under Prepayment and Default Risks

Transcription

1 Analyzng Yeld, aon and Convexy of Mogage Loans nde epaymen and efal Rsks Sz-Lang Lao * Mng-Shann sa ** Sh-Ln Chang *** * ** *** ofesso, epamen of Money and Bankng, Naonal Chengch Unvesy, ape, and Naonal Unvesy of Kaoshng, Kaohsng, awan, E-mal: laosl@ncc.ed.w. el:(+886) ex 85. Fax: (+886) Asssan ofesso, epamen of Bankng and Fnance, Naonal Ch-Nan Unvesy, l, awan, E- mal: mssa@ncn.ed.w. ocoal sden, epamen of Money and Bankng, Naonal Chengch Unvesy, ape, awan, E- mal: g3553@ncc.ed.w.

2 Absac: In hs acle, we consc a geneal model, whch consdes he boowe s fnancal and non-fnancal emnaon behavo, o deve he closed-fom fomlae of he mogage vale fo analyzng he yeld, daon and convexy of he sky mogage. Snce he sks of pepaymen and defal ae easonably exponded n o model, o fomlae ae moe appopae han adonal mogage fomlae. We also analyze he nflence he pepaymen penaly and paal pepaymen have on he yeld, daon and convexy of a mogage, and povde lendes wh an ppe-bond fo he mogage defal nsance ae. O model povdes pofolo manages a sefl famewok o moe effecvely hedge he mogage holdngs. Fom he esls of sensvy analyses, we fnd ha hghe nees-ae, pepaymen and defal sks wll ncease he mogage yeld and edce he daon and convexy of he mogage. Keywods: yeld, daon, convexy, defal nsance, pepaymen penaly, paal pepaymen

3 . Inodcon Mogage-elaed seces ae pevalen n he fnancal make as hey sasfy nvesos demands fo hgh yelds and sece ced qaly. e o ncean cash flows eslng fom boowes defal and pepaymen behavos, nvesos eqe a pemm o compensae fo poenal losses. eemnng an appopae pemm s mpoan fo pofolo manages and fnancal nemedaes. Fhemoe, hedgng he nees-ae sk of mogages s an exemely dffcl assgnmen. Becase boowe s pepaymen behavo nflences he daon, he pce sensvy of mogages o changes n nees aes becomes hghly ncean. ageng daon and convexy can be one of he mos momenos appoaches o managng pofolos of mogage-elaed seces. he nenon of hs acle s o lze he nensy-fom appoach o pce a mogage and hen nvesgae he nflence of nees-ae, pepaymen and defal sks on he yeld, daon and convexy of a mogage. Moeove, we also dscss how he yeld, daon and convexy of a mogage change nde vaos saons mogages wh a pepaymen penaly, paal pepaymen and defal nsance.mogage yeld speads eflec pemms ha compensae nvesos fo expose o pepaymen and defal sks. Mos mogage make pacones and academc eseaches ae concened wh he mpac of pepaymen and defal sks on mogage yeld, and how o mease he mogage emnaon sk n decdng hedgng saeges fo he pofolos. Leae shows he pemm and he emnaon sk of mogages have ypcally been analyzed sng he conngen-clam appoach, nensy-fom appoach and empcal analyss.

4 Unde he conngen-clam appoach, eseaches se he opon pcng heoy o nvesgae he pemms of pepaymen and defal. hey age ha boowes ae endowed wh he opon o pepay (call) o defal (p) he mogage conac. he vales of pepaymen and defal opons ae calclaed hogh specfyng elevan vaable pocesses sch as nees aes, hose pces and so foh (see e.g., Ka e al., 993; Yang, Bs and Megbolgbe, 998; Ambose and Bme, ; Azevedo-eea, Newon and axson, 3). Some sdes se he opon pcng heoy o deemne pepaymen and defal pemms. Fhemoe, hey nvesgae he effecs of elevan vaables (sch as nees ae volaly, yeld cve slope, ec.) on he vales of pepaymen and defal opons (see, Chlds, O and Rddogh, 997). he nensy-fom appoach evalaes he pobables of pepaymen and defal based on hazad aes nfomaon (see e.g., Schwaz and oos, 989, 993; Qgley and Van Ode, 99; Lambech, eadn and Sachell, 3; Ambose and Sandes, 3). Some eseaches nvesgae mogage sk pemms sng he nensy-fom appoach. hey nse he emnaon pobably no he model and deve he eqlbm mogage ae by calclang he sky mogage yeld. Compang he mogage ae of he sk-fee mogage and he sky mogage can deemne he sk pemm eqed o compensae fo expeced losses (see e.g., Gong and Gyoko, 998). Recen leae sng mogage make daa demonsaes ha ndvdal chaacescs ae elaed o pepaymen and defal sks. Some sdes se empcal analyses o expess elaonshps beween he mogage sk pemm and vaos obsevable vaables specfc o he boowe, sch as loan-o-vale ao, ncome, pay-o-ncome ao and so on (see, Bege and Udell, 99; Chang, Chow and L, ). 3

5 he daon, whch smply eflecs he change n pce fo a gven change n yeld, s wdely appled n nees ae sk managemen. When pepaymen o defal occs, cash flows of he mogage conac ae aleed. hs, he daon wll be dffeen a vaos pce levels as he pepaymen and defal expecaons change. Measng mogage daon s moe complcaed and nceases he dffcly n hedgng mogage-elaed seces. Valaon of mogage daon can be classfed no wo mehods: heoecal and empcal. As fo he heoecal aspec, O (986) povded a fondaon by devng he daon of an adjsable-ae mogage (ARM) nde a dscee me famewok. He evealed ha he ndex sed o adjs he mogage ae ends o be moe mpoan han he adjsmen feqency n deemnng he daon of an ARM. Haensly, Spnge and Walle (993) sed a connos paymen fomla o deve he fxed-ae mogage daon. hey fond ha daon monooncally nceases wh may when he make ae of nees s a o below he copon ae. On he ohe hand, daon nceases wh may, peaks and sbseqenly declnes as he make nees ae exceeds he copon ae. Empcal mease s anohe way o deve daon. I descbes he elaonshp beween changes n mogage pces and changes n make yelds as meased by easy seces. hs mehod ages hee s a make consenss on he mpac of yeld changes efleced n he behavo of make pces (see e.g., eosa, Goodman and Zazzano, 993). When consdeng hedgng mehods fo mogages, daon-machng s he mos 4

