multi-asset class (mac) ii risk model

Multi-Asset Class (MAC) II Risk ModelIvan MitovDirector of Risk Research [email protected]

Tom DavisDirector of Derivatives Research [email protected]

Viviana VieliAssociate [email protected]

Chris DesmaraisProduct Manager [email protected]

Antti HarjuSenior Quantitative Researcher [email protected]

Jevgenij KusakovskijQuantitative Researcher [email protected]

www.factset.com

mailto:ivan.mitov%40factset.com?subject=

mailto:todavis%40factset.com?subject=

mailto:vvieli%40factset.com?subject=

mailto:cdesmarais%40factset.com?subject=

mailto:aharju%40factset.com?subject=

mailto:jevgenij.kusakovskij%40factset.com?subject=

https://www.factset.com/

White Paper

Multi-Asset Class (MAC) II Risk Model

Ivan MitovDirector of Risk [email protected]

Tom DavisDirector of Derivatives [email protected]

Viviana VieliAssociate [email protected]

Chris DesmaraisProduct [email protected]

Antti HarjuSenior Quantitative [email protected]

Jevgenij KusakovskijQuantitative [email protected]

February 8, 2021

Copyright © 2021 FactSet Research Systems Inc. All rights reserved. 1 FactSet Research Systems Inc. | www.factset.com

White Paper

Contents

Introduction 4

Model Overview 6

Risk Models 101 Equity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1 Coverage Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Active/Style Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Regional and Industry Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Macroeconomic Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Commodity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Fixed Income Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 FI Pricing Methodology and Risk Factor Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Interest Rate Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Spread Risk–Contingent Claim Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Corporate Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Convertible Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Bank Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Other Covered Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Spread Risk–Linear Regression Model of Relative Spread Changes . . . . . . . . . . . . . . . . . . . 23Euro-Sovereign Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24U.S. Municipal bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Spread Risk–Linear Regression Model of Spread Levels . . . . . . . . . . . . . . . . . . . . . . . . 26Quasi-Governmental (Agency) Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Mortgage Backed and Mortgage Related Securities in USD . . . . . . . . . . . . . . . . . . . . . . . . 27U.S. and Non-U.S. ABS and CMBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Japanese MBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Inflation Protected Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1 Equity Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

The Impact of Options on Return Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Equity Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Equity and Equity Index Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31The Equity Volatility Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Fixed Income Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Eurodollar Futures (EDFs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Government Bond Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Option on Government Bond Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


White Paper

Option on Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Interest Rate Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Interest Rate Cap and Floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Callable Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Credit Default Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Currency Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Currency Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Currency Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Return Based Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Evaluation of Risk 381 Covariance Matrix Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.1 EWMA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.2 Covariance Matrix Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2 Monte Carlo Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.1 Simulating Distribution of Portfolio Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.2 Estimating Portfolio Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3 Marginal Contributions to Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Scenario Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1 Factor Stress Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Extreme Event Stress Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Extreme Event Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Example of Risk Report 50

Conclusion 54

Bibliography 55

Appendix 571 Principal Component Analysis of Yield Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 Fixed Income Model Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.1 Linear Regression Model of Relative Spread Changes–U.S. Municipal Bonds . . . . . . . . . . . . . 582.2 Linear Regression Model of Relative Spread Changes–Euro-Sovereign Bonds . . . . . . . . . . . . . 592.3 Linear Regression Model of Spread Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 Return Based Model Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.1 Alternative Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3 Real Estate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4 Fixed Income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5 Commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.6 Currencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65


White Paper

Introduction

A growing number of investment managers today regard the function of the enterprise risk managementsystem as moving above and beyond simple compliance into the heart of the investment process. Risk modelsare no longer simply used to monitor risk, but to actually drive portfolio performance. They are used forrisk forecasting (as forward-looking tools to predict future volatility), risk analysis (as tools to understandcurrent risk exposures), and risk attribution (as guides to past performance, evaluating actual returns inrelation to risk exposures of the portfolio).

A typical portfolio contains hundreds or thousands of securities and quantifying the risks of each portfolioholding and their complex interplay can be quite a daunting task. Factor risk models provide portfoliomanagers with a convenient way of reducing this complexity to a manageable set of precise market riskexposures. They supply a detailed understanding of the sources of risk in each portfolio, help portfoliomanagers decompose risk into contributions from well-defined market factors, and quantify correlationsbetween these contributions. Factor risk models are indispensable for understanding the contributions ofdifferent 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 issimple 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 eachinstrument as a nonlinear function of the factors. For example, when the model is based on full and exactrepricing of the instruments or a nonlinear interaction between factors and sources of valuation as in the caseof structural spread models. Such nonlinear models will be more difficult to implement and will require morecomputational power to run, but they will provide more precise estimates of tail and complex derivativesrisk.

In both cases, the risk factor model estimates the risk characteristics of the portfolio using a covariancematrix of factor returns. For most portfolios, the number of factors in the model is much smaller thanthe number of securities in the portfolio. Thus, the factor covariance matrix requires fewer parameters toestimate, leading to a much more stable and robust estimation of portfolio risks. Moreover, the factors canbe designed in such a way as to increase the stability of the covariance matrix and the quality of the riskestimators.

The factors of the model are designed on one hand to capture systematic market fluctuation and on theother hand, to be able 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 detailsof the portfolio exposures to the back and front ends of the curve or the slope of the curve. In this case, it isevident that more interest rate risk factors are required to quantify the portfolio exposure to these risks. Atthe same time, the total number of interest rate factors should remain limited to keep the dimensionality ofthe problem under control. Thus, factor risk model construction is, among other things, an exercise in theoptimal choice of the set of risk factors.


White Paper

Multi-asset class risk factor models are especially difficult because of the complexity of calculating risks acrossdifferent asset types. However, they are becoming absolutely necessary now, as more and more investmentmanagers are operating on global, multi-asset type markets. Historically, the risks of such portfolios havebeen 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 errorsin estimated risk because this methodology does not take into account correlations between different assetclasses. A multi-asset class risk factor model that is based on a set of factors that capture systematicshifts in all markets (equities, fixed income, commodities, etc.) remedies this deficiency and allows theportfolio manager to accurately estimate portfolio risk and its components stemming from different assettype markets.

In this paper we discuss the structure of the FactSet multi-asset class (MAC) factor risk framework and thetypes of factors used in its models, and describe estimation techniques used to compute risk factors and thecorresponding security returns.


White Paper

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 relativeto a benchmark). The model is based on Monte Carlo simulation of the joint distribution of the futureportfolio returns and allows for the calculation of various risk statistics such as Tracking Error Volatility(TEV), Value-at-Risk (VaR), and Expected Tail Loss (ETL) as well as other characteristics of the returndistribution (e.g., kurtosis, skewness). The framework can be described as consisting of the following basiccomponents:

• A set of factor models that capture systematic market risk components of returns across different assetclasses.

• A set of models that accurately forecast the volatilities and correlations of risk factors. The models areused to construct the forecast for the joint distribution of the factors.

• A Monte Carlo framework that is used to simulate the forecasted joint risk factor distribution at agiven horizon.

• A set of pricing algorithms connecting risk factors with security returns. The algorithms are used inthe Monte Carlo framework to simulate the future joint distribution of security returns and the totaldistribution of the portfolio returns.

• A framework for estimating portfolio risk metrics (e.g., total VaR, factor and security contributions toVaR) from Monte Carlo simulated distribution of security returns.

• FactSet’s flexible reporting platform, which allows for detailed reporting of the results of risk compu-tations 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 factorshocks using any of FactSet’s extensive data libraries.

Traded securities are exposed to many different types of risk and factor risk models are estimated by iden-tifying common systematic sources of these risks. There are multiple ways of defining the systematic riskfactors. For example, one can use the existing economic or financial time series as common risk drivers. Inthis case, the sensitivities of each security to these factors are estimated using time series regression of thesecurity returns on factor returns. Alternatively, one can first define the factor sensitivities; for example, asdummy variables having values of either 0 or 1 or as in fixed income models, using the analytics calculatedfrom pricing models such as duration or convexity. In this case, the factor returns are estimated using across-sectional regression of security returns on corresponding sensitivities. FactSet employs both methodsto construct its set of multi-asset class factor models.

The FactSet MAC framework uses separate factor risk models for equities, fixed income, commodities, andalternative assets and combines them into one multi-asset class model through a common factor covariance


White Paper

matrix. Each risk model is described in detail in the Risk Models chapter of the paper. The equity model(Section 1, Risk Models chapter) is a linear risk model that has four blocks of factors–currency factors, stylefactors, industry factors, and macroeconomic factors (oil and gold). (Sensitivities of style factors representa stock’s exposures to the attributes most commonly used for stock selection, e.g., momentum, volume,liquidity. It is a hybrid model, meaning that it utilizes both cross-sectional and time series regressions toestimate factor returns as well as factor sensitivities (betas). This structure allows the model to recognizethat each stock may have significant exposure to multiple regions or industries. At the same time, thismethodology permits application of Bayesian priors and statistical significance tests to determine relevantfactors for each stock. FactSet receives daily equity factor returns and exposures from Northfield InformationServices.

The FactSet commodity model (Section 2, Risk Models chapter) is a hybrid linear model that consists offive cross-sectional factors predicated on short-term momentum measures, along with 10 time series factorschosen to be most common across assets. The model factors are estimated using a universe of highly liquidcommodity indexes, continuous front month futures contracts, ETFs, and mutual funds. First, the stylefactors are computed using cross-sectional regression. The residual returns from that regression are usedto estimate sensitivities to the time series factors that include several FX rates, a set of major commodityindexes, a volatility index, and the three-month Treasury Bill rate. The model is constructed to be simpleand have a fully transparent methodology that complements the multi-asset class model, and it works wellwith balanced portfolios of equities, fixed income, currencies, and commodities.

The fixed income (FI) model (Section 3, Risk Models chapter) is the most complex and extensive of the setof risk models used by MAC framework. It is a fundamental model (i.e., based on cross-sectional regressionof known factor exposures on security returns) that uses pricing models for different types of FI securitiesavailable in FactSet as its foundation. The pricing models precompute a set of analytics that are used assensitivities to the FI risk factors. The model is based on a general decomposition of the return of each FIsecurity into the part driven by the yield curve movements and the part defined by the changes of the optionadjusted spread (OAS) of the security. (For particular asset classes, additional terms are considered in theexpansion.) Thus the FI risk model can be usually represented as a combination of several submodels—theyield curve (interest rate risk) model that applies to all FI securities and a number of spread models, eachspecific to a particular asset class. The interest rate risk component is calculated using a number of key ratesalong sovereign yield curves (every day FactSet computes key rate durations of several million FI securitieson more than 60 sovereign yield curves). A number of bond asset types (i.e., agency bonds, asset-backedsecurities, mortgage-backed securities) have their linear spread models that represent the change of bondOAS as a linear combination of various factors. Cross-sectional models of relative OAS changes are employedfor U.S. municipal and European sovereign bonds. The credit risk of corporate bonds is modeled using astructural credit model (Merton model) that relates the spread of these bonds to the return and volatilityof the underlying equity. These models are described in detail in subsections of Section 3 of the Risk Modelschapter.

FactSet calculates factor returns and exposures for all models overnight and archives the results, creatinga continually growing database of historical data that is used in model construction, historical simulations,and scenario analysis.

The ultimate goal of every risk model is the ability to accurately forecast the distribution of portfolio returnsat a given horizon. A factor risk model links the portfolio return distribution to the joint distribution offactor returns through security pricing algorithms, making the task of forecasting the return distributionequivalent to the task of forecasting the joint factor distribution. In practice, multifactor risk models employa factor covariance matrix to compute that distribution. The quality and accuracy of the model covariancematrix define the accuracy of the risk measures produced by the model, making the factor covariance matrix acornerstone of every multifactor risk model. The FactSet MAC risk framework employs the covariance matrix


White Paper

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 andknown to evolve stochastically over time. Any volatility model is always an attempt to forecast a set oflatent variables based on a series of historic observations of other variables, in the case of a risk model–ofobserved or computed risk factor returns. MAC framework employs the classical, exponentially weightedmoving average (EWMA) method of covariance matrix estimation that produces stable, slowly varyingportfolio volatility and VaR forecasts. The method is designed as a short-term model, meaning a one-dayto two-month horizon. The details of the implementation of the method are described in Section 1 of theEvaluation of Risk chapter. The covariance matrix of the MAC model is calculated daily, resulting in a richdatabase 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 spuriouscorrelations between factors and ultimately will translate into inaccurate risk predictions. In addition, ifthe number of time observations is less than the number of factors in the model (which is possible forlarge multi-asset class models), the matrix will be “rank deficient,” meaning that it is possible to use it toconstruct apparently riskless portfolios. In practice, such a matrix cannot be used to construct Monte Carlosimulations of the factor distribution and will result in model failure. As a result, it is always necessaryto apply to every covariance matrix estimator a regularization procedure that is designed to remedy thesedeficiencies. The methodology employed in the FactSet MAC framework to construct a robust, invertible,positive definite sample covariance matrix that is the closest possible estimator of the “true” factor covariancematrix is described in Section 1.2 of the Evaluation of Risk chapter. The methodology is based on randommatrix theory and employs spectral decomposition to regularize the distribution of the matrix eigenvaluesand produce a well-conditioned matrix.

Immediate calculations of risk measures for multi-asset class portfolios always present a trade-off betweenspeed and accuracy. The fastest methods rely on simplifying assumptions that a portfolio’s value changeslinearly with changes in market risk factors. Greater realism in measuring changes in portfolio value generallycomes at the price of increased complexity and nonlinearity of the pricing functions. For linear models, therisk metrics can be computed by simple algebraic manipulations of the factor covariance matrix and vectorsof portfolio weights and factor sensitivities. More realistic, nonlinear pricing functions make it impossibleto use this simple computational method. In contrast, Monte Carlo simulation is applicable with virtuallyany model of changes in risk factors and any mechanism for determining a portfolio’s value in each marketscenario. The FactSet MAC risk framework employs linear factor models for commodity and equity markets,but uses more accurate, nonlinear pricing algorithms for fixed-income securities and derivatives. This makesit necessary to use Monte Carlo simulation methods for computation of portfolio risk metrics.

The MAC Monte Carlo simulation engine creates a large number of possible scenarios for all model factorsbased on the joint distribution of the factor returns that is described by the factor covariance matrix,under the assumption that such a distribution is multidimensional normal. Under each simulated scenario,every security is repriced and a portfolio return is calculated. For equity and commodity securities, thesecurity returns under each scenario are computed as a linear function of the simulated factor returns. Forfixed-income securities and derivatives, MAC framework employs analytical approximation as well as fullrepricing to revalue securities in Monte Carlo simulations. The analytical approximation uses the Taylorexpansion of the return of an asset as a function of risk factors (e.g., rates, spreads). In this case, the changein the value of the instrument is not approximated by the first order term alone (delta) but uses higher orderterms when necessary to achieve greater accuracy of the computation. Full pricing models used for derivativescompute security returns using exact analytical formulas for their prices as functions of market risk factors.As a result, under each scenario, the total portfolio return can be represented either as a weighted sum ofthe simulated individual security returns or as a linear combination of the simulated factor returns. Both


White Paper

representations are used to compute a set of risk measures (TEV, VaR, ETL) reflecting contributions of eachsecurity and/or factor to the total portfolio risk, as well as the risk of the portfolio as a whole. The MonteCarlo simulation framework and the methodology for computing the risk measures from simulated returndistributions are described in Section 2 of the Evaluation of Risk chapter.

We should note that, while historical data in general allows risk models to create reasonable forecasts ofthe volatility and correlation of factors (and thus provide the estimates of future portfolio returns), theseare only the estimates of risk. Major market events such as a financial crisis, are usually followed by aperiod of extremely high volatility and significantly increased and sometimes reversed correlations betweendifferent asset classes. Because correlations and volatilities during these periods are so different from historicalpatterns, no risk model can ever predict when such events will occur. Thus, portfolio managers should not relyexclusively on risk models to manage the risk of the portfolios. It is always desirable to complement the riskmodel analysis with what-if analysis, i.e., scenario analysis which estimates future portfolio returns understressed conditions, with extreme values of market volatilities and correlations. Methodologies of variousscenario analysis capabilities of the FactSet risk framework are described in Section 3 of the Evaluation ofRisk chapter.


White Paper

Risk Models

1 Equity ModelThe equity risk model of the MAC framework is a hybrid linear model that combines both cross-sectional andtime series regression to maximize its explanatory power. This structure allows the risk model to leverage thebest techniques for the types of factors being estimated and to recognize that each stock may have significantexposures to multiple markets or industries. In addition, this methodology permits application of Bayesianpriors and statistical significance tests to determine relevant factors for each stock.

