Pricing and Modeling Risk of Agency MBS - RiskMetrics Group

Pricing and Modeling Risk of Agency MBSVersion 1.0October 2010Copyright © 2010 by RiskMetrics Group.All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic ormechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from thepublisher. Requests for permission to make copies of any part of this work should be sent to: RiskMetrics Group MarketingDepartment One Chase Manhattan Plaza, 44th Floor, New York, NY 10005 RiskMetrics Group is a trademark used herein underlicense.www.riskmetrics.com

RiskMetrics Groupwww.riskmetrics.com1 IntroductionRecently, RiskMetrics introduced a new, internally-developed model for pricing of mortgage-backedsecurities, and a set of new risk analytics based on the pricing model. This analytics replaces the oldsensitivities-based model. This document provides a brief, high level introduction to the new MBS analytics.For more details, please see additional readings referenced at the end of this document.2 BackgroundThe first generation of analytics for Agency mortgage-backed securities (MBS) introduced by RiskMetrics,utilized pre-computed interest rate sensitivities (key-rate durations and convexities) for Generic MBS. Weused sensitivities as parameters in a two-term Taylor series expansion to re-price securities for calculation ofVaR, or in stress tests. We obtained these sensitivities from two 3 rd party providers: Derivative Solutions andCMS Bondedge.Recently, RiskMetrics introduced a new, internally-developed model for pricing of mortgage-backedsecurities. In this pricing model, we explicitly simulate the behavior of interest rates, prepayment speeds,and (for CMOs and ABSs) the cash flows delivered to each security. The cash flows estimated by the modelare discounted and summed to yield the present value of the security. The instruments priced using the newmodel can participate fully in all risk analyses, including computations of sensitivities, VaR, and stress tests.The new approach offers clients the accuracy and advanced analytics features not possible with a simplesensitivity-based model. However, this approach also introduces a high computational cost, especially forcalculation of VaR. Therefore, we offer two flavors of the MBS analytics: “full-valuation”, and “quadraticapproximation”. The full-valuation approach features complete re-pricing of a security in each VaR scenario.With the “quadratic approximation” approach, we fully re-price a security in a reduced number of scenariosto compute sensitivities (key rate durations and convexities), and then use the sensitivities to re-price thesecurity in the remaining scenarios. Clients that require a high degree of precision at high confidence levelsshould choose the full-valuation approach. For other clients, the quadratic approximation approach offers anexcellent trade-off between precision and computational performance.RiskMetrics has also developed a “delta-gamma” approach – an enhanced sensitivity-based risk solution forGeneric MBS. With this approach we compute key rate durations and convexities using the new pricingmodel, and then use these sensitivities to re-price the security for VaR calculations, and in stress tests.While Delta-Gamma is similar to the original sensitivity-based approach, it offers clients more flexibility, andseveral advanced analytics features. Delta-Gamma was developed specifically to be used for Generic MBS. Insection 4 of this document, we provide a brief overview of the RiskMetrics approach to creating Generic MBSaggregates.3 Agency MBS Analytics3.1 Pricing ModelThe major difference between mortgage-backed securities and standard bonds, such as government orcorporate bonds, is the uncertainty of interest and principal cash flows due to prepayments by mortgageholders. Therefore, an MBS pricing model must include a method for estimating prepayments. Mortgageprepayments are dependent on a number of sources of risk, but the dominant factor is the level of interestrates. If rates drop, mortgage holders are incentivized to refinance their mortgages, and thereforeprepayments rise. If rates rise, there is no benefit to refinancing, and prepayments slow. Therefore, a keyvariable in a prepayment model is the projected level of interest rates, and a required element of MBSpricing is a model of interest rates. However, once cash flows of a bond are estimated, the final step in theTitle of Paper 3

