DRAFT Axioma Global Multi-Asset Class Risk Model Fact Sheet AXGMM Version 2.0 March 2017 1 Introduction Axioma’s Global Multi-Asset Class Model offers users an intuitive decomposition of a multi–asset class portfolio. The model is designed for a broad-based analysis of global multi–asset class holdings. As such, the res- olution of the factors targets understanding sources of risk across a broad portfolio, rather than, for example, analyzing smile risk across a volatility surface. The model contains factors derived from an integrated set of pricing models calibrated to tradeable asset prices for a wide range of asset classes. This allows portfolio managers to analyze the risk and performance of their portfolios across all assets in a consistent and rigorous manner. The Global MAC Model combines the cover- age of Axioma’s global equity risk model with coverage of the fixed-income and commodities markets. Leveraging the expertise embedded in Axioma Robust Equity Models and an extensive universe of fixed-income factors, the Global MAC Model allows portfolio managers holding a wide range of assets to use a consistent set of models to analyze their risk and attribute performance. Axioma’s factor models are designed to sep- arate and quantify systematic and idiosyncratic components of risk. The systematic factors cap- tured in the mode include: • Yield curve • Currency • Volatility • Inflation • Credit • Commodity curves • Sector • Country • Equity Style The risk models facilitate both absolute and relative decompositions of risk into either mar- ket risk or risk relative to a benchmark (tracking error). Moreover, since exposures are specified as sensitivities, asset managers can better under- stand the interplay among different risk factors. The Global MAC Model is produced by Ax- ioma’s Enterprise Risk Platform Axioma Risk. This cloud-based platform allows extensive flex- ibility in factor definition, risk model construc- tion, stress testing, risk analysis, and reporting. The Global MAC Model provides for a stream- lined portfolio risk analysis, while Axioma Risk offers a fully customizable enterprise risk system. 2 Top Level Structure of the Global MAC Risk Model The Global MAC Model is a linear, parametric risk model composed of i) a global set of risk fac- tors for which factor returns and an associated covariance matrix are computed; and ii) a set of exposures to these risk factors for each asset in the covered universe. If there are N A assets in the portfolio and N F risk factors, we can define an N A × N F exposure matrix B with the i th row containing the exposures of asset i to all the fac- tors. The portfolio risk (i.e. the volatility of the portfolio return) is then computed as σ Π = w T BΣ F B T w 1/2 where Σ F is the covariance matrix of the fac- tor returns and w is the vector of dollar posi- tion weights for each asset. Axioma provides the factor return covariance matrix and the expo- sure matrix for a specific portfolio as the weekly Global MAC Model deliverable, while the user provides the universe and position data. All risk and performance measures are derived from these components. In the following sections, we first Axioma 1
Transcript
DRAFT
Axioma Global Multi-Asset Class Risk Model Fact Sheet
AXGMM Version 2.0
March 2017
1 Introduction
Axioma’s Global Multi-Asset ClassModel offers users an intuitive decompositionof a multi–asset class portfolio. The model isdesigned for a broad-based analysis of globalmulti–asset class holdings. As such, the res-olution of the factors targets understandingsources of risk across a broad portfolio, ratherthan, for example, analyzing smile risk acrossa volatility surface. The model contains factorsderived from an integrated set of pricing modelscalibrated to tradeable asset prices for a widerange of asset classes. This allows portfoliomanagers to analyze the risk and performance oftheir portfolios across all assets in a consistentand rigorous manner.
The Global MAC Model combines the cover-age of Axioma’s global equity risk model withcoverage of the fixed-income and commoditiesmarkets. Leveraging the expertise embedded inAxioma Robust Equity Models and an extensiveuniverse of fixed-income factors, the Global MACModel allows portfolio managers holding a widerange of assets to use a consistent set of modelsto analyze their risk and attribute performance.
Axioma’s factor models are designed to sep-arate and quantify systematic and idiosyncraticcomponents of risk. The systematic factors cap-tured in the mode include:
• Yield curve • Currency
• Volatility • Inflation
• Credit • Commodity curves
• Sector • Country
• Equity Style
The risk models facilitate both absolute andrelative decompositions of risk into either mar-ket risk or risk relative to a benchmark (tracking
error). Moreover, since exposures are specifiedas sensitivities, asset managers can better under-stand the interplay among different risk factors.
The Global MAC Model is produced by Ax-ioma’s Enterprise Risk Platform Axioma Risk.This cloud-based platform allows extensive flex-ibility in factor definition, risk model construc-tion, stress testing, risk analysis, and reporting.The Global MAC Model provides for a stream-lined portfolio risk analysis, while Axioma Riskoffers a fully customizable enterprise risk system.
