Dynamic Option Adjusted Spread and the Value of Mortgage Backed Securities
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1 Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao and Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics Previous Draf: 27 January 28 This Draf: 27 April 29 Absrac We exend a reduced form model for pricing pass-hrough morgage backed securiies (MBS) and provide a novel hedging ool for invesors in his marke. To calculae he price of an MBS, raders use wha is known as opion-adjused spread (OAS). The resuling OAS value represens he required basis poins adjusmen o reference curve discouning raes needed o mach an observed marke price. The OAS suffers from some drawbacks. For example, i remains consan unil he mauriy of he bond (hiry years in morgage-backed securiies), and does no incorporae ineres rae volailiy. We sugges insead wha we call dynamic opion adjused spread (DOAS), which allows invesors in he morgage marke o accoun for boh prepaymen risk and changes of he yield curve. Keywords: Asse pricing, Morgage Backed Securiies, Term Srucure. JEL classificaion numbers: C23, G34 We wish o hank John Crosby and Joe Byrne for consrucive commen. The usual disclaimer applies. Corresponding auhor: Mario Cerrao, Universiy of Glasgow, Deparmen of Economics, m.cerrao@lbss.gla.ac.uk 1
2 1. Inroducion Morgage Backed Securiies (MBS) are securiies collaeralised by residenial morgage loans. The MBS marke has grown o become he larges fixed income marke in he Unied Saes. Probably one of he reasons of his enormous growh was he higher reurn paid by hese securiies and he percepion ha mos of hem carried a lower risk han oher fixed income securiies 1. However, alhough he marke for MBS was very dynamic and many sudies have been ineresed in pricing hese financial insrumens (see for example Longsaff, 24 and Chen, 24), here are sill quie a few issues concerning he risk managemen of hese securiies. Because of he borrowers prepaymen opion in he underlying morgage loans, morgage-backed securiies have characerisics similar o hose of callable bonds. Unlike callable bonds for which he issuers refinancing sraegies are assumed o be close o opimal, morgage borrowers may be slow o refinance when i would be financially favourable and someimes prepay when i is financially unfavourable. Invesors in morgage-backed securiies hold long posiions in non-callable bonds and shor posiions in call (prepaymen) opions. The non-callable bond is effecively a porfolio of zero coupon bonds, and he call opion gives he borrower he righ o prepay he morgage a any ime prior o he mauriy of he loan. Therefore, he value of he MBS is he difference beween he value of he non-callable bond and he value of he call (prepaymen) opion. In he marke place, dealers generally price he morgage by pricing hese wo componens separaely. To evaluae he call opion, he Opion-Adjused Spread mehodology uses opion pricing echniques. When he opion componen is quanified and aken away from he oal yield spread, he yield o mauriy of a non-benchmark bond can be compared o a risk-free of a benchmark securiy 2. Any model used o value a MBS should be able o value he non-callable componen of a morgage and he call opion componen. Ceeris paribus, given ha ineres rae and prepaymen risks have been accouned for, and incorporaed in he heoreical model, one would expec he heoreical price of an MBS o be equal o is marke price. If hese values are no equal, hen marke paricipans demand compensaion for he unmodeled risks. 1 The marke urbulence of hese monhs shows ha i is no he case. However in his sudy we only consider pricing pass-hrough MBS and assume ha hese are guaraneed by he US Treasury. We leave credi-liquidiy risk and he pricing of MBS on he agenda for fuure research. 2 See he opion adjused spread applicaion in his paper. 2