6 commonly sed saegy. Howeve, hs saegy does no popely eflec he nees ae sk f hee s a gea change n he nees ae. hs, manages shold make a hedgng analyss by measng he convexy of a mogage. evos leae seldom nvesgaed he sse concenng he convexy of he mogage, b s essenal fo he hedgng analyss of he mogage. We nend o consc a geneal model o deve he fomlae fo he daon and convexy of he sky mogage, and o dscss he nflence of nees aes, pepaymens and defal sks on hem. Mos sdes examne he emnaon sk of he mogage and evalae he mogage by he conngen-clam appoach. Moeove, when nvesgang he mogage nde conngenclam models, emnaons feqenly occ when he opons ae no n he money (.e., opons ae execsed nde sbopmal condons) (see e.g., nn and McConnell, 98a, b; Ka, Keenan and Km, 993). Sbopmal emnaon occs as a esl of gge evens. eng, Qgley and Van Ode (996) fond he mpoance of gge evens, sch as nemploymen and dvoce, n affecng mogage boowe s emnaon behavo. nn and McConnell (98a, b) model sbopmal pepaymen as a jmp pocess and embody no he pocess of he mogage vale o ovecome he sbopmal emnaon nde conngen-clam models. Howeve, s had o deal wh he pepaymen sk as well as he defal and nees-ae sks nde hs model. Wh hs appoach, s also dffcl o denfy he ccal egon of ealy execse and embody he elevan vaables no he models. By applyng he nensy-fom appoach, we no only avod hese poblems, b also can consde opmal and sbopmal emnaons abo boowe s pepaymen and defal behavo o moe accaely mease he vale, he daon and he convexy of he sky 5

7 mogage. In hs sdy, we se he backwad ecson mehod o expess an mplc mogage vale fomla, and hen deve he closed-fom solon of he mogage vale, yeld, daon and convexy nde he connos-me nensy-fom model. he key pon fo accaely pcng he mogage vale and measng he yeld, daon and convexy of he mogage s appopaely modelng he pepaymen and defal sks. In o model, he hazad aes of pepaymen and defal ae assmed o be lnea fncons of nflenal vaables sch as nees aes. Fhemoe, becase gge evens, sch as job loss o dvoce, nflence a boowe s ably o flfll monhly paymen oblgaons and he mogage emnaon ncenve by pepaymen o defal, he lkelhood of a boowe s pepaymen and defal wll change nde hese saons. o easonably expond mogage pepaymen and defal sks, we model he occence of non-fnancal evens as jmp pocesses no he specfcaon of hazad aes of pepaymen and defal. We deve a geneal closed-fom fomla fo sky mogages, whch consdes he boowe s nonfnancal emnaon (.e., sbopmal emnaon) behavo and can negae elevan economc vaables nde hs famewok.. O yeld fomlae, daon and convexy ae moe appopae han adonal fomlae as o fomlae ae moe sensve o changes n pepaymens and defal sks. hs pon s vey mpoan fo sk managemen saeges. Addonally, lendes can se loss avodance saeges o poec hemselves fom pepaymen and defal sks. epaymen penales ae wdely sed o elmnae he pepaymen sk n a vas majoy of 6

8 mogages. Mogage nsance s also sally eqed o edce defal losses (see, Rddogh and hompson, 993; Ambose and Capone, 998; Kelly and Slawson, ). Fale o consde he effecs of hese wo facos cold lead o an neffcen mmnzaon saegy. We ake no accon he effecs of he pepaymen penaly, paal pepaymen and defal nsance on he sk-adjsed yeld, daon and convexy of a sky mogage. hs acle s oganzed as follows: a wo pesens he valaon model, whch denfes he mogage conac componens; defnes he pobables and ecovey aes of pepaymen and defal; and deves he closed-fom solon of he mogage vale. In pa hee, we develop he yeld, daon and convexy of a mogage; condc a sensvy analyss o nvesgae he mpac of nees aes, nensy aes and loss aes of pepaymen and defal, and he nensy ae of non-fnancal emnaon on he yeld, daon and convexy of he mogage. a fo analyzes he nflence of paal pepaymen, pepaymen penaly and nsance on he yeld, daon and convexy of a mogage. he fnal secon s he conclson.. he Model. A Geneal cng Famewok he focs of o nvesgaon s on a fxed-ae mogage (FRM) he mogage make s basc bldng block. We consde a flly amozed mogage, havng an nal mogage pncpal M, wh a fxed copon ae c and me o may of yeas. he paymen Y n each peod can be wen as follows: Y Y c = M. () c e 7

9 he osandng pncpal a me, M, s gven by M c( ) e = M. () c e We le A and V epesen he vale of he skless mogage and he vale of a mogage wh pepaymen and defal sks a me, especvely,. We have A Y exp( ds) d. Fom he sk pemm pon of vew, he vale of a sky = s mogage s less han he skless mogage fo he nveso becase he sky yeld ms be hghe han he skless yeld. hs, V epesens a dsconed popoon fo he A emnaon sk. We assme ha he boowe s endowed wh he opons o pepay, defal o manan he mogage. he opmal saegy can be decded by he opon ha povdes he geaes benef. If M > Y + V, a aonal boowe wll no pepay becase hee s no pof. Howeve, nde he condon of M < Y + V (sch as when he nees ae declnes), boowes wll pay M o edeem he loan, as he benef s geae o pepay he mogage. he lende has a loss fom pepaymen becase he pesen vale of he balance of he mogage s less han he pesen vale of he mogage loan. We defne he pepaymen loss ae a me o be δ, < δ, whch s a andom vaable epesenng he faconal loss of he mogage make vale a pepaymen. he loss of pepaymen ( Y + V M ) s he oppony cos of he lendes. 8

10 he bank eqes ha he vale of he collaeal ms be geae han he vale of he mogage a he nal me of he mogage n ode o avod a hge loss n he case of defal. If H Y + V >, whee H s he make vale of collaeal a me, he aonal boowe wll no defal snce hee s no pof. If he collaeal vale s less han he mogage vale ( H < Y + V, sch as n a depessed make), he boowe wll pof by defalng on he mogage. I also means ha he boowe pays H o by back he mogage conac. Unde hs ccmsance, he bank has a loss fom defal. We denoe he loss ae of defal a me as η, < η, whch s a andom vaable epesenng he faconal loss of he mogage make vale a defal. In o famewok, he pobables of pepaymen and defal exs a each me pon po o he may dae. Le s denoe andom vaables τ and τ as he me of pepaymen and defal dng he peod fom o, especvely. he condonal pobables of pepaymen,, and defal,, can be expessed as follows: ( Δ < τ < τ > Δ) wh nal pobably. (3) ( Δ < τ < τ > Δ) wh nal pobably. (4) We se a dscee me appoxmaon (see, Boade and Glasseman, 997) o calclae he vale of he mogage, and hen deve he lmng fom of he coespondng fomla. Le n =, he valaon s a he pon of, =,, n, whee epesens he paymen Δ dae. he sk neal pcng mehod s sed o evalae he mogage ncldng he sochasc pocesses of nees ae, and defal and pepaymen sks. he mogage vale a 9

