Modeling Defaults in Residential Mortgage Backed Securities An

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Modeling Defaults in Residential Mortgage Backed Securities An Powered By Docstoc
					     
            Modeling Defaults in Residential
            Mortgage Backed Securities: An
              Intensity Based Approach  
                                                              
                                                            by

                                               Toma Donchev
                                                        August 2009
     
     
     
     
                                                        Supervision:




            P r o f. d r . A a d va n d e r V a a r t                    Jurgen Pe ters

            Dr. Federic o Camia                                          Barbara Bakker




    Submitted to the Vr ije Universiteit Ams ter da m as a req uiremen t fo r the success ful
                                             g rad ua tion in th e de gre e :

        Mas ter o f Sc ienc e in B us in es s M a th e ma t ics a nd In f or ma t ics – s p eci al iz a t io n
                                           F i nanc ia l Risk Mana geme n t




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          Modeling Defaults in Residential Mortgage


    Backed Securities: An Intensity Based Approach




                                                 Abstract 

    I n Ma y 2 00 8 the ou ts ta nd ing is s u ance o f E ur ope an A s s e t Bac k e d S ec ur it i es

was mo re th an €11 50 b illion a nd Res ide n tia l Mor tg age Backed Sec urities (R MBS)

acc oun ted for 77 % o f th is amo unt. Na tura lly, g i ven th ese figures , man ag ing d e fau lt

r is k o f th e c ol la t era l p oo l b ec om es o f c r uc ia l im por ta nce t o f ina nci al i ns titu t io ns

a nd i n ves to r s . In th is pa per w e p r ese n t a n i n t ens i t y b as ed a ppro ac h f or mo de li ng

r esiden tia l mor tg age d e fau lts . More s pecific ally, we w ill fit a Co x propo rtiona l

hazard rate model to describ e the prob a b i l i t y o f d e fa u l t ( PD ) fo r r e s i d e n ti a l

mor tg ages a nd the u ncer tain ty a ro und the e xp ec ted PD . O nce we are a ble to

m od el mo r tg age d ef aul t s w e w il l t ur n our a tt e n ti on t o m od el in g th e d is tr ib ut i on o f

l os s gi v en d e fa ul t ( L GD) a nd d e ter m ine t h e e f f ec t s o f de f au l ts to t he R M BS

t r a nc h es . W e ho pe th a t th e r es u l ts o f th is r es ea r c h w il l im pro ve N IBC ’s exi s ti ng

m e th ods fo r ma nag in g the c r e di t r is k o r i gi na t ing f r o m th e c o l la te r a l p oo l o f

Eur ope an R MBS tr ansac tio ns .




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    Contents:
           Abstract...............................................................................................................................................3

     I.             I n t r o d u c t i o n  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5  

           1.1                    A n   o v e r v i e w   o f   R e s i d e n t i a l   M o r t g a g e   B a c k e d   S e c u r i t i e s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5  

           1.2                    M o r t g a g e   T e r m i n a t i o n   –   D e f a u l t   o r   P r e p a y m e n t  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7

           1.3                    S t r u c t u r a l   v s .   I n t e n s i t y   B a s e d   A p p r o a c h  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8

           1.4                    L o s s   G i v e n   D e f a u l t   M o d e l s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 1

     II.            M a t h e m a t i c a l   t o o l s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 3

           2.1                    R a n d o m   T i m e s   a n d   H a z a r d   R a t e s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 3

     III.           M o d e l   f o r   P r o b a b i l i t y   o f   D e f a u l t   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 6

           3.1                    D u r a t i o n   a n d   T i m e   t o   D e f a u l t  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 6

           3.2                    T h e   C o x   P r o p o r t i o n a l   H a z a r d s   M o d e l  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 8

     IV.            A v a i l a b l e   D a t a   a n d   M o d e l   E s t i m a t i o n  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2 1

           4.1                    R e s i d e n t i a l   M o r t g a g e   H i s t o r i c a l   D a t a   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2 1

           4.2                    D e f a u l t   P r e d i c t o r s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2 3

           4.3                    P D   M o d e l   E s t i m a t i o n  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2 5

           4.4                    M o d e l i n g   L o s s   G i v e n   D e f a u l t  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2 9

     V.             R e s u l t s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3 2

           5.1                    C o x   P H   m o d e l   e s t i m a t i o n   a n d   r e g r e s s i o n   r e s u l t s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3 2

           5.2                    E x p e c t e d   L o s s   a n d   L o s s   D i s t r i b u t i o n  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3 5

           5.3                    S c e n a r i o   S i m u l a t i o n s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3 7

           5.4                    L o s s   D i s t r i b u t i o n   o f   R M B S   C o l l a t e r a l   P o o l  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3 9

           5.5                    L o s s   D i s t r i b u t i o n   a n d   D e f a u l t s   o f   R M B S   N o t e s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4 2

     VI.            C o n c l u s i o n  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5 0

     A p p e n d i x  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5 1

     R e f e r e n c e s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6 6  




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                                            I. Introduction 



    1.1       An overview of Residential Mortgage Backed Securities 

    Res id en tia l Mor tg age Back ed Secu rities (RMBS) are fin anc ia l secu rities backe d

b y a poo l o f r esiden tia l mo r tgag es . Th e pr ocess o f cr ea ting R MBS (as we ll as a ll

o ther types o f s tr uctur ed credits) is c alle d sec ur i tiza tio n . In th is sec tio n w e

p r ese n t a s h or t des c r ip t io n o f the R MB S tra ns ac t ions an d the r is k s as s oc ia t ed w i t h

them.


    Res id en tia l Mor tga ge Backe d Sec ur i ties ar e s tr uc ture d c red its tha t ca n b e

ch arac terize d b y th e fo llow in g : th e or ig in a tor ( usua lly a bank) has a p oo l o f

r esiden tia l mor tg ages o n its ba la nce s hee t. Th e orig ina tor sells those to a so

called Spec ia l Purp ose Veh icle (SPV), a company created so l e l y f o r t h e p u r p o s e o f

sec ur i tiz a tio n . The SPV r ais es fun ds to p urch ase th ese mo r tg ages b y issu ing

n o tes t o inv es t ors . I n t h is w a y t h e in v es tors on l y bea r th e r is k a r isi ng f r o m t he

p oo l o f mor tg ag es (co l la te ra l po ol) and are gen era lly ind epe nden t fr om th e cre dit

r is k o f th e r es pec t i ve ( fo r m er ) own er o f th ose as s e ts ( e . g . or ig ina t in g ba nk) .


    T he asse ts (in this case res id en tia l mor tgages ) o f th e co lla te ra l po ol g ene rate

in te res t and p rincipa l pa yme n ts . Th ese p a ymen ts as we ll as po te n tial losses , th at

may occur in case the under lying borr owe rs do n o t ser ve their ob lig ations , are

d is tr ib u ted t o the in ves t ors ac c o r d ing to th e s tr uc ture o f th e s ec u r i t iz a t io n. I n t his

w a y t h e c r e di t r is k o f t h e c o ll a te r a l p oo l is t r ans f er r e d t o th e in v es tors . Th e no tes

are divided into seve ral classes with different seniority, varying from AAA to

Eq uity. In g ene ra l the n o tes with th e low es t ra tin g are th e firs t to a bsor b losses in




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    th e und er lying p oo l o f ass e ts . Na tur a lly, the n o tes with low es t ra ting ha ve the

    h ig hes t r isk an d acco rd ing l y ge ner a te the h i ghes t re turn . Resp ec tively, th e mos t

    senior notes are the leas t r isky ones and produce the lowes t r e tur n . The set of

    r ul es , w h ich d is tr ibu t es c as h f lows ( an d los s es ) fro m the c o ll a ter al t o t h e n o tes , is

    ca lled the w a t e r fa l l o f the s truc tur ed cr ed it. Eac h R MBS deal has its sp ecific

    w a t er fa l l . T her e for e , i n ves t ors in R MB S ha v e t o f oc us o n bo t h t he u nd er ly i ng r is k

    o f th e s ecu ritize d por tfo l io (co l la tera l po ol) an d the r ules th a t d e ter mine wh ich

    co nseq uences in ves to rs ha ve to face in cas e cer ta in e ven ts occu r. Th e fac t tha t

    d i ffe ren t no tes ha ve d i ffe ren t r isk pr o files, tho ug h th e y a ll r e fe renc e the sa me

    u nde r l y in g p or t fo l io , is b as ed o n t h e r es p ec t i ve s p ecia l tra ns ac t io n s tr uc tu r e . This

    e nab les inves tors to sa tis fy the ir ind i vidua l r isk ap pe tites and ne eds . Figur e 1 .1

    d ep ic ts the g ene ra l s tr ucture o f a typic al RMBS tr ansac tio n .




    Figure 1.1 RMBS General Structure




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    Bec ause the wa ter fa ll o f each RMBS de al is un iq ue an d has bee n de ter m in ed

a t or ig ina t io n , the e f fec t o f a g iv e n ( e x pect e d) los s fro m th e poo l to t he no t es is

sp ecific for e ach RMBS b u t de ter minis tic . N IBC ’s T rad in g D epartmen t has its own

m od el to de t er m in e th is e f f ec t d e ter mi nis ti c a ll y . On t h e o th er h and , fr om a r is k

m an age men t p oi n t o f v i ew i t is i mp or tan t t o h a ve a mod el th a t s t ochas t ic a ll y

d es c r ibes   the    u nc er ta in t y   of   th e   los s es   o r ig ina t ing   from   the   po ol   i .e .   t he

u ncer tain ty ar oun d the e xpec ted PD . In th is p aper we w i ll p rese n t a s toch as tic

appr oach fo r modeling th e c redit r isks assoc ia t e d w i th the c o l l a te r a l p o o l o f

r esiden tia l mor tg ages .




          1.2         Mortgage Termination – Default or Prepayment 

    A gr ea t dea l o f r es ea r c h e x is ts t o da y on m od el in g m or tg age t e r m ina t io n . A

g ene ra l c ons ensus e xis ts in the litera ture – a mor tg age is ter mina ted if it is e i ther

p rep aid or th e borr owe r has de fa ulted from h is p a ymen t o bliga tio ns (Deng [ 3 ] and

De ng , Q uigley & Van Or der [ 4 ]) .


    T he g oa l o f th is p ap er , how eve r , is to m ode l the u nc er t ain t y a r ou nd t he

e xpec te d loss ass ocia ted with R MBS secu rities . In s truc ture d cr ed its th e proc eeds

fro m pr epaid (a nd a lso p aid on th eir le ga l matur i ty) mor tg ages ar e used b y th e

SPV to eith er replenish the mortgage pool (purc hase new mortga ges), or to repay

so me o f the ou ts ta nding no tes . In the firs t cas e there is pr ac tica lly no e ffec t o f

p rep aid mor tg ag es to th e c ash flows o f th e n o tes . In th e s econ d cas e this effec t is

d e ter mined b y the sp ecific R MBS con tr ac t - r epa ymen t of no tes cou ld b e for

e xample pro por tio na l to th e no tes tra nches ( in th is c as e a ga in th ere is prac t ic a ll y




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    n o impac t to th e cr ed it en hanc emen ts o f the n o tes) or it co uld o nly a ffec t the most

    s e ni or no tes . T o s um ma r iz e - i n R M B S t r a ns ac t ions, pre pa y men ts e i th er h a ve no

    impac t on th e cas h flows to the n o tes or th is impac t is d e ter minis tic (d etermined

    b y th e w ate rfa ll) . T her efore in th e sc ope of th is p ap er , the cre dit r isk assoc ia ted

    w i th mor t g a g e p o r t f o li o s is e s s e n t ia l l y th e r is k t h a t b o r r o w e r s w i l l d e fa u lt a n d f a i l

    t o me e t in te r es t r a te p a ym en ts o n th e o u ts t a nd ing ba la nce pl us th e r is k th at g i v en

    d e fau lt, the c olla tera l va lu e o f th e de fau l ted mor tgag e is less than the ou ts ta nd ing

    b alance plus unp aid in teres t.




                        1.3        Structural vs. Intensity Based Approach 

        T he cr ed it r isk mode lin g lite ra ture has b een ess en tia lly d e ve lop ed in tw o ways –

    th e s truc tur al appr oach and the re duce d- form ap proach . The s truc tur a l ap pro ach is

    a lso s ometimes c alle d op tio n-b ased appr oach . T he ances tor o f a ll struc tura l

    m od els is th e Me rton Mode l [5 ]. T he m ai n i de a is to us e t he evo lu t io n o f f i r ms ’

    ( borr owers ’) s truc tural var iab les, such as asse t (ho use) a nd de b t valu es , to

    d e ter mine th e time to de fau l t or to pre pa y. De fa ult is viewed as a pu t op tio n ; the

    b orrow er se lls his ho use back to the len der in e xch ang e for e limina ting th e

    mor tg age ob liga tio n . Wh ere as , pr epa ymen t is viewed as a c all op tion ; the b orr ower

    e xc han ges the un pa id b alance o n the de bt ins tru men t for a releas e fr om fur th er

    o bl ig a ti on . I n t he s tr uc tur a l a ppro ac h i t is as s u me d tha t t he r e are no t r a ns ac t io na l

    o r r epu ta tio n c os ts fo r d e fa ult or pr epaymen t an d tha t, b orr owers are we ll-

    i n fo r me d an d m ak e t he r a t ion al c h oic e to exe r c is e e i th er th e c a ll or t he pu t o p t ion

    wh en the y c an increas e th e ir we alth . Thes e assump tio ns ma y lo ok app ro pr ia te

    wh en dea lin g w i th c omme rcial bo rrowe rs bu t are n ot r ea l is tic wh en c onside ring




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r esiden tia l mor tga ges . T he beh a vior o f p riva te individua ls , wh ose p urpos e is to

fina nce the ir pro per ty w i th the lo an , is no t a lwa ys r ationa l in th e s ense o f the

ec ono mic th eor y.