The model can be broken down into four major factor blocks that are estimated in a step-wise manner:

• currency factors,

• region-specific active style factors,

• regional market and industry factors,

• and common macroeconomic factors.

Additional details of the model and the full list of factors can be found in [1]. Hereunder, we only summarizethe estimation procedure and provide more details on the last three factor blocks in respective sections.

The currency factors are FX rates of 64 unique currencies and stock sensitivities to these factors are binary.The remaining factors are constructed in local currency terms, which preserves the property of base currencyinvariance. As a result, risk of any portfolio can be expressed in any one of the covered currencies.

All active style factor returns, except for Style, are estimated first with cross-sectional regressions usingstandardized and winsorized sensitivities based on fundamental data. Size factor returns are estimated afterthe market and industry factor estimation to ensure that this factor captures relative size effects net ofthe market as a whole. Next, market and industry factor returns are constructed and stock sensitivities arecalculated via time series regressions with priors and significance tests on a stock-by-stock basis. Priors areimposed on stock’s home market factor and its regional sector. Sensitivities to other factors in these groupsare statistically checked for significance to ensure that each stock has non-zero sensitivity only to factorsthat actually affect its returns. Finally, sensitivities to macroeconomic factor–gold and oil are obtained witha combination of time series regressions and statistical tests.

The final residual returns of all stocks that are not explained by any of the systematic factors are consideredas stock-specific and independent. The annualized standard deviation of these residual returns yields thestock residual risk.


White Paper

1.1 Coverage UniverseFor a security to be represented in the equity risk model, it first must be available in FactSet’s equity universe.Additional criteria are applied and can be summarized as follows:

• asset type is listed as common equity, preferred equity, depository receipts, alien, unit, or investmenttrust;

• current price is available;

• current market capitalization is available;

• country of risk is available;

• Global Industry Classification Standard (GICS) class is available.

This universe consisted of around 52,000 securities at the end of 2005 and it grew to over 64,000 securities bythe end of 2017. The growing number of securities leads to a dynamic number of market and industry factors,which require additional filtering for estimation. The latter details are described in the section dedicated tothis factor block.

1.2 FactorsActive/Style Factors

Stock returns in their local currencies are used to estimate the regional active (style) factor blocks withcross-sectional regressions. There are eight types of these factors and their sensitivities are computed fromdata in FactSet’s Fundamentals and Estimates databases. The latter can be summarized as follows.

• Growth is a composite created from normalized annual earnings per share growth over the last 12months (LTM), lagged by 90 days, and forecast annual earnings per share growth, using the forecastearnings per share over the next year.

• Value is a composite of normalized book to price, earnings per price, cash flow to price, and dividendyield from the LTM, lagged by 90 days, and divided by the closing price from the previous night.

• Momentum is the normalized slope of an ordinary least squares regression of last 14 closing prices atfour-weekly intervals, i.e., from 20 trading days ago up to 280 trading days ago, against time:

Pti = βti + α where ti ∈ {20, . . . , 280}.

• Leverage is a composite of net debt to equity and net debt to assets ratios.

• Change in Liquidity is defined as the normalized time-weighted average of trading volume in the last10 days, divided by the equally weighted average of the trading volume over the past 30 days. Thetime weighting is linear and such that the observation from 10 days ago has half the weight of the mostrecent observation. Mathematically this is expressed as

Li =30

135

18V−1 + 17V−2 + . . .+ 9V−10∑−30i=−1 Vi

,

where V−t is the trading volume t trading days ago.

• Quality is an equally weighted combination of normalized return on assets, return on equity, and cashflow to sales ratio. Return on assets is the latest reported net income for LTM, divided by last year’sassets lagged by 90 days. Return on equity is the latest reported net income before extraordinaries for


White Paper

LTM, divided by last year’s common equity lagged by 90 days. Finally, cash flow to sales is cash flowfrom operations per share, divided by sales per share for LTM lagged by 90 days.

• Volatility is the normalized natural logarithm of the standard deviation of daily returns over the last65 trading days.

• Size is the normalized base 10 logarithm of market capitalization in U.S. Dollars.1

Prior to sensitivity calculation, all raw and fundamental data are subject to elementary filters to removeobvious data errors, such as negative dividends or book values. Afterward, the extreme observations arewinsorized to retain them in the dataset, but in a way that prevents them from distorting the style factorreturn estimates.

In the UK, Japan, USA, and Canada, the styles that use fundamental data (growth, value, leverage, andquality) are normalized by sector, while the other styles (momentum, change in liquidity, volatility, and size)are normalized by country. In the other multi-country regions, all the styles are normalized by country.

Regional and Industry Factors

The regional and industry factors are calculated from active factor residual returns by combining securityreturns into capitalization-weighted portfolios of all securities in a particular region or industry that passthe following screen:

• annualized volatility is greater than 10%,

• annualized volatility is less than 300%,

• more than 60 daily return history,

• local currency stock price is greater than 0.10,

• stocks in a market that is used in a market factor, (i.e., tax shelters like the Cayman Islands areexcluded,)

• ADRs, GDRs, and other foreign stocks where the stock’s country of risk is the same as the exchangewhere it is traded to avoids double counting.

The regional and industry sensitivities for individual securities are derived by regressing each security’sreturns on these factors. Industries within the model are based on the FactSet industry classification systemand regions are based on major economic market groups, (see [1] for more details).

The model does not assume that a security will have a binary exposure on its industry or region. Instead,the model recognizes that not all securities in an industry or region will necessarily have the same exposureto the industry or region factor, and that securities may have significant exposure to other industries orcountries/regions as well. Therefore, regression is employed rather than simple binary selection.

Variation of a t-test is then performed to ensure that each stock has only significant sensitivities to secondaryindustries and regions. A higher level of significance is required for negative exposures, which imposes a beliefthat these are rare and likely to have deleterious effects on both optimization and risk analyses if incorrect.For negative exposure, the significance level is 99% confidence, while positive exposure is 97%. If more thanone factor is deemed significant, then the one with the highest (absolute) t-statistic is chosen.

As was mentioned in Section 1.1, the structure of this model factor block evolves over time as more stocksenter the universe and it becomes possible to create well-diversified factors for certain markets and sectors.A maximum permissible constituent weight of 10% is employed and the total weight of the top 10 largest

1A reminder that this factor is estimated after the regional market and industry factors.


White Paper

holdings is monitored for every factor. Factors with fewer than 30 stocks or where the top 10 holdingsconstitute more than 80% of the factor weight may be dropped from the model. All such cases are reviewedby model creators individually and periodically every two years.

Macroeconomic Factors

A time series regression is performed on the residuals from prior blocks against a 520-day time series of theprice changes of oil and gold, represented by Brent Crude and London Afternoon Fix. Similar to the marketand industry factor block, sensitivities are tested for statistical significance and removed if the stock failssuch a test.


White Paper

2 Commodity ModelThe FactSet commodity model is a factor-based hybrid model, employing multiple stages of regression tocapture risk across a wide variety of commodity sectors. The model is designed to capture systematic riskfrom exposures to factors common across all commodities, as well as idiosyncratic risk specific to eachcommodity. The FactSet commodity model covers roughly 100 indexes, ETFs, and mutual funds as well asapproximately 500 unique commodities. Each specific contract for a future is mapped to its front monthcontract, 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. Certaincommodities, such as those with an open interest below a minimum threshold or those with exorbitantor missing returns for periods in which we use to model common factors, are removed from the estimationuniverse 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 stylefactors are computed in a cross-sectional regression of the forward daily return of all assets in the calibrationuniverse on factor sensitivities β:

r = β10mf10m + β30mf30m + β5βf5β + β20βf20beta + βvolfvol + ε. (1)The model uses five style factors: 10-day and 30-day momentum factors f10m and f30m, five-day and 20-daybeta factors f5β and f20β , and a volatility factor fvol.

The sensitivities of each security i to the momentum factors are represented by a measure of the correlationof the security price with time. Specifically, the short period (T = 10 days) and the long period (T = 30days) correlation coefficients of security price with time are computed as:

ρ =

T∑i=1

(pi − p̄)(ti − t̄)

(T − 1)σpσt

and the t-statistics of these coefficients are then used as a measure of the security momentum and a sensitivityto the momentum factors:

βTm = ρ

√T − 2

1− ρ2.

Sensitivities to the β factors are computed using a slope of the regression line of prices on time, with theregression 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:

βTβ =βT

PT

And finally, the sensitivity to the idiosyncratic volatility factor is represented by the standard deviation ofthe asset return measured over 22 trading days:

βvol =

√√√√ 1

N − 1

22∑i=1

(rti − r)2.


White Paper

Before performing the factor regression (1), each of these sensitivities is checked for outliers greater thanthree standard deviations from the mean and then normalized with a z-score.

Residuals from the first stage of the regression are then used as dependent variables in a second stage. Thesecond stage employs a time series regression of these residuals against the returns of several market indexesfor 250 days:

ε = β6f6 + β7f7 + β8f8 + β9f9 + β10f10 + 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’sexposure to these separate composite indexes offers differing perspectives. This is a result of the S&P GSCIIndex and the Dow Jones-UBS Composite Index being long-only commodity indexes, while Reuters-CRBTotal Return Commodity is a long/short index. Also, the asset allocations across the commodities that makeup the indexes are very different. Additionally, VIX was selected to represent equity volatility and the U.S.3M 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 dailycurrency return. The exposure for each asset is set as a binary value to each currency: 1 for the local currencyand 0 for all others. When a security is priced in a currency different from the model’s base numeraire, thesecurity’s residual is then adjusted by the factor return for that currency:

ϵnumeraire(t) = ϵlocal(t)− rFX(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. When performingMonte Carlo simulations of the FactSet MAC risk model, a future factor return distribution is simulatedand multiplied against each asset’s sensitivity to that factor to generate a returns distribution for eachcommodity.


White Paper

3 Fixed Income Model

3.1 FI Pricing Methodology and Risk Factor DefinitionsFactSet provides terms and conditions on over three million fixed income (FI) securities. Given the sizeof the fixed-income universe and large number of FI portfolios, the computational burden posed by thefull pricing of FI securities is a challenge faced by any simulation-based integrated risk solution. To dealwith this complication, FactSet uses an efficient estimation method based on a multivariate Taylor seriesapproximation of fixed-income returns. The idea of this approach is to reduce the computational burden of fullpricing under the multiple Monte Carlo-generated scenarios by using a robust set of risk factor sensitivitiesfor each fixed income security, pre-computed daily in a separate process. Below we describe the details ofthis methodology.

The market value MVt of a fixed-income security at time t can be expressed through the security price P ft

and its face value Ft as:MVt = P f

t Ft.

Here P ft 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 fluctuationsthat are not actually related to any risk but are there because of deterministic coupon payment. So, for riskpurposes, it is better to distinguish between the random (risky) portion of the price and deterministic couponpayment–accrued interest AIt. This is done through the definition of clean price P c

t as the portion of thesecurity’s price that does not include accrued interest:

P ft = P c

t +AIt.

If the face value of the security changes in time due to paydown (for example, because of prepayments as inthe case of an MBS), the return of the security (relative change of its market value) between the time t andt+∆t can be expressed as:

r =∆MV

MVt=

(P ct+∆t +AIt+∆t)Ft+∆t − (P c

t +AIt)Ft

P ft Ft

.

Introducing a paydown ratio:γ =

Ft+∆t

Ft,

we obtain the following equation for the FI security return:

r =∆AI

P ft

γ

rai

+ γ − 1rpdwn

+∆P c

P ft

γ, (2)

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 discountedcash flows. Thus, the main factor that underlies all FI prices (and returns) is the yield curve that is used tocompute the discount factors. When the price of other, more complex FI securities is computed using thosediscount factors, the result does not coincide with their market price because those securities are riskier thanthe sovereign bonds and their holders demand higher yields as compensation for the extra risk. This riskpremium is quantified by a spread (OAS in case of FI securities with optionality) that is added to the yieldcurve to compute discount factors used to price these securities. Thus, the price of every FI security can be


White Paper

represented as a function of interest rates (yield curve), spread, and time. The dependence on time reflectsthe fact that even when the curve and spread remain unchanged, the price of the FI security will changewhen its maturity shortens (roll 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

∂t∆t+

∑i

∂P c

∂ri∆ri +

1

2

∑i

∂2P c

∂r2i∆r2i + ...+

∂P c

∂OAS∆OAS +

1

2

∂2P c

∂OAS2∆OAS2 + ... (3)

The first term in this expression represents the time dependence of the security price and thus defines the roll-down return. The next two terms (as well as the omitted higher-order terms) include partial sensitivities tokey rates ri along the security-specific yield curve. When computing the security return, these sensitivitiesturn into familiar key rate durations KRDi = − 1

P f∂P c

∂riand key rate convexities KRCi = 1

P f∂2P c

∂r2i. The

next set of terms accounts for spread risk with spread duration defined as DOAS = − 1P f

∂P c

∂OAS and spreadconvexity as COAS = 1

P f∂2P c

∂OAS2 . We also use higher-order terms (through fourth order) of the expansionto compute spread returns. This allows the repricing formula to remain accurate over large spread changessuch as those seen over the 2007-2009 period of the credit crisis. Finally, additional components need to beconsidered in the expansion of certain complex instruments for accurate risk evaluation. In particular, theseare dependencies of convertible bonds on underlying equities and of callable municipal bonds on interest ratevolatility. These are introduced and described in respective sections.

The expansion equation (3) can be combined with the full return equation (2) to obtain the expression usedin the MAC Monte Carlo process to compute fixed income security returns:

r = rai + rpdwn +γ

P f

∂P c

∂t∆t

rroll

− γ∑i

KRDi∆ri +γ

2

∑i

KRCi∆r2i

ryc

− γDOAS∆OAS +γ

2COAS∆OAS2 + ...

rOAS

(4)Here, in addition to the accrued interest and paydown returns, we have introduced the roll return rroll aswell as the yield curve return ryc and the spread return rOAS .

The pricing function in equation (4) naturally defines the FI risk factors. For securities without principalpaydown (γ = 1) or in models with deterministic paydown, rai and rroll are deterministic. The projectedvalues of these components are known with certainty and do not need to be included in Monte Carlosimulations of returns carried out for risk modeling. Thus, FI risk factors can be divided into two groups:the yield curve (interest rate) factors that affect returns across all asset classes and sets of spread factorsthat model spread returns specific to a particular asset class.

Calculation of the interest rate risk component of VaR is based on pre-computing partial durations at 17 keymaturities (key rate durations or KRD) along the security-specific discount curve. To compute the value ofryc for each Monte Carlo scenario, the 17 KRDs are multiplied by the simulated changes of those key rates∆ri.

FactSet employs different credit spread models for different asset classes. The contingent claim analysis(CCA) framework is used for securities with credit and default risk primarily driven by the balance sheetof the issuing entity. Examples of these are corporate bonds from both developed and emerging markets,bank loans and more. The main advantage of the approach is that it can capture the non-linearities observedduring crisis that are not captured by the traditional regression-based spread models.

Euro-sovereign and U.S. municipal bonds are modeled with cross-sectional models that exploit a featureof spread dynamics that the spread volatility is proportional to the level of security spread regardless of


White Paper

their credit rating ([2]). The main advantage of the approach is that it can be used to construct a simpleyet precise linear model that allows for a meaningful decomposition of portfolio risks into the credit risk ofcountries and regions without having to introduce multiple credit rating classes.

Other security classes with credit risk that is not directly linked to the ratio of underlying company assetsand liabilities are modeled using a linear regression-based model of OAS. Examples of these are mortgagebacked securities (MBS), asset backed securities (ABS), inflation protected bonds and more. Although themain factors of the linear spread model are of the same type, each class has additional factors tailored forthe specific characteristics of the asset type.

The list of factors for all spread models employed by FactSet’s MAC model can be found in Section 2 of theAppendix.

3.2 Interest Rate RiskFactSet’s analytic platform is computing yield curves daily for more than 30 sovereign bond markets. Using17 key rates from all of these curves would amount to more than 500 yield curve factors. Adding additional FIfactors 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 increasingsize (see Section 1 of the Evaluation of Risk chapter for details). Besides, interest rate movements at similarmaturities tend to be highly correlated and this high correlation 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 employa principal component analysis (PCA) algorithm to determine the most important aspects of yield curvebehavior. (A good review of the PCA approach to modeling yield curve dynamics is given in [3].)