11 pon nde he sk-neal mease s V E V ( Y V )], (5) = [ whee V ( ) and E [ ] epesen he pesen vale and expeced vale condonal on he nfomaon of pon nde he sk-neal mease, especvely. Fge shows he composon of he mogage vale n each peod. If he mogage conac s no emnaed befoe he paymen, he lende wll eceve he pomsed amon, Y, and he mogage vale s Y + V. he pobably s. Ohewse, n he even of pepaymen o defal, he cash flow s assmed o be α Y + V ) o β Y + V ), whee α and ( ( β epesen he ecovey aes of pepaymen and defal a pon, especvely. Le α = δ and β = η. Consdeng he emnaon pobably, and he losses of pepaymen and defal n he model, he mogage vale can be wen as V [( ) = E + + )exp( + Δ)( Y+ + V+ + + exp( + Δ) α + ( Y+ + V + ) + + exp( + Δ) β + ( Y+ + V + )], (6) whee + s he annalzed skless nees ae beween o +. In Eqaon (6), he fs em s he expeced vale of a mogage conac, whch does no emnae nl pon +. he second em s he expeced vale of a mogage ha s pepad beween pons and +. he hd em s he expeced vale of a mogage ha defals beween pons and +. Eqaon (5) can also be ewen as = E[( Y + + V+ )[( + + )exp( + Δ + + α + exp( + Δ) V ) β + exp( Δ)]]. (7)

12 = =+ + + Y V V α Y V ) + ( β ( Y V ) Fge : Evolon of he mogage vale + + Replacng α + and β + wh δ and η, we have V = E [( Y+ + V+ ) Q+ ], (8) whee Q + )exp( Δ ) = ( ( δ + )exp( + Δ) ( η + )exp( Δ). Fo small me peods, sng he appoxmaon of exp( c) wh c, gven by c, we can ewe he expesson of Q+ as Q + [ + Δ + + δ + ( + + Δ) + ( Δ)] + η exp( [ + Δ + + δ + ( + Δ) + + η + ( + + Δ)]). (9) + he mogage vale a any me, ( ) Δ, s eqal o s dsconed expeced vale a me Δ, so ha V = E [ Q ( Y V )]. () ( ) Δ ( ) Δ Δ Δ + Δ

13 Accodng o Eqaon (), he mogage vale a me Δ can be expessed as A me Δ, he mogage vale s V V Δ = E Δ[ Q ( Y + V )]. = E [ Q ( Y V )]. () Δ Δ Δ Δ + Δ Snce he FRM s a flly amozed and fxed-ae, we have V = and Y = Y. hs, we sbse V Δ no Eqaon () and oban V Δ = E Δ[ Q ΔY + Q jy ]. j= Δ Ieaed backwads nl he nal me and sng an eaed condon,.e. E [ E [ ]] = E [ ], we oban he nal vale of mogage as follows: n Q j = j= + V = YE [ ( )]. () heefoe, sbsng Eqaon (9) no Eqaon (), we oban he mogage pcng fomla as follows: n V = YE exp( ( jδ + j j ( + jδ) = j= j= [ δ + η ( + Δ) ))]. (3) j= j j j When he me neval Δ appoaches, he dscee me sees s ansfomed no a connos me pocess, hs allowng s o appase he mogage n a connos me famewok. We defne he nensy ae of pepaymen and defal as λ and λ. he condonal pobables of pepaymen and defal ae hen epesened as = λ d and = λ d, especvely. We can oban he mogage pcng fomla as follows:

14 lmv Δ = Y E [exp( ( d + λ δ ( + ) d + λ η ( + ) d))] d. (4) Accodng o o foegong defnon fo he loss aes of pepaymen and defal, he loss aes ae andom vaables. Snce he loss ae can be esmaed by sng make daa, s sally assmed o be an exogenos vaable and eaed as a consan o a deemnsc vaable n edced-fom models whn sdes (see e.g., Jaow and nbll, 995; ffe and Sngleon, 999). Moeove, some empcal evdence shows cases no sgnfcan dffeence on pcng mogages no mae f he loss ae s assmed o be a sochasc vaable o a consan (e.g., Jokvolle and ea, 3). heefoe, he loss aes have been smplfed and eaed as a consan n o model o deve a closed-fom solon of he mogage. Unde he assmpons of δ = δ, η = η, leng λ δ d and λ η d (snce hey ae qe small n he eal wold), he mogage vale can be expessed as follows: + V = Y E [exp( ( ( + δλ ηλ ) d) d. (5) ] Accodng o hs fomla, he vale of he mogage can be expessed as he pesen vale of he pomsed payoff Y dsconed by he sk-adjsed ae + δλ + ηλ. hs s smla o he fomla obaned n ffe and Sngleon (999).. A Closed-Fom Fomla fo Mogage Vales nde Some Specfc Assmpons o oban he closed-fom solon of he mogage valaon pesened n Eqaon (5), we need o specfy he sochasc pocesses of he nees ae and hazad aes n o model. We 3

15 se he exended Vascek (977) model fo he em sce. he exended Vascek nees ae model s a sngle-faco model wh deemnsc volaly and can mach an abay nal fowad-ae cve hogh he specfcaon of he long-n spo nees ae (Vascek (977), and Heah, Jaow and Moon (99)). Unde he sk-neal mease, he em-sce evolon s descbed by he dynamcs of he sho nees aes d = a( ) d + σ dz ( ), (6) whee a s he speed of adjsmen, a posve consan; σ s he volaly of he spo ae, a posve consan; s he long-n spo nees ae, a deemnsc fncon of ; and Z () s a sandad Bownan moon nde a sk neal mease. he spo nees aes follow a mean-eveng pocess nde a sk-neal mease based on Eqaon (6). As shown n Heah e al. (99), o mach an abay nal fowad-ae cve, one can se ( ) = a f (, ) σ ( e f (, ) + ( + a a ( ) ) ). Combnng he above wo eqaons, he evolon of he sho nees ae can be shown as = a( ) σ ( e ) f (, ) + + σ e a a( v) dz ( v), (7) whee f (, ) s he nsananeos fowad ae. Le Θ d, he evolon of he can be wen as Θ Some eseach has shown ha he Vascek model (and hence he exended Vascek model) pefoms well n he pcng of mogage-backed seces (Chen and Yang (995)). 4

16 Θ = b(, ) ( ρ ( v, ) dz ( v) d. (8) f, ) d + d + he followng expessons show he expeced vale and vaance of and Θ (see, Appendx A): σ a a μ E[ Θ ] = f (, ) + ( ( e ) + ( e )), (9) a a a a a Σ = Va( Θ ) = σ ( ( e ) + ( e )). () a a a Mos empcal models show ha he pepaymen and defal nensy aes ae hghly sgnfcan o he change n nees aes (see, Schwaz and oos, 989; Colln-fesne and Hadng, 999). Accodng o he specfcaon n Colln-fesne and Hadng (999), nensy aes of pepaymen and defal ae desgnaed o depend on he pacla vaable n he model ha s elaed o he emnaon sk, sch as he nees ae. Moeove, pepaymen and defal evens occ fo boh fnancal easons (sch as a change n he nees ae) and non-fnancal easons (sch as job change, dvoce and seasonng). evos sdes demonsae ha gge evens (non-fnancal saes), sch as employmen and dvoce, can affec he pobables of a boowe s emnaon decson (e.g., eng, Qgley and Van Ode, 996). In ode o make hs model moe easonable who loss of genealy, we no only assme ha he hazad aes of pepaymen and defal ae lnea fncons of sho nees aes, b also se he osson pocesses o model he andom avals of nonfnancal pepaymen and defal evens. O specfcaon of hazad aes fo pepaymen and defal easonably depcs he emnaon sk becase s eadly ecognzed ha pepaymen and defal occ n boh fnancal and non-fnancal ccmsances. When a 5