   An o ther sh or tco ming o f s truc tur al mod els ar ises w he n cons ide ring the le ga l

as pec ts o f m or t gag e c o n tr ac ts . T he ma jor it y o f s tr uc tu r a l m ode ls w er e de ve lo ped

i n a t te mp t t o d es c r i be t he c r e di t r is k o f t he m or tg age m ar k e t in th e U ni t ed S t a tes .

W h i le in the U S , bor r ow er ’s ob l ig a ti ons to th e or ig in a to r o f th e l oa n ar e ter m i na te d

i n t he c as e o f de f au lt ( t he ba nk o n l y has r ig h ts o n th e p r op er ty, n o ma t te r i f i ts

ac tua l marke t value is less than th e va lu e o f the mor tg age con trac t) , th is is n o t th e

cas e in Eur ope . In mos t Eur opea n r eside ntia l mor tgag e c on tracts , if a borr ower

d e fau l ts he l os es h is pro per t y a n d i f t he m a r k e t v al ue o f t he p r op er t y do es not

c o v er t he p r ese n t va lu e o f t he ou ts t and in g i n te r es t paym en ts t he b or r ower is als o

o bl ig ed to c o v er t his d i f fe r enc e . I n t his c as e , th e a pp r oac h of m od el in g de f au lt

b eha vio r as a pu t o p tio n on the ho use va lu e is qu ite unr ea lis tic .


   A n e x te nsive l i te r a ture e x is ts em p lo y ing t h e s tr uc tura l app r oach i n t he v alu a ti on

o f mor tga ges (see , fo r ins tance , T i t ma n & T or o us [6 ], K au e t a l [ 7 ] or K a u an d

K e ena n [8 ]) . W h il e the o p ti on bas ed v i ewp oi n t has y i e lde d c ons id era bl e i n s i gh ts

in to th e wo rkings o f ide alize d mor tg ages , it has pr ove n difficu lt to employ such

m od els f or t h e pu r pose o f em p ir ic a l es t i ma ti on .


    T he r ed uc e d- for m mo de ls are a ls o c a l led i n te n s i ty o r haza r d r a t e models .

Co mp ared to s truc tu ra l reas on ing , the re duce d- for m po in t o f view is a g ood de al

less ec onomic al: d e fa ult or pr ep a ymen t is n o long er in ter na lly d e ter mined , but

r at her , e x te r na l l y imp os ed o n th e mod el acc or d in g t o s o me r a ndo m process . I n

i n te nsi t y bas ed mo del s , th e d e fa ul t t i me is mo de led a s a f i r s t j u mp t i me o f a n




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     e xoge nous ly g i ven jump process . In th e lite ra tur e se ve ra l e xplana tor y variab les for

     a de f au l t o f a mor tg age c on tr ac t ha v e b ee n iden t i f ied . Smi t h , S anch ez a nd

     L aw r e nc e [9 ] an d Deng [ 3 ] s elect mor tgag e s pec ific an d ec ono mic ch arac ter is tics

     f or pr ed ic tin g d e fau l ts a nd f or c a lcu la t in g th e p r ob ab il i t y o f inc ur r in g a l os s on a

     d e fau l te d lo an . San tos Silva and Mur teira [1 0 ] us e b orr ower ’s cha rac ter is tics , s uch

     as the D ebt- To- Inc ome ra tio (DTI) , w hic h is usua lly on ly obser va ble b y the iss ue

     o f t he mor tg age . I n the ir mo de l , F o ll ia n , H ua ng , and On dr ich [1 1 ] i nc lu de d ur at i on ,

     loca tio n , de mo grap hic a nd ec ono mic var iab les as co va ria tes to e xplain de fa ult.


         Co mb in a tions o f the struc tura l an d r educ ed fo rm mode ls a lso e xis t. To mod el

     time to default, Deng [3 ] a nd Deng and Quigley [ 1 2 ] p ropos e c ombin in g the

     fina ncial va lu e o f the pu t o p tion in the s truc tur a l appro ach , with n on- op tion r ela ted

     va riab les , suc h as u ne mp lo yment or divorce r a tes .


         In th is p ape r we c ons ide r res ide n tial mor tg age de fau l t as a n e ven t w hich is

     t r ig gere d b y mo r tgag e s p eci f ic , mac r o-eco no mic an d b y s om e p er s o na l “ no n-

     fina ncial" re asons , mo re than by a ra tion al econ omic dec is ion (se e als o De ng an d

     Q u i g l e y [12 ] and D e G io r g i [ 1 3 ]) . O ne co mmon ( macro -eco nomic) ca use for d e fau lt

     is un employmen t; anoth er is d i vo rce . In the c ase o f u ne mp lo yme n t the inco me of

     the borrower can dramatically decrease and th e c o n s e q u e n c e w i l l b e th e in a b il i t y

     to p a y the in ter es t on th e ou ts ta nd ing ba la nce . T here fore , co ns ide ring an RMBS

     c o l la te r a l po ol , w e t r y t o m ode l t he d is tr ibu t io n o f t h e exp ec ted n um ber o f defa ul ts

     acc ord in g    to   the    ec ono mic     e n viron men t,     i.e .   to   eco nomic     fac tors     suc h   as

     u ne mp lo yme n t a nd int er es t r a t es , o r to s oc ia l and d em ogra phi c d e ve lopm en ts ,

     s uc h as the inc r eas e o f t he n umb er o f d i v or c es.




10
    We pr opose an in te ns ity b ased a pp roac h fo r mod eling the time to d e fa ult,

w h ich w e tak e to be th e f ir s t- jum p- t im e o f a n inh om og ene ous Po is s o n pr ocess w i t h

s toc has tic in te nsity, also c alle d a do ub ly s toc has tic Po isso n or Co x Process . The

m a in ide a c o nsis ts in c o nd i ti on in g o n a s et o f e xp la in in g var ia bl es ( e .g . loa n- to-

v a lu e ( L TV) r a ti o or D T I ) , w hic h a f f ec t bor r ow er s ' c r ed i t q ua li ty a nd b eha vio r, and

t o c ons i der bor r ow e r d e fa u lts as in de pen de n t g i v en th e s e t o f in f or m a ti on a bo u t

t h e c o mm on eco no mic e n v iro nmen t . Th e in te nsi t y pr ocess is d ir ec t l y r e la t ed t o t h e

u nde rlyin g e xplana tory va r iab les, a s in the p ro por tiona l haza rd ra te mod el ( PHR)

o f C ox a nd O akes [14] . T he s pec ific c har ac ter is tic o f th e mode l a re p resen ted in

t h e ne x t c ha p ters .



                              1.4      Loss Given Default Models 

    A s a lr ea d y m en t ion ed ab o ve – a m or tga ge c o n tr ac t w il l c a use los s es i f gi ve n

d e fau lt, the c olla tera l va lu e o f th e de fau l ted mor tgag e is less than the ou ts ta nd ing

b alance p lus un pa id in te res t. We ther e fore a lso n eed a wa y to mo de l the loss

g iv e n de fau l t ( LGD) of th e res ide n tial mor tg ages in the c olla tera l p oo l .


    In th e e xis ting cre dit risk litera tur e, in itial ap pro aches for L GD estimation we re

d e ter mi nis ti c in n a ture . N e ver t he les s , nowa da ys i t h as b ec o me w ide l y ac c e p ted t o

tre a t LGD as a loss se ver i ty dis tr ib u tion ra ther th an to re gard eac h es tima te as

d e ter minis tic , s inc e a n umber o f fac tors play a ro le in th e ultima te rec o ver y, an d to

es timate thes e determinis tic ally is a difficult task .


    T her e are two ma in a ppr oach es for mode ling loss se ver i ty. Th e firs t o ne is

d e ter minis tic – it s i mp ly assu mes th a t a l l reco ver ies are fixed va lu es th a t are

k n own in a d vanc e . T he ar gu me n t f or th is s i mp l i fica t io n is the fac ts t ha t the




                                                                                                                        11
     u nc er t ai n t y o f the r ec o v er y r a tes do es no t c on tr ib u te s i gn i fic an tl y t o t he r is k o f

     l os s es , w he n c o mpa r e d w i th t he d e fa ul t r a te v o la t il i t y . I n o th er w o r ds , th e de f au l t

     r ate es tima te do minates th e LGD es tima te, whe n es tima ting th e e xp ec ted loss o f

     a n e xp os ur e .


         T he seco nd me tho d mod els th e reco ver y ra tes as a r and om var ia ble be twe en

     0 % a nd 100 % . T he LGD o f a mo r t ga ge is t h en g i ven as 1 m in us t he r ec o ver y r a t e .

     Mos t o fte n in th e lite ra tur e , a U-s hap ed b e ta d is tr ib u ti on is use d to m od el the

     r eco ver y va lu es . Th is d is tr ibu tion is ver y use ful becaus e it ca n b e bo und be tw een

     t w o po in ts a nd c an as s u me a w i de r a nge o f s ha pes . M a n y p opu la r c o mme r c i al l y

     a vaila ble po rtfo lio man age me n t a pp lic a tions us e a beta d is tr ib u tion to mo de l th e

     r ec o ver y va lu e i n t he e ve n t o f de f au l t . I n th is pa per w e a dop t th is m e tho do lo g y t o

     mod el the L GD o f R MBS’s co lla te ra l po ol o f mortg ages . The e xac t es timation

     p r oce dure a nd th e c al i br a t in g o f t h e be t a d is tr ib u ti on a r e pr esen t ed in the ne x t

     ch ap te r .


         T he res t o f th e pap er is or ga niz ed as fo llows : sec tio n I I g i ves a s h or t

     m a th ema t ica l b ac k gr o und o f r an do m t i mes , h az ar d r at es an d ju mp p r oces s es ; in

     sec tion I II th e mod el for p rob ab ility o f de fau l t is in trod uced ; section IV descr ib es

     t h e es t im at i on me t hod ol og y , the a va il ab i li t y o f d a ta a nd th e los s - g i ven-d e fau l t

     m od el ; in sec tio n V we p rese nt the res u lts o f our s imula tions an d s ec tio n VI

     co ncludes th e s tud y. T ech nic al res ults are r epor ted in th e App end ix.




12
                                        II.    Mathematical tools 

    L e t us f ir s t s ta r t b y p r ese n ti ng s o me ma th em a tic al t o ols f or th e a na l ysis o f

r educ ed- for m mo de ls. In p ar ticu la r w e will focus on r and om times and h azard

rates . We s tart with random times w i th de te rm in is tic h azar d ra tes and a fter tha t we

co nsider s itu a tions wh ere the on ly o bser va ble qu an tity is the d e fau lt time itse lf.

T h is forms th e bas is for an a na lys is o f a mo re re alis tic s i tua tio n wh ere a dditiona l

in fo rma tio n,       g ene ra te d    for    ins ta nce     by     ec ono mic    e xplana to r y    var iables ,    is

a v ai la bl e , s o th e haza r d r a te w il l t yp ic a l l y b e s t ochast i c . W e g i ve a d es c r ip t io n o f

th e do ub ly s toc has tic ra ndo m times. D oub ly s toc has tic ran dom times ar e the

s im pl es t exa mp le o f r and om t im es w i th s toc has t ic h az ar d r a t es an d a r e thus

f r e qu en t l y us ed i n dyn am ic c r e di t r is k m od els . W e ass um e t h a t t he r e ade r is

fa miliar w i th th e b asic no tio ns fr om the the or y o f s toch as tic pr ocess es , such as

filtration, stopping times or basic mar tingale theor y.



                           2.1         Random Times and Hazard Rates 


    L e t us cons ide r a prob ab ility spac e                (Ω, F, P )   and a ra ndo m time        τ   de f in ed on

th is spac e i.e.        τ : Ω → (0, ∞)       is a p os i t i ve ,   F -meas urab le ra ndo m va riab le tha t is

in te rpre ted as the time to d e fau lt o f a mor tg ago r . We d eno te b y                   F (t ) = P(τ ≤ t ) t h e

cu mu la tive d is tribu tion func tio n o f           τ    a nd b y F (t ) = 1 − F (t ) = P (τ > t ) t h e s ur v iva l

fu nc tion o f     τ   . W e as s um e that F (0) = P (τ = 0) = 0 a n d tha t F (t ) > 0 f o r a l l t < ∞ .

W e c an n ow d e fi ne the ju mp or de f au l t i nd ic a t or pr ocess (Yt ) ass oc i a te d w i t h            τ   by




                                                                                                                            13
     Yt = I {τ ≤t } f o r t ≥ 0 . Note that (Yt ) is a r ig h t con tin uous pr ocess wh ich ju mps fr om

     0 t o 1 a t t he d e fau l t t ime τ a nd t ha t 1 − Yt = I {τ >t } .


          A f i l tr a t io n   (Ft ) on (Ω, F ) is an increasing family {Ft : t ≥ 0} o f s u b- σ -a lg ebras

     of     F:     Ft ⊂ Fs ⊂ F             for   0 ≤ t ≤ s < ∞ . F or a g ene ric filtr a tion               (Ft )    we     set


     F∞ = σ (Ut ≥0 Ft ) . In prac tice filtrati ons are typi c a ll y use d t o m od el t h e f low o f

     i n fo r ma t io n . T h e fi l tr a t i o n Ft r e prese n ts t he s t a t e o f k now led ge o f a n obs er ver a t

     time     t a nd A ∈ Ft m e ans t ha t a t t i me t t h e obse r v e r is a bl e t o d e ter mi ne i f an

     even t      A occurred.

          In th e follow in g w e ass ume tha t th e on ly obs er va ble qu an tity is th e random

     time    τ   a n d eq ui v al en t l y t h e as s oc ia t ed ju mp p r ocess      (Yt ) . Let (H t ) b e g i ven b y


                                                         H t = σ ({Yu : u ≤ t})                                            (2.1)



     (H t ) is the filtra tio n ge ner ated b y de fau l t ind ica tor pr ocess i.e . the h is tor y o f the

     d e fau l t in f or ma t ion up t o and i nc lu di ng t im e       t . B y d e fi n i ti o n , τ i s a n (H t ) -s to pp in g

     time as      {τ ≤ t} = {Yt        = 1}∈ H t f o r a l l t ≥ 0 .


