EUR_EUROPE_OAS
EM_EUR_OTHER_OAS
USD Emerging Market Africa: Spread
USD Emerging Market Americas: Spread
USD Emerging Market Argentina: Spread
USD Emerging Market Asia: Spread
USD Emerging Market Brazil Corporate: Spread
USD Emerging Market Brazil Sovereign: Spread
USD Emerging Market Chile: Spread
USD Emerging Market China: Spread
USD Emerging Market CIS Ex Russia: Spread
USD Emerging Market Colombia: Spread
USD Emerging Market Eastern Europe: Spread
USD Emerging Market India: Spread
USD Emerging Market Indonesia: Spread
USD Emerging Market Israel: Spread
USD Emerging Market Malaysia: Spread
USD Emerging Market Mexico: Spread
USD Emerging Market Middle East: Spread
USD Emerging Market Peru: Spread
USD Emerging Market Philippines: Spread
USD Emerging Market Russia: Spread
USD Emerging Market South Africa: Spread
USD Emerging Market Turkey Corporate: Spread
USD Emerging Market Turkey Sovereign: Spread
USD Emerging Market UAE Corporate: Spread
USD Emerging Market UAE Sovereign: Spread
USD Emerging Market Venezuela: Spread
EUR Emerging Market Americas: Spread
EUR Emerging Market China: Spread
EUR Emerging Market Europe: Spread
EUR Emerging Market Other: Spread
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3.4 Structured Credit Products
U.S. Mortgage-Backed Securities and Mortgage-Related Securities
Factor ID
Factor Name
US_MBS_SPR_15Y
US_MBS_SPR_20Y
US_MBS_SPR_CONVENT_30Y
US_MBS_SPR_GINNIE_30Y
US_MBS_SPR_VEGA
US_CMO_SPR_IOPO_REFI
US_CMO_SPR_OTHER_REFI
US_CMBS_SPR_AGENCY
US_CMBS_SPR_OTHERS
US_CMBS_SPR_SENIOR_OAS
US_CMBS_SPR_NON_SENIOR_OAS
US_CMBS_SPR_FZIN_SHORTDUR
US_CMBS_SPR_FZIN_LONGDUR
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
U.S.
MBS 15yr
MBS 20yr
MBS 30yr Conventional
MBS 30yr Ginnie
MBS Vega
CMO IO/PO Refi Duration
CMO Non IO/PO Refi Duration
CMBS Agency Low Spread
CMBS Non-Agency Low Spread
CMBS Senior Spread
CMBS Non-Senior Spread
CMBS Short Spread Duration
CMBS Long Spread Duration
ABS, Non-U.S. CMBS, and Japanese MBS
Factor ID
Factor Name
US_ABS_CMBS_INTERCEPT
US_ABS_CMBS_SEASONING
US_ABS_CMBS_RATING_AA
US_ABS_CMBS_RATING_A
US_ABS_CMBS_RATING_BBB
NON_US_ABS_CMBS_INTERCEPT
NON_US_ABS_CMBS_SEASONING
NON_US_ABS_CMBS_RATING_A-BBB
MBS_JP_MEAN
U.S. ABS/CMBS Intercept
U.S. ABS/CMBS Seasoning
U.S. ABS/CMBS Rating AA
U.S. ABS/CMBS Rating A
U.S. ABS/CMBS Rating BBB
Non-U.S. ABS/CMBS Intercept
Non-U.S. ABS/CMBS Seasoning
Non-U.S. ABS/CMBS Rating A-BBB
Japanese MBS Mean
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4 Relative Spread Change Model Factor Groups
4.1 Revere Business and Industry Classification System Sector and Industry
Factor Definitions
RBICS Category
RBICS Subcategory
Finance
Banking
Investment Services
Real Estate
Specialty Finance and Services
Insurance
Business Services
Consumer Services
Consumer Cyclicals
Consumer Noncyclicals
Healthcare
Industrial
Energy
Non-Energy Materials
Technology
Telecom
Utilities
Sector
Financials
Industrials
Utilities
Table 6: Mapping of RBICS categories/subcategories to currency-specific relative spread model sectors
RBICS Category
and Subcategory
Banking
Investment Services
Real Estate
Specialty F&S
Insurance
Business Services
Cons. Services
Cons. Cycl.
Cons. Noncycl.
