Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao, Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics 27 January 2008 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). The laer 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: C23, G34 We wish o hank john Crosby for consrucive commen. The usual disclaimer applies. Corresponding auhor: Mario Cerrao, Universiy of Glasgow, Deparmen of Economics, m.cerrao@lbss.gla.ac.uk 1
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. The reason of his enormous growh is probably due o he higher reurn and lower risk profile compared o oher fixed income securiies. However, alhough he marke is growing very quickly, neverheless here are sill quie a few issues concerning he pricing and 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, however, for which he issuers refinancing sraegies are assumed o be close o opimal, morgage borrowers may be slow o refinance when i would financially favourable and someimes prepay when i is financially unfavourable. Invesors in morgage-backed securiies hold long posiions in noncallable bonds and shor posiions in call (prepaymen) opions. The noncallable bond is a 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 noncallable 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 1. Any model employed o value a MBS should be able o value he noncallable 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. 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 2
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). However, hese models alhough helping o clarify a number of issues concerning he pricing of MBS, are no used in pracice. On he oher hand, many researchers working in financial insiuions, and amongs hem op academics, have insead oped for economeric models o esimae he parameers of ineres o calibrae reduced form models and price MBS (see for example Chen (2004)). Therefore, from a praciioner s poin of view 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. This 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. 1 See our applicaion of opion adjused spread in his paper. 3
2. The Morgage Backed Securiy Model Consider he following probabiliy space ( Ω, F, P), and suppose he process ψ (, D, C, z), represening he price process of a morgage backed securiy, is adaped o he filraion F. The price process depends on he risk neural vecor of discoun bond price D i 0 < i < N, wih Q being he risk neural probabiliy measure, and he sae variable z. Also denoe wih C he cash-flow paid by he morgage a. Define he price process for a morgage a imet when z = 0 as he expeced value of he discouned fuure cash-flows: T Q i i = 0 E(ψ ) = E [ C D ] (1) i The main problem when deermining he price of his securiy is ha i is no simply deermined by discouningc, since he borrower can a each ime consider a i prepaymen acion. In he inroducion we have already menioned differen ways of modelling he prepaymen opion when pricing MBS. In his paper we shall follow Chen (2004) and implemen a reduced form model 2. In general, when pricing MBS one has o, firs, generae he morgage cash flows C( D, z) using, for example, a reduced form model. Once cash-flows have been generaed, he value of he morgage can be obained by discouning he simulaed cash flows beween1 < i < N : ~ ~ T ~ Q i i = 0 E(ψ ) = E [ C D ] (2) i If we use Mone Carlo o generae m pahs for m, we have ha ~ C E Q ~ 1 T M ~ ~ (ψ ) = [ C i Di ], limm C i m C m i = 0 m= 1 and he soluion of (2) gives he value of he morgage. Using Equaion (2) one can also esimae he opion adjused 2 Refer o he Appendix for a descripion of he model. 4
spread z in he following way. Define wih P he observed marke price of he morgage. We can compue z using a roo finding mehod o solve (3) below: ~ ~ ψ (, C, D, z) = Ρ (3) 0 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 is 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 model 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 a laer imet, F (, T ). In differenial form he model can be wrien as: df (, T ) m(, T ) d + (, T ) dw ( ) N = σ for 0 T (4) k = 1 k k Or also in inegral form Here ( 0, T ) F ( T ) = F( 0, T ) + m( v, T ) dv + ( v, T ) 0 N, σ (5) 0 k = 1 k dwk ( v) F is he fixed iniial forward rae curve, ( T ) m, is he insananeous forward rae drif, σ (,T ) is he insananeous volailiy process of he forward rae curve, and W 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. 5
The drif process is specified as: N T m (, T ) = k (, T ) σ (, s) ds k = 1 σ (6) The hardes problem when using he HJM model o simulae F(, T ) is ha he model is specified in erms of insananeous forward raes and he laer are no observable in he marke. 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 (, τ ) m(, τ ) d + σ (, τ ) dw ( ) = (7) Wih he drif specified as: τ m(, τ ) = σ (, τ ) σ (, s) ds + F(, τ ) (8) τ 0 We use he above parameerisaion when simulaing he forward raes. The spo rae z () used o discoun he cash flows can be deermined from (7) as follows: z( ) lim df (, τ ) τ 6