     De fi niti on 2.1  (hazard  r a tes  and  cumulative  h az ar d  fun c tio n) 


     T he fu nc tio n           Γ(t ) := − ln( F (t )) is c alled the cumu la tive haz ard fu nction o f the

     r and o m time             τ   . If   F is abs olu tely c on tin uo us w i th d ens ity             f , the func tion

     γ (t ) := f (t ) /(1 − F (t )) = f (t ) / F (t )   is ca lled th e haz ard r ate o f      τ   .




14
     By       d e fi ni ti on   we       h a ve    F (t ) = 1 − e − Γ (t )   and     Γ ′(t ) = f (t ) / F (t ) = γ (t ) ,     so

          t
Γ(t ) = ∫ γ ( s ) ds . W hen w e c ons ider a v er y s ma l l in ter v a l o f t i me , th e h az ar d r ate
          0


γ (t )   ca n be inter pre ted as th e ins tan tane ous ch ance o f de fau l t a t time t , g i ven

survival up to time t . For h > 0 w e h a ve :


                                                                     F (t + h) − F (t )
                                         P (τ ≤ t + h | τ > t ) =                                                           (2.2)
                                                                         1 − F (t )


a nd th ere fo r e


                               1                       1         F (t + h) − F (t )
                          lim P(τ ≤ t + h | τ > t ) =        lim                    = γ (t )                                (2.3)
                          h →0 h                      F (t ) h→0         h


     T he haz ard ra te         γ (t )    can b e in terp re te d as th e e xpec te d nu mber o f fa ilures

( de fa ults) in a un it o f time . Since in te gra tio n is pr actic ally su mma tion – the

                                                        t
c u mu la t i ve h az ar d func t io n Γ(t ) =         ∫ γ (s) ds
                                                       0
                                                                       can be und ers to od as th e e xpec te d


n umber o f fa ilur es in th e pe riod of time be twe en 0 to t.


     T her e ar e s e v er a l a dv a n tag es in l earn in g t o t h ink in t er ms o f h az ar d r at es ,

r ather than th e tra ditio na l d ens ity fu nc ti ons a nd c umu la tive dis trib u tion fu nctions .

Haz ard fu nc tions g i ve a more in tu itive wa y to in te rpre t an d un ders ta nd the pr ocess

th a t ge nera tes fa ilures . Th is is wh y in su r vival ana lys is re gress io n mod els are

m or e eas i ly g r asp ed b y e xp la in in g h ow d i ffer en t v ar ia bl es ( c o va r i a tes ) a f fec t t he

h azar d ra te .




                                                                                                                                    15
                             III.      Model for Probability of Default 

          A s a lr e a d y s t a te d , i n t h is paper we will tr y to describe the probability of defa ult

     o f r es id en ti a l mor tg ag es v ia an i n t ens i t y b as ed m ode l . Mor eo ve r , o ur g oa l is t o

     q uan t i f y the dep end enc e an d s e nsi t i v i t y of t h e PD on s om e e xp la na to r y fac t ors .

     T hes e fac to rs can be mor tga ge s pec ific ( LT V a nd/or DT I r atios) or exter nal

     ( une mp lo ym en t      and /o r    i n ter es t   r a tes ) .    M os t    of   th e    in te nsity - b as e d   m od els ,

     i nc lud in g ou r s , ma in tai n a do ub ly s toch as tic c ha r ac ter , w h ich m ea ns t ha t no t o nl y

     i t is u n c e r t a in w h e th e r a n o b l ig o r w i l l d e f a u l t a t a p a r t icu la r ti m e , b u t th a t the

     in te nsity (h azar d ra te ) b y wh ich this e ven t occurs is a lso unce rta in b e for eha nd .

     T her e for e we tr y to mod el th e d e fau lt time o f a res id en tia l mo rtg age as a ra ndom

     time with a s tochas tic h azar d ra te . Th is le ads to th e us e o f the s o-ca lled d oub ly

     s toc has tic Po isso n Proc ess to mo de l the pro bab ility o f d e fau lt.



                                   3.1      Duration and Time to Default 


          Co nsider th e fo llow ing se tting . L e t P                = {( si , Bi ,Vi ), i = 1,..., n} b e a p or t fo l io o f n

     r esiden tia l mor tgag es . For mortg age i , s i d e no tes the time o f issu e (c alen dar

     t i m e ) , Bi   = (Bi ,t )t ≥ s is a pr ocess g i ving the o u ts tan ding b alance a t time t and
                               i




     Vi = (Vi ,t )t ≥ s is a s tochas tic proc ess g i ving th e hous e va lu e a t time t . W e s upp ose
                      i



     t h a t the mor tg ag e por tfo l io is to tal l y c har ac te r iz ed b y P .




16
Figure 3.1 Duration and time to default.



     Now let Di : Ω → (0, ∞) b e a pos itive ra ndom variab le g i ving the d ura tio n or

l i f e ti me o f a mor tg age c on tr ac t      i a nd l e t τ i : Ω → (0, ∞) b e als o a pos itive ra nd om

v a r i ab le g iv i ng th e tim e to de f au l t o f a m or tg ag e i i . e . τ i is t he per io d o f ti me f r o m

n ow ( t 0 ) t i l l th e o b l ig o r i d e fa ul ts . W e as s u me t ha t f or al l mo r tga ge c on tr ac ts in P

w e h a ve P ( Di         = 0) = 0 . M o r e o ve r , P( Di > d ) > 0 , ∀d > 0 a n d a ls o P(τ i = 0) = 0 a n d

P(τ i > t ) > 0 , ∀t > 0 . W e h a ve Di = ∞ ( or equ iva le n tl y τ i = ∞ ) if mor tgag e i d o es

n o t de f au l t. A ls o l e t d i d eno t e the d e fau l t tim e ( c a le nda r t i me) o f o bl ig or i and    θi

b e the per io d o f t ime t h a t the mo r t ga ge h as b een ou ts ta nd ing ( pe r i od o f t im e from

issue till now – curr en t life time o f the mor tg age ) . See figu re 3 .1.


     S i n c e s i i s k n own in adva nc e f or a n y g i ve n t i m e t 0 , w e c a n c a lcu la t e      θi    a nd

s ince    θi        i s k n own an d d e te r m in is t ic , th e di s tr ibu t io n ( hazar d r at e) o f t i me t o

d e fau l t    τi     i s c omp l e te l y de te r mi ne d b y t h e d is tr ibu t io n ( haza r d r a te) o f t he

d ur a t io n Di . M a th ema t ica l l y w e h av e :



                                       P (τ i ≤ t ) = P( Di ≤ θ i + t | Di > θ i )                               (3.1)




                                                                                                                         17
         I n o th e r w o r d s – the p r o b a b i l i t y th a t mo r tg a g e i will de fa ult in a c er ta i n in ter va l

     o f t im e t is e qua l to the p rob ab ility tha t th e life time o f th e mor tgage                   i is less th an
     o r e qua l to th e c urre nt life time of th e mor tga ge plus t , g i ve n th a t t h e m or tg ag e is

     s till o u ts tan ding a t time t 0 ( h a s s u r vi v e d t il l t 0 ) . More o ver w e ha ve :



                                                                       P( Di ≤ θ i + t ) − P ( Di ≤ θ i )
                      P (τ i ≤ t ) = P ( Di ≤ θ i + t | Di > θ i ) =                                                      (3.2)
                                                                               P( Di > θ i )


     a nd if we kn ow the h azar d ra te or th e cu mu la tive haza rd fu nc tion fo r the

     d is tr ib u ti on o f Di t h en all t he v al ues o n the r ig h t ha nd-s id e o f ( 3 . 2) a r e k n own .


         We will now use Cox Proportio n a l H a z a r d R a t e Mod e l t o m o d e l t h e h a z a r d r a te

     o f mor tgage dur a tion .



                          3.2       The Cox Proportional Hazards Model 

         In this c hap ter we p res en t a w ay to mod el (a nd la ter es tima te) th e haz ard ra te

     o f mo r tga ge du ra tion . W e b orrow a mod el typ ica lly use d in me dica l sc ience in th e

     f i e ld o f s u r v i v a l a na l ys is . T he C o x M ode l is a w e l l- r ec o gn ized s t a t is t ic a l tec h ni que

     f or e xp lor in g the r e la t io ns h ip b e tw ee n th e s ur v i val o f a pa ti en t and s e v er a l

     e xplana tory var i ab les (a ls o ca lled c o var ia tes) .


         In o ur c ase we assu me th a t mor tg age d e faults are trigg ere d b y so me mor tga ge

     sp ecific and /or b y s ome e xter nal ( en viro nmen t spec ific) fac tors. W e s uppos e tha t

     w e c an f ind a s e t o f p r ed ic tor s f or t he de fa ul t e ve n t o f ob l igor           i . M a th em a tica l l y

     we ha ve a mu lti-d ime nsiona l s toc has tic p roc ess X i                    = ( X 1 ,..., X p ) , s uch that each




18
c o mp one n t X i , q ( q     = 1,..., p) r e prese n ts an e xp la in in g fac t or f o r t he e ve n t o f de f au lt

o f ob ligor     i , as f or e xam p le th e r eg io na l un em p lo ym en t r a t e .


    Let     λ (t | X i )   be th e haz ard r at e o f m or tg age d ur a t io n , g i ve n a pa r t ic ul ar

r ea liz a ti on o f th e de f au l t fac to r s   X i = ( X 1 ,..., X p ) . N o te t h a t λ (t | Xi ) simply s tates

that   λ (t )   i s a fu nc ti on o f   X i = ( X 1 ,..., X p ) . Co x Pro por tio na l H aza rd Mode l ass umes

t h a t the r ela t io ns h ip be tw een      λ (t | Xi )   a nd t he e x pl an a tor y fac to r s   X i i s g i v en b y :


                                        λ (t | X i ) = h(t ) * exp(β ′X i )                                           (3.3)



where      h(t ) is th e baseline haza rd (e ffec t o f (e la psed ) time t o n mor tgage

d ur a t io n) an d β i s a v ec t or o f c oe f fi c i en ts gi v in g th e s e ns i ti v i t y o f th e h az ar d r at e

to cha nges in the e xplana tory va r iab les . N o te tha t the base l ine haza rd                                    h(t )

co rresp onds to the ins ta n tan eous p rob ab ility o f de fa ult, g i ve n sur viva l (n o d e fau lt)

up to time t w hen a l l the co varia tes ar e ze ro . Th e mod el a lso s ta tes tha t th e

b ase line func tio n is th e same fo r all mor tg ages in co nsider a tion i.e . th e d e fau lt

r at es o f m or tg age           i a nd mor tg age           j d i f fe r on l y in t h e r e al iza t io ns o f t h e

covar iates       X i an d X j . T his f ac t p la ys a c r uc ia l r o le in the es t i ma t ion p r oce dure ,

as w e w i ll s e e la te r . F r om ( 3 . 3) a nd t he de fi n i ti on o f th e h az ar d r at e , i t fo l lows t ha t

t h e c um u lat i v e h az ar d f u nc ti on o f m or tg age d ur a t io n is e qua l to :


                           t                                              t
          Λ (t | X i ) = ∫ h( s ) * exp( β ′X i ) ds = exp( β ′X i ) ∫ h( s ) ds = exp( β ′X i ) H (t )               (3.4)
                           0                                             0




                                                                                                                               19
                             t
     w h e r e H (t ) =     ∫ h(s) ds
                             0
                                             is th e bas eline c umula tive haza rd fu nc tion .



           T he mode l e xp la i ns th e follow in g beh a vio r o f the d e fau lt in te nsity p rocess :

     su ppos e tha t a t the be ginn in g of the mor tg age a gree me n t a n e xpec te d in tens ity

     ( de fa ul t r at e)        λ0     c an be as s oc ia te d to ob li go r     i . I f t he o bl ig or ’s be ha v ior is n ot

     a ffec ted       by         an y     pre dic tors    X 1 ,..., X p , t he n w e e xp ec t n o c on tr ibu t io n o f

     X i = ( X 1 ,..., X p ) to th e in te nsity process , mean in g tha t β i = 0 , f o r a l l i = 1,..., p .

     M or e o ver , i f t he e lapse d ti me do es n o t c on tr i bu te t o t h e d e faul t i n te nsi t y, t h en

     λ (t ) = h(t ) ≡ λ0         is cons tant, wh ich w ou ld imp l y a h omoge nous Poiss on pr ocess .


           How e ver , in pr ac tic e we o bser ve tha t ob lig or ’s beh a vior cha nges d ur ing th e life

     of th e mortgage, meaning that the probab i l i t y o f i n c u r r i n g a d e f a u l t inc r e a s e s o r

     d ec r e as es . S o me f ac to r s X 1 ,..., X p a f f e c t t h e a b i l i t y o f o b l ig o r   i to pay the inte res t

     r ate on a mor tg age , ch ang in g stoc has tica lly the d e fau lt in te nsity. Equ a tion ( 3 .2)

     su gges ts th a t p red ic to rs X 1 ,..., X p a nd t im e t a f f ec t t h e r ea l iza t io ns o f             λ (t )   in a

     m u l ti p l ic a t iv e w a y .


           A n o ther th in g w or th m en t io ni ng is d e fau l t c o r r e la t ion, w h ich is c e r t a in l y low e r

     for    res i den tia l           mo rtg ages     compar ed      to    c ommerc ia l        mo rtgag es .     D epen denc e

     b e tw ee n r es ide n ti al mo r t ga ge defa ul ts c a n b e e xp la ine d , t o a la r g e e x ten d , o nl y b y

     th e macr oec ono mic e n viron men t (e .g . un emplo ymen t ra te and /or in teres t ra te ).