17 non-fnancal emnaon even occs, hee s a jmp fo he hazad ae of pepaymen o defal. We allow he jmp sze o be a andom vaable becase gge evens esl n an ncean change n he emnaon nensy ae. he hazad aes of pepaymen and defal ae se as follows: = λ λ dn() λ + + ξ, () and λ + = λ λ ξ dn() +, () whee N () epesens he andom aval of non-fnancal pepaymen and defal, whch s a osson pocess wh nensy ae of ϑ ; ξ and ξ especvely, epesenng he andom jmp magndes of pepaymen and defal ha ae assmed ndependen of he osson pocess N (). As menoned n Colln-fesne and Hadng (999), he escon of he gh hand sde of Eqaons () and () o only a sngle fnancal vaable smplfes he fndng of a closed-fom solon. Howeve, ohe me dependen vaables can be added o he model. he osson pocess N () cons he nmbe of jmps ha occ a o befoe me. If hee s one jmp dng he peod [, + d] hen dn ( ) =, and dn ( ) = epesens no jmp dng hs peod. We assme he dffson componen, dz (), and he jmp componen, dn (), ae ndependen. In addon, he andom vaable ξ l, l =,, epesens he sze of he h jmp, =,, ha s a seqence of dencal dsbons assmed o be ndependen of each ohe. 6

18 Accodng o he above specfcaons of λ and λ, we oban λ d = Θ + J, and (3) λ d = Θ + J, (4) l l l l l whee Θ = λ + λ d and J = ξ, l =,. he J l s defned as a compond N ( ) osson pocess. As menoned n Sheve (4), he jmps n = l J occ a he same me as he jmps n N (). Howeve, he jmp szes n N () ae always n each jmp; he jmp szes n l J ae of andom sze. hen, he mogage vale ha conans elevan vaable and non-fnancal facos can be descbed as follows: V = Y (exp( ( δλ + ηλ ) ) E [exp( (( + δλ + ηλ ) d) ] E [exp( ( δ J + J ))]) d. (5) η Snce Θ s nomally dsbed wh mean μ and vaance Σ, we can se he momen geneang fncon mehod o oban he vale of E [exp( (( + δλ + ηλ ) d) ]. ha s E[exp( (( + δλ + ηλ ) d)] = exp( ( + δλ + ηλ ) μ d + ( + δλ + ηλ ) Σ )). (6) he closed-fom solon of a mogage conac can be obaned when gven he dsbon of ξ l, sch as a nomal dsbon (see, e.g., Meon, 976) and a doble 7

19 exponenal dsbon (see, e.g., Ko and Wang, ). In o model, we assme ha and ξ follow a nomal dsbon wh mean μ ξ and covaance μ ξ, vaances Σ ξ and, Σ ξ. hs, we can oban he followng esls (see, Sheve, 4): ξ Σ ξ and, E[exp( δ J ηj )] = exp( ϑ(exp( ( δμξ + ημξ ) + ( δ Σξ + η Σξ + ηδσ ξ )) )). (7) Sbsng Eqaons (6) and (7) no Eqaon (5), we have Y ) V = [exp( (( δλ + ηλ ) + ( + δλ + ηλ μ ( + δλ + ηλ ) Σ ϑ (exp( ( δμ ξ + ημ ) ξ +, ξ ( δ Σξ + η Σξ + ηδσ )) ))] d. (8) he above eqaon s he closed-fom solon of he mogage. Accodng o hs esl, we calclae he yeld, daon and convexy of he mogage, and dscss he nflence of nees aes, pepaymens and defal sks on hem. 3. Yeld, aon and Convexy Analyses of Rsky Mogages 3. Yeld Mease fo Rsky Mogages he yeld of a fxed-ncome secy s he dscon faco ha eqaes he pesen vale of a secy s cash flows o s nal pce. hs, n ode o calclae he yeld, one needs o know he amon and he mng of he cash flow of he mogage and he pobables of pepaymen and defal. hese vaables lead o dffeen yeld speads and andom changes of he yeld ove me, snce a mogage has dffeen degees of sk ove me. efnng R as he sk-adjsed yeld of a sky mogage eqed by he mogage holdes a me, he mogage vale a me can be expessed as 8

20 V = Y exp( R) d. (9) Accodng o Eqaons (8) and (9), nde he assmpon of no abage oppones (see, Jacoby, 3), he yeld of a sky mogage s R = δλ + ηλ ) + ( + δλ + ηλ ) f (, ) ( + δλ + ηλ )( δλ + ηλ σ A ( ), ϑ (exp( ( δμ ξ + ημ ξ ) + ( δ Σξ + η Σξ + ηδσξ )) ), (3) whee A = a ( ( ) ( a e + e )), A > and a a a A a <. 3 he above fomla gves he lende a bee ndesandng of mogage analyss. Noe ha when he pobables of pepaymen and defal ae zeo (.e., λ = λ = ϑ = ), hee s no sk pemm o he lende. Ohewse, f posve emnaon pobables exs b ecovey aes ae fll (.e. δ = η = ), he sk pemm s also zeo. We condc he sensvy analyses o nvesgae how dffeen vaables (sch as he nees ae, he pobables of pepaymen and defal ncldng fnancal and non-fnancal saes) nflence he yeld of a mogage. he paal devave of he nsananeos yeld wh espec o dffeen vaables can be shown as f (, ) = + δλ +ηλ, (3) σ = ( + δλ + ηλ )( δλ + ηλ ) A, (3) A 3 Accodng o Eqaon (), we have A >. Wh egad o he esl of < a by condcng nmecal analyses., we checked hese esls 9

21 a A = ( + δλ + ηλ )( δλ + ηλ ) σ, (33) a λ = δ, (34) λ = η, (35) λ = δ ( f (, ) ( + δλ + ηλ ) σ ), (36) A λ = η( f (, ) ( + δλ + ηλ ) σ ), (37) A = λ + λ ( f (, ) + σ A) λ ( + δλ + ηλ ) σ δ A ) = λ η,, + ϑ(μ ξ δσξ ησξ ) exp( ( δμ ξ + ημ ξ ) + ( δ Σξ + η Σξ + ηδσξ )), (38) + λ ( f (, ) + σ A) λ ( + δλ + ηλ ) σ A) and ϑ,, + ϑ(μ ξ ησξ δσξ ) exp( ( δμ ξ + ημ ξ ) + ( δ Σξ + η Σξ + ηδσξ )), (39), = exp( ( δμ ξ + ημ ξ ) + ( δ Σξ + η Σξ + ηδσξ )). (4) By obsevng he above paal devave, one canno enely jdge whehe he mpac of he paamee on he mogage yeld s posve o negave. hs, we dscss he decon egadng he nflence of he paamee on he mogage yeld based on some condons. We analyze he nflence of he nees ae on he mogage yeld based on Eqaons (3) o (33). Common knowledge sggess hee s negave elaon beween he mogage vale