2 Top Level Structure of theGlobal MAC Risk Model
The Global MAC Model is a linear, parametricrisk model composed of i) a global set of risk fac-tors for which factor returns and an associatedcovariance matrix are computed; and ii) a set ofexposures to these risk factors for each asset inthe covered universe. If there are NA assets inthe portfolio and NF risk factors, we can definean NA×NF exposure matrix B with the ith rowcontaining the exposures of asset i to all the fac-tors. The portfolio risk (i.e. the volatility of theportfolio return) is then computed as
σΠ =(wTBΣFB
Tw)1/2
where ΣF is the covariance matrix of the fac-tor returns and w is the vector of dollar posi-tion weights for each asset. Axioma provides thefactor return covariance matrix and the expo-sure matrix for a specific portfolio as the weeklyGlobal MAC Model deliverable, while the userprovides the universe and position data. All riskand performance measures are derived from thesecomponents. In the following sections, we first
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describe the set of risk factors available in themodel and then describe how the exposures tothe risk factors are determined.
3 Risk Factors in the Global MACModel
The set of risk factors for the Global MACModel is comprised of the Axioma Global Eq-uity Fundamental Model (the World-Wide Fun-damental Model, version 2.1), the Axioma GlobalFixed-Income Factor Model, and a commoditymodel based on constant maturity futures curves,as well as volatility factors derived from swap-tion data. Details of the models for specific assetclasses are described here.
3.1 Global Equity Fundamental FactorModel
For equities, the Global MAC Model imposesa structure on the asset returns by identifyingcommon factors within the market that is, fac-tors that drive asset returns. Returns can thenbe modeled as a function of a relatively smallnumber of parameters, and estimating poten-tially millions of asset covariances can thus besimplified to calculating a much smaller set ofnumbers.
The factors used in the Global Equity Funda-mental Factor Model fall into several broad cat-egories:
• Industry factors reflect a company’s line ofbusiness.
• Country factors reflect a company’s domi-cile.
• Style factors encapsulate the financial char-acteristics of an asset e.g. a company’s size,debt levels, liquidity, etc.
Calculation of the exposures to these factors isdescribed in the Section 4. Once the exposuresto these factors have been determined, factor re-turns are then calculated off a cross sectional re-gression model carried out on an estimation uni-verse.
3.2 Global Fixed-Income Factor Model
The Global Fixed-Income Factor Model de-rives rate curves, spread curves and volatility sur-faces from market prices and estimates factorsthat drive changes in the curves. The factors aredesigned to separate and quantify systematic andidiosyncratic components of risk. The systematicfactors within a typical fixed-income portfolio in-clude changes in:
• Sovereign rates
• Swap spreads
• Spot FX rates
• Corporate credit spreads by rating and rat-ing/GICS sector
• Break-even inflation rates
• Asset-backed security spreads
• Covered bond spreads
• Other government, GRE and Supranationalspreads
• Derivatives: Swaption Volatilities, CDSspreads
The table in Appendix B lists the number ofsuch curves/factors grouped by category.
The interest rate and spread factors have a hi-erarchical organization made up of spreads ontop of the government yield curves; i.e. swaps,supranational, ratings and industries. The factorstructure differs by asset class; the hierarchicalstructure for the spread factors is summarized inAppendix A. As the reporting currency of theGlobal MAC Model is US dollars, the return of afixed-income asset is decomposed into a local re-turn and a currency return. Thus there exists anadditional risk factor for non-dollar denominatedassets.
3.2.1 Sovereign Bonds
At the root of the hierarchy are factors derivedfrom sovereign zero-coupon yields, extracted atkey rates on each currency’s sovereign yield
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curve. The risk of sovereign bonds, excludingthose of Eurozone countries, is driven solely bychanges in these key rates. The key points onthe curve are the 6m, 1y, 2y, 5y, 10y and 30ytenors. Interest-rate models are supplied for 37currencies worldwide.
For the Eurozone, a base Eurozone sovereignzero-coupon curve is created using the lowestyielding sovereign issuances from Eurozone mem-bers. Individual Eurozone countries each havetheir own sovereign spread over the Eurozonebase curve, and each sovereign spread is againbased on the benchmark bonds.
For Emerging Market countries issuing debtin USD, separate spread factors over the USsovereign curve are estimated, distinct from thelocal currency sovereign curve.
3.2.2 Interest Rate Swaps
In addition to the movement of governmentyield curves, the risk in swap spreads (over theyield curve) is also captured. The risk factorsare changes in swap spreads at maturities cor-responding to the key rates on the governmentcurve.