To use he wo facor model above, one has o specify he iniial forward raes and volailiies. In his applicaion we have used Bloomberg o obain he forward raes necessary o iniiae he process. Also, we have used Bloomberg o obain implied volailiies on ineres rae caps necessary for he calibraion of our model. Two volailiies are used. 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 cap wih mauriies 1 o 30 years. An Euler discreizaion scheme, wih 360 ime seps and 5000 simulaions, is used. 4. Numerical Example Table 1 shows a sample of simulaed prices for he morgage backed securiy using he model described above wih heir sandard errors. Table 1: Morgage Backed Securiy Valuaion wih 5% coupon rae ~ ψ 0 % 102.1375 102.1236 102.1786 102.1993 102.1504 102.1547 SE 0.070569 0.069554 0.063124 0.06940 0.073031 0.063799 Consider, for example, he morgage wih value equal o 102.1786%. Suppose he size of he underlying morgage pool is $1,000,000.00, he price of a morgage-backed securiy issued from he underlying pool will be $1,021,786.00. The observed marke price is assumed o be 100% of he par value. One can herefore compue, using a roo finding mehod, he opion adjused spread ha in his example is 46 basis poins. 7
1 MBS Cash Flow 5% Coupon 0.9 Presen Value of Fiure Cash Flows 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 Time o Mauriy (monhs) Figure 1. Figure 1 above shows 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. 5. Dynamic Opion Adjused Spread The opion-adjused spread (OAS) above 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. This implies ha raders will have o compue i and recalibrae heir models frequenly. 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 risk as well as changes in he yield curve. A poenial invesor holding a morgage can use he DOAS as a hedging ool. 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 8
prepaymen risk is higher. As he prepaymen risk becomes less accenuae, he spread decreases. Presen Value of Fuure cash Flows 0.14 0.12 0.1 0.08 0.06 0.04 0.02 CPI CPR PV RI 0 0 50 100 150 200 250 300 350 400 Time o Mauriy Figure 2. From an invesor poin of view he DOAS can be viewed as an invesmen 3. 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. 4 To compue he dynamic opion adjused spread, we used he following procedure. Use simulaions o simulae he cash-flows, a each, over he lifeime of he morgage. Compue he opion adjused spread (i.e z ) a 0 and use i o adjus he cash-flows of he bond a each.you have compued he adjused cash-flows. The difference, a each, beween he plain vanilla bond cash flows and he morgage cash flows, is he dynamic opion adjused spread in. The summaion of hese up o 0 is he porfolio value n ~ Q 0 = E [( ψ ) E( ψ i= 0 PV )] i (9) Equaion (9) describes he way we compued he porfolio value. Therefore he porfolio value is jus he difference beween a non-callable bond and a callable bond. 3 We call his invesmen a porfolio value (PV). 4 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. 9
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. 5.1 Numerical Example Table 2 shows esimaed porfolio values using Mone Carlo simulaions. We also repor sandard errors. Table 2: Porfolio Values (5% Coupon) and Their Sandard Errors. PV % 2.071118 2.07150 2.07006 2.06917 2.07241 2.07217 SE 0.00344 0.003308 0.00289 0.00324 0.003322 0.002969 The DOAS we use in our example is 2.07006% par value. If we assume ha he pool size of he morgage is $1,000,000.00, he porfolio value will be $ 20,700.60. The invesor can buy his opion o hedge ineres rae risk. In he nex secion, we show his wih an example. 0.012 Porfolio Value 5% Coupon 0.01 0.008 0.006 0.004 0.002 0 0 50 100 150 200 250 300 350 400 Time o Mauriy (monhs) Figure 3. 10
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 0 a 30-year morgage-backed securiy wih he price of he MBS being 100% of he face value. The invesor receives Treasury rae plus 46 basis poin (OAS). We assume he pool size o be $1,000,000. Anoher invesor, say Invesor B, buys a ime 0 he same morgage and also buys a DOAS opion. The DOAS opion is 2.07006% of he par value. Therefore he value of his invesmen will be 102.07%. 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 99.8534 % or $ 998,534.00. 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 0 + (porfolio value a ime 1 - porfolio value a ime 0 ) Pay-off = 99.8534 100 + (2.08289 2.07006) = - 0.1337 or $1,337 Example2: 6% coupon rae: We repor below anoher example choosing a coupon rae ha is above he iniial ineres rae used in he simulaion. Invesor A buys a ime 0 he morgage and receives ineress plus 227.70 basis poins. 5 Invesor B buys he same morgage bu also invess ino a DOAS opion whose price is 9.9080% for a oal of 109.908%. 5 OAS has been calculaed as in (5). 11