The PCA approach allows us to parsimoniously summarize yield curve dynamics with a reduced number offactors that explain the greatest percentage of the curve movement. FactSet uses four principal componentsof each curve as risk factors. The covariance matrix of the MAC model is computed using the PCA factorsand the future realizations of the components are simulated in the Monte Carlo process. The results of theMonte Carlo simulation are then converted back to key rate projections by reversing the principal componentscomputation algorithm and the fixed income repricing formula (4) is applied to the simulated rate changes.Thus, even though principal components are used to simulate yield curve movements, users will see output interms of the 17 partial points of the curve. 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.

3.3 Spread Risk–Contingent Claim Analysis ModelThis section describes the principles of the CCA approach as it is applied to valuing credit risk of fixedincome securities. Specifics of the application of the framework to different classes of securities are describedin the following subsections.

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, a financial institution, or a sovereign entity. From the balancesheet point of view, default occurs when the assets of the entity become insufficient to meet the amount ofdebt owed to creditors at maturity; that is, when assets fall below a distress barrier defined by the totalvalue of the company’s liabilities. Thus, the uncertainty in the changes in the future asset value relative tothe promised payment on debt is the driver of the credit and default risk. As total assets decline toward thedistress barrier or if the asset volatility increases such that the value of the assets becomes more uncertain,


White Paper

the value of the risky debt declines and credit spread of the risky debt rises. In other words, credit spreadsin this model are attributed entirely to the risk-neutral expected default loss, which is positively related toleverage and volatility in the firm value.

The contingent claim approach ([4]) to analyzing the credit risk of FI securities is based on three basicassumptions: (i) values of liabilities are derived from assets; (ii) liabilities have different priorities and, in thesimplest model, can be divided into senior claims (debt) and junior claims (equity); and (iii) assets follow astochastic process. In this framework, both risky debt and equity can be modeled as contingent claims onthe assets. Equity is modeled as a call option on the company’s assets, while risky debt is viewed as thedifference between the book value of debt and a put option on the firm’s assets. Basically, CCA assumesthat owners of corporate equity hold a call option on the firm value after outstanding liabilities have beenpaid off. Moreover, they have the option to default if their firm’s asset value falls below the present value ofthe notional amount of outstanding debt at maturity. Thus, bond holders receive a put option premium inthe form of a credit spread above the risk-free rate in return for holding risky corporate debt (and bearingthe 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 theCCA and has a great impact on the model results. The definition of the distress barrier depends on the typeof asset being modeled and will be described below in sections dedicated to each asset type modeled usingthe CCA approach.

The FactSet CCA spread modeling framework is based on the Merton model developed in [4]. The modelstarts with the balance sheet equation stating that, at any time t, the total market value of assets A(t) isequal to the market value of the claims on the assets–equity E(t)–and risky debt D(t) maturing at timeT :

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 promisedpayments, B, maturing in T − t periods. Thus, the value of the equity can be computed using the Black-Scholes-Merton formula for the value of a call:

E(t) = A(t)N(d1)−Be−r(T−t)N(d2), (5)

where N(x) is the cumulative probability of the standard normal density function below x,

d1 =ln

(AB

)+(r +

σ2A

2

)(T − t)

σA

√T − t

, (6)

d2 = d1 − σA

√T − t (7)

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 andequity volatility are connected to the values of asset and asset volatility through the following relation (see[4] for details):

EσE = AσAN(d2). (8)

Neither assets value nor assets volatility is directly observable. However, given the values of E(t) and σE ,the equations (8) and (5) can be used to get an implied estimate of the asset value and volatility. Thecollective view of many market participants is incorporated in the observable market prices of liabilities, andthe change in the market price of these liabilities will determine its volatility. The contingent claims approachimplicitly assumes that market participants’ views on prices incorporate forward-looking information about


White Paper

the future economic prospects of the company. This assumption does not imply that the market is alwaysright about its assessment of credit risk, but that it reflects the best available collective forecast of theexpectations of market participants. An important aspect of this type of model is that equity needs to beexpressed in market value and the volatility of equity has to be computed by means of the price evolution ofthe 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 andvolatility 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 (8) and (5) to obtain the starting values of the company assets A(t) and assetvolatility σA. Monte Carlo simulation is then used to forecast the distribution of equity values at the time ofrisk model horizon t+∆t using the equity factor model, and the starting value A(t) is used in conjunctionwith the relation (8) to obtain the distribution of values of the company assets from the terminal values ofthe equity E(t+∆t) in Monte Carlo simulation as:

A(t+∆t) = A(t) +E(t+∆t)− E(t)

N(d2)

σE

σA.

The simulated distribution of asset values at the risk model horizon is then used to compute the expectedvalue of the risky debt and derive the corresponding distribution of the spread changes. Under the CCAapproach, the risky debt is equivalent in value to default-free debt minus a guarantee against default. Thisguarantee can be calculated as the value of a put P (t) on the asset with an exercise price equal to B. Atthe 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). (9)

The values of the spread at the risk horizon can be computed from equation (9)

s(t+∆t) = − 1

T − (t+∆t)ln

(1− P (t+∆t)

Ber(T−(t+∆t))

), (10)

where the value of the put option P (t + ∆t) is computed using the simulated terminal values of the assetA(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 significantrole. The first one is the well-known concern that arises when one applies the Merton model to study defaultrisk: the probabilities obtained are in the risk-neutral measure and, consequently, one needs a model to gofrom risk neutral probabilities to risk natural probabilities. This is not a concern for the systematic spread riskmodel since we focus on the valuation of the contingent claims themselves and not on probabilities of default.The positive characteristic of the approach–that it strives to use information in various assets and marketsto gather information about systemic risk–is at the same time a concern since the implementation of themodel requires measurements of a number of variables that are not directly observable. This leads to a fairlycomplex calibration procedure that relies on a number of heuristic assumptions. Also, in practice, structuralmodels tend to underestimate credit spreads (this is known as the “spread puzzle”). Both these issues aremitigated by the fact that we use the approach not to measure the exact relationship between various assetcharacteristics, but as a framework for quantifying a range of uncertainty in those characteristics. We do notneed to accurately model the absolute spread levels and default probabilities but rather to produce a robustestimation for the volatility of spread changes over a given horizon.


White Paper

Specifically, the FactSet credit spread model computes the distribution of the changes in the credit spreadgiven the changes in the stock price of the issuer from the Monte Carlo simulation as described above andthen passes these changes to the pricing equation (4) to obtain simulated distribution of returns of themodeled credit security.

One aspect of the CCA spread model worth noting is the assumption of a 100% correlation between acompany’s spreads with the stock price, less the impact of interest rates. As a result, there are no contributionsfrom corporate bonds to the cross-asset covariance matrix; the interaction between spreads and other factorsin the risk model is expressed through the equity factor covariances with all other factors.

Corporate Bonds

Country Group

Sector Subgroup

Industry Subgroup

A Rating Group

AAA AA

B Rating Group

BBB BBB

Figure 1: Hierarchy of Equity for a Specific Country toDetermine Corporate Proxy

To be able to use the CCA framework to value cor-porate bonds, one first needs to identify the equityto which the credit relates. The FactSet corporatebonds model can use either a FactSet formula or client-supplied inputs to identify the underlying equity. If theissuing corporation is publicly traded or has a corpo-rate parent that is publicly traded, the framework relieson public equity information. For private corporationswith no public parent, a corporate proxy is created todetermine suitable equity inputs. FactSet maintains anumber of global corporate fixed-income indexes. Theseindexes are used daily to create a hierarchy of equityproxy groups as shown in Figure 1. Each bond issuedby a private corporate entity is assigned to one of thegroups based on the information available for the bond.The hierarchy of sectors and industries is created usingFactSet’s classification system. The rating of a bondis determined as a minimum value from three ratingagencies–S&P, Moody, and Fitch. Each proxy group isrepresented by an index of the corporate bonds from publicly traded entities. These indexes are used tocalculate the average equity model exposures, as well as Merton model inputs such as equity market cap-italization, implied equity volatility, and debt level for each proxy group. Risk statistics of a bond fromthe non-publicly traded entity are then modeled using FactSet’s corporate credit modeling framework withproxy values as inputs.

Once the equity value and volatility are determined for a given corporate bond, the model runs Monte Carlosimulations and computes the distribution of asset values at the risk model horizon as described in Section3.3 of this chapter. However, in the case of corporate bonds, we cannot use equation (10) directly becausestructural models are known to underestimate the value of the corporate bond’s spread (the effect known asa “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 barrieris in reality also a random variable. The model assumes that the default point is described by the firm’sleverage–the ratio of its assets and liabilities, where only assets are changing. But firms often adjust theirliabilities as they near default. It is common to observe the liabilities of commercial and industrial firmsincrease as they near default while the liabilities of financial institutions often decrease as they approachdefault. The difference is usually just a reflection of the liquidity in the firm’s assets and thus their ability toadjust their leverage as they encounter difficulties. Unfortunately, the Merton model is unable to specify thebehavior of the liabilities or include other sources of risk in the spread’s computation, and thus these effects


White Paper

must be captured by the calibration procedure. The FactSet corporate spread model take these effects intoaccount by calibrating an effective risk-neutral probability of default.

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 caseof no default, the payout equals just the total value of the debt. Thus, if p(t) is the risk-neutral probabilityof default at time t, the present value of the risky debt at that time is the risk-neutral expectation of thepayout:

Be−(r+s)(T−t) = p(t)RBe−r(T−t) + (1− p(t))Be−r(T−t). (11)

At the same time, if we assume that the firm’s leverage L is a geometric Brownian motion with volatilityσL, it can be shown that the risk-neutral probability of default is:

p = N(−ln(L) +

(r − σ2

L

2

)(T − t)

σL

√T − t

), (12)

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 defaultprobability p(t) using equation (11) and the starting effective value of leverage L(t) using equation (12).It then computes Monte Carlo simulated distribution of leverage using the simulated distribution of assetvalues obtained in the previous step, assuming that the relative change of leverage between current time andrisk model horizon is the same as the relative change of the asset value:

L(t+∆t)− L(t)

L(t)=

A(t+∆t)−A(t)

A(t)

and uses the simulated values of leverage to compute probabilities of default at the risk model horizonp(t+∆t) using equation (12). The final step is the computation of the distribution of spreads at time t+∆tas:

s(t+∆t) = − 1

T − tln

(1− p(t+∆t)(1−R)

).

Convertible Bonds

Convertible bonds are securities where the bond holder has the option to surrender the bonds to the issuerand receive a pre-specified number of equity shares of the issuer. Like corporate bonds, these securities areexposed to government yield curve risk and corporate spread risk. However, unlike standard corporate bonds,convertible bonds are exposed to the additional direct risk of equity price movements of the issuer. This riskis captured by two additional terms based on the change from the base case equity price versus the MonteCarlo simulated equity price–δE. The risk is reported in the convertible bond factor and the return due tothat factor rCB is computed as:

rCB = ∆CBδE +1

2ΓCBδE

2.

The delta ∆CB and gamma ΓCB are computed nightly by the FactSet fixed-income calculation engine andrepresent the first and second order numerical derivatives of convertible bond price versus small changes inthe stock price, holding OAS constant. FactSet’s option pricing model for convertibles is based on the 1998works of Tsiveriotis and Fernandes [5], which incorporates credit risk into the convertible bond valuation bysplitting the bond price into two parts:


White Paper

1. Cash-only part of the convertible bond (COCB), a hypothetical security that pays all of the cashpayments that an optimal holder of the convertible bond would receive

2. Equity part of the convertible bond (EPCB); the corresponding equity payments are much like anequity option

FactSet augments this model by adding another factor to account for stochastic interest rate movements andhas chosen the Hull-White model (mean reverting normal model of the short rate).

Bank Loans

For bank loans, FactSet uses the Merton framework as described above. For cases where a publicly tradedcorporation or parent is not identified, the equity proxy method is also used. Using the same hierarchy andthe terms and conditions information available, a corporate proxy is assigned.

Other Covered Bonds

For securities not modeled specifically by the covered asset classes, FactSet assigns an equity proxy using thehierarchy of equity groups described in the Corporate Bonds section. Using the same hierarchy and the termsand conditions information available, a corporate proxy is assigned. In these cases, however, it’s likely thatmost detailed information on sector, industry, and ratings may not be available. Thus these miscellaneoussecurities would more likely be assigned to the country proxy group.

3.4 Spread Risk–Linear Regression Model of Relative Spread ChangesAn alternative approach to modeling spread risk of credit securities is based on research in [2] and [6]. In thatresearch, authors observed a linear relationship between spread change volatility σ∆s and spread level s ofUSD-denominated corporate and sovereign bonds. This result had far-reaching implications for the design ofspread 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 precise linear models that allowa meaningful decomposition of portfolio risks into credit risk of different countries, regions, or sectors. Tosimplify the notation of the following discussion, we will suppress the time dependence of variables that donot require forecasting such as spreads, durations, convexities, and more.

It was suggested in [2] that spread change volatility σ∆s(t) of a particular security at time t is proportionalto the current spread level s:

σ∆s(t) = s×(β(t) + σε

), (13)

where the proportionality factor β(t) is common for all bonds in a given market segment regardless of theircredit rating. This relationship can be obtained assuming that the relative change of spread ∆s(t)/s of acredit security can be modeled as

∆s(t)

s= ∆f(t) + ε(t), (14)

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 (13) and (14) become equivalent and β(t) equates to the volatility of ∆f(t). The latter can bedenoted 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 tothat risk will be quantified by Ds× s, where Ds denotes the spread duration. More specifically, we can writethe volatility of the portfolio excess return as a sum of systematic and idiosyncratic volatilities as

σ2excess(t) = σ2

syst.(t) + σ2ε(t).


White Paper

Volatility of the systematic part can then be written as:

σsyst.(t) = Ds × σ∆s(t) = [Ds × s]σ∆s(t)

s= [Ds × s]σ∆f (t),

which confirms the statement above.

Our research presented in [7], [8] and [9] supported the empirical observations of the aforementioned workand observed the effect in other market segments. We also identified additional factors, which govern thedynamics of spread curves and improve model accuracy.

All of our relative spread change models are based on OAS computed by the FactSet analytics engine andfactor 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-specificattributes for different asset classes. Particulars of βj are going to be described in the following subsectionsas they are introduced.

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 governmentbonds are modeled as government bonds with zero spread relative to the euro yield curve and bonds fromother governments are priced relative to this curve. As a result, a broad range of spreads is observed andthis universe exhibits the behaviour described in [2] for corporate bonds and in [6] for USD-denominatedsovereign bonds. Our research for the European sovereigns has been presented in [8].

Country Group S&P RatingAustria, Luxembourg AA+ ,AAA

Finland AA+France AAItaly BBB

Netherlands AAAIreland A+Spain BBB+

Portugal BBB-Belgium AA

Latvia, Lithuania A-, A-, A, A+Slovenia, SlovakiaGreece, Cyprus B-, BB

Table 1: Country groups for regional factors in the spreadmodel of Euro-sovereign bonds.

For our estimation universe, we consider a union ofIG and HY indexes that track EUR-denominatedsovereign debt: Merrill Lynch All Euro GovernmentIndex (MLEZAS), the Merrill Lynch Euro Govern-ment Index (MLEG00), and the Merrill Lynch Slo-vak Republic Government Index (MLG0SL). The re-sulting universe includes all relevant countries andcovers a broad range of credit ratings as is shown inTable 1.

After choosing the above set of indices, we apply fil-ters to guarantee that the bonds we use for factorreturn estimation are highly liquid and have reli-able analytics. In particular, for most countries weconsider only noncallable fixed coupon bonds, whileallowing both fixed and variable coupon bonds forboth Greece and Cyprus. The latter is due to thesparsity of bonds and the nature of the debt issuancefollowing the Greek debt crisis and the subsequentdebt restructuring. Details of analytics filters can befound in [8].


White Paper

Details on regression methodology and model selection can be found in [8], while here we only summarizethe final results. Factor returns ∆fj(t) are estimated daily with weighted least squares (WLS) using

∆OAS(t) = ∆fLS(t) +OAS∆fc/r(t) +OASβSD∆fRSD(t) + βSD∆fASD(t) + ε(OAS,DOAS), (15)

where βSD = max{DOAS−DOAS , 0} is the short duration factor sensitivity that will be described shortly.