Healthcare
Industrial
Energy
Non-Energy Mat.
Technology
Telecom
Utilities
USD
EUR
GBP
Banking
Banking
Banking
Investment
Services
Investment
Services
Consumer
Cyclicals
Consumer
Cyclicals
Consumer
Noncyclicals
Consumer
Noncyclicals
Investment
Services
Insurance
Consumer
Services
Cons. Cycl.
Cons. Noncycl.
Healthcare
Industrial
Energy
Non-Energy Mat.
Non-Energy Mat.
Technology
Utilities
Industrial
JPY
AUD/CAD/CHF
KRW/SGD
Finance
Finance
Consumer
Industrial
Industrial
Technology
Technology
Technology
Utilities
Utilities
Utilities
Industrial
Table 7: Mapping of RBICS categories/subcategories to currency-specific relative spread model industries
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4.2 Emerging Market Regional Factor Definitions
Regional Group Name
Peru
Philippines
Russia
South Africa
Turkey Corp.
Turkey Sov.
UAE Corp.
UAE Sov.
Risk Country
Ivory Coast, Egypt, Nigeria, Zambia,
Tunisia, Senegal, Rwanda, Namibia,
Mozambique, Togo, Morocco, Angola,
Ghana, Cameroon, Gabon, Kenya,
Ethiopia
Costa Rica, Dominican Republic,
El Salvador, Guatemala, Paraguay,
Ecuador, Honduras, Uruguay
Argentina
Pakistan, Thailand, Vietnam, Taiwan
Brazil
Brazil
Chile
China
Armenia, Azerbaijan, Belarus,
Kazakhstan, Mongolia, Ukraine,
Uzbekistan
Colombia
Bulgaria, Croatia, Czech Republic,
Hungary, Latvia, Lithuania, Slovenia,
Slovakia, Poland, Romania
India
Indonesia
Israel
Malaysia
Mexico
Bahrain, Lebanon, Oman, Qatar,
Kuwait, Jordan, Iraq, Saudi Arabia
Peru
Philippines
Russian Federation
South Africa
Turkey
Turkey
United Arab Emirates
United Arab Emirates
Venezuela
Venezuela
Africa
Americas
Argentina
Asia
Brazil Corp.
Brazil Sov.
Chile
China
CIS ex. Russia
Colombia
Eastern Europe
India
Indonesia
Israel
Malaysia
Mexico
Middle East
ML Asset Type
Corporate
Quasi & Foreign Government
Corporate
Quasi & Foreign Government
Corporate
Quasi & Foreign Government
Corporate
Quasi & Foreign Government
Corporate
Quasi & Foreign Government
Corporate
Quasi & Foreign Government
Table 8: Country groups for the EM USD model factor loading
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Regional Group Name
Americas
China
Europe
Other
Risk Country
Argentina, Brazil,
Chile, Mexico
China
Bulgaria, Croatia,
Czech Republic,
Estonia, Hungary,
Latvia, Lithuania,
Poland, Romania,
Slovakia, Slovenia
Russian Federation,
Israel, Turkey,
United Arab Emirates,
Saudi Arabia
ML Asset Type
Corporate
Quasi & Foreign Government
Table 9: Country groups for the EM EUR model factor loading
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5 Return-Based Model Factors
A hundred factors in time series have been chosen as a representative suite of assets that covers a diverse
selection of what FactSet’s clients might own and have returns for. Limitations due to data availability,
frequency, and model methodologies have led us to choose an appropriate set of factors, but it is certainly
not all encompassing. In particular, the factor selection is constrained for linear type assets only, meaning
it’s not well suited for nonlinear assets like derivatives. It is currently limited to factors that have a daily
return stream. The factor universe encompasses Alternative Investment, Equity, Real Estate, Fixed Income,
Interest Rates, Commodities, and Currencies.
5.1 Alternative Investments
The first class of factors we include comprises 17 HFR indexes representing major sub-categories of hedge
funds. They include Absolute Return, Event Driven, Fund-of-Funds, Global Macro, Long/Short, and Relative
Value.
