SIMULATION-BASED PRICING OF MORTGAGE-BACKED SECURITIES. Jian Chen. Fannie Mae 3900 Wisconsin Ave. N.W. Washington, DC 20016, U.S.A.

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1 Proceedings of he 4 Winer Simulaion Conference R.G. Ingalls, M. D. Rossei, J. S. Smih, and B. A. Peers, eds. SIMULATION-BASED PRICING OF MORTGAGE-BACKED SECURITIES Jian Chen Fannie Mae 39 Wisconsin Ave. N.W. Washingon, DC 16, U.S.A. ABSTRACT Morgage-Backed-Securiies (MBS), as he larges invesmen class of fixed income securiies, have always been hard o price. Because of he following reasons, normal numerical mehods like laice mehods, or finie difference mehod for solving PDEs are hard o apply: 1) he pah dependence of morgage pool cash flows. ) he embedded American call opion o prepay. 3) he American pu opion o defaul. 4) he fac ha morgage borrower do no/canno exercise hese opion opimally. And hose reasons make Mone Carlo simulaion he bes approach o price MBS. A sandard MBS pricing framework would consiss he following pars:1) Ineres Rae model. ) Prepaymen model, which consiss house urnover model and refinance model. 3) OAS model, which capures risk facors from he marke price. Those facors are no accouned for in he previous wo models. In order o hedge MBS efficienly and effecively, we need o calculae hedging measures quickly and correc. Chen and Fu (1,, 3) has developed some efficien hedging algorihm in he pas o perform his ask. 1 INTRODUCTION TO MORTGAGE-BACKED SECURITIES A morgage-backed securiy (MBS) is a securiy collaeralized by residenial or commercial morgage loans. An MBS is generally securiized, guaraneed and issued by hree major MBS originaing agencies: Ginnie Mae, Fannie Mae, and Freddie Mac. The cash flow of an MBS is generally he colleced paymen from he morgage borrower, afer he deducion of servicing and guarany fees. However, he cash flows of an MBS are no as sable as ha of a governmen or corporae coupon bond. Because he morgage borrower has he prepaymen opion, mainly exercised when moving or refinancing, an MBS invesor is acually wriing a call opion. Furhermore, he morgage borrower also has he defaul opion, which is likely o be exercised when he propery value drops below he morgage balance, and coninuing morgage paymens would no be economically reasonable. In his case he guaranor is wriing he borrower a pu opion, and he guaranor absorbs he cos. However, he borrower does no always exercise he opions whenever i is financially opimal o do so, because here are always nonmoneary facors associaed wih he home, like sheler, sense of sabiliy, ec. And i is also very hard for he borrower o ell wheher i is financially opimal o exercise hese opions because of lack of complee and unbiased informaion, e.g., hey may no be able o obain an accurae home price, unless hey are selling i. And here are also some oher fixed/variable coss associaed wih hese opions, such as he commission paid o he real esae agen he cos o iniialize anoher loan, and he negaive credi raing impac when he borrower defauls on a morgage. All hese facors conribue o he complexiy of MBS cash flows. In pracice, he cash flows are generally projeced by complicaed prepaymen models, which are based on saisical esimaion on large hisorical daa ses. Because of he complicaed behaviors of he MBS cash flow, due o he complex relaionships wih he underlying ineres rae erm srucures, and pah dependencies in prepaymen behaviors, Mone Carlo simulaion is generally he only applicable mehod o price MBS. MBS have become increasingly imporan fixed income insrumens, boh because of heir volume and he role hey play in fund invesmen and porfolio managemen. The oal MBS issuance from he hree agencies opped $,131.9 billion in 3, which is only second o he US reasury bond marke which is abou wice he size. However, if we look a he ousanding MBS volume, which is $349.1 billion in 3, and i is rough equal o ousanding reasury securiies wih volume of $ billion. Combined wih non-agency MBS issues, he MBS capial marke is he larges of all fixed income securiies, exceeding Treasury securiies. (Above numbers are quoed from <hp:// 1.1 MBS Invesors Wih his huge marke volume, any serious invesor in he fixed income marke need o consider he benefis and risks associaed wih MBS, in order o make he opimal inves-