     T h is allo ws us to assu me co nditiona l inde pen denc e o f r esiden tia l d e faults . We

     ass ume tha t o blig ors wh o de fa ult up to time t ar e c ond ition ally in dep end ent, g i ven

     th e his tor y o f the pr ed ic to rs up to time t . Th is assu mp tio n s ee ms re ason able for




20
t h e k ind of p or t fo l io w e a r e c o nsi der in g i n t h is pa per – a p or t fo l io of p r i va te

ind i vidu a ls ( fo r comp an y d e fau lts a nd co mmerc ia l mor tg ages th is assu mp tio n

w o ul d n ot b e r e al is t ic ) . In f ac t th is c o nd i tio na l i nd epe nde nce im p li es tha t, g i ve n a

sce nar io th rou gh the pr ed ic tors, ob lig or de fau l ts occu r in dep en den tly, me an ing

t h a t th e d ep end ence s tr uc t ure is f ul l y des c r ib ed b y th e e v ol u t ion o f t he c o mm on

( macr oecon omic) co va ria tes .



               IV.         Available Data and Model Estimation 


                     4.1     Residential Mortgage Historical Data 

    N IBC Bank N.V. has ma in tained a sign ifica n t da ta bas e o f Du tch res id entia l

mor tg ages . T he da tab ase con tains ap pro xima te l y 92 tho usan d r ecords a nd was

recorded betw een 01/01/2002 and 6/1/2008 s o a l l s t i l l e xis t in g c o n t r a c ts h a ve

“ end _da te ” - 6 / 1 /20 08. S ee t ab le 4 . 1 for a s a mp le o f t he da ta base.


                                                                                              Right censored
     Original LTV            Original DTI          Loan start date       Loan end date             data
                                                                                              (0=defaulted)

     0.390438728             0.218330602              7/1/2000              6/1/2008                 1

     1.154709643             0.141826843              3/1/2005              6/1/2008                 1

     1.14553197              0.269474451              3/1/2005              6/1/2008                 1

     1.158583728              0.37568185              3/1/2005              6/1/2008                 1

     0.808333333             0.126808706              3/1/2008              6/1/2008                 1

     1.285714286               0.1235684              3/1/2005             10/1/2006                 0

     0.966666667             0.286158458              3/1/2005              6/1/2008                 1

           1                 0.170747839              3/1/2005              6/1/2008                 1




                                                                                                                       21
          1.343134328            0.367103444             1/1/2006             6/1/2008                1

          0.642487047            0.121560317             3/1/2007             6/1/2008                1

          0.898305085            0.224445451             3/1/2005             6/1/2008                1

           0.423536                0.235642              3/1/2005             10/1/2007               0

          1.229367273            0.303740859             3/1/2005             6/1/2008                1

          1.167959184            0.248989162             3/1/2005             6/1/2008                1

     Table 4.1 Sample from the Mortgage Database



         E ac h r ow r e pres en ts o ne m or tg ag e c on tr ac t. T he las t c o lu mn s hows th e de fa ul t

     s ta t us o f a mor tg age loa n      i – i t is 0 if obligor i has d e fau lte d (a nd con trac t

     s e ize d to exi s t in t he d a tab ase at i t s end _da te) ; a n d i t i s 1 if mo r tg age   i is eith er
     s t il l e xis t in g or i t w as t er mi nate d d ue t o p r ep a ym en t or r ep ay m en t ( and w as

     r emo v ed fro m t he d ata base o n th is c or r espo nd ing e nd_ da te ) . A m or tg age c o n tr ac t

     is co nsider ed to ha ve de fau l ted whe n it h as b een in arre ars fo r more th an 3

     mon ths i.e. the o blig or has mad e no in te res t or pr inc ip al pa yme n ts on h is

     mor tg age ob liga tio n for mo re than 3 mo n ths. As de fau lt is an e xtre me l y r ear e ven t

     ( espec ia lly in th e Ne th er lan ds) , a l mos t all o f the o bse r va tions ar e ce nsor ed ( las t

     c o lu mn is 1 ) . I n fac t fo r the 6 yea r s i n c onsi de r a t ion th ere w er e o nl y 15 58 d e fau l ts

     o u t o f 9 2 th ousa nd mo rtga ge loans .


         T he d ur a t io n ( l i fe t i me ) o f mo r tg age c on tr ac t   i is ob tained b y tak in g the

     d i f fe r enc e b e tw ee n en d_d a te i an d s tar t_ date i , e xpresse d in months .




22
                                        4.2       Default Predictors 

     A s w e a lr e ad y s ta ted , th e r eal iz a ti on o f t he c o v ar i a tes ( de fau l t pr ed ic tin g

f ac t ors ) has s ig ni f ican t i mpac t o n th e r e ali z a t ion o f th e h az ar d r at e o f m or tg ag e

d ura tio n . To c ons tr uct o ur mode l we c hose 4 d e fau lt factors , na me ly :


   X 1,i      Or igina l Loa n-T o- Va lue ( LT V) R atio o f lo an i


   X 2 ,i     Or i gi na l D eb t- To- I nc om e R a t io ( D T I ) o f loan i


   X 3 ,i     Q uar te rly un emplo ymen t r a te ( a t co n trac t’s e nd d a te)


   X 4 ,i     3 - mo n th Eu r i bor in te r es t r a te ( a t c o n tr ac t ’s e nd d a te)

Table 4.2 Default Predictors



     L T V an d D T I are a c o mm on c ho ic e f or fac tors e xp lai n ing mo r tg age de f au l ts .

L T V s tands for loa n- to - v a lue r a ti o an d g i ves the r a ti o o f th e s iz e o f the mor tg ag e

l oa n to t he v a l ue o f th e r ea l es ta t e p r op er ty – o r s imp l y – lo an v a lu e o v er h ous e

v a lu e . T he o r i gi na l LT V is th e lo an- t o- va lue r a t io o f b or r ow er i a t or ig ina t io n o f

t h e m o r tg a g e c o n tr a c t . A s w e w i l l s h o w la te r L TV h a s v e r y s m a l l a n d s t a tist i c a ll y

i ns ign i f ican t imp ac t o n t he haza r d r a te o f t i me to de f au l t . The D T I r a t io o n t he

o ther h and has a s ig nific an t e xp la na to r y powe r in o ur mod el. It sta nds for de b t- to-

i n c o me r a tio (some times als o ca lled PT I (p ay men t- to- inco me ) ra tio) and e xpress es

th e ra tio o f mon thly pa yme n ts du e o n the mor tg age lo an to the r epor ted inco me o f

borrow er i i.e . it d ir ec tly r ela tes th e pa yment we igh t to the ab ility o f pa yment for

o bl ig or i . As on e w ou ld e xpec t, the D TI r atio has a s ign ifican t impor tanc e in

e xplaining th e haz ard r ate o f time to de fau lt.




                                                                                                                                 23
            W e a ls o c ho os e Quar te r l y U ne mp lo y me n t R a t e a nd E ur ib or In te r es t R a t e ( w i th

     th e     appr op ria te     la g   –    this     w ill   be     exp la in ed      la ter) ,   beca use       the y     a re

     macro econo mic variab les th a t sh ou ld h a ve an impac t o n ob ligors’ ab ility to p a y the

     i n te r e s t o n t h e ir mo r tg a g e o b l ig a t io n s . A r is e in u n e mp lo y me n t w i l l me a n tha t m o re

     p eop le lose the ir pr imar y so urce o f inc ome wh ich w ill a ffec t th eir ab ility to p a y

     i n te r es t o n t h e ir loa ns . Sa me is t r ue f or int er es t r at es – for mos t D u tch r es ide n ti al

     mor tg ages ( and mos t o f the mor tg ages in ou r da tabase ) the in teres t pa ymen ts due

     to th e ob lig or are d e te rmine d b y a bas e in te res t ra te (Eur ib or) p lus a marg in . T his

     m ea ns th a t a l ar ge inc r ease i n E ur ib or r a tes w il l i nc r e as e the pa y me n t w ei gh t o f

     m o r tg a g o r s a n d c o n s e q u e n tl y w i l l mak e s ome obli gors incapable o f p a y in g t h e s e

     p a ym en ts .


            W e ha v e to po i nt o u t h er e th a t t h ere are t w o de f i ni t io ns o f u nem p lo ym en t r a te

     in the Ne the rlands . On e is e xpr esse d as a p erce n tag e o f to ta l pop ula tion and o ne

     – as a p erce n tag e o f the lab or forc e ( tha t is p opu la tion b e tween 1 6 and 65 ye ars

     o f a ge) . As e x pec te d t he y are a l mos t pe r f ec t l y c or r e la t ed w i th e ac h o th er a nd it

     m ak es prac t ic a ll y no d i f f er ence w h ich on e w e use i n o ur m od el . The on l y

     d i f fe r enc e i s th e   β -c oe ffic ien t   for un emp l oymen t in th e Co x reg ress ion mod el

     ( 3. 3) . W e a r e go in g to us e the o ne th at is m or e fre qu en t l y used in th e me di a an d

     n amely the o ne th a t is e xpress ed as a perc en tage o f the la bor force .


            F or c o n tr ac ts th a t ha v e be en te r m ina t ed d ur ing t he p er iod o f o ur s tu d y, w e

     ass ig n une mp lo yment ra te a nd the in teres t ra te at the month o f ter mina tio n .

     L ook ing a t th e d a ta we see th a t we ha ve a l ar ge am oun t o f mor tga ges tha t ar e s till

     o u ts tan ding (ap pro xima te ly two th ir ds o f th e rec ords) and the ir ac tua l cova ria tes

     ( X 3,i a nd      X 4,i ) are n o t obse r ved . We ha ve no ac tua l end_ da te fo r mor tg age




24
c o n tr ac ts th a t s t i ll e x is t . To o v erc o me th is p r ob le m w it h m is s in g v a lu es , w e s i mp l y

ass ig n a 0 f o r X 3,i a nd X 4,i f o r c on tr ac ts th a t a r e s t i ll e x is t ing . I n t he ne x t s ec tion

w e w i l l e x p l a in w h y w e m a k e th is c h o ic e a n d i ts im p a c t o n t h e e s t im a t i o n o f t h e β

c o e f fic ie n ts .




                                  4.3           PD Model Estimation 

     In th is section we e xp la in th e ma th ema tics be hind th e es timation o f th e Co x

Mod el. A no n-p ara me tr ic me thod for es timating th e                     β    c o e f fic ie n ts w as d e ve lop ed

b y Cox [1 5] h ims elf an d is ca lled p ar tia l l ike l iho od es timation. The es timation is

n on- para me tr ic , me an in g t ha t th e b as e l ine haz ard c a n b e le f t u ns p ec ifi ed . T his

m ea ns th a t w e d o n o t h a ve t o as s u me a c er t a in s ha pe f or t he base l ine f unc t io n . In

t h is w a y t he es ti ma t ion is n o t b iase d b y th e c h oic e o f a b as e li ne haz ard .


     Su ppos e we ha ve a da ta se t w i th n o bse r vations and k d is t inc t fa i lu r e ( e v ent

o r d e fau lt) times . W e firs t sor t th e orde red fa ilure times suc h tha t t1                     < t 2 < ... < t k ,

w h e r e t i d eno t es the f ail ur e ti me of t h e       i - th mo rtga ge . No te tha t t1 < t 2 < ... < t k a r e

th e ac tua l times wh en d e fau lt happ ene d i.e . o nly unc enso red cases ( l ast co lumn in

o ur d a tab ase = 0 ) . We n ow wa n t to e xpr ess th e e ve n t times as a fu nc tion of the

c o v a r ia te m a tr i x X .


     T he      pa rtial   like lihood    func tion      is    der ived    by       tak in g   th e   pro duc t   of    the

co nd itiona l pr ob ab ility o f a fa ilu re a t time t i , g i v en th e nu mb er o f mor t gag es a t




                                                                                                                               25
     r isk a t t ime t i . I n o the r w or ds , g i ve n tha t a de fa ul t h as oc c ur r ed , w ha t i s the

     p rob ab ility th a t it occurr ed to the i - th mor t gag e fro m a r i s k s e t o f s i z e N ?


          L e t R (t i ) de no te th e nu mb er o f mo r tgag es tha t are a t r isk o f fa iling ( defa ultin g)

     a t t i me t i i . e . R (t i ) i s t he r el e van t r is k s e t . The n th e pr ob ab il i t y t ha t t he              j - th

     m or tg age w il l d e fau l t a t t im e t i is g i ve n b y :



                                           λ (t | X i )            h(t ) * exp( β ′X i )       exp(β ′X i )
              P (t j = t i | R(t i )) =                   =                               =                                    (4.1)
                                           ∑ λ (t | X j )
                                          j∈R ( ti )
                                                                   ∑ h(t ) * exp(β ′X j )
                                                                  j∈R ( t i )
                                                                                               ∑ exp(β ′X j )
                                                                                              j∈R ( t i )




     s ince t he bas el in e haza r d h(t ) is the s ame for a l l mor tgag es . Th e de no mina tor in

     t h e a bo ve e x press io n i s t he s umm a t ion o ver a ll mor t ga ges t ha t a r e at r is k a t t i me

     t i . T a k in g t h e p r o d u c t o f th e s e c o n d i tio n a l p r o b a b i l i ti e s y ie ld s t h e p a r tial

     l ik el ih oo d fu nc ti on :


                                                                 ⎡                 ⎤
                                                              k
                                                                 ⎢ exp( β ′X i ) ⎥
                                                       Lp = ∏ ⎢                                                                (4.2)
                                                            i =1    ∑ exp(β ′X j ) ⎥
                                                                 ⎢ j∈R ( t )       ⎥
                                                                 ⎣        i        ⎦


     w i th co rresp ond in g log -like liho od func tion :


                                                      k ⎡           ⎛                 ⎞⎤
                                          log L p = ∑ ⎢ β ′X i − log⎜ ∑ exp( β ′X j ) ⎟⎥
                                                                    ⎜ j∈R ( t )       ⎟                                        (4.3)
                                                    i =1 ⎢
                                                         ⎣          ⎝        i        ⎠⎥⎦


          By ma ximizing the log- like lihoo d fu nc tion ( 4.3) , es timates o f                           β   are o b ta ine d .