3 The difference in values migh be due o unmodeled risks which are aribuable o he srucure and liquidiy of he bond. One of hese unmodeled risks is he forecas error associaed wih he prepaymen model. For example, he acual prepaymen may be faser or slower han wha he model predics. In his case, he OAS is he marke price for he unmodeled risks. Because here is no agreemen on how o model prepaymens among morgage holders and many differen ineres rae models exiss, opion-adjused spread calculaion suffers from he lack of a sandard erm. The academic lieraure in his area has mainly focused on modelling OAS dynamics such ha he embedded morgage call opion price can be esimaed and consequenly he morgage priced (see for example, Dunn and Spa (1986), Liu and Xu (1998), Schwarz and Torous (1992) amongs ohers). Alhough helping o clarify a number of issues concerning he pricing of MBS, hese models are no used in pracice since in many cases hey are unable o fi he observed marke prices. On he oher hand academics and praciioners have insead oped for economeric models o esimae he parameers of ineres o calibrae reduced form models and price MBS (see for example Chen, 24) 3. Therefore reduced form models seem o be he ideal way of pricing MBS. However, since mos of hese models are proprieary models heir funcional form is no known in he marke. Furhermore as menioned above, hese models may be miss-specified since hey assume a consan opion adjused spread over he lifeime of he morgage. This paper makes wo imporan conribuions o he lieraure on pricing passhrough MBS. Firsly, we propose he so called dynamic opion adjused spread (DOAS) which, as we shall explain in Secion 5, accouns for volailiy shifs in he ineres raes erm srucure. Secondly, we show ha he DOAS can also be used as a hedging ool by invesors in his marke. The paper is organised as follows: we discuss he MBS model used in his sudy in Secion 2, Secion 3 discusses he ineres rae model and is calibraion, Secion 4 presens a numerical example, Secion 5 he dynamic opion adjused spread, Secion 6 presens he empirical resuls finally Secion 7 concludes. 3 Obviously praciioners have been able o develop reduced-form models since, generally, hey also dispose of proprieary daa needed o calibrae he model. 3
4 2. The Morgage Backed Securiy Model Consider he following probabiliy space ( Ω, F, P) and he process p (, D, C, z) adaped o he filraion F. The price process depends on he risk neural vecor of discouned bond prices D() < < T, wih Q being he risk neural probabiliy measure. z is a sae variable ha will be defined shorly and C () is he cash-flow paid by he morgage a. In his model z may represen he opion adjused spread used o mach he heoreical and he marke prices of he MBS. Define he price process for a morgage a ime as he expeced value of he discouned fuure cash-flows: T Q p( ) = E [ C( ) D( )] (1) = The main problem when using equaion (1) o price a MBS is ha he borrower can a each ime consider a prepaymen acion. In he inroducion we have already menioned differen ways of modelling he prepaymen opion when pricing MBSs. In his paper we shall follow Chen (24) and implemen a reduced form model 4. In general, when pricing a MBS one has o generae he morgage cash flows C () using, for example, a reduced form model. Once he cash-flows have been generaed, he value of he morgage can be obained by discouning he simulaed cash flows and summing hem up. E Q Using Mone Carlo o generae m pahs for C ( ) m, we have ha T M 1 ( p) = [ C( ) D( )], lim m C( ) m C, and herefore he simulaed price, m i = m= 1 say * p, will converge o he rue price p. Using Equaion (1) one can also esimae he opion adjused spread z. Suppose ha p is he observed marke price of he morgage. As we do when we wan o obain implied volailiies for plain vanilla opions, we can compue z using a roo finding mehod o solve: 4 Refer o he Appendix for a descripion of he model. 4