2 men decision. There are several special ype of invesor worh menioning because of he specific benefi brough by MBS: Pension/Reiremen Funds: Because of he longerm invesmen horizon, and consan cash flow requiremen for his ype of invesors, MBS is an ideal ype of invesmen wih long payback period, and regular amorized paymens Morgage Originaors: Because he capial requiremen for MBS is much lower han for morgage loans (8%), his ype of financial insiuion could swap heir loans for MBS, and uilize heir capial more efficienly Morgage Servicers: Because he servicing income is highly sensiive o morgage prepaymen invesing in sripped morgage Principal Only securiies would be a naural hedge. 1. MBS Srucures There are many differen ypes of MBS. Based on paymen srucure, hey could be classified ino hree major caegories: Pass-hrough MBS, Collaeralized Morgage Obligaions (CMOs), and sripped MBS. Pass-hrough MBS is he mos common MBS. As he name suggess, i pass-hrough principal and ineres paymen colleced from morgage borrowers, afer subracing he servicing and guarany fee, o he invesor. This is he building block for mos oher srucured MBS. Pass-hrough MBS could be srucured ino several CMOs, wih he mos common case, sequenials. In his srucure, he original pass-hrough MBS is divided ino several classes, which are called ranches, and generally named A, B, C, D, ec. Tranche A will ge he principal paymen firs and class B will ge principal paid back only when ranche A has been paid off. Tranche C will ge principal paymen afer ranche B is being paid off, so on and so forh. The ineres paymen for each ranche is proporional o is remaining balance. The reason for his srucure is ha invesors have differen preferences. Tranche A is paid off firs wih less risk, and shorer life, and he reurn is lower. While ranche D is paid off afer all oher ranches have been paid off, so i is more risky wih longer life span, he invesor would require a higher reurn on his class of MBS. Sripped MBS is differen from he above CMOs in his approach: his srucure divides he paymen of pass-hrough MBS ino wo classes: Ineres Only (IO), and Principal Only (PO). Each piece is more risky han he original pass-hrough. In some scenario, if morgages prepay real fas he IO invesor could no even recover her iniial invesmen. However, as we have poined before, he PO class could be a naural hedging insrumen for he morgage servicing income. 1.3 Challenges in MBS Analysis The challenges in MBS analysis lies in wo aspecs: pricing and hedging. Pricing of MBS is o esimae he NPV of uncerain fuure cash flows. This basically involves wo fundamenal models: ineres rae model, which deermines he discouning facor and he overall ineres rae environmen and prepaymen model, which deermines he prepaymen behavior of he morgage borrower. Because all agency MBS and mos of non-agency MBS are insured agains principal loss, so defaul is generally considered par of prepaymen and i generally only makes 1% of oal prepaymen mos MBS invesors do no model defaul separaely from prepaymen. To have a good prepaymen model in of paramoun imporance in he pricing and hedging of MBS. Hedging of MBS is o esimae he price sensiiviy o risk facors, generally ineres raes. The mos common sensiiviy measures used are he following: Saic duraion, which measures he price change wih respec o ineres rae change, wihou prepaymen consideraion Zero duraion, which measures he price change wih respec o ineres rae change, wih fixed prepaymen assumpion Opion adjused duraion, which measures he price change wih respec o ineres rae change, wih prepaymen rae adjused for ineres rae change Key raes duraion, which measures he price change wih respec o key ineres raes change, wih prepaymen rae adjused for ineres rae change Principal componen duraion, which measures he price change wih respec o ineres raes change driven by he principal componens of ineres rae, wih prepaymen rae adjused for ineres rae change. The above hedging measures are discussed in Chen and Fu (1,, 3) in deail. Also Chen and Fu(3) demonsraed ha he principal componen duraion are much more efficien in MBS hedging.