RiskMetrics Groupwww.riskmetrics.comMBS pricing model is the same as for any other bond – the estimated cash flows are discounted using theappropriate set of zero coupon rates, and summed to yield a present value of the bond.In the RiskMetrics framework, the pricing of an Agency mortgage-backed security consists of three steps:1. Generation of a set of future interest rate paths using a one-factor Hull-White interest ratemodel2. Estimation of prepayment vectors using the Andrew Davidson prepayment model3. A cashflow modelIn the first step, we simulate the interest rates using a one-factor Hull-While model. Each interest rate pathis simulated 30-yrs forward and consists of 360 monthly time steps. Typically, it is necessary to generate atleast 50 interest rate paths. To calibrate the model, we use Swaption volatilities.In the second step, we estimate prepayments using the Andrew Davidson prepayment model. AndrewDavidson is one of the leading mortgage analytics companies, and its prepayment model has been widelyadopted by the industry. The Andrew-Davidson model takes as input the following information:• Terms and Conditions of the Mortgage Poolo Loan type (i.e. FNAM fixed-rate, FHLMC hybrid ARM, etc)o Current weighted average maturity (WAM)o Current weighted average coupon (WAC)o Current weighted average loan age (WALA)o Additional parameters for ARMs and Hybrids, such as initial reset date and reset frequency• Interest Rate Paths: Forecasted LIBOR rates (2 and 10 year) for each month are required. Theseinterest rate paths are estimated in the first step.• MBS current coupon computed as a weighted average of 2yr and 10yr swap rates plus a pool-typespecific spread.The Andrew Davidson prepayment model also provides a number of tuning parameters that allow modificationof the model’s behavior. This may be necessary when a mortgage pool exhibits unusual borrowercharacteristics that are not captured by the base model. We expose these parameters in the application, sothat they can be modified by the client. The prepayment model returns a monthly path of forecastedprepayment speeds.In the third and final step, the terms & conditions of the security and the prepayment speeds provided by theprepayment model are used to estimate the future cashflows. These cashflows are then discounted andsummed to yield the present value of the security.3.2 Quadratic ApproximationA full-valuation approach to pricing and risk measurement of mortgage-backed securities is highlycomputationally intensive, in particular for computing VaR. A typical VaR calculation requires 1,000 repricingsin order to estimate stable distribution parameters. Each re-pricing requires generation of 50 ormore interest rate paths, each with 360 time steps. As a result, the total number of simulation pathsrequired to compute VaR for a single MBS is over 50,000. Quadratic Approximation alleviates theTitle of Paper 4

RiskMetrics Groupwww.riskmetrics.comcomputational burden by reducing the number of simulations required to compute VaR for a mortgage-backedsecurity.Quadratic Approximation involves an estimation of a security’s price by means of Taylor series expansion,where the terms of the expansion represent changes in the MBS price due to duration (delta) and convexity(gamma). To obtain deltas and gammas we first perform a number of full valuations - typically 200 or less.We then perform a least-squares regression of the prices on the model variables (interest rates and swaptionvolatilities), to obtain delta and gamma sensitivities.Quadratic Approximation reduces the computational burden when computing VaR, while producing resultsthat closely approximate those obtained using a full valuation approach. It’s important to note that we usequadratic approximation only for computing VaR. Base price of a security, as well as values under stress testsare recomputed using full valuation. For additional details see “Quadratic VaR Approximation for SecuritizedProducts” (October 2010).3.3 Delta-GammaDelta-Gamma is a sensitivity-based approach to MBS analytics that is based on using the sensitivitiescomputed with the Quadratic Approximation described above. Delta-Gamma was developed for use withGeneric MBS. As in the computation of VaR using quadratic approximation, a Taylor series expansion is usedto re-price a security, using deltas and gammas as inputs. The main difference between Delta-Gamma andQuadratic Approximation is how we reprice a security in stress tests:• Delta-Gamma: We revalue a security using Taylor series expansion.• Quadratic Approximation: We re-price the security using the full pricing model.4 Generic MBSIt is a common industry practice to use Generic MBS as proxies for Agency MBS. A generic MBS security is afictitious security that represents a group of real Agency mortgage-backed securities with certain commoncharacteristics. These characteristics include: Agency (FNMA, FHLMC, and GNMA), program (30-yr, 20-yr, and15-yr), coupon, and year of origination. The motivation for using Generic MBS is to reduce the computationalcost of pricing individual MBS. There are over 1 million Agency MBS outstanding; pricing a large number ofindividual securities would be too computationally intensive. Using Generic MBS is justified, because thecollateral underlying different Agency MBS pools with common characteristics is very homogeneous, whichresults in minimal differences in the valuation and risk exposure.RiskMetrics has developed a Generic approach to the pricing and modeling of risk of mortgage backedsecurities. To create Generic MBS, the Agency MBS universe of approximately 1,000,000 individual agencyMBS pools is partitioned into a parsimonious and more manageable subset of approximately 10,000 genericaggregates. The partitioning criteria (defined by agency, program, coupon, and origination year) leads tomutually exclusive groupings of CUSIPs. For each Generic security, we compute aggregate level weightedaverage pool statistics: Weighted-Average Maturity (WAM), Weighted-Average Coupon (WAC), Weighted-Average Loan Age (WALA), and Weighted Average Price (WAP). The weights are based on the currentoutstanding balances of the pools selected for the given generic.Each generic is assigned a ticker that serves as a unique identifier for the security, and also describes thecharacteristics of the pools it represents. For example: FH-PT-1998-30Y-6.25. (FH stands for Freddie Mac,PT stands for “pass-through”, 1998 is the origination year, 30Y is maturity, and 6.25 is coupon)Title of Paper 5