22 and he fowad ae. hs, he fowad ae posvely nflences he mogage yeld eslng n = + δλ + ηλ >. Empcal esls fom some pevos leae demonsae f (, ) ha he nflences of nees aes on pepaymen and defal pobables ae negave (see, Schwaz and oos, 993). hs, we can nfe ha λ and λ < <, obanng < f (, ) <. hs esl shows ha f we consde he pepaymen and defal sks n pcng a mogage vale, he nflenal magnde of a fowad ae on yeld wll decease. Accodng o he esls of + δλ + ηλ and δλ + ηλ, we can also nfe ha > < σ > and a <. he vaance (he speed of adjsmen) of a sho nees ae has a posve (negave) effec on he mogage yeld. λ and As fo he nflence of he emnaon sk on mogage yeld, we fnd ha changes n λ affec he mogage yeld n he posve decon based on Eqaons (34) and (35). he mpac of λ and λ on he mogage yeld depends on whehe he vale of f (, ) s lage o less han he vale of ( + δλ + ηλ σ A. Unde he condon of ) μ > Σ, we have f(, ) > ( + δλ + ηλ ) σ A, eslng n λ > and λ >. Obsevng Eqaons (38) and (39) one can noe he affecs loss aes of emnaon sk have on yeld. Becase he hazad aes ae a posve vale n pacce, we have E[ Θ ] = ( λ + λ ( f (, ) + σ A)). heefoe, we nfe λ + λ ( f (, ) + σ A) > >. If we easonably assme ha μ ξ >, δσ ξ + ηδσξ, hen we have >. By he same way, we δ

23 also have >. hs, boh loss aes of pepaymen and defal posvely nflence η mogage yeld. Moeove, he esl of > ϑ can be obaned nde he condon of, δμξ + ημξ > ( δ Σξ + η Σξ + ηδσ ξ ). he above dscssons show ha when he fowad ae goes p, he expeced pesen vale of he amon eceved wll decease eqng a hghe yeld. As long as hee s a gea degee of change n he sho nees ae (.e. σ s lage), he mogage becomes sky. Lendes wll eqe a hghe yeld. Fhemoe, snce an ncease n he vale of a wll lead o sably n he sho nees ae lendes wll eqe a lowe yeld. No mae when he fnancal o non-fnancal nensy aes of pepaymen and defal go p, he mogage becomes ske. Lendes wll hen eqe a hghe yeld o compensae fo he hghe emnaon sk. If he loss aes of pepaymen and defal ncease, he amon eceved when pepaymen o defal occs wll be lowe and he lende wll eqe a hghe yeld. heefoe, we conclde ha hee ae posve elaonshps beween he sk pemm of a sky mogage and he facos ncldng he fowad ae, he vaance of sho nees ae, he fnancal and non-fnancal nensy aes of pepaymen and defal, and pepaymen and defal loss aes. Moeove, a negave elaonshp exss beween he sk pemm of a sky mogage and he speed of adjsmen fo he sho nees ae. hese esls also confom o o economc nsons. 3. aon and Convexy Mease fo Rsky Mogages

24 he man challenge fo pofolo manages s deemnng he daon and convexy of he mogage holdngs. Becase he hedge aos n mogages have o be adjsed fo changes n elevan vaables, pofolo manages also need o nvesgae how he sensvy of daon and convexy. Snce he nflence of a boowe s pepaymen and defal decsons has o be consdeed, qanfyng daons and convexes n mogage seces canno be saghfowadly deemned as n non-callable easy o copoae seces. o begn wh we defne he sk-adjsed daon fo a sky mogage as = V V. (4) he paal devave of mogage vale wh espec o he yeld s: V = Y exp( R) d. (4) he daon can be obaned as follows: = W d, (43) whee W F exp( R) = = V exp( R) d, whch epesens he wegh of cash flows a me. F = Y exp( R). he emnaon nensy ae, loss aes of pepaymen and defal, and ecoveed amon shold all be consdeed n ode o mease he sk-adjsed daon of a sky mogage. Eqaon (43) shows ha any decease n F wll esl n a edced daon of he mogage becase he vale of W becomes smalle as appoaches he may dae. 3

25 hee s a posve elaonshp beween F and daon. Appaenly, he ncease n he nees ae cases F o decease; a decease n he daon of he mogage conac hen follows. he nflence of elevan vaables on he daon of he mogage s moe complcaed as he change n a faco ncs a ade-off beween posve and negave effecs. We heefoe se sensvy analyss o show how he change n daon wll be affeced by dffeen vaables. he paal devaves of he daon wh espec o vaablesφ, whch epesens f (, ), a, Appendx B): σ, λ, λ, λ, λ, δ, η and ϑ, can be developed as follows (see = φ φ G, (44) whee G = W ( )( ) d <. he above paal devave expessons show ha he nflences of vaables φ on he mogage daon ae conas o he nflences of φ on he mogage yeld. Unde he foegong condons, whch ae sed fo he mogage yeld dscsson, we have he esls as follows: When he fowad ae ( f (, ) ), he vaance of sho nees ae ( σ ), and nensy aes of fnancal o non-fnancal pepaymen and defal ( λ, λ, λ, λ and ϑ ) go p, he daon of he mogage becomes smalle. Smlaly, as he loss aes of pepaymen and defal (δ and η ) ncease, he daon of he mogage wll also decease. Fhemoe, an ncease n he speed of adjsmen of sho nees ae (a) wll case he mogage daon o become lage. hese esls nfe ha a hghe sk of nees ae, pepaymen o defal wll lead o smalle mogage daon. hs nfeence s smla o he 4

26 agmens n Chance (99), and eosa e al. (993). he defnon of he convexy fo a sky mogage s: V C =. (45) V V Snce = Y exp( R) d, we can oban he convexy of he mogage as C = W d. hen, we have he followng: C = φ φ G, (46) whee G W ( ) d. < Accodng o Eqaon (46), one can fnd ha he nflences of vaables φ on he mogage convexy ae smla o he nflences of vaables φ on he mogage daon. Neveheless, he affecs of vaables φ on he mogage convexy ae less han he affecs of vaables φ on he mogage daon becase G < G. 4. Applcaon of he Model In he mogage make, paal pepaymen and mogages sbjec o pepaymen penaly clases ae common phenomena. When boowes have spls money, b no enogh o pepay he whole amon of he mogage loan, hey ofen decde o make a paal pepaymen n ode o edce nees expense. Howeve, mos leae ofen only focses 5