Different terms in model (15) capture different dynamics of a given OAS distribution. Factor fLS , referredto as the low spread factor, captures parallel shifts of the entire distribution when spreads rise or decrease alltogether and it mainly drives the systematic spread volatility of bonds with the lowest spreads. Spread factorfc/r is specific to every country or region listed in Table 1 and these factors capture how the distributionstighten (or widen) during rallies (or corrections).

Factors fASD and fRSD are referred to as absolute short duration and relative short duration factors. Theycapture the volatility component due to changes in curvature of the spread curve of the asset class becauseof their non-linear sensitivities βSD and OASβSD, respectively. These two factors can also be thought of asrespective 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 fromdifferent sectors, regions and interest rate volatility. The latter factor proved to be of high significance as thismunicipal bond market consist predominantly of callable bonds. Our research on this asset class is describedin [9], while only the main results 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: states of New York, California, New Jersey, Illinois,Pennsylvania, Florida, Connecticut, Texas, Massachusetts, Washington, territories of Puerto Rico, and agroup that consists of Guam and Virgin Islands. These regions were selected because the factors related tothem were shown to add explanatory power to the model.

The municipal bond class is distinguished from the ones introduced above by a much narrower distribution ofspreads and a greater presence of callable bonds. To consider the interplay of these peculiarities, the Taylorseries approximation in equation (4) is augmented as follows

r = rai + rpdwn + rroll + ryc + rOAS + rvega.

The accrued interest, paydown, roll, curve, and spread return components were introduced previously. Thecomponent rvega arises from changes of interest rate volatility and is proportional to the vega durationDvega. The vega duration of a noncallable bond is zero. The spread and vega returns together form theexcess return

rexcess = r − (rai + rpdwn + rroll + ryc) = rOAS + rvega,

which captures the effects of callable bonds. To be compatible with the current model framework, ourmunicipal bond factor model was built to explain the DOAS-scaled excess return, i.e., −rexcess/DOAS . In thecase 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 in order to remove securities that arelikely to exhibit large idiosyncratic movements and data outliers. Then, zero coupon bonds and pre-refundedbonds are removed from the universe. All bonds are removed from the universe six months before theirmaturity to exclude prices that are converging to the par value. Callable bonds and sinking-fund bonds are


White Paper

removed from the universe six months before the first call or sinking-fund date. Additional filters are appliedto every daily cross section of observations to ensure that most liquid bonds are selected and analytics areof the highest possible quality. Parameters of the filters were carefully tuned to ensure that all regions arerepresented when estimating the factor returns. For finer details of the filtering procedure, the reader isreferred to [9].

The model is defined by

− 1

DOASrexcess(t) = ∆fsector−LS(t) +OAS∆fsector(t) + ∆fregion−LS(t) +OAS∆fregion(t)

+ βSD∆fASD(t) + βLD∆fALD(t) + IcallDvega

DOAS∆fvega(t) + ε(OAS,DOAS).

where βLD = max{DOAS −DOAS , 0} is the long duration sensitivity, Icall is an indicator function that takesthe 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 factorsthat are common for all bonds. The former factor has been introduced while describing the model of Euro-sovereign bonds and the latter is its counterpart driving the long end of the spread curve. Finally, ∆fvegadenotes the vega factor return, which explains the changes in the interest rate volatility.

3.5 Spread Risk–Linear Regression Model of Spread LevelsSecurity classes whose credit risk is defined by processes that are not directly related to the ratio of theunderlying company assets and liabilities are modeled using linear regression models of spread. Althoughthe main factors of the linear spread model are the same for all asset classes it is used for, each class mayhave additional factors tailored for the specific characteristics of the asset type. Detailed descriptions of theseadditional factors are provided in the following subsections dedicated to specific asset types.

The regression model for spread is designed to capture the level and slope of the common spread factor asa function of the security’s duration (level and slope of the spread curve):

OAS(t) = flevel(t) +Dfslope(t) +∑

βjfj(t).

Here OAS and D are spread and duration of the security, flevel(t) and fslope(t), are estimated slope andlevel factors, and βj and fj(t) are additional sensitivities and factors, if any.

The returns of the estimated time series of all factors are used as input into the factor return matrixand eventually the overall covariance matrix. During Monte Carlo simulations of the future factor returndistribution, the systematic change of the security OAS between current time t and forecast horizon t+ 1 isestimated from simulated factor values as:

∆OAS(t+ 1) = ∆flevel(t+ 1) +D∆fslope(t+ 1) +∑

βj∆fj(t+ 1).

The simulated change of OAS is then used in the pricing formula (4) to obtain simulated security re-turns.

Quasi-Governmental (Agency) Bonds

Quasi-government securities, also known as agencies, are securities issued by non-sovereign entities that haveclose ties to, or the backing of, their sovereign governments and are not properly considered to be corporate


White Paper

debt. Further complicating the issue is the fact that these entities sometimes issue outside their homecurrency. Thus, for each currency, we have a home currency spread model and another model for externalquasi-governments issuing in that currency. The spread risk is estimated using a regression model for spreadthat is designed to capture the level and slope of the common spread factor as a function of the security’sduration (level and slope of the spread curve). For each time t the spread factors are computed by runningthe cross-sectional regression on the universe of the agency securities issued in a given currency:

OAS(t) = flevel(t) +Dfslope(t),

where OAS(t) is the level of spread of the security at time t, and D is the effective duration.

For markets where there is great dissimilarity between the issuers (regression results show statistically in-significant value of fslope) or an insufficient number of unique issues, only the level factor flevel is used.These reduced models are employed for Australian and European home currency models as well as all for-eign currency models (Supranational in USD, EUR, GBP, and AUD). The full models are used for U.S. andCanadian home currency agencies and Canadian USD agencies.

Mortgage Backed and Mortgage Related Securities in USD

Mortgage backed securities (MBS) are securities where the principal and interest payments on a pool oflike mortgages are passed through to the investor. They are issued by the government agencies or quasi-governmental agencies of the Government National Mortgage Association, Federal National Mortgage Associ-ation, and Federal Home Loan Mortgage Corporation. Mortgage related securities also include collateralizedmortgage obligations (CMOs), which are structured securities backed by agency MBS and residential ABS,which are structured securities backed by non-agency MBS.

The spread risk of these securities is measured using FactSet’s U.S. MBS prepayment models and FactSet’sMonte Carlo option pricing model. Sensitivity of the MBS to prepayment speeds is captured by the turnoverduration DT and refinancing duration DR. Turnover duration is the change in price for a change in theturnover component of the prepayment model. Refinancing duration is the change in price for a change inthe refinancing component of the prepayment model.

To calibrate the MBS spread factors daily, we regress

OAS(t) = flevel(t) +DT fturnover(t) +DRfrefi(t)

over a portfolio of liquid interest-only and principal-only mortgages.

U.S. and Non-U.S. ABS and CMBS

The spreads of the ABS/CMBS securities from the universe that includes all U.S., UK, and Euro-zoneABS and CMBS (excluding MBS related securities–manufactured housing ABS, home equity loan ABS, andCMOs) are modeled using a linear regression model that is designed to capture the spread sensitivity torating levels and seasonality. Boolean ratings sensitivities I(rating) are used to capture spread sensitivityto general ratings classes. The sensitivity value would be set to 1 if the rating class applies, 0 otherwise.For instances when rating information is not available, a spread rating is assigned based on FactSet’s initialspread calculation. The seasoning sensitivity is captured by the weighted-average loan age WALA. The modelfor the U.S. ABS/CMBS includes additional factors for AA, A, and BBB ratings:

OAS(t) = flevel(t) +WALAfseason(t) + I(AA)fAA(t) + I(A)fA(t) + I(BBB)fBBB(t) (16)

and the model for non-U.S. ABS/CMBS distinguishes only the above-AA and below-AA ratings:


White Paper

OAS(t) = flevel(t) +WALAfseason(t) + I(A−BBB)fA−BBB(t). (17)

Daily cross-sectional regressions (16) and (17) are computed separately for U.S. ABS/CMBS and non-U.S.ABS/CMBS. Different universes of securities created from multiple ABS and CMBS indexes are used todetermine the level, rating, and seasonality factors.

Japanese MBS

Figure 2: The historical movement of the Nomura BPIfor 2010 to 2012

The Japanese MBS securities are issued by the JapanHousing Finance Agency and its predecessor entities.There are approximately 100–150 Japanese MBS cur-rently outstanding. The overall spread of JPY MBS se-curities (Figure 2) is low and not volatile. The spreadmovements of individual Japanese MBS are highly cor-related with each other as well as the index. In fact, theaverage correlation between the changes in OAS of eachindividual security versus the index is 0.90. For thesereasons, we opted for a simple model where the system-atic change of each JPY MBS security is estimated asthe change in the mean OAS of a representative uni-verse. The OAS of the securities in the calibration uni-verse is calculated using FactSet’s Japanese MBS pre-payment model and FactSet’s Monte Carlo MBS pricingmodel.

Inflation Protected Securities

Inflation protected securities are securities where the principal balance is tied to an inflation index. FactSetincludes future inflation assumptions to convert these securities to a nominal yield basis, allowing one toestimate the interest rate risk of these securities relative to the nominal government curve.

The models of the spread risk of inflation protected securities run separately for each country. The modelscover U.S. TIPS, as well as inflation protected securities from the UK and France. Given the fewer inflationprotected securities in the following markets, we use a simple mean across the issuance to model systematiccomponents of the spread in inflation protected securities in Canada, Germany, Italy, and Japan.

The linear regression model for inflation protected securities is based on effective duration and effectiveconvexity:

OAS(t) = flevel(t) +Dfdur(t) + Cfconv(t),

where D and C are effective duration and convexity.


White Paper

4 Derivatives

4.1 Equity DerivativesThere are two aspects to equity derivatives to consider when calculating risk. First, the return distributionsof portfolios that include equity derivatives will not be normal, necessitating full repricing for each futurescenario. Second, the dynamics of the volatility surface will need to be modeled to determine the impliedvolatility on each future scenario.

The Impact of Options on Return Distributions

Figure 3: Simulated return distribution of a portfolio ofa single equity

To gain some intuition around the resulting non-normaldistributions and its impact on VaR, we will examinetwo standard option strategies.

We start with a portfolio containing one share of a fundthat tracks the S&P500, initially at $2, 085.51 with anassumed volatility of 17.8%. The return distribution ofthis portfolio is assumed to be normal as seen in Figure3.

The VaR can be analytically calculated in this case tobe V aR5% = zσ

√Thoriz = 10.727%.

A protective put is the combination of a long stock anda long put option positions. It is a strategy intendedto protect the investor from downturns in the stock,the put acting as insurance, paying precisely when theportfolio is otherwise adversely affected by the stockmovement. The return distribution is then capped onthe downside, resulting in a highly non-normal distri-bution. The portfolio VaR is significantly reduced bythis option strategy. In the present case with simplify-ing assumptions2 the VaR is determined by simulationto 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 tothe 99.5% VaR, a scaling that cannot be explained by a normal distribution.

A covered call is the combination of long a stock and short a call option. The resulting option premium willgenerate return at the expense of giving up possible future appreciation of the underlying stock. This arisesin an opposite looking distribution as compared to the protective put. As one can see in Figure 5, the upsidegains are eliminated, and the median return is above zero.

In the case of the covered call, there is no true risk reduction of downside measures like VaR, since thestrategy sells upside potential for increased revenue–in the present case, the simulated V aR5% = 10.25%.3The slight decrease is due to the increased return of the portfolio by selling the option and receiving theoption premium.

These two simple examples demonstrate that the VaR of portfolios containing even simple equity option2The future volatility is held at the 17.8% level and the option expiry is less than the horizon time with a strike value of

$1900.3The option strike in this case was $2300.


White Paper

strategies can lead to drastically non-normal returns distributions and therefore full repricing of these optionsis required to adequately measure risk.

Equity Option Pricing

Figure 4: Simulated portfolio distribution for aprotective put option strategy

The pricing for equity options is based on the Black-Scholesmodel and the assumption of a normal distribution of equityreturns.

For European-style options (those that can only be exercisedat the maturity date), the price of a call option with strike Kcan 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 maturingon the same date as the option expires

d1 =lnF (t,T )

K + v√v

,

v =

∫ T

t

σ2(s)ds,

and N(x) is the cumulative normal distribution.4

Figure 5: Simulated portfolio distribution for acovered call option strategy

The situation is more difficult when the option owner can ex-ercise at any time up until maturity (American-style options).These options have no fixed payoff condition; exactly where theowner exercises the option is a dynamic function of the stockprice and cannot be determined a priori.

One possible solution is to use a binomial tree to numericallysolve for the price of an American-style option. The binomialtree is a discretized approach where the final distribution issampled on a finite set of points, and the risk-neutral probabil-ities are used to determine the option price at an earlier time.This is then iterated until the starting level of the equity isreached at time t.5

The benefit of this approach is that at every “node” the con-tinuation value can be calculated and therefore the optimalexercise time for American options is handled naturally. Themain drawback of this approach as it relates to Monte CarloVaR is that on each forward simulation, the number of calcu-lations required for each option increases the complexity of thecalculation by an order of magnitude.

At FactSet we have consequently used an approximate closed form approach to value American optionsbased on an early paper by Barone-Adesi and Whaley [10]. In this paper, they examine commodity options

4Note that the volatility does not have to be constant in time; the Black-Scholes formula uses the total realized variance ofthe underlying equity.

5This is the approach taken in the Portfolio Analysis product.


White Paper

and determine an approximate closed form solution based on the difference between the cost of carry for theunderlying and the risk-free rate.

The starting point is noting that the same PDE applies to both American and European options and thereforeto the American option premium as well. To see this, simply note that the PDE can be written as a linearoperator LC(S, T ) = 0, where:

L =∂

∂t+

1

2S2σ2(t)

∂2

∂S2+ rS

∂

∂S− r.

Since CA(S, T ) and CE(S, T ) both solve the PDE, so does their difference representing the American optionpremium, which we will denote ϵ(S, T ) = CA(S, T ) − CE(S, T ). The next step is to use the parameteriza-tion6:

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 thePDE is given by:

CA(S, T ) = CE(S, T ) + g(T )α1Sβ , (18)

where β depends on the parameters of the stock (volatility, risk-free rate, and dividend yield) and α will bedetermined 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 greaterthan that of a European option since the owner has more optionality. The value of a call option cannot bemore than the value of the underlying stock. The value of an American call option can never be less than theintrinsic value (S −K). This is subtly, but importantly, different than European options that have a lowerbound of (P (t, T )F (t, T )−P (t, T )K). In the case of a non-dividend paying stock, this lower bound is higherthan the American lower bound, implying that the owner of an American option on a non-dividend payingstock 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 risk-free rate), then this lower bound can be less than the European lower bound. In this case, early exercise ispossible. If we denote the S∗ as the stock price at which early exercise first becomes optimal, then α1 canbe solved by setting the left-hand side of equation (18) equal to the exercise value (S∗ −K) and equatingthe first derivative. This leads to the Barone-Adesi and Whaley approximation for the value of an Americancall option:

CA(S, T ) = CE(S, T ) + α2

(S

S∗

)β

,

where:α2 =

(S∗

β

)[1− e−qTN

(d1 (S

∗))]

,

and q is the constant dividend yield of the stock. Barone-Adesi and Whaley demonstrate that S∗ can besolved 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 differencebetween the reference security/index price on a specific future date and a price struck at the initiationof the contract. EIFs are marked-to-market daily, which results in no net exposure to default of eithercounterparty.

6This parameterization seems to be ubiquitous in American option pricing; see [11].


White Paper

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 is simulated in the local currency usingthe joint probability distribution of these equity risk model factors. The new price is then subtracted fromthe 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 forwardscenarios.

The implied volatility of options is seen empirically to strongly depend on the strike, a phenomenon known asthe volatility smile. For equities and equity indexes, the longer maturity option volatilities are monotonicallydecreasing functions of strike (sometimes referred to as the volatility skew) whereas short-dated maturitiesdo 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 [12]. The volatilitysurface is first fit with a five-factor regression in the variance and time variables:

σrel ≡σ

σATM= β1ν

2 + β2ν + β3t+ β4νt+ β5, (19)

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 usingequation (19). The coefficients are fit nightly on the number of options.

4.2 Fixed Income DerivativesFor all fixed-income derivative types, in contrast to the equity derivatives, FactSet employs the analyticalapproximation based on Taylor expansion (see Section 3.1 of the Risk Models chapter) with appropriateadjustments. The first and second order KRDs used in the Taylor expansion formula (4) are computed usingstochastic models that take into account the optionality embedded in various FI instruments.