5 p (, C, D, z) = p (2) 3. The Term Srucure Model To solve Equaion (2) one has o simulae he erm srucure of ineres raes ou of he mauriy of he morgage. We exend he above model by using a wo facor Heah, Jarrow, and Moron (1992) model (HJM). The HJM model belongs o a class of models, and herefore one needs o specify he iniial forward raes and volailiies o specify he model iself. Below we explain he way we have deal wih his problem. The HJM aemps o consruc a model of he erm srucure of ineres raes ha is consisen wih he observed erm srucure. The sae variable in his model is he forward rae in ime for insananeous borrowing a T >, F (, T ). In differenial form he model can be wrien as: df (, T ) m(, T ) d + (, T ) dw ( ) N = σ for T (3) k = 1 k k Or also in inegral form F N k = 1 ( T ) = F(, T ) + m( v, T ) dv + ( v, T ), σ dw ( ) (4) k k Here F (, T ) is he fixed iniial forward rae curve, ( T ) forward rae drif, ( T ) k, m, is he insananeous σ is he insananeous volailiy process of he forward rae curve, and W k () is a sandard Brownian moion process. The model above is very general and encompasses all he shor rae models such as, for example, he Hull and Whie (1993) model. The drif process is specified as: N T m (, T ) = σ k (, T ) σ (, s) ds (5) k = 1 The hardes problem when using he HJM approach o simulae F(, T ) is ha he model is specified in erms of insananeous forward raes and he laer are no 5
6 observable. To overcome he problem we use he following deerminisic specificaion for he volailiies, and he Musiela parameerizaion: σ k (, T ) = σ k (, T ) Tha means ha our model belongs o he Gaussian class of models and mauriy is specified as ime o mauriy. Therefore if we se τ = T, i follows ha: d F = (6) (, τ ) m(, τ ) d + σ (, τ ) dw ( ) Wih he drif specified as: τ m(, τ ) = σ (, τ ) σ (, s) ds + F(, τ ) (7) τ We use he above parameerisaion when simulaing he forward raes. The spo rae r () used o discoun he cash flows can be deermined from (6) as follows: r( ) lim F(, τ ) τ To use he wo facor model above, one has o specify he iniial forward raes and volailiies. In pracical applicaions of our model we use Bloomberg o obain he forward raes necessary o iniiae he process. Also from Bloomberg one can obain implied volailiies on ineres rae caps necessary for he calibraion of he model. Two volailiies are used in his case. The firs is se fixed for all he mauriies and equal o he implied volailiy of a hiry year ineres rae cap opion. The second refers o implied volailiies of ineres raes caps wih mauriies beween 1 and 3 years. An Euler discreizaion scheme, wih 36 ime seps and 5 simulaions, is used. 6
7 4. Numerical Example In his secion we provide a preliminary numerical example and describe how he OAS is compued. The MBS price can be obained by simulaing he morgage`s cash flows (i.e. C ()) over he lifeime of he morgage using a prepaymen model. The prepaymen model used in his sudy is described in he Appendix. Furhermore one also needs o simulae he erm srucure dynamics over he relevan horizon. We use he model described in Secion 3. Finally equaion (1) gives us he bond`s price. The simulaed MBS price wih he respecive sandard error in bracke is (.63124). The value of he morgage is equal o %. Suppose he size of he underlying morgage pool is $1,,., he price of a morgage-backed securiy issued from he underlying pool will be $1,21,786.. For simpliciy we assume ha he observed marke price is 1% of he par value. Since all he elemens of equaion (1) are known and he marke price of he morgage (or a similar one) can be observed, one can now compue, using equaion (2) and a roo finding mehod, he opion adjused spread. The opion adjused spread in his example is 46 basis poins. 1 MBS Cash Flow 5% Coupon.9 Presen Value of Fiure Cash Flows Time o Mauriy (monhs) Figure 1: MBS Cash Flow Figure 1 shows he simulaed pahs of he monhly cash flows of he morgage. As he bond approaches mauriy he value of he prepaymen opion decreases and consequenly he morgage cash flow becomes less uncerain. 7