27 on fll mogage pepaymen. In hs secon, we analyze how paal pepaymen sk nflences he yeld, daon and convexy of he mogage. Fhemoe, snce he boowe owns he call opon o pepay he mogage, he lende always ses self-poecon saeges o hedge he execse of he opon. epaymen penaly clases ae appended o a vas majoy of mogages n ode o edce pepaymen sk. aal pepaymen and he pepaymen penaly ale boh he emnaon me and he amon of cash flow pomsed o mogage holdes. hs, accaely calclang he yeld, he daon and he convexy of a mogage s moe dffcl n hese saons. In addon, mogage nsance s sally eqed o edce losses n case of defal. Realscally, befoe ssng mogage seces o nvesos and n ode o enhance he ced of he conac, he ognao of a mogage may se nsance o edce he defal sk. Wha s a fa fee o pay nsance companes? How do elevan vaables nflence nsance ae levels? In hs secon, we nvesgae he nflence of all hese facos on he mease of mogage yeld, daon and convexy. 4. aal epaymen We assme he boowe s paal pepaymen amon s ( ϕ ) M a he me, whee ϕ <. hs, he paymen a evey pon and he osandng mogage pncpal a he nex pon wll be ϕ Y and ϕ M, especvely. If he lende edces he vale of he collaeal o he same pecenage as he mogage pncpal, he collaeal vale H changes o ϕ H smlaneosly. he mogage vale changes fom V o ϕ V nde he assmpon of he same pobables of pepaymen and defal. 6

28 Moeove, snce he boowe s ncenve fo decdng o make a paal pepaymen s dffeen fom he ncenve o make a fll pepaymen, hee wll be a change n he pepaymen pobably. hs, we assme ha he nensy ae of pepaymen changes fom λ o ~ λ, and hen λ, λ and ϑ become ~ λ, ~ λ and ~ ϑ. Becase he same popoonal changes n Y, H and V leads o he dencal ncenve fo he boowe s defal decson, (fo he ppose of smplcy) we assme hee s no change n he defal nensy ae. ~ ~ Replacng λ, λ and ~ ϑ no Eqaons (3), (43) and (46), we oban he yeld, he daon and he convexy of a mogage ha ncldes paal pepaymen, R ~, ~ and C ~. Accodng o he pevos dscsson, we know hee ae posve elaons beween he mogage yeld and he nensy aes of fnancal and non-fnancal pepaymen, and also negave elaons beween he daon and he convexy of a mogage and he nensy aes of fnancal and non-fnancal pepaymen. If ~ λ > λ, he yeld of a mogage sbjec o a paal pepaymen sk ( R ~ ) s hghe han he yeld of a mogage who paal pepaymen sk ( R ). he daon and he convexy of a mogage ha ncldes a paal pepaymen ( ~ and C ~ ) ms be smalle han he daon and he convexy of a mogage wh no paal pepaymen ( and C ). Alenavely, f ~ λ < λ, he yeld of a mogage ncldng paal pepaymen ( R ~ ) s lowe han he yeld of a mogage who paal pepaymen ( R ). he daon and he convexy of a mogage ha ncldes a paal pepaymen ( ~ and C ~ ) become lage han he daon and he convexy of a mogage wh no paal pepaymen ( and C ). 7

29 I s woh nong ha f he lende does no ale he collaeal vale o he same pecenage as he mogage pncpal (.e., hee s no change n he collaeal H ), he mogage vale nde he condon of paal pepaymen wll be geae han ϕ V becase of he nceasng defal ecovey ae. Addonally, he defal pobably wll decease becase he ao of he vale of collaeal o he loan wll ncease. Accodng o he analyzed esls n he pevos secon, we fond ha a posve elaonshp exss beween he yeld of a mogage and he defal loss ae, and he nensy aes of fnancal and nonfnancal defals. hee ae negave elaonshps beween he daon and he convexy of a mogage and he defal loss ae, and he nensy aes of fnancal and non-fnancal defals. hs, f ~ λ ~ <, we can nfe R < R λ ~, > and ~ C > C. Howeve, f ~ λ > λ, he nflence of he boowe s paal pepaymen on he yeld, he daon and he convexy of a mogage ae dffcl o esmae becase he ncease n pepaymen sk and he decease n defal sk lead o wo oppose effecs on he yeld, daon and convexy of a mogage. 4. epaymen enaly In geneal, pepaymen penaly clases wll dee mogage pepaymens. hs, pepaymen penales have he effec of edcng pepaymen sk (see, Kelly and Slawson, ). hs mples ha he yeld, daon and convexy ms change de o he changes n pepaymen pobably and pepaymen ecovey ae. o analyze hs poblem, we nodce he fxed penaly no o model. Assme a consan faconal penaly, μ, fo he pecedng k peods of he mogage conac, hen μ 8

30 faconal penaly fo he emanng peods of mogage conac, whee μ > μ. If k s eqal o he may dae, hen he fxed penaly becomes he pemanen penaly whch has a consan pecenage penaly fo he ene lfe of mogage. If he mogage s pepad a pon j, when j k, μ M j amon s chaged. Alenavely, μ M j amon s pad f he boowe pepays a pon j, when k < j <. hen one can ge he pepaymen ecovey ae ˆα and ˆα, whch epesen he ecovey ae a he pecedng k peods and he ˆ μ ˆ μ emanng peod especvely, whee α = α ( + ) and α = α( + ). he pepaymen penaly nceases he boowe s cos of elmnang he mogage lably and shold affec he ncenve fo he boowe s pepaymen decson. We assme ha pepaymen pobably changes fom λ o λˆ, and hen λ, λ and ϑ become ˆλ, ˆλ and ϑˆ. Becase he pepaymen penaly edces he pepaymen sk, mples ha λˆ < λ. Replacng ˆα, ˆα, ˆλ, ˆλ and ϑˆ no Eqaons (8), (44) and (47), we oban he yeld, he daon and he convexy of a mogage ha ncldes pepaymen penaly, ˆR, ˆ and Ĉ fo j k <, and ˆR, ˆ and Ĉ fo k < j <. Accodng o he dscsson n he pevos secon, we know ha hee s a posve elaon beween he mogage yeld and he nensy aes of fnancal and non-fnancal pepaymens and he loss ae of pepaymen. hese condons mply ha ehe ˆR o ˆR s lowe han he ognal yeld ( R ). ˆR s less han ˆR becase μ > μ wll lead o he geae decease n he loss ae and he pobably of pepaymen fo j k < han fo k < j <. 9