Eurodollar Futures (EDFs)

A Eurodollar Future is a contract where one party has the obligation to pay the difference between theLondon Interbank Offered Rate (LIBOR) on a specific future date and a rate struck at the initiation of thecontract. EDFs are marked-to-market daily, meaning that the counterparty that has lost will put money intoa 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 convexityadjustment. This term arises7 from the benefit that the long receives from the positive correlation betweenthe value of the contract and rising interest rates. As interest rates rise and the long gains value, the shortmust 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 interestrequired 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 Blackmodel of lognormal LIBOR rates.

7Technically, 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.


White Paper

Government Bond Futures

Government bond futures are run using a LIBOR Market Model (LMM) in Monte Carlo. Two hundredpaths are run and the cheapest to deliver (CTD) is determined on each path. The value of the CTD is thendiscounted using the stochastic discount factor on each path and the resulting present value from each pathis averaged.

Option on Government Bond Futures

The CTD bond is chosen as the CTD as of the valuation date and a closed-form solution in the Black modelwhere the bond price is assumed to be lognormal is used to price the option on this bond.

Option on Bond

For over the counter (OTC) options on bonds, FactSet employs the Hull-White model, a mean-revertingnormal model of the short rate with a mean-reverting parameter of a = 0.05. The short rate is technicallythe instantaneous rate of lending for an infinitesimal period of time; however, it can be proxied by theovernight rate, or more correctly just thought of as a mathematical construction representing the singlestochastic factor driving movements of the entire yield curve term structure.

The method is closed form; however, it is based on a trick by Jamshidian [13]. The reason special treatmentis required is that a bond option is not simply an option on a single cash flow, rather, the buyer of the optionis entitled to many cash flows, a fact that needs to be accounted for in the determination of the price. Itis not acceptable to simply treat the collection of cash flows as a collection of individual options, since thebuyer 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 allcash flow options (or optionlet, to borrow from the term caplet) are exercised at the same time.

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,8 the IRS market has a gross market valueof $8915 billion.

The bulk of the IRS market is in “vanilla IRS,” which are instruments that swap a fixed stream of cash flowsfor 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 cashflows is cτN , where c is the relevant coupon, τ is the accrual fraction between the coupon start and end date,and N is the notional principal value (referred to simply as the notional). Even though the cash flows arebased on the notional, this amount is not exchanged for the simple fact that this would have no economicimpact.

Other types of swaps include:

• Cross-currency swaps. These can be either fixed-float, float-float, or fixed-fixed (sometimes called FXswaps). 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 markethas started trading “mark-to-market” (or resetting notional) cross currency swaps. On each paymentdate, one counterparty makes a notional payment whose magnitude is set so that the ratio of thenotionals is equal to the spot foreign exchange rate.

8As of H2 2016.


White Paper

• Zero coupon swaps. Interim cash flows are not paid but start to accrue interest, and all cash flows arepaid at maturity.

• Inflation swaps. The floating rate of inflation swaps is linked to an inflation index such as U.S. CPIex-tobacco. The inflation leg is set in arrears (with an economy-dependent lag) and paid in arrears sothat the inflation protection is on the coupon period.

• Amortizing swaps. The notional changes over time according to an agreed-upon schedule.

No modeling is required to determine the price of IRS. For explicit information on how FactSet prices IRS,please refer to the white paper [14].

Swaption

Swaption is an option to enter a fixed-to-float interest rate swap at a specified time in the future. Thenomenclature for an interest rate swaption is determined by the direction of the fixed leg of the underlyingswap. 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 thelong 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 exercisedecision

3. Swap tenor–the length of the underlying swap

For instance, a 2x10 payers swaption struck at ATM+100 is a contract where, in 2y time, the long has theoption 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 LIBORrates. Each cap can be deconstructed into a single period cap, known as a caplet. By virtue of this decompo-sition, each LIBOR rate can be modeled in isolation, and therefore the correlated dynamics of neighboringLIBOR rates do not need to be taken into account.

Callable Bond

Again the Hull-White model, mean-reverting normal model of the short rate with mean reversion speeda = 0.05, is used as the model to value callable bonds. The calculation method chosen is the trinomialtree, which is a discretized backwards induction technique where the short rate is determined on a set ofnodes that expand around the expected value, which can be set to exactly reproduce the term structure asseen on the valuation date. The payoff at maturity is determined on each final timestep of the tree and thecontinuation value can then be determined backward until the valuation node is reached.


White Paper

Credit Default Swap

Ever since the credit “big bang” in 2011, there has been a standardized method to price credit default swaps(CDSs).9 FactSet uses this pricing methodology.

Credit default swaps name a “reference entity” in the contract, which one counterparty would like protectionfrom the adverse effects of a credit event on this entity. This reference entity can be a corporation, likeRadio Shack, or it could be a specific bond. CDS contracts have two legs, similar to IRS. The protectionbuyer makes fixed rate payments to the other counterparty, called the protection seller. The protection selleronly makes a payment after the occurrence of a “credit event” which is precisely defined and adjudicated bythe International Swaps and Derivatives Association (ISDA). The amount of the payment depends on theamount that was able to be recovered by the protection buyer as a result of the credit event (known as therecovery rate R).

In the pre-standardized CDS contracts, the fixed coupon that the protection buyer would pay was determinedby a breakeven analysis based on the market expectations of the probability of default and recovery rate ofthe reference entity. However, today’s current standard contract specifies that the coupon is either 100bpsor 500bps. This implies that contracts no longer have zero value at initiation and therefore current CDScontracts are quoted in “upfront fee” or “premium” that is the present value of the two legs, again takinginto account the market expectations of the probability of default and the recovery rate of the referenceentity.

FactSet employs the CCA model described in Section 3.3 of this chapter to compute the risk of CDScontracts. The CDS running spread that is used as an input to the model is computed from the termstructure of CDS Spreads of the reference entity. For single name CDS the financial information of thereference entity or suitable proxy is used. For CDS indexes the relevant market corporate proxy is used inthe Merton equation.

Currency Forward

Currency forwards require no modeling and are priced as two zero coupon bonds in different currencies wherethe 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 anotheron a specified future date at a rate specified at the initiation of the contract. In FactSet currency futures areloaded as “physical” security in the contract’s base currency. In the universe options of Portfolio Analysis theuser can choose to then “show currency exposure” that splits the position into two, with one leg representingthe base currency and the other the term currency. This option should be used for the purpose of riskanalysis.

Currency Options

Currency Options are securities that gives the purchaser the right, but not the obligation, to exchange moneydenominated in one currency into another currency at a rate agreed at the initiation of the contract. Theyare typically traded OTC and used to hedge currency exposures. To value currency options, FactSet uses amodified version of the same pricing algorithms as is used for equity or index options that are detailed inSection 4.1 of this chapter, which takes into account the risk free rate of each currency. FactSet does notcurrently have a currency volatility surface model and computes implied volatility based on the contract price

9www.cdsmodel.com


White Paper

supplied by the client. FactSet then re-prices the option in the Monte Carlo simulation process by holdingthe implied volatility constant. If no price is supplied by the clients, the historical volatility is used.


White Paper

5 Return Based ModelAt a time when markets are trending in ways to incorporate diversification amongst portfolios via non-traditional asset types, regulatory oversight has mandated such portfolios be viewed not from an esotericperspective 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 accessto 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. Inthe context of the RBR model, a dynamic approach is taken to modeling the risk of securities–exposuresare calculated at runtime from a regression against the predefined factors within the risk model. The modelemploys observed market returns as time series of risk factors, alongside a proprietary algorithm to modelvariance of almost any asset type. The RBR risk factors encompass a wide range of asset classes, includingequity, fixed income, hedge funds, commodities, and other alternatives, all based on market indexes. Therange of these asset types can be quite diverse, so a balance between an all-encompassing range of factorsspanning each asset class and limitations on data availability, frequency, methodology, and integration mustbe chosen. As a result, more than 100 time series factors have been selected as a representation of clientholdings, including 17 hedge funds from the SPAR HFR database, six real estate benchmarks, 11 equityindices, 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 3.)

The model makes very little assumption of the source or distribution on these returns. The only requirementis that the known return series must be daily and have a minimum of 180 days to calculate risk, thoughin future iterations the model will accept monthly returns series. The model is designed to work based onlimited 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 inthe model, leading to unstable results that are difficult to interpret. Since we are not making assumptions onthe underlying returns series and cannot limit ourselves to a small subset of factors given the variety of assetsemployed by portfolio managers, a programmatic approach is necessary to choose the correct predictors foreach asset. FactSet has developed a clustering algorithm based on the Hierarchical Agglomerative Clusteringapproach (see, for example, [15], Chapter 17), to allow this process to occur at runtime. Its goal is selectinga set of statistically significant factors with a high correlation to the asset, while at the same time avoidinginter-factor correlation. The result is a parsimonious selection of independent factors for a wide range ofassets that are generally uncovered by other models.

The factor selection algorithm is based on Spearman’s rank correlation coefficient between factors and be-tween factors and asset return time series. It chooses a subset of factors that have the highest correlationwith the asset, then defines clusters of highly correlated factors within the subset and uses a representativefactor from each cluster in the final model to ensure minimal cross-correlation among factors and thus avoidmulticollinearity in model regression. The final step of the algorithm is the stepwise regression of the assetreturn time series on all chosen factors. The Akaike information criterion corrected for sample size is used torank the regressions, and the subset of factors with the highest AIC and significant values of the T-statisticsis 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 largeadjusted R2.


White Paper

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 risk of a portfolio on an absolute basis orrelative 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 probability of 1 − α, and provides ameasure for tail risk of a portfolio. For example, if a portfolio’s V aR95 (at a 95% confidence level) is reportedas 10 bp, then there is 95% probability that the portfolio loss will be smaller than 10 bp (the portfolio returnwill be greater than −10 bp). Alternatively, one could say that the portfolio’s return is expected to be worsethan −10 bp 5% of the time. VaR measures only a threshold value and does not provide information aboutthe extent of the losses beyond that. ETL overcomes this shortcoming by measuring the average loss of allthe worst-case scenarios beyond the threshold, which gives a better representation of the potential losses ofthe portfolio at a given probability.

All these risk measures are derived from the probability distribution of the portfolio returns estimated by theMAC risk model. To obtain this distribution using a given risk factors model, three steps are necessary:

• Generate joint distribution of risk factors

• Apply pricing functions to transform the distribution of risk factors into the distribution of securityreturns at investment horizon and ultimately into the return distribution of a given portfolio

• Compute return distribution statistics (TEV, VaR, ETL) that serve as risk measures for the portfolio

The FactSet risk model represents the return of each security through changes in systematic risk factorsand the security’s exposures to each of them. Each systematic factor is treated as a random variable and aportfolio’s return distribution is ultimately modeled through the joint distribution of these variables. Usingthe wealth of data available at FactSet, we employ statistical techniques to generate time series of historicalobservations 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 isused to predict the volatilities and correlations of the factors, and thus represents a key component of therisk model.

Having specified the behavior of risk factors and their correlations, the risk model estimates the distributionof returns of each security by generating multiple samples from the factors’ joint distribution using MonteCarlo methodology. For each Monte Carlo scenario, the simulated factor returns are used to compute theindividual security returns. For linear models such as Equity and Commodity models, the factor returnsare multiplied by the corresponding sensitivities of each security. For non-linear fixed income models thepricing function (see Section 3.1) is applied to the simulated factors to obtain the FI security returns. Thesimulations are repeated several thousands of times to generate an entire distribution of possible returns foreach security in a given portfolio. Then the results of all scenarios are aggregated to get the entire returndistribution for the portfolio. To estimate the portfolio’s excess return (tracking error) distribution, we also


White Paper

perform the same simulation for the portfolio’s benchmark and take the difference in returns (portfolio minusbenchmark) for each simulation run. Once we have generated the entire distributions for total return andtracking error, we can easily calculate TEV, VaR, ETL, and any other risk measure at any chosen confidenceinterval. The risk model also provides the breakdown of portfolio risk into additive contributions of variousrisk factors, allowing portfolio managers to quickly identify sources of risk in the portfolio and see if itmatches their desired risk profile.

1 Covariance Matrix ComputationThe 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 jointdistribution of factors to estimate risk metrics of a portfolio. This distribution is defined through the factorcovariance matrix that combines forecasts for the volatilities and correlations of the model factors, and thusrepresents a critical component for constructing a high-quality risk model. The need for forecasting volatilityand correlations, not just estimating them based on past return data, determines the main components ofthe covariance matrix model employed in the FactSet MAC risk framework.

The difficulty of creating a good volatility (and correlations) forecasting model becomes apparent onceone realizes that volatility is inherently unobserved and evolves more or less randomly through time. (Inmathematical terms, the volatility is a latent stochastic process.) The recognition of the time-varying natureof volatility leads to the volatility models that take conditional flavor. In particular, having a return processwith observations available at equally spaced discrete points in time as rt ≡ r(t), t = 1, 2, ..., we define itsconditional variance as:

σ2t+1|t = V ar

[r2t+1|Ft

], (20)

where Ft is the information set that reflects all relevant information through time t. This conditional variancedoes not necessarily equal the unconditional variance V ar

[r2t+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 thevolatility is a latent process 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 oferror. Effectively, the latent nature of the volatility process turns the volatility estimation problem into afiltering problem on the set of past returns. But under the simplifying assumption that we are dealing withfull information sets, so that the conditional volatility is directly observable, the equation (20) provides anunbiased estimate of the future volatility.

The simplest way to provide a low biased estimate of time-varying volatility based on the actual return datais to use the exponentially weighted moving average (EWMA) filter that effectively became the market-widestandard for risk models since it was first employed by JPMorgan RiskMetrics [16]. This method is also usedin 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 inestimating the factor covariance matrix lies in the dimensionality of the problem. A multi-asset model caneasily contain more than 1000 factors, requiring estimation of more than a million independent elementsof the matrix. An estimator of such a matrix is susceptible to noise and spurious relationships that areunlikely to persist out-of-sample. If the covariance matrix is computed naively, by simply computing thepairwise correlations of the time series of all the factors taken over a short period of time, then it is likelyto be extremely ill-conditioned. For instance, if the number of time observations is less than the number of


White Paper

factors in the model, the matrix will be “rank deficient,” meaning in practice that it can be used to constructapparently riskless portfolios. Thus, an estimator of a covariance matrix obtained from observed time seriesof risk factors has to be regularized–a special procedure has to be applied to filter the excess noise in thematrix and ensure that the matrix is symmetric positive definite (i.e., well-conditioned).

1.1 EWMA ModelThis widely used method to forecast the conditional volatility of a factor return process rt employs themoving average estimator for the conditional volatility while assigning more weight to recent observationsthan to the observations farther in the past. In the EWMA model, the importance of past observationsdecreases smoothly as time passes with the speed of this decrease defined by the model decay constantδ:

σ2t+1|t =

(1− δ)

1− δM

M∑i=1

δi−1r2t−i+1.

In the EWMA model, the value of δ determines the trade-off between the responsiveness and persistenceof the estimator. Short half-life results in a very responsive estimator that can incorporate large volatilityshocks. On the downside, they tend to decrease too much in low-volatility periods, thus being surprised whenthat period comes to an end. Estimators with longer half-lives are not quite as responsive, but they do notdecrease as much in low-volatility periods. In the standard MAC daily model, the value of δ = 0.9944 isused, corresponding to approximately 125 days half-life of the decaying weights sequence δi−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 chosensuch that the memory of the farthest-away sample of the return time series was practically nonexistent. Inpractice, we chose the value of M = 570, which provides a long enough time series of data from the EWMAmemory point of view and, at the same time, supplies enough time points in the time series to construct arobust covariance matrix (see Section 1.2 below).

To construct a covariance matrix, the time series of factors from all MAC models (equity and commodityfactor returns, spread, and interest rate factors, etc.) are combined in a matrix F, where the factors formcolumns and the rows run along the time dimension. In theory, the covariance matrix estimator Σ can becomputed directly from the factor matrix F as:

Σ =(

W1/2F)T

W1/2F, (21)

where W is the diagonal matrix with exponentially decayed weights at the diagonal. However, sometimeshistorical factor time series do not have the same length, making it impossible to use equation (21) directly.In particular, this happens when new factors are introduced into the model (for example a yield curve froma country that was not available before a certain date). To be able to incorporate new factors into the modelas soon as possible, the factor can be included in the model when it has 250 days of history (this lengthstill provides enough data from the EWMA memory point of view). This makes it necessary to compute thefactor variances and covariances pairwise, column by column. Using factor time series of variable lengths doesresult in undesirable properties of the covariance matrix as a whole, but these negative effects are mitigatedto 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 beused to run historical computations of portfolio risks, as well as scenario analysis based on specific marketcorrelation events. The MAC model can also be run with custom historical look-back and decay parametersif required.