8 5. Dynamic Opion Adjused Spread The opion-adjused spread (OAS) can be viewed as a measure of he yield spread. I is consan over he benchmark curve chosen for he valuaion process. The reason why his spread is referred o as opion-adjused is because he cash flows of he underlying securiy are adjused o reflec he embedded opion. Mos marke paricipans find i more convenien o hink abou yield spread han price differences. One issue wih he opion spread is ha i assumes he yield spread o say unchanged over he mauriy of he bond. Therefore, if fuure ineres raes become volaile, he OAS remains unchanged. Clearly in his siuaion a prepaymen model, using an opion adjused spread approach, is miss-specified. Furhermore his implies ha raders will have o compue i and re-calibrae heir models frequenly. This may carry an addiional cos in erms of ime necessary for he re-calibraion. In his secion we propose a modificaion of he OAS ha we call Dynamic Opion Adjused Spread (DOAS). The DOAS allows one o capure prepaymen risks as well as changes in he yield curve. Furhermore, a poenial invesor holding a morgage can also use i as a hedging ool. From an invesor poin of view he DOAS can be viewed as an invesmen 5. The value of his porfolio can be posiive or negaive depending on he spread adjusmen. A bond having a posiive OAS has a posiive porfolio value. On he oher hand, a bond wih a negaive OAS will have a negaive porfolio value. 6 To compue he dynamic opion adjused spread, we use he following procedure. Simulae he bond`s cash-flows, a each, over he lifeime of he morgage. Compue he opion adjused spread (i.e z ) and use i o adjus he cashflows of he bond a each.we have in his way he adjused cash-flows. The difference, a each, beween a plain vanilla bond cash flow ( C ) and he morgage cash flow, is he dynamic opion adjused spread in. The summaion of hese up o is he porfolio value p T Q = E [( C p ) = PV ( C)] (8) 5 We call his invesmen a porfolio value (PV). 6 OAS can be negaive when he morgage coupon is low bu ineres rae volailiy is relaively high. In his case invesors in his marke migh no be very concerned wih he MBS opionaliy, a leas no in he shor run. 8
9 Equaion (8) describes he way he porfolio value is compued. Therefore he porfolio value is jus he difference beween a non-callable bond and a callable bond. I migh be worh noicing ha, by buying a MBS and invesing in he above porfolio, he invesor has indeed creaed a synheic non-callable bond bu wih he difference ha he is also hedging agains ineres rae risk 7. Figure 2 below shows he condiional prepaymen rae (CPR) funcion, he refinancing incenive (RI) and he porfolio value (PV). A he beginning of he morgage here is a posiive spread (i.e. he difference beween he value of he porfolio and he cash flow of he morgage). The difference would compensae he invesor if he opion is exercised by he borrower. The spread is paricularly relevan in he firs one hundred monhs which, in general, corresponds o he ime when he prepaymen risk is higher. As he prepaymen risk becomes less accenuae, he spread decreases. Presen Value of Fuure cash Flows CPI CPR PV RI Time o Mauriy Figure 2: Prepaymen Model 7 In effec he idea ha here is a posiive relaionship beween opion adjused spread and prices of non-callable securiies (in his cases Treasury securiies) was firs repored in Brown (1999). He also suggess, in line wih our model, ha he opion adjused spread is a noisy measure. 9
10 5.1 Numerical Example Using he same approach as explained in Secion 4 and using equaion (8) we can also compue he porfolio value (wih sandard error in bracke). The porfolio value is, in his case, 2.76% (.289) of he par value. The DOAS in our example is 2.76% par value. If we assume ha he pool size of he morgage is $1,,., he porfolio value will be $ 2,7.6. The invesor can buy his opion o hedge ineres rae risk. The nex secion furher clarifies his..12 Porfolio Value 5% Coupon Time o Mauriy (monhs) Figure 3: Porfolio Value 5.2 Numerical Example The invesor can use he porfolio described above as a hedging insrumen agains prepaymen risk in general and changes of he yield curve. The examples below show exacly his. Example1: 5% Coupon rae: Invesor A buys a ime a 3-year morgage-backed securiy wih he price of he MBS being 1% of he face value. The invesor receives Treasury rae plus 46 basis poin (OAS). We assume he pool size o be $1,,. 1