31 Snce he decease n he nensy aes of fnancal and non-fnancal pepaymens and he loss ae of pepaymen wll case he nceases n he daon and he convexy of a mogage, he daon and he convexy of a mogage wh penaly ( ˆ and Ĉ ) ms be lage han he daon of a mogage wh no penaly ( and C ). ˆ and Ĉ ae less han ˆ and Ĉ becase μ > μ. hese esls show ha he sk pemm deceases and he daon and he convexy of a mogage ae lage de o he pepaymen penaly deceasng he loss ae and he pobably of pepaymen. 4.3 Insance Rae Snce he md-99s, a gowng nmbe of cones have been neesed n mogage defal nsance (see, Blood, ). Mogage defal nsance can poec lendes and nvesos agans losses when boowes defal and he collaeals ae nsffcen o flly pay off he mogage oblgaon. Nmeos make pacones and academc eseaches focs he sdes on how o decde an appopae nsance ae. In o model, he nsance pemm s calclaed as an p-fon fee defned as he dffeence beween he vale of he mogage ( V ) and he ecoveed amon ( β j j M j ) n he even a defal occs a pon j. Becase he defal sk s avodable hogh nsance, he ppe-bond nsance ae s he spead beween he yeld of he defalable mogage and he yeld of he mogage wh no defal sk. Accodng o he deved mogage yeld (Eqaon (3)), he ppe-bond nsance ae, I, s descbed as follows: 4 4 Eqaon (47) can be obaned by calclang he vale of he yeld who defal sk mns he vale of he yeld wh defal sk. 3

32 I = ηλ +ηλ f (, ) ( + δλ +ηλ ηλ σ A ) ξ, + ϑ exp( δμ ξ + δ Σ ) ( exp( ημ ξ + ( η Σξ + ηδσξ )). (47) Snce he defal sk s no enely elmnaed, mples he fa nsance ae fo paal mogage defal nsance s less han I. Alenavely, he fa nsance ae fo fll mogage defal nsance s I as he defal-fee mogage shold have a zeo defal sk pemm. hs, he level of nsance ae pad depends on he degee of defal sk elmnaed by he mogage defal nsance. Moeove, n addon o he defal sk, pepaymen sk also nflences he nsance ae, I. hs s becase we consde he coelaon beween he sks of pepaymen and defal n o model. evos leae does no ake no accon he pepaymen sk o nvesgae he mogage defal nsance. I mples ha he mogage defal nsance ha we povde shold be moe appopae de o he consdeaon fo he elaonshp beween pepaymen and defal sks. Insance nsons can decde an appopae nsance ae hogh he ae ha we povde. 5. Conclson Measng yelds, daons and convexes of mogage conacs s qe complex de o he boowes pepaymen and defal behavos ha case nceany fo boh he emnaon me and pomsed cash flows. hs cases nceany n he changes n he yeld, daon and convexy of a mogage. heefoe, he mease of a mogage yeld, daon and convexy shold appopaely eflec pepaymen and defal sks n addon o he nees-ae sk. hs pape povdes a famewok fo nvesgang he effecs of vaos facos on he yeld, daon and convexy of a mogage. hese facos do no only nclde 3

33 he em sce of he nees ae, fnancal emnaon pobably, and pepaymen and defal loss aes, b also nclde non-fnancal emnaon pobably, pepaymen penaly and paal pepaymen sk. hs, o fomlae fo yeld, daon and convexy, whch pecsely accons fo pepaymen and defal sks, may help eglaos and fnancal nsons edce he solvency sk. Hedgng a mogage wh emnaon sk s an exemely dffcl endeavo. heefoe, he mease of a mogage yeld, daon and convexy shold appopaely eflec pepaymen and defal sks, n addon o he nees-ae sk. Snce o fomlae fo daon and he convexy moe sensvely eflec he mpac of pepaymen and defal sks, s moe appopae n he managemen of nees ae sk han adonal daon and convexy. heefoe, o model can povde an appopae famewok fo pofolo manages o moe effecvely hedge he mogage holdngs. Accodng o he sensvy analyses, we fnd ha hee ae posve elaonshps beween he yeld and he nensy aes of fnancal and non-fnancal emnaon, he loss aes of pepaymen and defal, he fowad ae and he vaance of he sho nees ae. Addonally, hee s a negave elaonshp beween he yeld and he speed of adjsmen of he sho nees ae. hese esls confm ha seces wh hghe emnaon and nees-ae sks have a hghe sk pemm. Moeove, he nflence of all hese facos on he daon conass he nflence hey have on yeld. We can nfe ha hghe nees-ae, pepaymen and defal sks wll edce he mogage daon. hs asseon s also conssen wh Chance (99) and eosa e al. (993). Fhemoe, o compae he 3

34 analyzed esls fo mogage daon, he nflence of hese facos on he mogage convexy have he same esls. he facs ha pepaymen penales and boowes paal pepaymen behavo sgnfcanly affec he yeld, daon and convexy meases of a mogage ae well known n he mogage make. Howeve, few sdes have nvesgaed mogage yeld and daon wh hese phenomena. In hs pape, we analyze he mpac of pepaymen penaly and paal pepaymen on he yeld and daon. O model shows ha yeld deceases and daon nceases when a mogage has a pepaymen penaly. Fhemoe, he nflence of a boowe s paal pepaymen behavo on he yeld and daon of he mogage ae ambgos de o he posve and negave effecs of paal pepaymen on sks of pepaymen and defal. Fnally, we analyze he poblem concenng mogage defal nsance and povde a efeence ppe-bond fo lendes and nvesos. hey can mease an appopae nsance ae by he mogage defal nsance we povde becase consdes he elaonshp beween pepaymen and defal sks, lendes and nvesos. 33

35 Refeences Ambose, B.W. and Bme, J., R.J.,. Embedded Opons n he Mogage Conac. Jonal of Real Esae Fnance and Economcs (), 95-. Ambose, B.W., Capone, C.A., 998. Modelng he Condonal obably of Foeclose n he Conex of Sngle-Famly Mogage efal Resolons. Real Esae Economcs. 6(3), Ambose, B.W., Sandes, A.B., 3. Commecal Mogage-Backed Seces: epaymen and efal. Jonal of Real Esae Fnance and Economcs. 6(/3), Azevedo-eea, J.A., Newon,.., axson,.a., 3. Fxed-Rae Endowmen Mogage and Mogage Indemny Valaon. Jonal of Real Esae Fnance and Economcs. 6(/3), 97-. Bege, A.N., Udell, G.F., 99. Collaeal, Loan Qaly, and Bank Rsk. Jonal of Moneay Economcs. 5, -4. Blood, R.,. Mogage efal Insance: Ced Enhancemen fo Homeoweshp. Hosng Fnance Inenaonal. 6(), Boade, M.. Glasseman,., 997. cng Amecan-syle seces sng smlaon. Jonal of Economc ynamcs and Conol., Chance,.M., 99. efal Rsk and he aon of Zeo Copon Bonds. Jonal of Fnance. 45(), Chen, R. R., Yang,.., 995. he Relevance of Inees Rae ocess n cng Mogage- Backed seces. Jonal of Hosng Reseach. 6(), Chang, R.C., Chow, Y.F., L, M.,. Resdenal Mogage Lendng and Boowe Rsk: he Relaonshp Beween Mogage Speads and Indvdal Chaacescs. Jonal of Real Esae Fnance and Economcs. 5(), 5-3. Chlds,.., O, S.H., Rddogh,.J., 997. Bas n an Empcal Appoach o eemnng Bond and Mogage Rsk emms. Jonal of Real Esae Fnance and Economcs. 4(3), Colln-fesne., Hadng, J.., 999, A Closed Fomla fo Valng Mogages, Jonal of Real Esae Fnance and Economcs, 9:,