White Paper

1.2 Covariance Matrix RegularizationThe methodology used by the FactSet MAC model to obtain the robust estimator of the model factorcovariance matrix is based on the random matrix theory. (The detailed description of the methodology canbe found in [17]). The advantage of the proposed method is that it not only guarantees that the resultingestimator will be positive definite, but also reduces the amount of noise in the estimator and minimizes thedifferences between the estimator and (generally unknown) true covariance matrix of the model.

Random matrix theory (a good review of the theory can be found, for example, in [18]) allows one to derivethe 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 awell-defined region between certain values of λmin and λmax–the support region of the eigenvalue probabilitydistribution. Since the random processes we are correlating are independent, the true correlation matrixshould be diagonal and should have only one eigenvalue. Thus, the width of the support region effectivelyidentifies 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 eigenvaluesthat is wider than in the case of a pure random matrix as there will be a number of large eigenvalues lyingoutside of the support region of a random matrix. A simple and intuitive assumption to make is that thecomponents of the correlation matrix that are defined by the small eigenvalues within the random matrixsupport region (i.e., are orthogonal to the space of the large eigenvalues) are dominated by noise. In otherwords, only the eigenvalues of the sample matrix that lie outside of the support region [λmin, λmax] containinformation relevant to the actual correlation matrix.

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

100

200

300

400

500

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

200

400

600

Figure 6: Distributions of correlation values in the sam-ple matrix before (top) and after (bottom) regularization

The underlying assumption we use to construct the op-timal estimator is that the eigenvectors of the optimalestimator are the same as the eigenvectors of the sam-ple estimator (for detailed discussion see [19]). This as-sumption allows us to construct the optimal matrix bydoing spectral decomposition of the original estimator,adjusting the eigenvalues, and reconstructing the opti-mal estimator using the same eigenvectors. This adjust-ment is similar in nature to principal component anal-ysis, when only the principal components that carryinformation are used, and the ones that carry only thenoise are discarded. In the same spirit we reconstructthe optimal estimator of the correlation matrix usingonly the eigenvalues outside of the RMT support re-gion λ > λmax. After the matrix is reconstructed, wereplace the diagonal elements of the new estimator with1. This will ensure that the estimator can be used as a correlation matrix–that it is positive definite and itsdiagonal elements are all 1.

To illustrate the effect of finite sampling and subsequent regularization on a correlation matrix, we con-structed an example (100 × 100) matrix with all correlations equal to 0.1. We use this matrix to constructthe sample matrix estimator using a (100 × 1000) set of random samples drawn from standard normal dis-tribution. Because of the finite number of samples, the resulting correlation coefficients vary between −0.1and 0.27. The top panel of Figure 6 shows the distribution of the elements of the sample matrix (excludingdiagonal elements) before regularization is applied. While the distribution of the true correlation matrixshould be a single vertical line at 0.1, the estimation noise results in the distribution approximately centeredaround the true value, but having a final width (standard deviation) of about 0.05. The bottom panel of that


White Paper

figure shows the same distribution after we applied the regularization procedure described in the previoussection. Clearly, the amount of noise in the correlation matrix (the width of the distribution) is significantlyreduced.


White Paper

2 Monte Carlo Risk Measures

2.1 Simulating Distribution of Portfolio ReturnsThe Monte Carlo process of estimating the portfolio risk measures starts by simulating the future distributionof the risk model factor returns. The FactSet MAC risk model assumes that all the model factors are jointlydistributed with a multidimensional normal distribution described by the factor covariance matrix. MonteCarlo process generates a number of samples from the factor joint distribution that are later converted intoa set of samples from the return distribution of the portfolio. Standard Monte Carlo run generates K = 5000samples for each factor of the model. Then the appropriate pricing function is applied to each factor scenario,resulting in a number of samples from joint distribution of contributions from each factor to the return of agiven security.

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

r1,1,1 . . . . . . . . . r1,K,1

......

......

rS,1,1 . . . . . . . . . rS,K,1

F Factors

SSe

curit

ies

K Samples

Figure 7: Results of Monte Carlo sampling with K samplesfor a portfolio of S securities, using the model of F factors

For a portfolio of S securities and a risk model ofF factors, the result of the Monte Carlo simulationcan be visualized as a 3D matrix that has F ver-tical slices; each slice is a matrix of S rows and Kcolumns (Figure 7). Each row of the slice matrixis the set of returns for one security/one factor,obtained on K Monte Carlo samples. This matrixcan be aggregated along different directions to ob-tain the distribution of returns of individual secu-rities rsk (a set of k = 1, ...,K samples from returndistribution of the sth security) or the distributionof returns from individual risk factors rfk (a setof k = 1, ...,K samples from return distributionof the f th risk model factor) or aggregated alongboth security and factor dimensions to obtain thedistribution of total portfolio returns rpk.

The generated portfolio return distribution is usedto estimate the total portfolio risk measures, while the distributions of security and factor returns allow themodel to produce marginal contributions to risk. Below we consider the process of estimating VaR andcomponent VaR as an example.

2.2 Estimating Portfolio Risk MeasuresThe VaR associated with probability 1 − α (e.g., α = 1%) is the lower bound for the loss incurred by aportfolio 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

I(x ≤ a) =

{1 if x ≤ a

0 if x > a


White Paper

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, using the Monte Carlomethod, a sample of independent and identically distributed portfolio returns rpk, k = 1, ..N . In this case theestimator for the expectation is:

α =1

N

N∑k=1

I(rpk ≤ −V aRα)

and in this form it can be used to find VaR from a sorted list of sample returns given a value of α. If thevalue of VaR equals the nth return in the sample V aRα = −rpn, then for every sample with k ≤ n in thesorted list the indicator function is equal to one, and for every sample with k > n the indicator function iszero. Thus, the confidence level for that value of VaR is:

αV aR =n

N.

So we can just find the value n in the list such that nN is closest to the given value of α and use the sample

−rn as an estimate of VaR.

2.3 Marginal Contributions to RiskThe top-level risk analysis relies on aggregate portfolio risk measures such as Value-at-Risk (VaR) or ExpectedShortfall (ES). Further decomposition of the risk measures allows the portfolio manager to understand thesources of risk in terms of specific market factors. For capital allocation, measurement of risk-adjustedperformance, developing hedging strategies, and in general, understanding the impact of different risk factorsand component on portfolio risk, it is useful to allocate the risk to elements of the portfolio based on theirmarginal contributions to total risk. If we represent the return of a portfolio as a weighted sum of returncontribution from different components (be it individual securities or risk model factors), the marginalcontribution to portfolio VaR from component i, MV aRα,i, can be defined as the change in portfolio VaRresulting from a marginal change in the ith component position:

MV aRα,i =∂V aRα

∂wi. (22)

This metric allows portfolio managers to find the components that can be used to significantly revise theoverall 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 sumto 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


White Paper

and thus the additive component of VaR in terms of the marginal contribution to VaR can be definedas:

CV aRα,i = wiMV aRα,i. (23)

It can be shown that the marginal contribution to portfolio VaR, defined by equation (22), is the conditionalexpectation of the component return, conditioned on the value of V aRα:

MV aRα,i =∂V aRα

∂wi= −E[ri|rp = −V aRα]. (24)

Here the index i can represent either the security index s (when the marginal contribution from individualsecurity is computed) or the factor index f (when contribution from individual risk factor is computed). Itis not possible to use this formula directly to estimate marginal VaR values in the Monte Carlo process–thesampling variability of the estimate is large and will not go down as we increase the number of samples forthe simulation. The problem is that the contribution to VaR from a given component depends on the singlereturn sample that happens to be the αth return observation for the portfolio (the simulated V aRα). Thecontribution to VaR depends on that single return observation in such a way that the sampling variabilitydoes not change with the number of trials in the simulation.

For example, an estimator for a conditional expectation (24) of the sth security return in this case willbe:

E[rs|rp = −V aRα] =

∑Kk=1 rskI(r

pk = −V aRα)∑N

k=1 I(rpk = −V aRα)

. (25)

Unfortunately, if we just generate a single sample of K values of rsk, the sums in the estimator for conditionalexpectation will only have a single non-zero term. In other words, we will have a single MC sample in theregion of interest.

We can remedy this situation by generating a number of samples of K portfolio returns such that we willhave multiple realizations of rpk = V aRα. This, however, is extremely inefficient. Instead, we can relax thecondition in the expectation (25) from rp = −V aRα to

∣∣rp + V aRα

∣∣ < ε so that the estimator of the marginalcontribution to VaR becomes:

MV aRα,s = −E[rs|∣∣rp + V aRα

∣∣ < ε] = −∑K

k=1 rskI(∣∣rpk + V aRα

∣∣ < ε)∑Nk=1 I(

∣∣rpk + V aRα

∣∣ < ε). (26)

Because the averaging region of formula (26)∣∣rp + V aRα

∣∣ < ε is located in the tail of the portfolio returnsdistribution, the weighted average of the conditional mean returns will be less negative than the portfolioquantile return. In other words, the estimator (26) is biased and the weighted sum of marginal contributionsis expected to be less than the estimated portfolio VaR:

n∑i

wiMV aRα,s < V aRα.

To correct that, we introduce the normalization factor ω defined as:

ω =V aRα∑n

s wsMV aRα,s


White Paper

and define the adjusted estimator for marginal contribution to VaR as

MV aRα,s = ωMV aRα,s.

The corresponding estimate for the CVaR follows from equation (23). Due to the adjustment factor ωthe sum of the CVaRs exactly equals the initially estimated overall portfolio VaR, as required by CVaRdefinition.

The introduction of the normalization factor ω is very similar to the method of control variates in MonteCarlo 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) quantitiesthat are the primary focus of the simulation. In our case, the portfolio VaR is used as a control variate forcomponent VaR estimators. The more detailed analysis of the adjusted estimator and justification of thenormalization procedure can be found in [20].

α V aRα σV aRσV aR

V aRα% ESα σES

σES

ESα%

90 4.79 0.13 2.66 6.55 0.11 1.6095 6.16 0.14 2.23 7.69 0.10 1.2697 7.06 0.12 1.63 8.44 0.10 1.1699 8.61 0.13 1.55 9.92 0.13 1.29

Table 2: Estimators and errors for VaR and ES of BarclaysEUR Aggregate Index for different confidence levels α

The size of the averaging region ε in formula(26) has to be chosen to have a sufficient numberof points in the neighborhood of VaR to bringthe variance of the Monte Carlo estimator down.At the same time, the width of the region hasto be restricted to limit the variability of theportfolio return within the neighborhood. TheMonte Carlo process can provide a measure ofthe variance of the estimator. From the samecomputation one can obtain both an estimatedresult and an objective measure of the statisticaluncertainty in the result. In our case, we use multiple Monte Carlo simulations of a portfolio VaR and itscomponents at different confidence levels with K = 5000 samples as described above. We run each simulationmultiple times and record the means and standard deviations of estimated VaR and MVaR values. We usethis data to evaluate the adequacy of the number of samples for our purposes and to establish an acceptableaveraging region ε for computing marginal contributions to VaR that provide reasonable balance betweenbias and variance of the estimators.

Mean σα 1% 5% 1% 5%90 0.15 0.07 0.11 0.0595 0.12 0.06 0.08 0.0497 0.1 0.04 0.08 0.0399 0.08 0.05 0.06 0.04

Table 3: Parameters of error distribu-tions

Table 2 shows the estimated Value-at-Risk (V aRα) and Expected Short-fall (ESα) values for different confidence levels α computed for BarclaysEUR Aggregate Index. Also shown are Monte Carlo standard deviationsof the estimators σV aR and σES and corresponding relative error foreach estimator, expressed as a percentage of the estimator itself. It isclear that with the employed number of samples (K = 5000), the errorof both risk measures never exceeds 3%.

The index we use for testing purposes contains around 4000 securities.We analyze performance of the Monte Carlo algorithm by comparingthe distributions of the errors of the components corresponding to eachsecurity obtained at different values of the averaging region width pa-rameter ε (eq. (26)).

Table 3 shows the mean and standard deviation of the distribution of errors obtained with different values ofaveraging region width parameter ε (1% and 5%) for different VaR threshold probabilities α. With the valueof ε equal to 5% of the total number of samples, the center of the error distribution is located at around


White Paper

6%, while its width never exceeds 5%. Thus, the value of ε = 5% is sufficient to keep the errors of the VaRcomponents within a 10% range.

To estimate the portfolio’s excess return (tracking error) distribution, we also perform the same simulation forthe portfolio’s benchmark and take the difference in returns (portfolio minus benchmark) for each simulationrun. 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. More detailed descriptionof the methodology and test results is provided in [21].

3 Scenario AnalysisThe risk measures discussed above–VaR, ETL, TEV–are effectively the measures of the parameters of theprobability distribution of the portfolio returns. As such they serve as good statistical, probabilistic measuresof risk, but provide little information on possible losses or gains of the portfolio under specific market condi-tions. Value-at-Risk informs portfolio managers of the probability of certain losses but gives little indicationunder what specific market scenario these losses might be realized. Thus it is very useful to complement therisk management framework by scenario analysis capabilities designed to quantify the potential losses underspecific, user-defined market events.

In the FactSet risk management framework, three types of scenario analysis, or stress tests, are avail-able:

• Factor Stress Testing

• Extreme Event Stress Testing

• Extreme Event Simulation

3.1 Factor Stress TestingIn factor stress testing it is possible to apply a shock to individual factors to see what impact the shock couldhave on the portfolio as a whole. Both endogenous factors (factors within the risk model) and exogenousfactors (factors that are not in the risk model) can be stressed.

Figure 8: Extreme event stress test definition

The FactSet framework can perform two typesof factor stress tests–time weighted and eventweighted. In the time-weighted algorithm, the modelcomputes the covariance matrix of the risk modelfactors together with the exogenous factors in es-sentially the same way as described above. In theevent-weighted algorithm, the model assigns higherweights to the periods that have factor returns sim-ilar to the predefined factor shock amount. First,the absolute difference between the shock amountand the historical returns of the stressed factor iscalculated and sorted in ascending order, and thenthe exponential weights are assigned to each periodso that periods with a factor return similar to theshock will have higher weights. For example, if thescenario 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 5%.


White Paper

Exactly how much more weight they are given is determined by the decay rate. In both time-weighted andevent-weighted modes, the user can define the amount of history used and the decay factor.

Figure 9: Extreme event simulation definition

The covariance matrix of all the factors is used tocompute the sensitivity β of the stressed factor fsto each risk model factor fi as:

βsi =Cov(fs, fi)

V ar(fs).

These sensitivities are then used to estimate theamount of each factor return ∆fi that is implied bythe defined shock of the stressed factor ∆fs:

∆fi = βsi∆fs.

For example, let’s assume that the beta between thestress factor (oil) and a risk model factor, e.g., thesize factor, is 0.5. This means that for each unit ofexposure to the size risk factor, every 1% change inthe oil factor will have a positive 0.5% impact. If you shock oil by 20% and assuming that the portfolio hasan 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 appropriatepricing function to each factor return and aggregating the resulting security returns (see Chapter “RiskModels” for discussions of the pricing functions of different asset class risk models).

3.2 Extreme Event Stress TestingExtreme event stress testing takes today’s portfolio and hypothesizes what its return would be if an extremeevent were to happen again. Current factor exposures are used with actual factor returns from historicalevents 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. Forexample, Figure 8 illustrates the definition of a credit crisis (10/2008) test for today’s portfolio. To computethe implied return of the portfolio under this test, the factor returns for the period between 09/30/2008and 10/31/2008 are used in the security pricing functions, together with the factor portfolio exposures as oftoday.

3.3 Extreme Event SimulationExtreme event simulation is similar to Monte Carlo VaR except it will use the covariances from the dateof the event with 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 covariancematrix as of this date. The extreme event simulation type is only available if you subscribe to Monte CarloVaR. Figure 10 shows an example of the extreme event simulation report.