Suppose ha a ime 1 he ineres raes volailiy increases, as before, from 0.00132 o 0.00264. 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 99.9825 % or $ 999,825.00. The loss for he Invesor A is herefore $ 175.00. As a consequence of he increase in ineres rae volailiy he value of he DOAS opion increases o 9.9275%. The payoff for he Invesor B is herefore given by: Pay-off = bond value a ime 1 - bond value a ime 0 + (porfolio value a ime 1 - porfolio value a ime 0 ) Pay-off = 99.9825 100 + (9.9275 9.9080) = 0.0020 % or $20.00 6. Empirical Resuls Table 3 shows MBS prices wih differen coupons and also he opion adjused spread. We noe ha he price of he morgage increases as he coupon rae increases. Table 3: Morgage-Backed Securiy Values and Dynamic Opion Adjused Spreads Coupon Rae % 5.00 5.50 6.00 6.50 7.00 MBS Price 102.17 106.28 110.26 114.21 117.58 SE 0.06312 0.06204 0.07211 0.06606 0.05265 OAS bp 46.18 135.66 227.70 321.05 412.33 DOAS % 2.0700 6.0034 9.9080 13.7460 17.0849 SE 0.00289 0.00837 0.01223 0.01690 0.02427 The highes price is reached when he coupon is 7% and i is 117.58. Such a high premium clearly canno be explained jus by par plus a number of refinancing poins. These high prices are consisen wih wha generally is observed in he marke where morgage prices can easily reach hese levels (see also Longsaff, 2004, for a discussion on his issue). 12
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 compensaion for he prepaymen opion. In fac our model suggess a spread on he Treasury curve of more han 400bp when a 7% coupon is considered. We have also compued sandard errors from he simulaion by using 100 independen rials of he model in secion 2. A he boom of Table 3, 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 we showed above an invesor migh decide o buy his opion, and pay a higher price for he morgage, if he wishes o be hedge agains prepaymen risk and changes in he slope of he yield curve. 13
Conclusions The Morgage Backed Securiies marke is he larges fixed income marke in he Unied Saes. These asses are collaeralised by a pool of morgages and allow invesors o gain higher ineres raes wih a relaively lower risk compared o oher fixed income insrumens. Given he imporance of hese securiies, in he las decade, here has been a proliferaion of models rying o explain he opimal prepaymen behaviour of he borrower. The main problem wih mos of hese models is ha hey canno always explain, wihin a raional model, how borrowers decide o refinance heir loans. Therefore, some of hese models have ried o model he prepaymen acion as an endogenous problem (see Sanon and Wallace, 1998 amongs he ohers), bu MBS prices obained by using hese models canno generally mach marke prices. If on one hand various differen models have been proposed in he lieraure o price MBS. On he oher hand here has been very lile done in erms of he 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. We show ha an invesor in his 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 non-callable bond and a callable bond over he mauriy of he morgage. The concep of DOAS can be easily exended o oher fixed income securiies such as callable bonds and a variey of exoic swaps. 14
References Brace, A., D., Gaarek, and M., Musiela, 1997, The Marke Model of Ineres Rae Dynamics, mahemaical Finance, 7, 127-155. Chen Jian, 2004, Simulaion Based Pricing of Morgage Backed Securiies, Proceeding of he 2004 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. 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 60 (1), 77-105. 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), 235-254. Liu Jian Guao and Eugene Xu, 1998, Pricing of Morgage Backed Securiies wih Opion Adjused Spread, Managerial Finance, vol. 24, No. 9/10. Longsaff, F., 2004, 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, 221-239. Sanon, Richard and N., Wallace, 1998, Morgage Choice: Wha`s he Poin?, Real Esae Economics, 26, 173-205. 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 = 0.28 + 0.14 an 10 AGE = min 1, 30 B 1 BM = 0.3 + 0.7 B0 [ 8.571+ 430( WAC r ( ) )] MM akes he following values, which sar from January and end in December: (0.94, 0.76, 0.74, 0.95, 0.98, 0.92, 0.98, 1.1, 1.18, 1.22, 1.23, 0.98), r is 10-year Treasury rae, and WAC is he weighed average coupon rae. 10 0.0735 Refinancing Incenive 5% Coupon 0.073 0.0725 Refinancing Incenive 0.072 0.0715 0.071 0.0705 0.07 0.0695 0.069 0.0685 0 50 100 150 200 250 300 350 400 Time o Mauriy (monhs) Figure 3. 16
0.12 Refinancing Incenive 7% Coupon 0.115 0.11 Refinancing Incenive 0.105 0.1 0.095 0.09 0.085 Figure 4. 0.08 0 50 100 150 200 250 300 350 400 Time o Mauriy (monhs) Figure 3 and 4 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 4. 17
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