RiskMetrics Groupwww.riskmetrics.comWhile it is possible to model individual Agency MBS pools, we recommend that clients use Generic MBS. Thissignificantly reduces computational time for large portfolios, and produces results that are not materiallydifferent from full valuation. For additional details on our approach to Generics see “Generic MBSParameterization” (Nov 2009).5 Comparing Different ApproachesThe new Delta-Gamma solution offers significant advantages over the approach that uses sensitivities fromDerivative Solutions and CMS Bondedge. The following table compares key analytics features of the twoapproaches:AnalyticsMethodFull Valuation orQuadraticApproximationDelta-GammaOriginal SensitivitiesbasedApproachOAS and OASbasedstatisticsand stress testsGeneric OAS computedfrom weighted averagepool pricesGeneric OAS computedfrom weighted averagepool pricesOAS not computedRisk FactorsSwap rates & swaptionvolatilitiesSwap rates & swaptionvolatilitiesSwap rates onlyPricing ModelFull valuation for PV,stress tests, andgreeks. Full valuationfor VaR, or regresseddelta-gamma withquadraticapproximationModel calibrated toweighted average poolprices. Delta andgamma (includingcross order terms)regressed from fullvaluation sample spaceDelta and gammacomputed from first,second, and crossorder termsPrepayment ModelTuningBase prepaymentmodel tunings exposedat position levelBase prepaymentmodel tunings exposedat position levelNo access to baseprepayment modeltuningsPrepayment ModelStress TestsCompute stressedpresent value due toshifts in tuningparametersCannot computestressed presentvalues due toprepayment modelshiftsCannot computestressed presentvalues due toprepayment modelshiftsCoverageSpecific securities andGeneric MBSGeneric MBSGeneric MBSTitle of Paper 6

RiskMetrics Groupwww.riskmetrics.com6 RiskManager InstrumentsThe following table lists three RiskManager instruments that implement MBS analytics, and thepricing/modeling options provided by each instrument. Access to these instruments is based on subscription.RiskManager InstrumentPricing ModelUS MBSThis is a full-valuation MBS pricing model. User must specify allrequired parameters of the security.Generic Mortgage Backed SecurityThis instrument provides several options for pricing Generic MBS.User must specify/select one of the available Generic MBSaggregates. Three pricing options are available:- Full-valuation- Quadratic Approximation- Delta-GammaAgency MBS Sensitivity InstrumentThis instrument implements pricing of Generic MBS using Delta-Gamma approach only.Title of Paper 7

RiskMetrics Groupwww.riskmetrics.comAdditional ReadingRiskMetrics Research Department: RMBS in RiskServer Using a Statistical Model. June, 2009RiskMetrics Research Department: Generic MBS Parameterization: November, 2009RiskMetrics Research Department: TBA Parameterization within DataMetrics. June, 2009Andrew Davidson & Co: Tuning the Prepayment Model. November, 2007Andrew Davidson & co: Fixed Rate MBS Prepayments & Model Enhancements. May, 2006RiskMetrics Research Department: Quadratic VaR Approximation for Securitized Products. October, 2010Title of Paper 8