White Paper

Figure 10: Extreme event simulation example


White Paper

Example of Risk Report

Figure 11: Summary section of risk report

FactSet’s multi-asset class risk model is fully integratedacross FactSet’s portfolio analytics suite. The suite in-cludes a flexible, highly customizable risk reporting plat-form that allows for detailed reporting of the results ofthe risk computations and scenario analysis tightly in-tegrated with data from other FactSet systems such asfixed income analytics, performance attribution, and oth-ers. The framework can be used to generate consolidatedanalytics and risk reports in a client-ready PDF presenta-tion or to view dynamic asset and factor-level data witha customizable level of detail. The user can define customrisk reports that allow her to analyze risk exposures ofthe portfolio from different points of view (e.g., portfolioholdings, analytics, risk factor exposures) and study thedecomposition of risk measures of the portfolio in bothfactor and asset space to a desired degree of granularity.This section will walk the reader through several layersof a sample risk report to illustrate the process of riskanalysis that the portfolio manager can carry out usingthe FactSet platform.

We will consider a sample risk report for a global balancedfixed-income portfolio managed against a global aggregatebond index. The FactSet risk framework allows the userto analyze the portfolio both on an absolute basis andrelative to a benchmark. The following example assumesthat a portfolio is actively tracking a benchmark, i.e., itis allowed to deviate from the benchmark to a certainextent to obtain superior returns. The degree of the de-viation from the benchmark is usually specified in termsof the Tracking Error Volatility that provides an estimateof the magnitude of the expected difference between theportfolio and benchmark returns. At the same time, the portfolio manager might want to monitor the tailrisk 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 themanager to judge whether the TEV of the portfolio is within the manager’s mandate and the tail risk iswithin the acceptable range to understand how the factor and asset composition of the portfolio relates tothe benchmark and affects the total risk, and to make decisions regarding portfolio compositions that are


White Paper

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 exposuresto major sources of market risk and looks at their risk on absolute and relative levels. The report showsthe higher-level decomposition of risk, displaying aggregate risk components corresponding to the main riskdrivers of the portfolio.

The first section of the summary report (Portfolio Characteristics) shows the extent to which both portfolioand 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 only0.2% of benchmark securities (46 out of more than 19000) have problems. The user can generate a detailedreport 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 benchmarkand 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 largerconvexity, and about 7 bp lower yield. In other words, even on an aggregated level, one can see that theportfolio tracks the benchmark closely but not perfectly, and one should expect nontrivial contributions toportfolio risk from both yield curve and spread components.

Figure 12: Risk decomposition by factor contributions

The next section of the summary report shows abso-lute risks of both portfolio and benchmark and theirdecomposition into major factor contribution thatcan be described in terms of categories or groups ofrisk factors, e.g., Yield Curve, Muni, MBS, CreditSpread. Each asset class is exposed to a specific set ofrisk factors in addition to factors common to all as-sets in that market (for example, Yield Curve factorsare common to all fixed income assets). The analysisof the factor contributions to risk allows the portfo-lio manager to understand the relationship betweencommon and specific sources of risk in the portfolioand benchmark. In our example, one can see thatin both portfolio and benchmark, the yield curveis by far the largest contributor to total risk, as itis 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 cus-tomizable by the user and can show different thresh-old VaR as well as portfolio and benchmark returnvolatilities.) Since both portfolio and benchmark areglobal, i.e., contain securities denominated in differ-ent currencies, the second largest contributor to riskis the currency exposure. Its sources will be analyzedfurther when we examine the factor level report thatshows, in particular, the exposures of both portfolioand benchmark to different risk factors. The thirdand fourth largest risk contributions on an absolute level are credit spreads and mortgage-backed securityspreads.

As indicated by the duration exposures, the benchmark effectively hedges the majority of the portfolio yield


White Paper

curve risk. As a result, seen in the last section of the summary (Relative Risk), the yield curve component ofthe tracking error volatility of the portfolio/benchmark pair becomes much smaller (only about 2 bp) thancontributions from other risk factors. The total monthly TEV of the portfolio is 75 bp (annualized) and ismainly composed of the currency, credit spread, and MBS spread components with FX exposure giving thelargest contribution to risk and the 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 majorfactor group described above can be presented as a set of detailed risk factors and the risk report can beconfigured to look at all levels of granularity of risk exposures down to contributions from individual factorsor 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 showsthe details of exposure and risk contribution from Yield Curve and Currency factors and the aggregatecontribution from other factor groups. (The portfolio is exposed to more than one sovereign curve, but inthe figure we have shown only the USD curve to save space.)

Figure 13: Asset level risk decomposition

The Yield Curve factors allow the portfolio man-ager to analyze the duration risk of the portfolioto a much greater detail than the simple durationover/underweight view in the summary report. Thefactors show exposure of the portfolio to each of the17 key maturity points along the yield curve, lettingthe user examine the risk of the portfolio that comesfrom various possible curve movements. One can seefrom the report that the portfolio is effectively longthe middle of the curve (six-, seven-, and nine-yearpoints have major exposures) and short the long endof the curve (20-, 25-, and 30-year points). Thus theportfolio effectively has a curve flattener in zero toten-year region and a curve steepener in the 10 to30-year region. The fact that the points of the shortduration result in a positive contribution to risk,while the regions that long key rate durations con-tribute to risk negatively is the consequence of thecorrelation structure of the fixed income market cap-tured 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), whilethe 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 theportfolio 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 portfoliorisk from different asset classes, asset groups, or individual assets in the portfolio. Figure 13 shows an exampleof 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 (notshown in the figure), and the contributions can be analyzed on individual asset level. The report comparesthe market value of portfolio holdings with the benchmark and shows contribution to risk from each securitygroup. This kind of information would be especially useful for an equity portfolio where the weights of assetsin various groups (like industries or sectors) effectively define portfolio risk exposure. Applied to a fixedincome portfolio, this report illustrates the shortcomings of market weight decomposition for analysis of


White Paper

risk. For example, the analyzed portfolio is overweight Industrial Corporate bonds by 6.18% and only veryslightly underweight the Financial sector by -0.32%. Yet the largest contribution to risk (40 bp) comes fromthe Financial sector. Thus, the market weight information alone, while important, tells us very little aboutthe actual risk composition of a fixed-income portfolio, and has to be combined with the factor exposuresand factor risk decomposition to produce an adequate picture of portfolio risk. Taking the above exampleof Corporate bond risk, one should look at the factor decomposition report (Figure 12) and the currency ofdenomination of individual assets, to realize that most of the risk attributed to the Financial sector actuallycomes from the FX exposure of the bonds in the sector. A detailed analysis of this kind can be performedusing a report showing various factor exposures (e.g., key rate durations, spread durations) grouped by assetclass, sector or industry. However, even that kind of report will give only a basic understanding of riskcontributions as it will not take into account correlations between different factors, which brings us backto the factor decomposition in Figure 12 that is based on risk contribution computations that use all theinformation available in the factor correlation matrix (see Estimating Portfolio Risk Measures section).

Figure 14: Stress test results

A factor-based risk model al-lows for deep analysis of therisk imbalances in the port-folio. Each of the risk cate-gories is modeled with a de-tailed set of risk factors thatare designed to capture theparticular sources of risk theasset class is exposed to. Theabove examples are just a cur-sory demonstration of how thefactor model can offer use-ful insights from risk manage-ment and portfolio construc-tion perspectives.

A complementary view of theportfolio risk is provided bystress tests (or scenario anal-ysis) that can be performed inseveral ways. One may wantto reprice the whole portfoliounder a particular scenario onrisk factors, such as interestrates or spreads, and look at the hypothetical return under that scenario. Another way is to evaluate howthe portfolio would have performed under particular historical scenarios. And finally, one might want toexamine 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 resultsof four different factor scenarios–depreciation of EUR, U.S. Treasury and U.S. stock market sell-off, andoil price rally. In all four cases, the absolute and relative portfolio returns are computed by simulating thechanges in risk factors that correspond to a specific perturbation of the tested market variable. Thus, theresults include all information on market correlations and volatilities that is stored in the model covariancematrix and represent the most likely actual outcome of the event that implicitly includes all hedging anddiversification strategies built into the portfolio.


White Paper

Conclusion

Factor risk models help portfolio managers to describe risk exposures and imbalances of different portfoliosusing a relatively small set of common, intuitive, and easily understandable factors. The analysis is performedwith the help of risk reporting tools that can present a coherent view of portfolio risk across differentportfolios, asset classes, and asset management styles. These reports can be used to assess risk/return trade-offs in a given portfolio, to guide the portfolio risk balancing process, and to assist in the portfolio constructionprocess through risk/return optimization.

The FactSet multi-asset class risk model provides detailed information about risk exposures of portfolios andcan be very valuable for portfolio management. It is built to be reactive to the current market environmentand to reflect the true risks a portfolio faces. By using a Monte Carlo approach, along with repricing of eachsecurity, one can correctly estimate the risk of non-linear securities (such as derivatives). Full repricing ofderivatives allows for correct simulation of their non-normal return distributions, even though the underlyingrisk factor distribution is assumed normal. Combining factors from global equities, commodities, currencies,and fixed income creates a single framework to investigate portfolios and their risks. The resulting riskstatistics 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 multipleoptions to analyze the risk of their portfolios. These reporting tools allow the manager to merge all her viewsand constraints into a final portfolio simply and objectively. The manager can then compare the risk withthe expected return of each of the views and decide on the optimal portfolio allocation.


White Paper

Bibliography

[1] J. MacQueen and R. Young. Construction Methodology for the FactSet/Northfield Global Eqiuty RiskModel. FactSet White Paper, 2018.

[2] A. B. Dor, L. Dynkin, J. Hyman, P. Houweling, E. van Leeuwen, and O. Penninga. DTSSM (DurationTimes Spread): A New Measure of Spread Exposure in Credit Portfolios. The Journal of PortfolioManagement, 33(2):77–100, 2007.

[3] B.W. Golub and L.M. Tilman. Risk Management. Approaches for Fixed Income Market. John Wiley& Sons, Inc., Hoboken, NJ, 2000.

[4] R. Merton. On the pricing of corporate debt: The risk structure of interest rates. The Journal ofFinance, 29(2):449–470, 1974.

[5] K. Tsiveriotis and C. Fernandes. Valuing convertible bonds with credit risk. Journal of Fixed Income,8:95–102, 1998.

[6] A. B. Dor, L. Dynkin, J. Hyman, and B. D. Phelps. Quantitative Credit Portfolio Manage- ment:Practical Innovations for Measuring and Controlling Liquidity, Spread, and Issuer Concentration Risk.Wiley, 1st edition edition, 2011.

[7] D. Mossessian and C. Westenberger. FactSet Corporate Spread Model. FactSet White Paper, 2017.

[8] D. Mossessian and C. Westenberger. FactSet European Sovereign Spread Model: Foundations. FactSetWhite Paper, 2018.

[9] A. J. Harju and D. Mossessian. FactSet MAC U.S. Municipal Bond Risk Model. FactSet White Paper,2019.

[10] G. Barone-Adesi and R. E. Whaley. Efficient analytic approximation of American option values. TheJournal of Finance, 42(2):301–320, 1987.

[11] Ju N. and Zhong R. An approximate formula for pricing American options. The Journal of Derivatives,7(2):31–40, 1999.

[12] Tompkins R. G. Implied volatility surfaces: uncovering regularities for options on financial futures. TheEuropean Journal of Finance, 7(3):198–230, 2001.

[13] Farshid Jamshidian. An exact bond option formula. The Journal of Finance, 44(1):205–209, 1989.

[14] T Davis. Interest Rate Swap Valuation. FactSet White Paper, 2015.

[15] C.D. Manning, P. Raghavan, and H. Schütze. Introduction to Information Retrieval. Cambridge Uni-versity Press, 2008.

[16] JP Morgan. RiskMetrics: Technical Document, 1996.


White Paper

[17] D. Mossessian and V. Vieli. Robust Estimation of Risk Factor Model Covariance Matrix. FactSet WhitePaper, 2016.

[18] Bai Z. Methodologies in spectral analysis of large dimensional random matrices, a review. StatisticaSinica, 9:611–677, 1999.

[19] O. Ledoit and M. Wolf. Nonlinear shrinkage estimation of large-dimensional covariance matrices. TheAnnals of Statistics, 2(40):1024–1060, 2012.

[20] W. G. Hallerbach. Decomposing portfolio value-at-risk: A general analysis. Journal of Risk, (5):1–18,2002.

[21] D. Mossessian and V. Vieli. Decomposing Portfolio Risk Using Monte Carlo Estimators. FactSet WhitePaper, 2016.


White Paper

Appendix

1 Principal Component Analysis of Yield CurvesPCA is a powerful statistical technique for simplifying a set of data in which a linear transformation isapplied to the original variables such that the new, transformed variables become independent (have zerocorrelation), while all the information about the volatility of the original variables is maintained. The newvariables, called principal components (PC), have different explanatory power in terms of the volatility ofthe original system. The purpose of PCA is to select those PCs that explain the variability of the datato a required accuracy. A typical threshold criterion to select principal components is for their cumulativevolatility to exceed a certain percentage of the original volatility of the data set. This is equivalent to selectingthe largest eigenvalues of the original data covariance matrix such that their sum exceeds a certain percentageof the total sum of that matrix eigenvalues.

To perform PCA we take the historical time series of rate changes for the 17 key rates (tenors) for each spotrate curve for the most recent T trading days. The returns are z-scored by subtracting the mean of eachtenor 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

C17×17

=1

TR

17×T× RT

T×17

can be decomposed as

C17×17

= QΛQT ,

where Λ is a diagonal matrix of the eigenvalues of matrix C. The principal component projection matrix Qconsists of eigenvectors of the curve covariance matrix. The matrix of all principal components of the curveP (17× T ) is computed as

P17×T

= QT

17×17× R

17×500.

Based on our research, we pick four components out of the 17 as sufficient to explain more than 99% of theyield curve dynamics. These first four time series of principal components form a (4× T ) matrix P̂. The P̂matrices for all curves are combined with the time series of all other model factors for the purpose of creatingthe MAC covariance matrix used in model Monte Carlo simulations. When the portfolio risk is computedduring Monte Carlo simulations, the components’ future values for each simulated scenario are recorded ina (4× 1) vector P̂. Also, the first four columns of the PC projection matrix Q are stored in a new (17× 4)

matrix Q̂. This matrix is used to convert the simulated values of the PCs into the projected rate changes ofthe 17 maturities as

R̂17×1

= Q̂17×4

× P̂4×1

.