3 PREPAYMENT MODEL FOR MBS As discussed earlier, prepaymen model is he mos imporan par in pricing MBS, and here are four main ypes of prepaymen funcions (Fabozzi ()): Arcangen Model: (An example from he Office of Thrif Supervision (OTS).) WAC CPR( arcan(5.9518(1.89 r )) ( ) CPR(S,A,B,M) Model: CPR ( RI( AGE( MM ( BM ( 1 RI( is refinancing incenive AGE( is he seasoning muliplier MM( is he monhly muliplier, which is consan for a cerain monh BM( is he burnou muliplier. Prepaymen models incorporaing macroeconomic facors, i.e., he healh of economics, housing marke aciviy, ec. Prepaymen models for individual morgages. For he las wo ypes of prepaymen models, we do no have any explicily saed funcional forms, mainly because hey are proprieary models in he morgage indusry. Bu since our approach is general for any ype of prepaymen funcion, we can derive he derivaives once we are given an explici form for he prepaymen funcion. 3 SIMULATION-BASED PRICING OF MBS In his secion, we are going o give he implemenaion of he algorihm o price an MBS in deail. 3.1 Problem Seing Generally he price of any securiy can be wrien as he ne presen value (NPV) of is discouned cash flows. Specifying he price of an MBS (here we consider only he passhrough MBS) is as follows: M M P E[ V ] E PV ( E c(, P is he price of he MBS, V is he value of he MBS, which is a random variable, dependen on he realizaion of he economic scenario, PV( is he presen value for cash flow a ime is he discouning facor a ime c( is he cash flow a ime M is he mauriy of he MBS. Mone Carlo simulaion is used o generae cash flows on many pahs. By he srong law of large numbers, we have he following: [ ] E V N 1 lim, N V i N i 1 V i is he value calculaed ou in pah i. The calculaion of is found from he shor-erm (risk-free) ineres rae process,,1) 1,) 1, 1 i exp( r( i) exp{ [ 1 i r( i)] }, i, i+1) is he discouning facor for he end of period i+1 a he end of period i r(i) is he shor erm rae used o generae i, i+1), observed a he end of period i is he ime sep in simulaion, generally monhly, i.e. 1 monh. An ineres rae model is used o generae he shor ermrae r(i) hen is insanly available when he shorerm rae pah is generaed. The difficul par is o generae c(, he pah dependen cash flow of MBS for monh which is observed a he end of monh. From chaper 19 of Fabozzi (1993), we have he following formula for c(: c( MP( + PP( TPP( + IP( MP( SP( + IP( TPP( SP( + PP( MP(: Scheduled Morgage Paymen for monh TPP(: Toal Principal Paymen for monh IP(: Ineres Paymen for monh SP(: Scheduled Principal Paymen for monh PP(: Principal Prepaymen for monh. These quaniies are calculaed as follows: WAC /1 MP( 1 (1 + WAC /1) WAC IP( 1 PP( SMM ( ( SP( ) WAM + TPP( SMM ( 1 1 CPR( ) 1 (1)

4 is he principal balance of MBS a end of monh WAC is he weighed average coupon rae for MBS, weighed by he balance of each morgage WAM is he weighed average mauriy for MBS, weighed by he balance of each morgage SMM( is he single monhly moraliy for monh observed a he end of monh CPR( is he condiional prepaymen rae for monh observed a he end of monh. In Mone Carlo simulaion, along he sample pah, CPR( is he primary variable o be simulaed. Everyhing else can be calculaed ou once CPR( is known. Differen prepaymen models offer differen CPR(, and i is no our goal o derive a new prepaymen model or compare exising prepaymen models. Insead, our concern is, given a prepaymen model, how can we efficienly esimae he price sensiiviies of MBS agains parameers of ineres? Generally differen prepaymen models will lead o differen sensiiviy esimaes, so i is a he user s discreion o choose an appropriae prepaymen funcion, as our mehod for calculaing he Greeks is universally applicable. 