11 Anoher invesor, say, Invesor B buys a ime he same morgage and a DOAS opion. The DOAS opion is 2.76% of he par value. Therefore he value of his invesmen will be 12.7%. Suppose a ime 1 he ineres rae volailiy increases from 13bp o 26bp. Wha is he impac of his increase on he MBS price and he invesor`s porfolio? A ime 1, he price of he morgage drops o % or $ 998,534.. Therefore ha implies a $1,466 loss on he morgage for Invesor A. On he oher hand, he value of he invesmen for he Invesor B, is given by: Pay-off = bond value a ime 1 - bond value a ime + (porfolio value a ime 1 - porfolio value a ime ) Pay-off = ( ) = or $1,337 Example2: 6% coupon rae: We now show anoher example choosing a coupon rae ha is above he iniial ineres rae used in he simulaion. Under his scenario he prepaymen risk is more relevan han in he previous example. Invesor A buys a ime he morgage and receives ineress plus basis poins. Invesor B buys he same morgage bu also invess ino a DOAS opion whose price is 9.98% for a oal of 19.98%. Suppose ha a ime 1 he ineres raes volailiy increases, as before, from 13bp o 26bp. Wha is he impac of his increase on he bond price, and he invesor`s porfolio? A ime 1 he price of he morgage drops o % or $ 999,825.. The loss for he Invesor A is herefore $ As a consequence of he increase in ineres rae volailiy he value of he DOAS opion increases o %. The payoff for he Invesor B is herefore given by: Pay-off = bond value a ime 1 - bond value a ime + (porfolio value a ime 1 - porfolio value a ime ) Pay-off = ( ) =.2 % or $2. 11
12 6. Empirical Resuls In his secion we use he model described in Secions 2, 3 and 5 o price morgage backed securiies wih differen coupon raes 8. Table 1 shows he MBS prices and he opion adjused spread. As expeced he price of he morgage increases as he coupon rae increases 9. This is because of he refinancing incenive for he borrower when he coupon rae is above he marke ineres rae. Table 1: Morgage-Backed Securiy Values and Dynamic Opion Adjused Spreads Coupon Rae % MBS Price SE OAS bp DOAS % SE Noe: SE are sandard errors obained by 1 rials. OAS bp is he opion adjused spread in basis poin. The highes price is reached when he coupon is 7% and i is Such a high premium clearly canno be explained by par plus a number of refinancing poins 1. These high prices are consisen wih wha generally is observed in he marke where morgage prices can easily reach hese levels (see also Longsaff, 24, for a discussion on his issue). Condiionally on he ineres rae level used in our simulaion, we noe ha higher coupon raes will increase he incenive for he borrower o repay he morgage and his clearly will affec he spread ha an evenual invesor would require as a 8 Noe, we are unable o challenge our model agains markes prices. In fac daa needed o calibrae our model are generally proprieary daa and we were no able o obain hese informaion. However as we shall menion, our empirical resuls are in line wih wha he lieraure would predic and our MBS prices can, in cerain cases, reach he values observed in he marke place. 9 Noe ha he iniial rae used for he simulaion is 5%. 1 This is generally he conclusion reached by many praciioners and academic models. 12
13 compensaion for he prepaymen opion. In fac our model suggess a spread over he Treasury curve of more han 4bp when a 7% coupon is considered. We have also compued sandard errors from he simulaion by using 1 independen rials of he model in secion 2. These empirical resuls are in line wih heoreical and empirical sudies in his area (see for example Gabaix e al., 27). A he boom of Table 1, we repor he simulaed dynamic opions adjused values. As we see, given he ineres rae level used in he simulaion, he value of he opion increases as he coupon increases. This is consisen wih a higher prepaymen risk implici wih higher coupons. As menioned an invesor can buy his opion, and pay a higher price for he morgage, o hedge he prepaymen risk and changes in he slope of he yield curve Conclusions Morgage Backed Securiies are asses collaeralised by a pool of morgages and allow invesors o gain higher raes of reurn (wih a relaively lower risk) compared o oher fixed income insrumens. Given he imporance of hese securiies 11 in he las decade here has been a proliferaion of models rying o explain he opimal prepaymen behaviour of he borrower. Their main problem is ha hey canno