26
     No te tha t th e prod uc t in ( 4 .2) an d the s um in (4 .3 ) a re o ver all mor tg ages i

t h a t h a ve ac tu al l y de fa ul t ed i . e .   i = 1,....k a n d k = 1558 in our case. Those 1558 a r e

e xac tly th e con trac ts th a t are no t ce nsor ed , th ere fore fo r all i = 1,....k , w e h a ve

X 3,i ≠ 0 a nd X 4,i ≠ 0 ( a s m e n t io n e d b e fo r e , for a l l mor tg a g e s th a t s ti l l e xis t - n o

d e fau lt e ven t has occu rred - we se t X 3,i = 0 a nd X 4,i = 0 ) .



     Of c ours e the r isks sets R (t i ) , i = 1,....k c on ta in a ll mor t ga ges t ha t ar e a t r is k of

d e fau l t ing a t t ime t i wh ich inc ludes ce nsored cas es a nd co nseq ue n tly mor tg ages

t h a t a r e s ti l l e x is t in g ( fo r w h ich w e h a ve n o a c tu a l o b s e r v e d u n e m p l o ymen t a n d

in te res t   ra te    values) .      Fr om         (4 .1)   we   see   th a t   cens ore d   cases      co n tr ibu te

in fo rma tio n on ly re le va n t to the risk s e t (den omin a tor o f ( 4.1) and ( 4 .2)) . Th ere fore

b y s e t t in g X 3, j = 0 a n d X 4, j = 0 fo r all th ose s till e xis ting mor tg ages we ac tua lly


set exp( β ′X j ) ≈ 1 (b ecaus e             β ′X j ≈ 0 )    and the d enomina tor in (4 .1) and (4 .2) is

s im pl y inc r e as ed b y 1 f or e ac h m or tg age c o n tr ac t th at s ti l l e x ists . In th is w a y w e

tr y to min imiz e an y bias co ming fr om th e fac t tha t we are una ble to obse rve the

u ne mp lo yme n t an d i n te r es t r a t e v a lu es o f th e mo n th of t er m ina t io n o f a l l t hos e s ti l l

e x is t ing l oa ns . O n th e o t her h and w e c a n n o t s im p l y r e mov e t he m fr om t he

es tima tion b ecaus e we do no t wa n t to lose a n y in forma tion a bou t th e occu rrenc e

o f d e fa ul ts ( a nd th e fac t th a t defa ul t is an e x tr eme l y r e ar e ve nt ) . M a the ma t ica l l y

b y s e t t in g   X 3,i = 0 and X 4,i = 0 , we r emo ve a n y e ffec ts o f thos e un obse r ved

v a r i ab les to t he w e igh t of t he r is k s e t – the den om ina t or o f ( 4 . 1) a nd ( 4 . 2) , a nd w e

l e t th e h az a r d r at e for t h ese c on tr ac ts be deter m ine d on l y b y the ir b as e l ine haz ard




                                                                                                                           27
     a nd th e o th er two c ons t an t ( a nd k nown f or a l l c on t r ac ts ) c o var i a tes X 1 ( lo an - t o-

     v a lu e r a t io) a nd X 2 ( debt- to -inco me ra tio) .


          O nc e w e ha v e es t im at e d the          β     c o e f fic ie n ts w e a ls o n ee d a n es t im a te o f the

     b ase line haz ard            h(t ) to fina lly ob ta in an es tima te o f the h az ar d r a t e o f f a ilu r e

     λ (t ) .   In the liter ature there are a number o f app roac hes th a t ha ve be en a dop ted to

     es tima te th e b ase line haz ard                 h(t ) . T he simp les t an d mos t fr eq uen tly used

     a ppr oach was pr op ose d b y Bres low [1 6 ]. He der ive d a maximum like liho od

                                                                                                           t
     es tima tor        of    th e     base l ine   cumu la tive           h azar d   func tio n   H (t ) = ∫ h( s ) ds , after
                                                                                                           0


     ass um in g th a t th e fa il ur e t ime d is tr ib u ti on h as a h az a r d r a te w h ich is c o ns tan t

     b e tw ee n eac h p air o f s uc c es s i ve obs er ve d f a i lur e t imes - a r easo nab le as s u mp t io n

     in our c ase . Th e es tima te o f             h(t ) in the interval [t i −1 , t i ] b e tw ee n two s uc c es s ive

     failure ti mes t i −1 a nd t i i s gi v en b y :



                                                     ~             di
                                                     hi =                                                                (4.4)
                                                            δ i ∑ exp( β ′X j )
                                                              j∈R ( ti )




     where        δ i = t i − t i −1   is t he len g th o f th e ti m e in t er va l an d d i is t he n um be r of

     d e fau l ts tha t oc c ur in t i me t i ( note th a t usu al l y d i = 1 , but this es timation also

     a l low s for m u l ti p l e f a i l ur es a t th e s am e tim e t i - these failur es a re a lso ca lled

     ties ) . Equation (4 .4 ) ca n b e in terp re te d as th e ra tio betwe en the n umber o f e ven ts

     a nd the w ei gh t ed nu mb er o f ‘perso n- time ’ u n i ts a t r is k , w he r e the w e ig h t of eac h

     i nd i v idu a l j i n t h e r is k s e t R (t i ) is exp( β ′X j ) . A r oug h es t im a te o f H (t i ) − H (t i −1 )




28
    ~
is hi δ i a nd if we su m all thos e terms o ver a ll t i ≤ t , we ob tain wha t is ca lled the

Bres low ’s es tima tor of t h e c um ula t i ve base li ne haza r d f u nc ti on a t t i m e t :


                                      ~                            di
                                      H (t ) =   ∑                                                        (4.5)
                                                 t ( i ) ≤t    ∑ exp(β ′X j )
                                                              j∈R ( ti )




                         4.4          Modeling Loss Given Default 

    Res id en tia l mor tga ge loans a re a lwa ys back ed b y so me k ind o f re al es ta te

co lla te ra l . If an ob ligo r de fa ults o n his paymen t o blig a tions the n the lend er ge ts

h ol d o f t he c ol la t eral . T he r ec o v er y va lue – i.e . pr ocee ds fro m s e l li ng t h is

co lla te ra l e xpress ed as a p ercen tage o f loa n’s o u ts tan ding ba lanc e , are use d to

co ver losses ar ising fr om d e fau lts o f ob ligors .


    As we a lre ad y men tio ned w e wa n t to mo de l the r eco ver y ra tes as a ra nd om

va riab le be tw een 0 % a nd 100 %. T he loss- giv en-d e fau lt ( LGD) o f a m or tga ge is

t h en g i v en as 1 m in us t h e r ec o ve r y r a te . Mos t o f ten in l i ter a tur e, a U - s ha pe d b eta

d is tr ib u ti on i s use d to mo de l t he r ec o v er y v a lu es . Th e be t a d is tr ib u ti on is ver y

us e fu l for m od el in g r ec o ver y r at es bec ause i t pro duces v a lues b e tw ee n 0 a nd 1

a nd ca n have a larg e va rie ty o f sh apes (see fig .4 .1) .


The probability density fu n c ti o n o f t h e Be ta d is tr ib u ti o n is g i ve n b y :




                                                                                                                    29
                                                              x α −1 (1 − x) β −1
                                       f ( x; α , β ) =   1
                                                          ∫u
                                                               α −1
                                                                      (1 − u ) β −1 du
                                                          0

                                                       Γ(α + β ) α −1
                                                     =               x (1 − x) β −1                      (4.6)
                                                       Γ(α )Γ( β )
                                                          1
                                                     =           x α −1 (1 − x) β −1
                                                       Β(α , β )


                     ∞

                     ∫
                           x −1 −t
     w h e r e Γ( x) = t      e      dt is th e ga mma fu nc tion .
                     0




     Figure 4.1 The PDF of the beta distribution for different values of the parameters        α   and    β:




         As we see fro m fig . 4 .1 the s hape o f th e Beta d is tribu tion is de ter mined b y th e

     parameters     α      a nd      β.   Thes e par ame ters ar e us ua lly es tima ted in th e fo llowing

     way:




30
                                  ⎛ μ ⋅ (1 − μ ) ⎞                       ⎛ μ ⋅ (1 − μ ) ⎞
                         α = μ ⋅⎜               − 1⎟ a nd β = (1 − μ ) ⋅ ⎜             − 1⎟                (4.7)
                                  ⎝ σ              ⎠                     ⎝ σ              ⎠
                                           2                                      2




where    μ    a nd   σ   ar e t he me an a nd s ta nda r d d e vi a t ions o f th e r ec o v er y r a t es . N IBC

Ba nk N .V. has a data se t o f r esiden tia l mor tga ges fro m w hich we o b ta in th e

fo llow ing ch arac teris tics o f r eco ve r y ra tes :


               Number of Losses            Average RR         Std. Dev. RR      Average LGD
                              860               89.59%              19.19%               10.41%



                                                                             ⎛ μ ⋅ (1 − μ ) ⎞
a nd   t he    α     and      β      p a r am e ters   b ec o me :   α = μ ⋅⎜              − 1⎟ = 1.7185     and
                                                                             ⎝ σ              ⎠
                                                                                      2



               ⎛ μ ⋅ (1 − μ ) ⎞
β = (1 − μ ) ⋅ ⎜             − 1⎟ = 0.4537 .
               ⎝ σ              ⎠
                        2




    O nce w e ha ve ca librate d th e corr ect Be ta dis tr ibu tio n we can use it to s i mu la te

r and om r ec o v er y r a tes a nd c om b in ing thos e w i th the d is tr ib ut io n o f th e e x pec te d

p rob ab ility o f de fau lt, ob tained fr om our PD mo de l, we can d e ter min e the

d is tr ib u ti on o f t he e x pec t ed los s d ue to de fa ul t o f a s in gl e m or tga ge c o n tr ac t or o f

a por tfo l io o f mor tgage lo ans .




                                                                                                                     31
                                                    V.         Results 


                 5.1       Cox PH model estimation and regression results 

          I n t he pre v io us c ha p ter w e p r op osed us in g 4 di s ti nc t d e fa ul t pre dic tor s

     (O r i gi na l LT V , Ori g ina l D T I , U ne mp loy ment r at e a nd Interes t rate ) for b u i l d i n g t h e

     mos t su itab le p rop ortion al h aza rd ra te mod el. Afte r r unn ing a ser ies o f Co x

     r egress io ns w e fo und ou t tha t th e firs t d e fau lt factor – Or ig in al L TV – h as no

     s ta tis tica lly s ign ifica nt e xplan ato r y pow er (s ee App end ix for ac tua l resu lts) .

     T her e for e we w ill remo ve it from o ur mo de l an d fr om n ow on we w ill use o nly 3

     fac tors for mode lin g th e haz ar d ra te o f time to d e fau lt. Le t us na me the 3

     r ema in in g co var ia tes as follows :


      X DTI ,i    Or i gi na l D eb t- t o- Inc om e r a t io o f a borr ow er i


      X UN ,i     Un emplo ymen t r a te o f the q uarter prec ed in g de fau l t e ven t o f borr ower i

                  E ur ib or 3-m on t h I n ter es t R a te ( m on t hl y a ve r ag e o f the mo nth prec ed ing
       X IR ,i
                  d e fau l t e v en t o f bor r ow e r i )

     Table 5.1 Covariates

          Rec all that in our mod el the h azar d ra te for the d is tribu tio n o f mo rtg age

     d ur a t io n ( l ife t im e) is gi v en b y the f o l low ing :


                                                   λ (t | X i ) = h(t ) * exp(β ′X i )


          T o ob tain es tima tes o f th e be ta coe ffic ien ts an d the b ase line haz ard fu nction ,

     we use the bu ild- in Co x pro por tio na l haza rds r egress ion function – ‘coxp h fi t ’,

     w h ich is inc lud ed in th e Sta tis t ic a l T oo lb o x® o f M a tl ab ® .




32
     T he ma x i mu m l ik e li hoo d es t i ma t io n o f t he b e ta c oe f f ic ien ts has p r o duce d t he

f o l lo w ing r e s u l ts :


              Covariate:         DTI - X DTI ,i     Unemployment - X UN ,i         Euribor 3m - X IR ,i
           Beta coefficient      2.792984442              37.66940855                  60.52762111
                p-value          8.02754E-49              2.00197E-84                  1.0386E-148
            standard error       0.190191281              1.934816002                  2.330536474
              z-statistics       14.68513399              19.46924592                  25.97153994
Table 5.2 Coefficient estimates


                                                                    t
     A n d t he C u mu la t i ve H az ar d Func t io n     H (t ) = ∫ h( s ) ds f or th e d is tr i bu t io n o f
                                                                    0


mor tg age du ra tion has th e fo llowin g sha pe :




Figure 5.3 Baseline Cumulative Hazard



     We ca n s ee th a t the cu mu la tive bas elin e h azar d func tion h as ve r y low va lu es

e ven for high du ra tions . Th is of c ourse is w ha t w e e xpec ted s inc e mor tg ag e

d e fau lt is a ver y rare e ven t. O ur es tima te o f the cu mu la tive base l ine h azard

fu nc tion is o nly given for dur a tions less th an or eq ua l to 305 months , w hic h is the




                                                                                                                    33
     ma xima l du ra tion o f a d e fa ulted lo an in ou r da ta base . Th is does n o t cons titu te a

     flaw in the mod el s ince in practice we a lmos t n e ver ha ve to a na l yze mo rtga ge

     co n trac ts th a t ha ve be en o u ts tan ding fo r mo re th an 25 years ( 300 mo nths) .


          Ex amp le:

          L e t us now us e o ur haz ard r a te m od el t o c o mp u te th e p r ob ab il i t y t h a t a s pec i fic

     m or tg age c o n tr ac t w il l de fa ul t in t he n e x t y e ar ( i. e . t h e e xp ec te d 12- mo nth PD) .