3. Simulaion Framework In his secion we are going o describe he building blocks of our simulaion framework: ineres rae model and prepaymen model. We choose our ineres model o be he onefacor Hull-Whie model (Hull and Whie 1993), for is simpliciy and easy calibraion o marke erm srucure. For he prepaymen model, we consider a CPR(S,A,B,M) model Ineres Rae Model In he one-facor Hull-Whie ineres rae model, he underlying process for he shor-erm rae r( is given by dr( ( ϕ ( ar( ) d + σd, is a sandard Brownian moion a is he mean revering speed, consan σ is he sandard deviaion, consan ϕ( is chosen o fi he iniial erm srucure, which is deermined by f (, σ ϕ ( + af (, + (1 e a a f(, is he insananeous forward rae, which is deermined by ), Differeniaing boh sides, wih respec o we have R(, f (, + R(,, R(, is he coninuous compounding ineres rae from now o ime i.e. he erm srucure. In order o simplify he simulaion process, he model can be re-parameerized from is original o he following: dx( a( x( d + σ d, x() x( is deermined by σ a. a( r( x( f (, + (1 e a The soluion for process x( is given by x a au ( σ e e d u), which is a Gaussian Markov process, and can also be represened as a a e 1 x( σ e W ( ), a {W(, } is also a Brownian moion. In his case, he ineres rae r( can be represened in he following form: r +, ( ( a( g( W h ( ) σ a( f (, + a a g( σe a e 1 h(. a (1 e a To simulae r( given by above, we will firs simulae x(, which is a Gaussian Markov process, and hen compue he shor-erm ineres rae by (). For calculaing he price of MBS, he shor-erm rae is no sufficien he long-erm rae process is also required, especially he 1-year Treasury rae, which is a deerminisic funcion of r( in he Hull-Whie model. Generally his is he case for shor-erm rae models, bu no rue for more complicaed ineres rae models, e.g., he HJM (Heah, Jarrow and Moron (199)) model and he LIBOR forward ) ) () 1 R (, f (, u) du,

5 rae model (Jamshidian(1997)). The long-erm rae R(T) is calculaed from he following, : P( T ) e R( T )( T a( T 1 e T ) a P(, T ) ln P(, ln A( T ) ln T ) P(, σ 3 4a A( T ) e ( e at e T ) r( a ) ( e a 1). P(T) is he zero coupon bond price a ime wih face value $1, maured a T. Thus we can derive he R(T) as following: BM ( ) MM( akes he value from [.94,.76,.74,.95,.98,.9,.98, 1.1, 1.18, 1., 1.3,.98], saring from January, ending in December r 1 ( is he 1-year rae, observed a he end of period a quaniy ha is highly correlaed wih he prevailing 15-year and 3-year fixed morgage raes. The MBS we price is a fixed-rae morgage pool, wih a WAC of 6.6%, and pool size of $4,,. Once he 1-year rae is simulaed, he prepaymen rae can also be deermined also wih he characerisics of he morgage pool. Figure shows one pah of simulaed morgage prepaymen rae. ln A( T ) T ) r( R( T ). ( T Figure 1 gives one simulaed pah of he shor rae r( and long erm rae r1(, which is laer used o generae he prepaymen rae. Figure : Simulae Morgage Prepaymen Rae (1 Pah) Once he prepaymen rae is acquired, he cash flow from he morgage pool is deermined. Figure 3 shows one simulaed pah of he cash flow and is corresponding ne presen value. And he sum of he presen value would he ne presen value in his scenario. Figure 1: Simulaed Ineres Rae (1 Pah) 3.. Prepaymen Model We use he second ype of prepaymen funcion, among he four described in secion 3. An example for his ype of prepaymen model is available from he sample code a hp:// CPR(RI(AGE(MM(BM( RI(.8+.14an -1 ( (WAC-r 1 (-1))) AGE ( min(1, ) 3 Figure 3: Simulaed Cash Flow and Presen Value