White Paper

2 Fixed Income Model Factors

2.1 Linear Regression Model of Relative Spread Changes–U.S. Municipal Bonds

Factor ID Factor NameUS_MUNI_GENERAL_OBLIGATION_IV U.S. Muni General ObligationUS_MUNI_AUTHORITY_IV U.S. Muni AuthorityUS_MUNI_BUILDING_IV U.S. Muni BuildingUS_MUNI_CERT_OF_PART_IV U.S. Muni Certificate of ParticipationUS_MUNI_ECONOMIC_DEV_IV U.S. Muni Economic DevelopmentUS_MUNI_EDUCATION_IV U.S. Muni EducationUS_MUNI_HEALTH_IV U.S. Muni HealthUS_MUNI_HOUSING_IV U.S. Muni HousingUS_MUNI_SCHOOL_DISTRICT_IV U.S. Muni School DistrictUS_MUNI_TAX_IV U.S. Muni TaxUS_MUNI_TRANSPORTATION_IV U.S. Muni TransportationUS_MUNI_UTILITY_IV U.S. Muni UtilityUS_MUNI_OTHER_IV U.S. Muni OtherUS_MUNI_GENERAL_OBLIGATION_OAS U.S. Muni General Obligation OASUS_MUNI_AUTHORITY_OAS U.S. Muni Authority OASUS_MUNI_BUILDING_OAS U.S. Muni Building OASUS_MUNI_CERT_OF_PART_OAS U.S. Muni Certificate of Participation OASUS_MUNI_ECONOMIC_DEV_OAS U.S. Muni Economic Development OASUS_MUNI_EDUCATION_OAS U.S. Muni Education OASUS_MUNI_HEALTH_OAS U.S. Muni Health OASUS_MUNI_HOUSING_OAS U.S. Muni Housing OASUS_MUNI_SCHOOL_DISTRICT_OAS U.S. Muni School District OASUS_MUNI_TAX_OAS U.S. Muni Tax OASUS_MUNI_TRANSPORTATION_OAS U.S. Muni Transportation OASUS_MUNI_UTILITY_OAS U.S. Muni Utility OASUS_MUNI_OTHER_OAS U.S. Muni Other OASUS_MUNI_PUERTO_RICO_IV U.S. Muni Puerto RicoUS_MUNI_ILLINOIS_IV U.S. Muni IllinoisUS_MUNI_NEW_JERSEY_IV U.S. Muni New JerseyUS_MUNI_CALIFORNIA_IV U.S. Muni CaliforniaUS_MUNI_PENNSYLVANIA_IV U.S. Muni PennsylvaniaUS_MUNI_FLORIDA_IV U.S. Muni FloridaUS_MUNI_CONNECTICUT_IV U.S. Muni ConnecticutUS_MUNI_TEXAS_IV U.S. Muni TexasUS_MUNI_MASSACHUSETTS_IV U.S. Muni MassachussettsUS_MUNI_WASHINGTON_IV U.S. Muni WashingtonUS_MUNI_NEW_YORK_IV U.S. Muni New YorkUS_MUNI_PUERTO_RICO_OAS U.S. Muni Puerto Rico OASUS_MUNI_ILLINOIS_OAS U.S. Muni Illinois OASUS_MUNI_NEW_JERSEY_OAS U.S. Muni New Jersey OASUS_MUNI_CALIFORNIA_OAS U.S. Muni California OASUS_MUNI_PENNSYLVANIA_OAS U.S. Muni Pennsylvania OASUS_MUNI_FLORIDA_OAS U.S. Muni Florida OASUS_MUNI_CONNECTICUT_OAS U.S. Muni Connecticut OASUS_MUNI_TEXAS_OAS U.S. Muni Texas OASUS_MUNI_MASSACHUSETTS_OAS U.S. Muni Massachussetts OASUS_MUNI_WASHINGTON_OAS U.S. Muni Washington OASUS_MUNI_NEW_YORK_OAS U.S. Muni New York OASUS_MUNI_SHORT_SPREAD_DURATION U.S. Muni Short Spread DurationUS_MUNI_LONG_SPREAD_DURATION U.S. Muni Long Spread DurationUS_MUNI_VEGA_IF_CALLABLE U.S. Muni Vega if Callable


White Paper

2.2 Linear Regression Model of Relative Spread Changes–Euro-Sovereign BondsFactor ID Factor NameEURO_SOV_SPR_LS Euro Sovereign: Low SpreadEURO_SOV_SPR_AT_LU Euro Sovereign Austria, Luxembourg: SpreadEURO_SOV_SPR_FI Euro Sovereign Finland: SpreadEURO_SOV_SPR_FR Euro Sovereign France: SpreadEURO_SOV_SPR_IT Euro Sovereign Italy: SpreadEURO_SOV_SPR_NL Euro Sovereign Netherlands: SpreadEURO_SOV_SPR_IE Euro Sovereign Ireland: SpreadEURO_SOV_SPR_ES Euro Sovereign Spain: SpreadEURO_SOV_SPR_PT Euro Sovereign Portugal: SpreadEURO_SOV_SPR_BE Euro Sovereign Belgium: SpreadEURO_SOV_SPR_CEE Euro Sovereign Central and Eastern Europe: SpreadEURO_SOV_SPR_GR_CY Euro Sovereign Greece, Cyprus: SpreadEURO_SOV_SPR_REL_SHORTDUR Euro Sovereign: Relative Short DurationEURO_SOV_SPR_ABS_SHORTDUR Euro Sovereign: Absolute Short Duration


White Paper

2.3 Linear Regression Model of Spread LevelsFactor ID Factor NameUS_AGENCY_INTERCEPT U.S. Agency InterceptUS_AGENCY_DURATION U.S. Agency DurationCAD_AGENCY_INTERCEPT Canadian Agency in USD InterceptCAD_AGENCY_DURATION Canadian Agency in USD DurationEUR_SUPRANATIONAL_MEAN Supranational in EUR MeanGBP_SUPRANATIONAL_MEAN Supranational in GBP MeanAUD_SUPRANATIONAL_MEAN Supranational in AUD MeanOTHER_SUPRANATIONAL_MEAN Supranational in USD MeanCAD_MUNI_INTERCEPT Canadian Agency InterceptCAD_MUNI_DURATION Canadian Agency DurationJPY_MUNI_MEAN Japanese Agency MeanAUD_MUNI_MEAN Australian Agency MeanNORTHERN_EUR_MUNI_MEAN Northern Europe Agency MeanSOUTHERN_EUR_MUNI_MEAN Southern Europe Agency MeanUS_TIPS_INTERCEPT U.S. Inflation Protected InterceptUS_TIPS_DURATION U.S. Inflation Protected DurationUS_TIPS_CONVEXITY U.S. Inflation Protected ConvexityUK_TIPS_INTERCEPT UK Inflation Protected InterceptUK_TIPS_DURATION UK Inflation Protected DurationUK_TIPS_CONVEXITY UK Inflation Protected ConvexityFRANCE_TIPS_INTERCEPT France Inflation Protected InterceptFRANCE_TIPS_DURATION France Inflation Protected DurationFRANCE_TIPS_CONVEXITY France Inflation Protected ConvexityGERMANY_TIPS_MEAN Germany Inflation Protected MeanITALY_TIPS_MEAN Italy Inflation Protected MeanCAD_TIPS_MEAN CAD Inflation Protected MeanJPY_TIPS_MEAN JPY Inflation Protected MeanCMO_INTERCEPT U.S. RMBS InterceptCMO_REFI_DURATION U.S. RMBS Refi DurationCMO_TURNOVER_DURATION U.S. RMBS Turnover DurationUS_ABS_CMBS_INTERCEPT U.S. ABS/CMBS InterceptUS_ABS_CMBS_SEASONING U.S. ABS/CMBS SeasoningUS_ABS_CMBS_RATING_AA U.S. ABS/CMBS Rating AAUS_ABS_CMBS_RATING_A U.S. ABS/CMBS Rating AUS_ABS_CMBS_RATING_BBB U.S. ABS/CMBS Rating BBBNON_US_ABS_CMBS_INTERCEPT Non-U.S. ABS/CMBS InterceptNON_US_ABS_CMBS_SEASONING Non-U.S. ABS/CMBS SeasoningNON_US_ABS_CMBS_RATING_A-BBB Non-U.S. ABS/CMBS Rating A-BBBMBS_JP_MEAN Japanese MBS Mean


White Paper

3 Return Based Model FactorsA hundred factors in time series have been chosen as a representative suite of assets that covers a diverseselection 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 certainlynot all encompassing. In particular, the factor selection is constrained for linear type assets only, meaningit’s not well suited for non-linear assets like derivatives. It is currently limited to factors that have a dailyreturn stream. The factor universe encompasses Alternative Investment, Equity, Real Estate, Fixed Income,Interest Rates, Commodities, and Currencies.

3.1 Alternative InvestmentsThe first class of factors we include comprises 17 HFR indexes representing major sub-categories of hedgefunds. They include Absolute Return, Event Driven, Fund-of-Funds, Global Macro, Long/Short, and RelativeValue.

Factor ID Factor Name IndexRBA_HFRXAR HF Rets: Absolute Return HFRXARRBA_HFRXMREG HF Rets: Multi-Region Index HFRXMREGRBA_HFRXEH HF Rets: Equity Hedge Index HFRXEHRBA_HFRXEMN HF Rets: EH: Equity Market Neutral Index HFRXEMNRBA_HFRXDS HF Rets: Event Driven Distressed Securities HFRXDSRBA_HFRXMA HF Rets: Event Driven Merger Arbitrage HFRXMARBA_HFRXED HF Rets: Event Driven Index HFRXEDRBA_HFRXM HF Rets: Macro/CTA Index HFRXMRBA_HFRXNA HF Rets: North America Index HFRXNARBA_HFRXEHG HF Rets: EH: Fundamental Growth Index HFRXEHGRBA_HFRXEHV HF Rets: EH: Fundamental Value Index HFRXEHVRBA_HFRXMLP HF Rets: MLP Index HFRXMLPRBA_HFRXGL HF Rets: Global Hedge Fund Index HFRXGLRBA_HFRXSDV HF Rets: Systematic Diversified CTA Index HFRXSDVRBA_HFRXMD HF Rets: Market Directional Index HFRXMDRBA_HFRXRVA HF Rets: Rel Value Arbitrage Index HFRXRVARBA_HFRXCA HF Rets: Rel Value Convertible Arbitrage HFRXCA


White Paper

3.2 EquityThe equity factors are daily returns of the following eleven indices:

Factor ID Factor Name IndexRBA_SP50.R Equity Rets: U.S. Large Cap SP50.RRBA_SML Equity Rets: U.S. Small Cap SMLRBA_183660 Equity Rets: Euro Large Cap 183660RBA_186745 Equity Rets: Australia 186745RBA_180461 Equity Rets: Japan 180461RBA_180721 Equity Rets: Korea 180721RBA_LFEYK Equity Rets: Emerging Mkts LFEYKRBA_BXM Equity Rets: U.S. Large Cap Buy Write BXMRBA_EAFE Equity Rets: EAFE see footnote1

RBA_FR0000R1 Equity Rets: World FR0000R12

RBA_EMIF Other Rets: Emerg Mkts Infrastructure EMIFVIX CBOE Volatility Index VIX

3.3 Real EstateThe Real Estate class of factors comprises six assets representative of various REIT investment styles.

Factor ID Factor Name IndexRBA_CWH Real Est Rets: Commercial REITs EQCRBA_URE Real Est Rets: Leveraged U.S. URERBA_NLY Real Est Rets: REITs NLYRBA_HTS Real Est Rets: Residential Single Family REITs CMORBA_ARR Real Est Rets: Residential REITs A ARRRBA_TWO Real Est Rets: Residential REITs B TWO

1The factor is a weighted sum of FactSet Market Indexes for Europe, Africa, and Asia2FR0000R1 is a FactSet Market Index of the world using a market value weighting


White Paper

3.4 Fixed Income18 Fixed Income and 15 Government Interest Rate global indexes comprise the Fixed Income universe offactors.

Factor ID Factor Name IndexRBA_MLHC00 FI Rets: Canadian High Yield MLHC00RBA_MLG0CP FI Rets: Canadian Provincials – Municipals MLG0CPRBA_MLEB00 FI Rets: EMU Corporates / Financials MLEB00RBA_MLEA00 FI Rets: EMU MBS MLEA00RBA_MLEJRE FI Rets: EUR Corporates Real Estate MLEJRERBA_MLE0LG FI Rets: Euro-Dollar & Globals Governments – Local MLE0LGRBA_MLE0GG FI Rets: Euro-Dollar & Globals Governments Guaranteed MLE0GGRBA_MLHPID FI Rets: European Currency HY; Constrained / Non-Financials MLHPIDRBA_MLHA00 FI Rets: Global HY & Emerging Market – Plus MLHA00RBA_MLUF00 FI Rets: Sterling Corporates / Financials MLUF00RBA_MLUIRE FI Rets: Sterling Corporates / Real Estate MLUIRERBA_MLUA00 FI Rets: Sterling MBS MLUA00RBA_MLCABS FI Rets: U.S. ABS & CMBS MLCABSRBA_MLCMBS FI Rets: U.S. CMBS Fixed Rate MLCMBSRBA_MLCF00 FI Rets: U.S. Corporates / Financials MLCF00RBA_MLCIRE FI Rets: U.S. Corporates / Real Estate MLCIRERBA_MLM0A0 FI Rets: U.S. Mortgages MLM0A0RBA_MLU0A0 FI Rets: U.S. Municipals MLU0A0RBA_MLG9D0 Int Rate: Euro Government: 10+ Years MLG9D0RBA_MLG2D0 Int Rate: Euro Government: 3-5 Years MLG2D0RBA_MLG4D0 Int Rate: Euro Government: 7-10 Years MLG4D0RBA_MLG9Y0 Int Rate: Japan Government: 10+ Years MLG9Y0RBA_MLG2Y0 Int Rate: Japan Government: 3-5 Years MLG2Y0RBA_MLG4Y0 Int Rate: Japan Government: 7-10 Years MLG4Y0RBA_MLG9L0 Int Rate: UK Government: 10+ Years MLG9L0RBA_MLG8L0 Int Rate: UK Government: 15+ Years MLG8L0RBA_MLG2L0 Int Rate: UK Government: 3-5 Years MLG2L0RBA_MLG4L0 Int Rate: UK Government: 7-10 Years MLG4L0RBA_MLG8O2 Int Rate: UST: 10+ Years MLG8O2RBA_MLG9O2 Int Rate: UST: 15+ Years MLG9O2RBA_MLG2O2 Int Rate: UST: 3-5 Years MLG2O2RBA_MLG4O2 Int Rate: UST: 7-10 Years MLG4O2RBA_SHORTTERM Int Rate: World: 0-3 Years Aggregate see footnote1

1The World Interest Rate factor is an equal weighted composite comprised of German Government 0-3Y (MLG1DB), UKGilt 0-3Y (MLGBL0), Japan Government 0-3Y (MLG1YA), and U.S. Treasury 0-3Y (MLG1QA)


White Paper

3.5 CommoditiesSix Commodity indices, managed by S&P GSCI, are available in the RBR model. Timber, whose base returnsseries is the near-term spot rate of timber, is also included.

Factor ID Factor Name IndexRBA_SPGSAG Commodity Rets: Agriculture SPGSAGRBA_SPGSEN Commodity Rets: Energy SPGSENRBA_SPGSIN Commodity Rets: Industrial Metals SPGSINRBA_SPGSLV Commodity Rets: Livestock SPGSLVRBA_SPGSNG Commodity Rets: Natural Gas SPGSNGRBA_SPGSPM Commodity Rets: Precious Metals SPGSPMRBA_LB00-USA Other Rets: Timber LB00-USA


White Paper

3.6 CurrenciesTwenty six global currencies are available as factors within the RBR model.

Factor ID Factor Name IndexEuro Euro EURNorwegian Krona Norwegian Krona NOKSwedish Krona Swedish Krona SEKSwiss Franc Swiss Franc CHFUK Pound UK Pound GBPRussian Rouble Russian Rouble RUBAustralian Dollar Australian Dollar AUDCanadian Dollar Canadian Dollar CADNew Zealand Dollar New Zealand Dollar NZDSouth African Rand South African Rand ZARBrazilian Real Brazilian Real BRLChilean Peso Chilean Peso CLPMexican Peso Mexican Peso MXNIsraeli Shekel Israeli Shekel ILSIndian Rupee Indian Rupee INRMalaysian Ringgit Malaysian Ringgit MYRSingapore Dollar Singapore Dollar SGDIndonesian Rupiah Indonesian Rupiah IDRPhilippines Peso Philippines Peso PHPSouth Korean Won South Korean Won KRWThai Baht Thai Baht THBTaiwanese Dollar Taiwanese Dollar TWDHong Kong Dollar Hong Kong Dollar HKDChina Renminbi China Renminbi CNYJapanese Yen Japanese Yen JPYU.S. Dollar U.S. Dollar USD


FactSet (NYSE:FDS | NASDAQ:FDS) delivers superior content, analytics, and flexible technology to help more than 155,000 users see and seize opportunity sooner. We give investment professionals the edge to outperform with informed insights, workflow solutions across the portfolio lifecycle, and industry-leading support from dedicated specialists. We’re proud to have been recognized with multiple awards for our analytical and data-driven solutions and repeatedly scored 100 by the Human Rights Campaign® Corporate Equality Index for our LGBTQ+ inclusive policies and practices. Subscribe to our thought leadership blog to get fresh insight delivered daily at insight.factset.com.

Learn more at www.factset.com and follow us on Twitter: www.twitter.com/factset.

A B O U T F A C T S E T

I M P O R TA N T N O T I C E

The information contained in this report is provided “as is” and all representations, warranties, terms and conditions, oral or written, express or implied (by common law, statute or otherwise), in relation to the information are hereby excluded and disclaimed to the fullest extent permitted by law. In particular, FactSet, its affiliates and its suppliers disclaim implied warranties of merchantability and fitness for a particular purpose and make no warranty of accuracy, completeness or reliability of the information. This report is for informational purposes and does not constitute a solicitation or an offer to buy or sell any securities mentioned within it. The information in this report is not investment advice. FactSet, its affiliates and its suppliers assume no liability for any consequence relating directly or indirectly to any action or inaction taken based on the information contained in this report. No part of this document (in whole or in part) may be reproduced, displayed, modified, distributed or copied (in whole or in part), in any form or format or by any means, without express written permission from FactSet.

https://insight.factset.com/

https://www.factset.com/

https://twitter.com/FactSet

multi-asset class (mac) ii risk model

Documents