     Co nsider a mor tg age loa n l w i th o r i gi na l D TI r a ti o – 3 0% t h a t has b een iss ue d on

     0 1 /07 /200 6 an d sup pos e tha t th e c urre n t qu ar ter l y un emplo ymen t ra te in the

     Ne ther la nds is 4 .8% an d the cu rren t mon thly a vera ge o f the 3- mon th Eur ib or

     i n te r es t     r at e      is      1 .5 % .     In     o th er     w or ds        we    h a ve :   X DTI ,l = 30% ,

     X UN ,l = 4.8% , X IR ,l = 1.5% . T he n ac c o r d ing t o (3.4) the cumulative hazard fu nc tion

     f or t he d is tr ib u ti on o f t h e l i fe t i me f or t his s pec i f ic l oan i s :


                       Λ (t | X l ) = exp(β ′X l ) H (t ) = exp(β1 * 0.3 + β 2 * 0.048 + β 3 * 0.015) * H (t )


     a nd th e       β ’s   a r e g i ve n in t ab le 5 . 2.


          M or e o ver , b y d e fi ni t ion 2 . 1 , t he c u mu la t i ve d is tr ibu t ion fu nc ti on fo r t he dur ati on

     of    l is g i ve n b y : Fl (t ) = P ( Dl ≤ t ) = 1 − exp(− Λ (t | X l ) ) an d it has the fo llow ing

     sh ape :




34
Figure 5.4 Cumulative Distribution Function



     N o t e tha t t h e s ha pe i s ver y s im i lar t o the s hap e o f t h e c um ula t i ve base li ne

h az ar d func t io n . T hi s is b ec a use o f th e mu l ti pl ic a t i ve           r e la t io ns h ip b e tw ee n

b ase line haz ard        H (t ) a nd spec ific haz ard Λ (t | X l ) and the fac t that 1 − exp(− x) ≈ x

f o r s ma l l x .


     Now s ince       θ l = 37    ( m on ths s i nce da te o f iss ue) an d follow in g ( 3 .2) , the

e x pec te d 12 - mo n th p r o bab i li t y o f d e fau l t for m or tg age c o n tr ac t l is:


                                             P( Dl ≤ 49) − P( Dl ≤ 37) 0.01603 - 0.01071
 P (τ l ≤ 12) = P( Dl ≤ 49 | Dl > 37) =                               =                  = 0.005377
                                                    P( Dl > 37)           1 - 0.01071




                     5.2         Expected Loss and Loss Distribution 

     T he e xpec te d los s due t o d e fau lt o f a mor tg age c on tr ac t i can be c harac ter ize d

b y t he fo l low ing :




                                                                                                                         35
                                Ε( Loss ) = PD * Ε( LDG ) + (1 − PD) * 0 = PD * Ε( LGD)                                     (5.1)


     a nd f or th e m or t gag e c o n tr ac t l t h a t w e c ons ide r ed in t h e pr e vi ous e xa mp le , w e

     g e t the f ol low in g e xp ec te d l os s :


                                    Ε( Loss ) = 0.005377 * (1 − 0.8959) = 0.056%


         As we men tio ned befor e , fro m r isk mana ge me n t’s poin t o f view, n o t on ly th e

     e xpec te d loss is impo rtan t bu t a lso th e unc er ta in ty a rou nd it – in other wo rds we

     a r e m or e in t er es t ed in t he w h ole pro bab i li ty d is tr ib u tio n o f th e exp ec ted los s .


         Since in ou r mode l th e PD depe nds on two unc er tain in the fu ture fac tors ,

     n am el y   une mp lo y ment       a nd    in te r es t   r at es ,   th e   s toch as tic   d is tr ib u ti on   of    t he

     e xpec te d PD will be de termin ed b y th e s toch as tic d is tribu tio n o f th ose fac tors . Or

     p u t in o t her w or ds , w e c a n der i ve th e dis tr ib u ti on o f th e e xpec t ed PD b y s im u la t ing

     a l ar g e n um ber o f poss ib le r ea l iza t io ns fo r u ne mp lo yme n t a nd in te r es t r a tes . I n t h e

     s a me w a y w e c an use t he be ta d is tr ib u ti on t ha t w e e x pl ai ne d a nd es ti mat e d in

     sec tio n 4 .4 , to s imu la te a numb er o f LGD re aliza tio ns . C ombin ing PD an d LGD

     s imula tions, w e a re ab le to der ive the whole dis tribu tio n o f the loss ar is ing fr om a

     d e fau lt o f a mor tg age con tr ac t (o r a p or tfo lio o f mo r tga ge con tr ac ts as w e w ill see

     in the next sec tion) .


         T he s imulate d pro bab ility dis tr ib utions o f PD, LGD as we ll as the dis tr ibu tio n o f

     th e e xp ecte d Loss fo r this spec ific mor tg age c on tr act are pr esen ted in th e

     Ap pen dix (figu res A11 , A 12 a nd A 1 3) .




36
                                  5.3         Scenario Simulations 

      T o s i mu la te in t er es t r a t e pa t hs w e use a tec hn iqu e w i de l y us ed i n m a the ma t ica l

f i na nce . W e as s um e t h a t t he e v ol u ti on o f in t er es t r a t es is a mea n r ev e r t i ng

p r ocess t ha t c an be d es c r ibe d b y t he C o x - I ng er s o ll - R os s ( C IR) mo de l (Cox,

In gers oll an d R oss [1 7 ]) .         The mode l spec ifies that the sh or t ter m in teres t r ate

fo llows th e fo llow ing s toc has tic d i ffe ren tia l e qua tio n :


                                         drt = a(b − rt )dt + σ rt dWt                                         (5.2)



w h e r e a is the mean revers ion parameter , b i s th e l ong - t er m mea n ( e qu il ibr iu m

l e ve l) ,   σ   is th e volatility and Wt is a s tan dar d Brow nian mo tion . T he C IR mo de l is

a n e x te nsio n o f th e w e l l- k n ow n V as ic ek mo de l ( Vas icek , Ol dric h [ 1 8 ]) .


      S i mu la t in g u ne mp lo yme n t r a tes is a b i t tr ick ier s ince in th e l i te r atur e ther e are

n o c lass ic a l mod els d es c r ib ing u ne mp lo ym en t e vo luti on as a s toc has t ic r and om

v a r i ab le . Th ere a r e , h ow e ve r , a num ber o f mod els t h a t use A R M A or A R C H

r egress io ns to for ecas t une mp loymen t ra tes . We ha ve to make it c lea r he re , th a t in

th is     p aper     we    a re   n ot    in ter ested   in    for ecas tin g    the     mos t      like ly   fu tu re

u ne mp lo yme n t r a te , bu t in a ll p os s ib le r e al iza t io ns o f u ne mp lo yme n t r a tes , i.e . w e

a r e in t er es t e d in t he w h ol e s t och as tic d is tr i bu t io n o f u ne mp lo yme n t . Tha t is w h y ,

to gen era te our theo re tica l une mp lo yment re a liz a tio ns , we use the h istor ic al

d is tr ib u ti on o f the r e l at i v e c ha nge o f u ne mpl o y men t ( qu ar ter l y) for the las t 9 y ea r s .

We ha ve de termin ed (s ee Ap pend ix) tha t for this time p er iod the d is tr ib u tion o f the

r el a ti v e c ha nge i n une mp lo y ment i s b es t fi tte d b y a no r ma l dis tribu tio n w i th me an




                                                                                                                         37
     μ un _ r = 0.00193    a nd s t an da r d d e vi a tio n   σ un _ r = 0.04034   r espective ly. T he id ea is to

     m od el a poss ib le e v olu t io n o f u ne mp lo y ment i n th e fo llow in g w a y :


                                              unt +1 = unt * (1 + un _ rt )                                           (5.3)



     w h e r e unt is t h e le ve l o f u ne mp lo ym en t a t t im e t a nd un _ r           ~ N ( μ un _ r , σ un _ r ) is a

     n or m al l y- dis tr ib u ted r a ndo m var ia bl e , w i t h th e ab o ve m ea n an d s ta nda r d de v ia t io n .

     un _ rt re pres en ts th e re la tive c han ge in une mp loymen t fr om time t to time t + 1 .


         We w i ll r un o ur simu lations w i th 3 bas ic une mp lo yment sce nar ios. T he firs t o ne

     we w i ll ca ll s ta nda rd . In this sc en ar io we assu me tha t th e mon thly re la tive ch ang e

     o f un emp lo y me n t w i ll k e ep i ts h is t oric al m ea n a nd s t an dard d e v ia t ion a nd t he

     f u t ur e e v olu t io n o f un em pl o ymen t w il l be d es c r ibe d b y ( 5. 3) w i t h the h is t or ic al

     m ea n   μ un _ r = 0.00193    an d s ta nda rd d e via tion      σ un _ r = 0.04034 .    In this sce nar io the

     e v ol u ti on of u ne mp lo ym en t is des c r ib ed b y p a ths t ha t a r e a l mos t e v en l y d is tr ib u ted

     a rou nd the c urre n t leve l o f unemp lo ymen t a nd h a ve a sligh t upwa rd tr en d ( me an is

     p ositive but small, see Appe nd ix) .


         O n th e o the r h and , in th e c urre nt cr ed it cr isis we e xp ec t tha t unemp lo ymen t w i ll

     r ise ( and it h as be en r is ing for so me time alre ad y) . Alth oug h we a re n o t tr yin g to

     f or ec as t the mos t l ik el y f u tur e un em pl o ymen t w e h a ve t o tak e in to acc oun t th a t a t

     th is momen t it is much mor e likely tha t unemp lo ymen t w ill incr ease s ubs ta ntia lly in

     t h e ne ar fut ur e . T he D u tc h C en tr al Ba nk ( D e N ede r l an ds c he B ank – D N B) h as two

     s tr es s tes ti ng s c e nar ios fo r t he u ne mp lo yme n t r a te in t he N e t her la nds . The f ir s t

     o ne – DNB base scenar io s u g g e s ts tha t t h e u n e mp lo y me n t r a t e w i l l b e a r o u n d

     5 . 5 % a t t he en d o f 20 09 and aro und 8 .7 % a t t h e e nd o f 2 010 . T he s ec o nd on e –




38
t h e DN B s tr ess scenar io s ug ges ts t ha t u nem p lo ym en t r a te w i ll be aro un d 5.7 % a t

t h e en d o f 2 009 and ar oun d 9 .7 % a t t he e nd o f 201 0 .


    In o rder to simu la te th e e vo lu tion o f u ne mp lo yme n t th a t c orresp onds to th ese

s c e nar ios w e s im pl y a dj us t t he d is tr ib u ti on o f the r el at i v e c ha nge in u nem pl o y men t

i n th e fo l low ing w a y – w e k e ep t h e m on thl y s tan dar d d e v ia t ion t h e s a me , b u t w e

i nc r ease t he m ean o f t h e n or m al ly d is t r i bu ted r a ndo m va r i ab le un _ rt i n ( 5. 3) . As a

co nseq uence we g e t s imula ted p a ths with much h ig he r u pwar d tre nd , co mp are d to

the s tand ard sce nar io (s ee Appen dix fo r actu al r esu lts figur es A 10 , A 1 1 and A12 ).

T hes e scen ar ios re flec t be tter th e curr en t macroeco no mic ou tlo ok and s ho uld

p r od uce mo r e r ea l is tic e xp ec ta t io ns o f t he f u tur e PD’ s and e x pec t ed l os s es for

r esiden tia l mor tg ages .


    An     o ver view     of    une mp lo yment        an d    in te res t   ra te   simu la ted   scen ar ios   is

p rese n ted in the Ap pen dix ( figures A 7 to A 1 2) .




               5.4          Loss Distribution of RMBS Collateral Pool 

   T he co lla ter al po ol o f a typ ica l RMBS transac tio n us ua lly co ns ists o f th ousa nds

o f mor t gag e c on tr ac t s . In o r de r t o ob t ai n th e l os s d is tr ib ut io n for th e w h ole

mor tg age p or tfo lio we jus t su m the e xpecte d abso lute losses o f a l l lo ans in the

p oo l an d exp r ess the r es u l t as a p er c e n tag e o f th e to tal ou ts t and in g ba la nce o f th e

p oo l . Aga in, b y s i mu lat i ng a lar ge nu mbe r of s ing l e m or tg ag e l os s es w e a r e a bl e to

o b ta in th e los s dis tr i bu t io n o f the w ho le c o ll a te r a l po ol.




                                                                                                                      39
          Example: Storm 2007-I B.V.


          Stor m 2 007 -I B.V. is a EUR 2bn tru e sa le sec ur i tiz a tion trans ac tio n o f mor tg age

     l oa ns , o r i gin a ted i n the N e th er lan ds b y Ob vi on N . V . T he C o ll a ter al P o ol t h is R MB S

     t r a ns ac t io n c a n be br ie f l y c h aract er iz ed b y th e fo ll owi ng :


                     Or igina l Princ ipa l Ba la nce :                                 2 047 484 1 81 €

                     Nu mb er o f Bor rowers :                                                 1 0 49 9

                     Aver age Loa n Per Borr ower                                            1 94 3 09 €

                     We ig h ted Aver age Se ason in g (mon ths)                                   6

                     W e ig h ted A v er age Ori g ina l L T V                                 8 3 .7 %

                     W e ig h ted A v er age D T I                                             2 9 .7 %

     Table 5.3 Collateral Pool



          U n f or tu na te l y w e d o n o t ha ve de t ai le d i n for ma t ion o n eac h s i ng le loa n c on tr ac t

     a nd w e w il l use t he w e ig h ted a ve r ag es in ou r PD m ode l . T his o f c o urs e m ea ns t ha t

     i n o u r s i m u l a t i o n a l l m o r tg a g e s w il l h a ve the same pr obability of default, which is

     n o t the c a s e in r e a l i t y. W e h o p e t h a t th is l a c k o f d e ta i le d i n for ma ti o n w i l l n o t h a ve

     a c r uc ia l im pac t o n an al yz in g the R M BS t r a ns ac t ion .