36 eng, Y., J., Qgley, M., Van Ode, R., 996. Mogage efal and Low ownpaymen Loans: he Coss of blc Sbsdy. Regonal Scence and Uban Economcs. 6, eosa,., Goodman, L., Zazzano, M., 993. aon Esmaes on Mogage-Backed Seces. Jonal of ofolo Managemen. 9 (), ffe,., Sngleon, K.J., 999. Modelng em Sces of efalable Bonds. he Revew of Fnancal Sdes. (4), nn, K. B., McConnell, J. J., 98a. A Compason of Alenave Models fo cng GNMA Mogage-Backed Seces. Jonal of Fnance. 36(), nn, K. B., McConnell, J. J., 98b. Valaon of GNMA Mogage-Backed Seces. Jonal of Fnance. 36(3), Gong, F.X., Gyoko J., 998. Evalang he Coss of Inceased Lendng n Low and Negave Gowh Local Hosng Makes. Real Esae Economcs. 6(), Haensly,.J., Spnge,.M., Walle, N.G., 993. aon and he ce Behavo of Fxed- Rae Level aymen Mogage: An Analycal Invesgaon. Jonal of Real Esae Fnance and Economcs. 6, Heah,., Jaow, R., Meon, A., 99. Bond cng and he em Sce of Inees Raes: A New Mehodology fo Conngen Clams Valaon. Economeca. 6, Jacoby, G., 3. A aon fo efalable Bonds. Jonal of Fnancal Reseach. 6(),9-46. Jaow, R., nbll, S., 995. cng Opons on Fnancal Seces Sbjec o efal Rsk. Jonal of Fnance. 5, Jokvolle, E., ea, S., 3. A Model fo Esmang Recovey Raes and Collaeal Hacs fo Bank Loans. Eopean Fnancal Managemen. 9(3), Ka, J.B., Keenan,.C., Mlle Ⅲ, W. J., Eppeson, J. F., 993. Opon heoy and Floang-Rae Seces wh a Compason of Adjsable- and Fxed-Rae Mogages. Jonal of Bsness. 66, Ka, J.B., Keenan,.C., Km, ansacon Coss, Sbopmal emnaon and 35

37 efal obably. Jonal of he Amecan Real Esae and Uban Economcs Assocaon :3, Kelly, A., Slawson, J., V.C.,. me-vayng Mogage epaymen enales. Jonal of Real Esae Fnance and Economcs. 3(), Ko, S.G., Wang, H.,, Opon cng Unde a oble Exponenal Jmp ffson Model, Wokng ape. Lambech, B.M., eadn, W.R.M., Sachell, S., 3. Mogage efal and ossesson Unde Recose: A Compeng Hazads Appoach. Jonal of Money, Ced and Bankng. 35(3), Meon, R., 974. On he cng of Copoae eb: he Rsk Sce of Inees Raes. Jonal of Fnance. 9, O J., R.A., 986. he aon of an Adjsable-Rae Mogage and he Impac of he Index. Jonal of Fnance. 4(4), Qgley, J.M., Van Ode, R., 99. Effcency n he Mogage Make: he Boowe s especve. AREUEA Jonal. 8(3), Rddogh,.J., hompson, H.E., 993. Commecal Mogage cng wh Unobsevable Boowe efal Coss. AREUEA Jonal. (3), Schwaz, E.S., oos, W.N., 989. epaymen and he Valaon of Mogage-Backed Seces. Jonal of Fnance. 44, Schwaz, E.S., oos, W. N., 993. Mogage epaymen and efal ecsons: A osson Regesson Appoach. AREUEA Jonal. (4), Sheve, S.E., 4, Sochasc Calcls fo Fnance II Connos-me Models, Spnge essed. Vascek, O., 977. An Eqlbm Chaacezaon of he em Sce. Jonal of Fnancal Economcs. 5, Yang,.., Bs, H., Megbolgbe, I.F., 998. An Analyss of he Ex Ane obables of Mogage epaymen and efal. Real Esae Economcs. 6(4),

38 Appendx A In hs appendx, we povde he devaon of Eqaon (9). Fom Eqaon (7), one can oban he followng esls (see, Heah e al., 99): and = a σ ( e ) f (, ) + + σ e a a( v) dz ( v). b( v, ) ( b( v, ) dz ( v), + d = f, ) d dv + whee gves: a( s) e b( s, ) = σ. o assme he nal yeld cve s fla, a dec compaon a μ = E[ ( ) d] = f (, ) d + b(, ) d σ a a = f (, ) + ( ( e ) + ( e )). a a a By Io s Lemma, we oban E [ dz ( ) dz ( )] d and E [ dz ( ) dz ( )] = hen, Σ =. = Va ( ) d) = E b( v, ) dz ( v) b( v, ) dz ( v) ] ( [ = a a b( v, ) dv = σ ( ( e ) + ( e )). a a a 37

39 Appendx B Fom Eqaon (44), we have = φ W d, (B) φ whee φ epesens f (, ), a, σ, λ, λ, λ, W wh espec o vaance φ can be ge as follows: Snce W F F V = φ φ V V φ = ( F = ( W F φ V V W φ λ, δ, η and ϑ. he paal devave of sw s sf s ds) φ ds). φ ae no vayng wh me (see Eqaons (3) o (4)), we have: φ W = W ( swsds ) φ φ = W ( ). (B) φ Sbsng Eqaon (B) no Eqaon (B), we have whee G W ( d. ) = φ φ φ W ( ) d = G, (B3) Moeove, we have: G = W ( d ) = < W ( ) d + W ( ) d + = W ( ) d W ( ) d W ( ) d 38

40 he neqaly = W d W d =. (B4) W ( ) d < W ( ) d holds becase W ( ) d and W ( ) d ae posve and < n hs egon. Moeove, anohe neqaly W ( ) d W ( ) d holds snce becase W ( ) d and < W ( ) d ae negave and > n hs egon. heefoe, he fs neqaly n Eqaon (B4) holds. Addonally, he las eqaly holds becase W d = and W d =. heefoe, we have he esl of G <. Wh egad o he analyses of convexy, by he same way, we have C = φ Moeove, we have G W d = W ( ) d G φ φ. (B5) φ W ( d < W ( ) d =. ) heefoe, we have he esls of Eqaons (44) and (46). 39