always explain, wihin a raional analyical framework, how borrowers decide o refinance heir loans. Therefore, some of hese approaches have ried o model he prepaymen opion as an endogenous problem (see Sanon and Wallace, 1998 amongs ohers) bu MBS prices obained by using hese frameworks canno generally mach marke prices. If on he one hand various differen models have been proposed in he lieraure o price MBS, on he oher hand here has been very lile work on hedging and risk managemen of hese securiies. In his paper we have ried o fill his gap. We exend a reduced form model o price MBS and propose a novel approach o managing ineres raes risk. Firsly we poined ou ha mos reduced form models, by relying on he opion adjused spread, migh be miss-specified. Secondly, we suggesed wha we call he dynamic opion adjused spread. We show ha an invesor in he MBS marke, by aking a long posiion on an opion (DOAS), can hedge ou ineres rae risk. The DOAS is simply he difference beween he cash flows of a noncallable bond and a callable bond over he mauriy of he morgage. The concep of 11 And he subsequen urmoil in he financial markes caused by he collapse of hese securiies. 13
14 DOAS can be easily exended o oher fixed income securiies such as callable bonds and a variey of exoic swaps. References Brace, A., D., Gaarek, and M., Musiela, 1997, The Marke Model of Ineres Rae Dynamics, mahemaical Finance, 7, Brown, D. 1999, The Deerminans of Expeced Reurns on Morgage-backed Securiies: An Empirical Analysis of Opion-adjused Spreads, Journal of Fixed Income 9, Chen Jian, 24, Simulaion Based Pricing of Morgage Backed Securiies, Proceeding of he 24 Winer Simulaion Conference. Dunn, Kenneh B., and Cheser S., Spa, 1986, The Effec of Refinancing Coss and Marke Imperfecions on he Opimal Call Sraegy and he Pricing of Deb Consracs, Working Paper, Carnegie-Mellon Universiy. Gabaix X., Arvind K., and O., Vigneron, 27, Limis of Arbirage: Theory and Evidence from he Morgage Backed Securiies Marke, Journal of Finance, Vol. 62, Heah, D., R., A., Jarrow, and A., Moron, 1992, Bond Pricing and he Term Srucure of Ineres Raes: A New Mehodology for Coningen Claims Valuaion, Economerica 6 (1), Hull, J., and A., Whie, 1993, One Facor Ineres rae Models and he Valuaion of Ineres Rae Derivaive Securiies, Journal of Financial and Quaniaive Analysis (28), Liu Jian Guao and Eugene Xu, 1998, Pricing of Morgage Backed Securiies wih Opion Adjused Spread, Managerial Finance, vol. 24, No. 9/1. Longsaff, F., 24, Borrower Credi and he Valuaion of Morgage Backed Securiies, UCLA Anderson School, Working Papers in Finance. Schwars, Eduardo, S., and Waler N., Torous, 1992, Prepaymen, Defaul, and he Valuaion of Morgage Pass-Through Securiies, Journal of Business, 65, Sanon, Richard and N., Wallace, 1998, Morgage Choice: Wha`s he Poin?, Real Esae Economics, 26,
15 Appendix 1 The model assumes ha four facors (i.e. refinancing incenive, burnou, seasoning, and seasonaliy) explain 95% of he variaion in prepaymen raes. These facors are hen combined ino one model o projec prepaymens: CPR = RI AGE MM BM where, RI represens he refinancing incenive; AGE represens he seasoning muliplier; MM represens he monhly muliplier; BM represens he burnou muliplier. Therefore, he prepaymen model is: CPR = RI AGE MM BM where: 1 RI = an 1 AGE = min 1, 3 B 1 BM = B [ ( WAC r ( ) )] MM akes he following values, which sar from January and end in December: (.94,.76,.74,.95,.98,.92,.98, 1.1, 1.18, 1.22, 1.23,.98), r is 1-year Treasury rae, and WAC is he weighed average coupon rae Refinancing Incenive 5% Coupon Refinancing Incenive Time o Mauriy (monhs) Figure 4: Refinancing Incenive (5% coupon) 15
16 .12 Refinancing Incenive 7% Coupon Refinancing Incenive Time o Mauriy (monhs) Figure 5: Refinancing Incenive (7% coupon) Figure 4 and 5 above show he refinancing incenive funcion for 5% and 7% coupon raes. Borrowers have a higher incenive o exercise he prepaymen opion and refinance he morgage when he coupon rae is higher han ineres raes. This is shown in Figure 5. 16
17 17
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