          T o o b ta in t h e los s di s tr ibu t io n o f t he P ool , w e f ir s t g en er a t e 5 000 r a ndo m

     r ea liz a tions o f une mplo ymen t and in teres t ra tes . We use those in o ur PD mo de l to

     obtain 5000 realizations of a single mor tgage contrac t pr obability of default. In

     e ac h d i f fere n t s i mu la ti on a l l c on tr ac ts ha ve t he s a me P D . Fo r each o f th ose 5 000

     s imula tions, we s imula te a ra ndo m LGD va lu e (as desc ribed in s ec tion 4 .4) for

     e ac h o f the 1 0 499 lo ans i n t he p oo l . The s u m o f th e abso lu te los s es o f a ll loa ns

     e xpress ed as a perc en tage o f the ou ts ta nd in g ba la nce is then the e xpec ted loss of

     the pool. In this example we will s imula te one ye ar per iod and calculate the loss




40
d is tr ib u tion for 1 yea r. L e t us firs t co ncen tr ate on the s tand ard sce nar io fo r the

s im ul a ti on o f u ne mp loym en t . Si mu la t io ns f or o t her s c en ar ios are p r ese n ted in the

A p pen di x . F i gure 5 .3 r epr esen ts th e e x pec t ed los s d is tr ib u ti on f or th e a bo ve

co lla te ra l po ol o f mor tg ages .




Figure 5.3 Expected Loss Distribution (Standard unemployment Scenario) – results for other

scenarios are presented in the Appendix




    T he    fo llow ing    ta ble   su mmarizes       some     imp or tan t   in    risk   ma nag emen t

ch arac teristics o f th e loss dis tr i bu tio n :


                                     M ea n                  0 . 042 2 %

                            Sta ndar d De via tio n          0 .016 6 %

                                9 5 %- qu an t il e          0 . 073 5 %

                                9 9 %- qu an t il e          0 . 099 1 %

Table 5.4 Loss distribution characteristics




                                                                                                               41
                5.5           Loss Distribution and Defaults of RMBS Notes 

          L e t us a gain foc us on Sto rm 20 07- I B.V. T he No tes s truc ture o f this RMBS

     t r a ns ac t io n i s the f ol low ing :


         C lass            Ra tin g              Size ( %)     Size ( EURm)           Cr ed it Enha nce men ts

           A1                AAA                   1 0 .0            2 00                       5 .00 %

           A2                AAA                   1 7 .0            3 40                       5 .00 %

           A3                AAA                   6 9 .0          1 380                        5 .00 %

            B                 AA                   2.0                40                        3 . 00 %

            C                 A+                   1.2                24                        1 . 80 %

            D                 A-                   0.8                16                        1 . 00 %

            E               BBB-                   1.0                20             Excess Spread – 0.5%

     Table 5.4 Notes Structure



          T he C r e di t E nha nce me n t ( C E) is l ink ed to c r ed i t q ua li t y – i t is a c us hi on t h at

     p ro tec ts inves tors (no tes ho ld ers) aga ins t losses tha t ar ise fro m th e und er lying

     p oo l .


          In th is RMBS tra nsaction ther e a re 3 types o f cre dit en hanc ement. T he firs t a nd

     m os t c o mmo n is – s u bo r d ina t io n . S u bor di nati on me ans t h a t a gi ve n tra nc he ( c las s

     o f no tes) b ears an y loss es on ly if th e tr anch es jun io r to it h a ve bee n fu lly

     e xhaus ted . In o ther wo rds loss es are pr opa ga te d fr om C lass E to Class A1 no tes.

     S u bord in a ti on is o ne o f t he g r o und in g pri nc ipa ls o f s ec ur i tiz at i on . I n th is R MB S

     tra nsac tio n s ubo rd ina tion pr o tec ts all the no tes e xc ept C lass E no tes . The sec ond

     typ e o f cred it en hance me n t is th e Excess Sp re ad ( XS) – it protec ts the C lass E

     n o tes . The X S i s th e d i f fe r enc e b e tw ee n int er es t p a ym en ts der ive d fro m th e p oo l




42
o f m or tgag es a nd t h e w e i gh te d a v era ge c ou po n p ai d on th e n o tes . F or t h is

t r a ns ac t io n t h e X S is g uar an te ed b y a s w a p ag r ee me n t an d is f i x ed on 5 0 basis

p oi n ts pe r a nnu m . Th e th ird c r e di t enh ance me n t is t h e r es er ve f und – th is is a

la ye r o f pro tec tion tha t h as to be e xhaus ted b e fore no te ho lders be ar an y loss es .

A t or i gi na t io n o f t he tr ansac t io n t h e r es er ve f un d is f u nde d b y th e is s ua nc e o f the

E no tes , i .e . th e S PV iss ues a n e x tra tr ance o f eq ui t y n o tes ( a lso s om e t imes

ca lled ‘tur bo no tes ’ - in th is cas e the E n o tes ) to c reate a bu ffer o f pr o tec tio n for

th e tranc hes se nior to th e eq uity n o tes .


    We c an think o f the cr ed it en ha nce men ts as th e thr esho lds th a t ne ed to be

cr ossed so tha t cer tain c lass is a ffec ted b y a loss ar ising fro m the co lla ter al po ol.

S i nce l os s es f r o m the po ol ar e t r ans f er r ed t o t he no t es fr om th e mos t jun io r ( in

this case C lass E notes) to the mos t s enior ( the A1 C lass) , a loss to a cer tain

c lass o f n otes on ly occu rs if the ac tua l loss is big e nou gh to fu lly e xha us t a ll the

tra nch es jun io r to it, i.e . if the loss cr osses the cr ed it en hanc emen t thr esh old o f

t h is tra nc he .


    F i gure 5 .4 rep rese n ts the loss d is tribu tion o f the no tes for the Stor m 20 07-I

B.V. R M BS tra nsac tion a fter a 50 00 s imu lations (w ith th e Standa rd scen ario for

u ne mp lo yme n t) for a 1 y ear per io d . H er e ag ai n w e use t he c ur r en t une mp loy m en t

o f 4.8 % an d curr en t 3 mon th Eu ribor 1 .5% as s tar ting values o f th e sce nar io

s imula tions.




                                                                                                                    43
     Figure 5.4 Loss Distribution of the Notes (1 year simulation period)



         We ca n s ee fro m figur e 5 .4 tha t in th is c ase th ere is n o ac tua l loss for th e no te

     h ol ders . Ac c o r d ing t o o ur mo de l, a l l th e los s es ar isi ng f r o m d efa ul ts o f mor tg ag es

     i n th e c o ll ater a l po ol a r e a bs or be d b y th e ava i lab le e xc es s s pr e ad .


         L e t us n ow conc en tra te o n a s i tua tio n w here u nemp loymen t will r ise ra pidly in

     t h e ne ar fut ur e – as pr ed ic ted b y t h e bas e an d s tr es s s c e nar ios o f t h e D N B .




     Figure 5.5 Expected Loss Distribution of the Collateral pool (DNB Base Scenario)




44
Figure 5.6 Expected Loss Distribution of the Notes (DNB Base Scenario)



    A s w e c an s ee fr om f i gu r e 5 .6 , acc or d in g to ou r m ode l , e ven i f u ne mp lo yme n t

incr eases ra pidly as pr ed ic ted b y th e D NB bas e in scen ar io ( to a le ve l o f 8 .7 % b y

t h e en d o f 2 010 ) , t here w i ll be no los s es to t h e N o tes o f t his par ti c u l a r R M BS de a l.

O n the o the r ha nd in th e DN B s tress sc en ar io th ere occ ur some losses to the E

No tes bu t ag ain thos e loss es ar e r elative ly rare - on ly in 10 o u t of 10 00

s imula tions th e loss is b ig eno ugh to a ffec t c lass E Notes (s ee fig 5 .8) .




Figure 5.7 Expected Loss Distribution of the Collateral pool (DNB Stress Scenario)




                                                                                                                   45
     Figure 5.8 Expected Loss Distribution of the Notes (DNB Stress Scenario)




          W e h a ve d one al l of t he abo ve s im ul a tio ns as s u mi ng th a t the r e w i l l be no

     s ign i f ican t d ec l in e i n th e r es ide n ti a l pro per ty m ar k e t in t he N e therl an ds , w h ich w i ll

     e ffec t th e p roce eds fr om s ellin g th e co ll a te ra l o f de fa ulted loans . T his so ca lled

     M ar k e t V a lu e D ec l ine ( M VD) w il l l ow er t he r ec o ver ies f r o m the c o ll a ter al a nd

     i nc r ease t he e xp ec te d L GD an d th ere f ore w il l h a ve an im pac t on t h e e x pec t ed los s

     o f the mortg age poo l a nd th e loss o f the RMBS Notes . For tun a te l y ou r mode l

     a l lows    us    to   ta ke    th is      poss ib le   market     va lue    dec line     in to   acc oun t     in   o ur

     s im ul a ti ons, s im p l y b y a d jus t ing t h e    α   a nd   β   p ar a me te r s o f t h e b e ta d is tr ibu t ion

     t h a t w e use fo r s imu la t in g r ec o v er y r a tes . Fo r e xa mp le w e c a n s im ul a te a n

     e x t r em e c as e w her e th e D N B s tr es s s c en ar i o f or u nem p lo ym en t w as to c om e tr ue

     to ge th er w ith a MVD o f 20 % fo r a 1- ye ar ho riz on . MVD of 20 % simply me ans tha t

     r ec o ver y r at es w i ll f al l w i th 2 0% o n a ver ag e. T he los s d is tr ib u ti on o f th e C o lla t er a l

     p oo l a nd th e los s to t h e N o t es f or t h is e x tr eme s c ena r i o ar e pres en te d in f i g ur es

     5 . 9 an d 5 .91 , r es p ect iv e l y .




46
Figure 5.9 Expected Loss Distribution of the Collateral (DNB Stress Scenario w ith 20% MVD)




Figure 5.91 Expected Loss Distribution of the Notes (DNB Stress Scenario with 20% MVD)




    We can see he re that in th is extre me case the re are sign ifi c an t nu mb er o f

s imula tions (7 67 o u t of 50 00 to be e xac t) in wh ich th e e xpec te d loss is b ig e nou gh

to h i t c lass E no tes . T h ere are e ven 5 cas es in w hich the c lass D no tes s u ffer

losses an d 1 case in wh ich the C n o tes are h it. So if th is sc enar io is a very lik e ly

s c e n a r io w e c o u ld a s s ig n a probability of default   767          = 15.34% f o r th e E n o t e s
                                                                        5000




                                                                                                                 47
     o f this R MBS trans action , mea nin g tha t th er e is a 15.34% c ha nc e th a t E no t es w i ll

     suffer losses .


         W e ha ve to men t io n a ga in h er e t h at t he go al o f t his p r oj ec t is n o t t o i den t i f y

     t h e mos t l ik e l y d e ve lo pm en t o f f u t ur e u nem p lo ym e n t or in t er es t r a t es or the f u tur e

     mo ve men ts o f th e res id en tia l p rop er ty marke t in th e N e the rlands . Wh a t is

     impor ta n t for us h er e is tha t our mode l a l lows th e user to inp u t his own

     e x pec ta t ions o f these var ia b les an d t o a dj us t th e a na l ys is ac c o r d ing to h is own

     view of th e macr o ec onomy. Our model presents a w a y to analyze the default r isks

     ass oc i a ted w i th R M BS t r a ns ac t io ns for any p o s s i b le e v o l u t ion o f une mp loy m en t ,

     in te res t ra tes and res id en tia l pr op er ty marke t.


         T ab le 5 s u mm ar izes t h e r es u l ts w e ha v e o b ta in ed a f t er r unn in g a s er ies of

     s imula tions w i th differ en t ass ump tions ab ou t th e e volu tions o f the s toc has tic

     fac tors in ou r mo de l in 1- year horizon .




48
                                                Notes Probability of Default
                                         MVD           Standard       DNB Base DNB Stress
                          A1 Notes                 0%       0.00%           0.00%   0.00%
                                                 10%        0.00%           0.00%  0.00%
                                                 20%        0.00%           0.00%  0.00%
                                                 30%        0.00%           0.00%  0.00%
                          A2 Notes                 0%       0.00%           0.00%   0.00%
                                                 10%        0.00%           0.00%  0.00%
                                                 20%        0.00%           0.00%  0.00%
                                                 30%        0.00%           0.00%  0.00%
                          A3 Notes                 0%       0.00%           0.00%  0.00%
                                                 10%        0.00%           0.00%  0.00%
                                                 20%        0.00%           0.00%  0.00%
                                                 30%        0.00%           0.00%  0.00%
                          B Notes                  0%       0.00%           0.00%  0.00%
                                                 10%        0.00%           0.00%  0.00%
                                                 20%        0.00%           0.00%  0.00%
                                                 30%        0.00%           0.00%  0.00%
                          C Notes                  0%       0.00%           0.00%  0.00%
                                                 10%        0.00%           0.00%  0.00%
                                                 20%        0.00%           0.00%  0.02%
                                                 30%        0.00%           0.00%  0.08%
                          D Notes                  0%       0.00%           0.00%  0.00%
                                                 10%        0.00%           0.00%  0.00%
                                                 20%        0.00%           0.04%  0.10%
                                                 30%        0.00%           0.12%  0.64%
                          E Notes                  0%       0.00%           0.06%  0.20%
                                                 10%        0.00%           1.12%  3.96%
                                                 20%        0.02%           6.42% 15.34%
                                                 30%        0.06%         16.86%  32.32%
Table 5 - Probability of default for Note’s tranches in different scenarios




      O ur mod el a l s o a ll ow s u s to s i mul a te t i me ho r iz ons di f fe r en t t h a n 1 - year , w h ich

is the s tand ard in man y c red it r isk man agemen t too ls . Simu la tion resu lts fo r time

h or izons o f 2 an d 3 ye ars are p res en te d in th e Appe nd ix ( figur es A18 to A21 ). We

o nly pr esen t th e s imu la tio n r esu lts for the DN B base sce nar io o f une mp lo yme n t

a nd 2 0 % m ar k e t v alu e dec l ine b ec aus e w e be li e ve t h is is t he m os t l ik e ly o ne in

the     c ur r ent   eco no mic      s i t ua t ion .   Of   c ou r s e   o th er   s c en ar ios   c an   be   eas i l y

ca lcu la ted . We no te th a t losses for long er p eriods ar e b igg er b ecause the

p rob ab ility o f de fau l t o f mor tga ge con trac ts fo r a lo nge r time h orizon is h igh er and

co nseq uently the e xpec ted loss o f the p or tfo lio is h ighe r.