Factor ID
Factor Name
Index
RBA_HFRXAR
RBA_HFRXMREG
RBA_HFRXEH
RBA_HFRXEMN
RBA_HFRXDS
RBA_HFRXMA
RBA_HFRXED
RBA_HFRXM
RBA_HFRXNA
RBA_HFRXEHG
RBA_HFRXEHV
RBA_HFRXMLP
RBA_HFRXGL
RBA_HFRXSDV
RBA_HFRXMD
RBA_HFRXRVA
RBA_HFRXCA
HF
HF
HF
HF
HF
HF
HF
HF
HF
HF
HF
HF
HF
HF
HF
HF
HF
HFRXAR
HFRXMREG
HFRXEH
HFRXEMN
HYS-US
HFRXMA
HFRXED
HFRXM
IWV-US
HFRXEHG
HFRXEHV
AMZ
HFRXGL
HFRXSDV
HFRXMD
HFRXRVA
HFRXCA
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Absolute Return
Multi-Region Index
Equity Hedge Index
EH: Equity Market Neutral Index
Event Driven Distressed Securities
Event Driven Merger Arbitrage
Event Driven Index
Macro/CTA Index
North America Index
EH: Fundamental Growth Index
EH: Fundamental Value Index
MLP Index
Global Hedge Fund Index
Systematic Diversified CTA Index
Market Directional Index
Rel Value Arbitrage Index
Rel Value Convertible Arbitrage
5.2 Equity
The equity factors are daily returns of the following twelve indices:
Factor ID
Factor Name
Index
RBA_SP50.R
RBA_SML
RBA_183660
RBA_186745
RBA_180461
RBA_180721
RBA_LFEYK
RBA_BXM
RBA_EAFE
RBA_FR0000R1
RBA_EMIF
VIX
Equity Rets: U.S. Large Cap
Equity Rets: U.S. Small Cap
Equity Rets: Euro Large Cap
Equity Rets: Australia
Equity Rets: Japan
Equity Rets: Korea
Equity Rets: Emerging Mkts
Equity Rets: U.S. Large Cap Buy Write
Equity Rets: EAFE
Equity Rets: World
Other Rets: Emerg Mkts Infrastructure
CBOE Volatility Index
SP50.R
SML
183660
186745
180461
180721
LFEYK
BXM
see footnote1
FR0000R12
EMIF
VIX
1 The
factor is a weighted sum of FactSet Market Indexes for Europe, Africa, and Asia
is a FactSet Market Index of the world using a market value weighting
2 FR0000R1
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5.3 Real Estate
The Real Estate class of factors comprises six assets representative of various REIT investment styles.
Factor ID
Factor Name
RBA_CWH
RBA_URE
RBA_NLY
RBA_HTS
RBA_ARR
RBA_TWO
Real
Real
Real
Real
Real
Real
Est
Est
Est
Est
Est
Est
Rets:
Rets:
Rets:
Rets:
Rets:
Rets:
Index
Commercial REITs
Leveraged U.S.
REITs
Residential Single Family REITs
Residential REITs A
Residential REITs B
EQC
URE
NLY
CMO
ARR
TWO
5.4 Fixed Income
Eighteen Fixed Income and 15 Government Interest Rate global indexes comprise the Fixed Income universe
of factors.
Factor ID
Factor Name
Index
RBA_MLHC00
RBA_MLG0CP
RBA_MLEB00
RBA_MLEA00
RBA_MLEJRE
RBA_MLE0LG
RBA_MLE0GG
RBA_MLHPID
RBA_MLHA00
RBA_MLUF00
RBA_MLUIRE
RBA_MLUA00
RBA_MLCABS
RBA_MLCMBS
RBA_MLCF00
RBA_MLCIRE
RBA_MLM0A0
RBA_MLU0A0
RBA_MLG9D0
RBA_MLG2D0
RBA_MLG4D0
RBA_MLG9Y0
RBA_MLG2Y0
RBA_MLG4Y0
RBA_MLG9L0
RBA_MLG8L0
RBA_MLG2L0
RBA_MLG4L0
RBA_MLG8O2
RBA_MLG9O2
RBA_MLG2O2
RBA_MLG4O2
RBA_SHORTTERM
FI Rets: Canadian High Yield
FI Rets: Canadian Provincials – Municipals
FI Rets: EMU Corporates / Financials
FI Rets: EMU MBS
FI Rets: EUR Corporates Real Estate
FI Rets: Euro-Dollar & Globals Governments – Local
FI Rets: Euro-Dollar & Globals Governments Guaranteed