6 Afer we have performed a cerain number of simulaions, we can calculae he mean of he NPV, and ha will be he price of he MBS we priced. Figure 4 shows he hisogram of he NPV disribuion for 3 simulaions. The price of he MBS is $4,439,67.73, and he 95% confidence inerval is $36, OAS could be viewed as excess reurn beyond he risk free reurn, adjused for he prepaymen opion. I capures he reurn required by he invesor communiy, o compensaed for risks associaed wih MBS, afer adjusmen for prepaymen. I could be viewed as he premium for a iny porion of credi risk from he MBS issuer, and he model uncerainy in he pricing framework, or marke liquidiy premium. 4 CONCLUSION As we have poined before, because of he complicaed naure of MBS, pricing and hedging of his ype of securiies remain difficul. To make hings more ineresing, here are a lo of new morgage producs, like skip-a-paymen morgage, which he borrower could skip 1 o 1 paymens, porable morgage, which he borrower could ake wih her when moving, auomoive refinance morgage, which he borrower could auomaically ge refinance when he morgage rae drops below a hreshold. As far as indusry praciioners are concerned, Mone Carlo simulaion is sill he only way o price and calculae price sensiiviies of MBS. So how o improve simulaion accuracy and efficiency will sill pose a grea challenge for fuure research. ACKNOWLEDGMENTS Figure 4: Hisogram of he NPV of MBS 3..3 OAS Adjusmen When we apply he ineres rae and prepaymen model o price an MBS, i is generally no in agreemen wih he marke price. In order o adjus our model, such ha i could produce a price equal o marke price, we need o inroduce he opion-adjused spread (OAS). If we believe our prepaymen model is accurae, i.e., he cash flow is correc in order o change he price, we can only adjus he discouning facor, and ha is exacly how OAS plays he role. If we change he discouning facor in (1) o he following:,1) 1,) 1, such ha 1 i exp[ r( i) + s) ] exp{ [ 1 i ( r( i) + s)] }, M E c( marke _ price, hen he spread s which make equaion (19) holds is called he OAS for his MBS. I is generally solved by some recursive algorihm. The auhor would like o hank Michael C. Fu for his inviaion o presen his paper a he winer simulaion conference. Any opinions expressed in his aricle are hose of he auhor and do no reflec he views of Fannie Mae. REFERENCES Chen, J., and M.C. Fu. 1. Efficien Sensiiviy Analysis for Morgage-Backed Securiies. Working paper, Universiy of Maryland, College Park, MD. Chen, J., and M.C. Fu.. Hedging Beyond Duraion and Convexiy. In Proceedings of Winer Simulaion Conference, ed. E. Yücesan, C.-H. Chen, J. L. Snowdon, and J. M. Charnes, Piscaaway, NJ: Insiue of Elecrical and Elecronics Engineers. Chen, J., and M.C. Fu. 3. Three essays on morgagebacked securiies: hedging ineres rae and credi risks. Docoral disseraion, Rober H. Smih School of Business, Universiy of Maryland, College Park, MD. Fabozzi, F. J Fixed Income Mahemaics. New York, NY: McGraw-Hill. Fabozzi, F. J... The Handbook of Morgage-Backed Securiies. New York, NY: McGraw-Hill. Heah, D., R. A. Jarrow, and A. Moron Bond Pricing and he Term Srucure of Ineres Raes: A New Mehodology for Coningen Claims Valuaion. Economerica 6 (1): Hull, J., and A. Whie One-Facor Ineres-Rae Models and he Valuaion of Ineres-Rae Derivaive Securiies. Journal of Financial and Quaniaive Analysis (8) : Jamshidian, F Libor and swap marke models and measures. Finance and Sochasics (1) 4:

7 AUTHOR BIOGRAPHY JIAN CHEN is currenly a financial engineer in Fannie Mae. He received his Ph.D. in compuaional finance in he Rober H. Smih School of Business, a he Universiy of Maryland. He received his B.S.E.E. and M.S.E.E. from JiaoTong Universiy, in Xi an and Shanghai, respecively. His research ineress include simulaion and compuaional finance, paricularly wih he pricing and hedging for ineres rae derivaives and credi derivaives.