3.2.3 Corporate Bonds
Corporate bonds are exposed to additionalspread-related risk factors. The spread factorsare layered so that Rating/Sector spreads arestated as spreads per sector in excess of the Rat-ing spreads. (See the corporate section of Figure1 in Appendix A and the Rating/Sector curvecoverage in Appendix E.) The additional risk fac-tors for corporate bonds are:
• Ratings spreads per currency for the G7
• Ratings/Sector spreads per currency whendata allow
• Individual issuer curves for issuers with suf-ficient data
Rating spread factors are generated for the G7currencies for the following Axioma compositerating buckets:
• AAA • AA • A • BBB
• SUB-IG (Sub-Investment Grade)
This common hierarchical structure allows for auniform analysis of assets across currencies andregions.
Ratings spreads for each of the G7 curren-cies are implemented as factors for all corporatebonds. For sectors in currencies for which suffi-cient data are available at various rating levels,additional excess Ratings/Sector spreads are im-plemented as factors. Issuer spread curves, de-fined as excess spread over the Ratings and Rat-ings/Sector spreads, are used to compute specificrisk (idiosyncratic risk uncorrelated to other fac-tors) for corporate bonds related to those issuers.Proxy logic is used to map corporate bonds thatare not captured explicitly by these factors (e.g.unrated bonds or bonds issued in non-G7 curren-cies).
Bonds with optionality, such as callable bonds,also have exposure to an interest-rate volatilityfactor. In addition to the fixed-income factors,convertible bonds have exposure to the equityrisk factors.
3.2.4 ABS (Asset-Backed Securities)
For US RMBS agency pass-through securi-ties, a dynamic prepayment model is used whichprojects asset cash-flows as a function of cur-rent interest-rate factors and swaption volatili-ties. For other US ABS securities, investment-grade spreads are estimated based on yields ob-served in the markets. Axioma produces spreadsfor eight different ABS securitization types cor-responding to the underlying collateral. For ex-
1GICS is a registered trademark of MSCI and S&PGlobal Inc.
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ample, there is a specific spread factor for autoloan securitization bonds. See Appendix A forthe full list of ABS categories covered.
3.2.5 Inflation-linked Bonds
The prices of inflation-linked bonds in individ-ual currencies are used to estimate real interest-rate term structures. Break-even inflation curvesare estimated as the difference between the nom-inal sovereign rate curve and the real rate curve.Inflation-linked assets are then modeled with ex-posure to nominal sovereign rate and break-eveninflation rate factors.
3.2.6 Covered Bonds
Spread factors for five European countries withsubstantial numbers of Covered Bond issuers areestimated from investment-grade Covered Bondmarket-yield data as spreads over each country’sswap curve. Spreads are available for Euro is-suance as well as Swedish Krona issuance.
3.2.7 Other Government, GRE and Suprana-tional Bonds
Additional spread factors are estimated forthree US agency bond issuers as spreads over theUS sovereign curve. Ten spread factors over theCanadian sovereign curve are available for Cana-dian provincial bond issuers. For supranationalbond issuers, spreads over the swap curve are es-timated for both USD and Euro issuance at theAAA and investment grade level. US munici-pal bond spread factors are proxied by the cor-responding US corporate rating spread factors.
3.2.8 Derivatives
Volatility factors are used to capture risk infixed-income derivatives, as well as bonds withoptions (callable bonds, pass-through mortgagesecurities, etc.). The model provides interest-rate volatility for 19 currencies computed as the2 year × 2 year at-the-money swaption volatility.
For credit derivatives, the model usesindividual-name CDS spreads and CDS indexspreads as risk factors.
3.3 Commodities
Commodity futures contracts are modeled inAxioma Risk using Constant Maturity Futures(CMF) curves. The model uses daily futures set-tlement prices and maturities to construct theCMF curves as a set of interpolated nodes atfixed maturities. The nodes of the CMF curvesare the risk factors, and the CMF curves are con-structed using the one day log returns of the fu-tures prices.
4 Exposure Calculation in theGlobal MAC Model
Calculation of the sensitivity of an asset pricereturn to the various risk factors is handled dif-ferently across the broad asset groupings: Equity,Fixed-Income, and Commodities. Additional de-tail is also provided for computing exposure tocurrency and volatility risk factors.
4.1 Exposure to Equity Risk Factors
Exposure to the industry and country factorsin the equity risk model are binary, either 0 or1. These are determined by the GICS industryclassification of the company and the countryof incorporation. Exposure to the style factorsis computed based on a Z-score transformationof a combination of financial ratios or perfor-mance measures associated with the definitionof the style factor. The models are constructedto maximize the explanatory power of the stylefactors driven by the definitions of these expo-sures. Industry and country exposures are gener-ally static, while most style factor exposures areupdated when new financial performance dataare released.