FI Rets: European Currency HY; Constrained / Non-Financials
FI Rets: Global HY & Emerging Market – Plus
FI Rets: Sterling Corporates / Financials
FI Rets: Sterling Corporates / Real Estate
FI Rets: Sterling MBS
FI Rets: U.S. ABS & CMBS
FI Rets: U.S. CMBS Fixed Rate
FI Rets: U.S. Corporates / Financials
FI Rets: U.S. Corporates / Real Estate
FI Rets: U.S. Mortgages
FI Rets: U.S. Municipals
Int Rate: Euro Government: 10+ Years
Int Rate: Euro Government: 3-5 Years
Int Rate: Euro Government: 7-10 Years
Int Rate: Japan Government: 10+ Years
Int Rate: Japan Government: 3-5 Years
Int Rate: Japan Government: 7-10 Years
Int Rate: UK Government: 10+ Years
Int Rate: UK Government: 15+ Years
Int Rate: UK Government: 3-5 Years
Int Rate: UK Government: 7-10 Years
Int Rate: UST: 10+ Years
Int Rate: UST: 15+ Years
Int Rate: UST: 3-5 Years
Int Rate: UST: 7-10 Years
Int Rate: World: 0-3 Years Aggregate
MLHC00
MLG0CP
MLEB00
MLEA00
MLEJRE
MLE0LG
MLE0GG
MLHPID
MLHA00
MLUF00
MLUIRE
MLUA00
MLCABS
MLCMBS
MLCF00
MLCIRE
MLM0A0
MLU0A0
MLG9D0
MLG2D0
MLG4D0
MLG9Y0
MLG2Y0
MLG4Y0
MLG9L0
MLG8L0
MLG2L0
MLG4L0
MLG8O2
MLG9O2
MLG2O2
MLG4O2
see footnote1
1 The
World Interest Rate factor is an equal weighted composite comprised of German Government 0-3Y (MLG1DB), UK
Gilt 0-3Y (MLGBL0), Japan Government 0-3Y (MLG1YA), and U.S. Treasury 0-3Y (MLG1QA)
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5.5 Commodities
Six Commodity indices, managed by S&P GSCI, are available in the RBR model. Timber, whose base returns
series is the near-term spot rate of timber, is also included.
Factor ID
Factor Name
Index
RBA_SPGSAG
RBA_SPGSEN
RBA_SPGSIN
RBA_SPGSLV
RBA_SPGSNG
RBA_SPGSPM
RBA_LB00-USA
Commodity
Commodity
Commodity
Commodity
Commodity
Commodity
Other Rets:
SPGSAG
SPGSEN
SPGSIN
SPGSLV
SPGSNG
SPGSPM
LB00-USA
Rets: Agriculture
Rets: Energy
Rets: Industrial Metals
Rets: Livestock
Rets: Natural Gas
Rets: Precious Metals
Timber
5.6 Currencies
Twenty-six global currencies are available as factors within the RBR model.
Factor ID
Factor Name
Index
Euro
Norwegian Krona
Swedish Krona
Swiss Franc
UK Pound
Russian Rouble
Australian Dollar
Canadian Dollar
New Zealand Dollar
South African Rand
Brazilian Real
Chilean Peso
Mexican Peso
Israeli Shekel
Indian Rupee
Malaysian Ringgit
Singapore Dollar
Indonesian Rupiah
Philippines Peso
South Korean Won
Thai Baht
Taiwanese Dollar
Hong Kong Dollar
China Renminbi
Japanese Yen
U.S. Dollar
Euro
Norwegian Krona
Swedish Krona
Swiss Franc
UK Pound
Russian Rouble
Australian Dollar
Canadian Dollar
New Zealand Dollar
South African Rand
Brazilian Real
Chilean Peso
Mexican Peso
Israeli Shekel
Indian Rupee
Malaysian Ringgit
Singapore Dollar
Indonesian Rupiah
Philippines Peso
South Korean Won
Thai Baht
Taiwanese Dollar
Hong Kong Dollar
China Renminbi
Japanese Yen
U.S. Dollar
EUR
NOK
SEK
CHF
GBP
RUB
AUD
CAD
NZD
ZAR
BRL
CLP
MXN
ILS
INR
MYR
SGD
IDR
PHP
KRW
THB
TWD
HKD
CNY
JPY
USD
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