4.2 Exposure to Fixed-Income Risk Fac-tors
For all fixed-income rate and spread risk fac-tors, the exposure is computed as the price returnsensitivity to changes in the factor. This requiresa pricing function PA(F1, . . . , Fk) for each assetA as a function of the relevant risk factors. The
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exposure to factor Fj is then defined as
βj =1
PA
∂PA
∂Fj(1)
For sovereign rates and swap spreads, this cor-responds to the negative of key rate durations(KRDs) at the six key rates. The exposures tothe other spread factors are computed as the neg-ative of spread durations. Axioma has imple-mented detailed pricing functions for all fixed-income assets covered and computes exposures tothe underlying risk factors based on these pricingfunctions.
4.3 Exposure to Commodity Risk Factors
As the risk factors for commodity future con-tracts are log-return CMF curves, the exposureof a contract with the same maturity as a nodepoint is exactly 1 (i.e. the volatility of the com-modity future contract with maturity T is thesame as the volatility of the CMF curve at pointT ). When the maturity of the contract lies be-tween two nodes, the contract has exposure tothe two surrounding nodes and each exposureis computed as the proportional linear distancefrom the maturity to the node maturity.
4.4 Exposure to Volatility and CurrencyRisk Factors
Because the volatility and currency risk fac-tors are computed as log returns, the exposure iscomputed as
β =X
PA
∂PA
∂X(2)
where X is either the volatility level σ or the spotexchange rate. For non-US dollar denominatedassets with no currency exposure other than thereporting currency exchange-rate risk, this for-mula corresponds to a currency risk exposure of−1.
5 Global MAC Model Parameters
The Global MAC Model includes a covariancematrix estimated from the time series of the rel-evant factors for a given portfolio. Details of the
parameter choices for the covariance estimationcalculation are given here.
5.1 Model History
Monthly covariance matrix history from Jan-uary 2012 onward; daily factor returns from Jan-uary 2007.
The covariance matrix for the model is esti-mated based on five years of daily factor-returndata using the following estimation procedure.The daily factor returns are aggregated into over-lapping weekly returns, thus creating a time se-ries of approximately 1250 observations of weeklyreturns. Note that weekly returns are usedto minimize the impact of asynchronous timingof returns reporting and potentially stale fixed-income pricing. Overlapping returns are used toensure that the model still responds quickly tomarket shocks, as well as to increase the effectivesample size used in the covariance estimation.An exponentially decaying weighting scheme isused to more heavily weight the recent return be-havior. A half-life of 250 daily observations (oneyear) is used to compute both variances and cor-relations.
5.4 Specific Risk
For assets with sufficient data and idiosyn-cratic drivers of risk, specific risk (i.e. additionalvolatility uncorrelated with the factors in themodel) is estimated and included in the model.The methodology implemented varies by assetclass in the model:
• For equity assets, specific risk is defined asthe volatility of the residual that is not ex-plained by the factors.
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• Specific risk for fixed-income corporate secu-rities is estimated, where the data are avail-able, from single-name issuer-curve volatili-ties.
6 Data Deliverables
The Axioma Global Multi-Asset Class Model isaccessible through Axioma’s SFTP site.
1. Availability: The covariance and exposurematrices are updated weekly and availableto download via SFTP.
2. Historical Coverage: A monthly historyof covariance and exposure matrices is avail-able going back to January 2012. Daily fac-tor returns are available going back to Jan-uary 2007.
3. Data Format: Delimited text files (flat-files)
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Appendices
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Appendix A Fixed-Income Factor Hierarchy
Figure 1: Fixed Income Factor hierarchy for the Global MAC Model
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Appendix B Global MAC Model: Curve and Factor Structure
Curve/Factor Type Definition CountReferenceCurve
CommentsReturnType
CO- Commodity Futures Constant Maturity Futures 63 N/A See Appendix D
Changein LogPriceReturn
Currency Currency Risk Factors 92 N/ASpot exchange rates for 92currencies. All exchangerates are relative to USD
Log FXRateReturn
EQ-CountryEquity Country Factors fromAXWW21 (Global Equity)Model
85 N/APercentageReturn
EQ-IndustryEquity Industry Factors fromAXWW21 (Global Equity)Model
68 N/APercentageReturn
EQ-LocalEquity Local Market Resid-ual Factor from AXWW21(Global Equity) Model
1 N/APercentageReturn
EQ-MarketEquity Global Market Factorfrom AXWW21 (Global Eq-uity) Model
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