                                                                                                                            49
                                               VI.           Conclusion 

          I n t h is p ap er w e h av e pr esen te d a n a ppr oach f or m od el in g th e d is tr ib ut i on

     functi ons of the Pr obability of Default, L o s s G i ve n D e f a u l t a n d c o n s e q u e n tl y t h e

     E x p ec t ed L os s fo r a m or t gag e c o n tr ac t or a p or t fo l i o o f mor t ga ge c on tr ac ts . W e

     h a ve c o nsid ere d the d is tr ib u ti on o f mor t gag e l i fe t i me as w el l as t h e d is tr i bu t io n o f

     t i m e to d e fa ul t a nd t he ass oc i a te d c ond i t ion al in te nsi ty proc es s es , g i ven the s e t o f

     p r ed ic tor s f or t he d e fa ul t e ven t . W e h a ve m od el ed the in t ens i t y ( h azar d r a t e) o f

     mor tg age du ra tion as a fu nc tion o f tw o macr oeco nomic co va ria tes (u nemploymen t

     a nd in ter est ra tes ) and o ne mo r tg age -spec ific variab le (de b t- to- inco me ra tio) . Our

     h az ar d r ate m ode l tu r ne d o u t t o be v er y s ui t ab le in t r a ns l a tin g th e s t och as tic

     b eha v io r o f t he mac r oeco nomi c var ia bl es to t he b eha v io r o f e xp ec t ed an d

     u ne xp ec ted PD a nd th e s toch as t i c d is tr i bu ti on o f t he E x p ec t ed Loss f or a po r t f o li o

     o f mor tg age lo ans . Ou r mod e l is fle xib le w i th res pec t to th e ch oic e o f d e fau lt

     p red ic tors ( as long as th is ch oice is ec ono mica lly s ou nd) . The mod el is a lso ver y

     we ll s uited fo r per forming s tress tes ting on o ne or all o f th e factors th a t influe nce

     th e cash flows o f RMBS tranc hes .


          We     ha ve      es tima te d   th e    mo de l    us in g    a    non- para me tr ic       pa rtial     lik eliho od

     a ppr oach b ased o n th e C o x Pr opor tio na l H azar ds mod el, wh ich e nab led us to

     es t i ma te the e f fec ts o f d e fau l t p r e d ic t or s to t h e de f au l t i n te nsi t y pr ocess .


          We ha ve th en r un a ser ies of s imula tions an d d eter m ine d th e s toc has tic

     d is tr ib u tion o f the Expec ted L oss o f a co lla te ra l po ol of mor tg ages a nd tr ans la te d

     t h ese e x pec t ed los s es t o the N o t es o f t h e a t yp ic a l R MBS t r a ns ac ti o n . Th is

     a ppr oach m ak es i t poss ib le t o a na l yze t h e los s d is tr ib u ti on o f d i f fer ent R M BS

     t r a nc h es   a nd     gi v es   i ns ig h ts   of    t he   d e fau l t   r is k s   as s oc i a te d   w i th   R MBS




50
t r a n s ac t io ns . T he m od el c an be us ed for th e an al ys is o f th e de f au l t r is k o f an y

D u tc h R MB S . A l th oug h the mod el w as des ig ned t o a na l yze R MB S tra ns ac t ions

o rigina te d in the Ne ther la nds , it ca n be e asily ad op te d for o th er c ou n tr ies ,

p r o v ide d th a t e no ugh d at a on t h e h is tor y o f r es id en t ia l mor tg ag e d e fa ul ts is

a vaila ble .


    F ur th er rese arch has to be do ne in the d ir ection o f b e tter hand ling missin g

co var ia te va lu es . Th is co uld impr o ve the es timation of th e c oe ffic ien ts for th e PD

mod el an d c onse quen tly - the s ens itivity a nd sign ificanc e o f cer ta in d e fa ult

p r ed ic tor s . A n o the r poss ib le i mp r o ve men t c ou ld b e t ak in g in to a c c ou n t a h az ar d

r at e mo de l t ha t a ll ow s for ti me- d epe nde n t c o v ar ia tes , w h ich w ou ld a ls o s ol v e the

p r o b le m w it h mis s i ng c o v ar ia te obs er va t io ns .


    O t her pos s i b le i mpr ove me n ts to t h e mo de l c an i nc l ud e mod el ing t he LG D i n

d epe nde nce o f LT V. We k now th a t the lo an -to- va lu e r atio has an impac t on the

r ec o ver y r a t es a nd c a n b e use d t o be t t er d es c r ibe r es ide n ti al mo r t ga ge L GD . This

c a n b e d one f or e xa mp le , b y es tim a t ing a d if f er en t be ta d is tr i bu t io n for pr ede f in ed

L T V bucke ts – b u t for re liab le es timates a lar ger ( th an th e on e we h ad) d ata base

o f L GD’s is re qu ired . T he d epe nd ence s tr uc tur e o f LGD a nd PD on o ne ha nd , and

u ne mp lo yme n t an d in te r es t r a t es o n th e o the r , s ho ul d a ls o be in ves t ig a ted . B as ed

o n th e da ta w e us e w e ha v e fou nd no s ign i f ican t c or r el a ti on b et w e en PD a nd LGD

b u t th is c ou ld be du e t o t he f ac t t h a t th e t im e per iod t h a t w e base ou r es t ima t es o n

( 200 2-20 08) is a re lative ly s table p er iod w i th n o e xtre me econ omic sh ocks . A

d a tab ase th a t s pans o ver a large r per iod of time a nd inc lud es da ta from eco no mic

d ow n tu r ns , c o ul d i mpro v e the ac c u r ac y o f ou r mo de l .




                                                                                                                     51
       Appendix 



                       Coefficient     LTV         DTI        Unempl.      Euribor 3m
                           Beta         -7.4E-08    2.80631   53.477971    71.41378169
                          p-value      0.989268    3.01E-49    3.06E-86    6.1286E-179
                         st. erorr     5.51E-06     0.19024   2.7170636    2.503752991
                        z-statistics   -0.01345    14.75144   19.682267    28.52269451
     Table A1 – Cox model coefficients statistics (w ith LTV)




                                            Beta Covariance matrix
                         3.04E-11        4.04E-10       6.53E-08          6.75915E-08
                         4.04E-10        0.036191      -0.040678          0.091384154
                         6.53E-08        -0.04068      7.3824345          2.378678034
                         6.76E-08        0.091384       2.378678           6.26877904
     Table A2 – Covariance matrix of coefficient estimates (w ith LTV)




     Table A3 – Cox Model Regression statistics (w ithout LTV)




     Table A4 – Covariance matrix of coefficient estimates (w ithout LTV)




     *Note that the coefficient estimate of LTV is practically 0 and its p-value is almost 1 (significantly

     different of 0) and also the parameter estimates for the other covariates are the same with or

     without LTV – i.e. LTV has no significant impact in modeling the hazard rate of default.




52
Figure   A5   –    Sur vi val   Function   of   mortgage   contract   l : X DTI ,l = 30% , X UN ,l = 4.8% ,
X IR ,l = 1.5% .




Figure   A6   –    Cumulative     Distribution    Function   of   mortgage   contract   l : X DTI ,l = 30% ,
X UN ,l = 4.8% , X IR ,l = 1.5% .




                                                                                                               53
     Figure A7 – Euribor 3month simulations – 1000 paths for 12 month period

     CIR Process:

                                        drt = a (b − rt ) dt + σ rt dWt

     with parameters (estimated on monthly data from the last 9 years):


     a = 0.014849660801059      (mean reversion parameter)


     b = 0.033987279169537      (long term mean)


     σ = 0.010397560084086      (volatility)




54
     Min
                                                          years.
                                                                                                                                                         20
                                                                                                                                                            00




      3.20%
                                                                                                                                                         20 De
                                                                                                                                                           01 c.
                                                                                                                                                                    /2
                                                                                                                                                         20 Ap 0 0
                                                                                                                                                            01 r./2 1
                                                                                                                                                                            F




     Max
                                                                                                                                                                                     0
                                                                                                                                                                                         1
                                                                                                                                                                                             2
                                                                                                                                                                                                 3
                                                                                                                                                                                                     4
                                                                                                                                                                                                         5
                                                                                                                                                                                                             6
                                                                                                                                                                                                                 7
                                                                                                                                                                                                                     8

                                                                                                                                                         20 Au 001 eb
                                                                                                                                                            01 g./         Ju .
                                                                                                                                                                     2




      7.00%
                                                                                                                                                         20 De 0 0 ne
                                                                                                                                                           02 c. 1
                                                                                                                                                                    /2 O
                                                                                                                                                         20 Ap 0 0 c t.
                                                                                                                                                            02 r./ 2
                                                                                                                                                                    2       F
                                                                                                                                                         20 Au 002 eb
                                                                                                                                                            02 g./               .
                                                                                                                                                                     2     Ju




     Mean
                                                                                                                                                         20 De 0 0 ne




                 U n em pl o ym en t s ta t ist i c s :
                                                                                                                                                           03 c. 2




      4.96%
                                                                                                                                                                    /2 O
                                                                                                                                                         20 Ap 0 0 c t.
                                                                                                                                                            03 r./ 3
                                                                                                                                                                    20 Fe
                                                                                                                                                         20 Au 03 b
                                                                                                                                                            03 g./               .
                                                                                                                                                                     2 Ju
                                                                                                                                                         20 De 0 0 ne
                                                                                                                                                           04 c. 3
                                                                                                                                                                    /2 O




     Median
      4.85%
                                                                                                                                                         20 Ap 0 0 c t.
                                                                                                                                                            04 r./2 4 F
                                                                                                                                                         20 Au 004 eb
                                                                                                                                                            04 g.
                                                                                                                                                                    /      Ju .
                                                                                                                                                         20 De 20 0 ne
                                                                                                                                                           05 c. 4
                                                                                                                                                                    /2 O
                                                                                                                                                         20 Ap 0 0 c t.
                                                                                                                                                            05 r./2 5 F




     Std. Dev.
                                                                                                                                                        20 A 00 eb




         1.13%
                                                                                                                                                           05 ug 5 J .
                                                                                                                                                                   .
                                                                                                                                                        20 De /20 un
                                                                                                                                                          06 c. 05 e
                                                                                                                                                                  /2
                                                                                                                                                        20 Ap 0 0 Oc t
                                                                                                                                                           06 r./2 6            .
                                                                                                                                                                           F
                                                                                                                                                        20 Au 006 eb
                                                                                                                                                           06   g.             .*
                                                                                                                                                                  /       Ju
                                                                                                                                                        20 D 20 0 ne
                                                                                                                                                          07 ec. 6 *
                                                                                                                                                                  /2 O
                                                                                                                                                        20 Ap 0 0 ct.
                                                                                                                                                           07 r./2 7 *
                                                                                                                                                                           F
                                                                                                                                                        20 Au 007 eb
                                                                                                                                                           07 g./         J u .*
                                                                                                                                                                    2
                                                                                                                                                        20 De 0 0 ne
                                                                                                                                                          08 c. 7 *
                                                                                                                                                                  /2 O
                                                                                                                                                        20 Ap 0 0 ct.
                                                                                                                                                           08 r./ 8 *
                                                                                                                                                                  2        F
                                                                                                                                                        20 Au 008 eb
                                                                                                                                                           08 g./              .*
                                                                                                                                                              D     20 J un
                                                                                                                                                               ec 08 e*
                                                                                                                                                                 ./2 O
                                                                                                                                                                      0 0 ct
                                                                                                                                                                         9 .*
                                                                                                                                                                           Fe
                                                                                                                                                                              b.
                                                                                                                                                                                *




                                                          Figure A8 – Evolution of the quarterly unemplo yment rate in the Netherlands for the last 9




55
     Figure A9 – Relative change in unemployment – Distribution and Normal fit.




     Re la tive c ha nge in une mp lo yment – his torica l s ta tis tics :
      Mean             Std. Dev.         Median
          0 .001 93      0.040335285                    0



     No rma l fit – es tima ted par ame ters :
      Mean                 Std. Dev.
             0.00192985         0.0403353




56
Figure A10 - Unemployment simulations – 1000 paths for 12 month period




Un emplo ymen t p a ths ( Sta nda rd Sce nar io) :

                                           unt +1 = unt * (1 + un _ rt )

where     un _ r ~ N ( μun _ r , σ un _ r ) is a ra ndom r ela tive cha ng e fo llow in g a no rma l-
d is tr ib u ti on w i th th e fol l ow i ng p ar a me t er s ( es t im ated on d a ta f or th e l as t 9 ye ars ) :


μun _ r = 0.00192985      ( mea n)


σ un _ r = 0.0403353    (s ta nd ard d e via tion)




                                                                                                                       57
     Figure A11 - Unemployment simulations – 1000 paths for 12 month period




     U n emplo ymen t p a ths (DN B Base Sc enar io) :

                                                unt +1 = unt * (1 + un _ rt )

     where     un _ r ~ N ( μun _ r , σ un _ r ) is a ra ndom r ela tive cha ng e fo llow in g a no rma l-
     d is tr ib u ti on w i th th e fol l ow i ng p ar a me t er s ( es t im ated o n da ta f or th e l as t 9 ye ars ) :


     μun _ r = 0.0335    ( mea n)


     σ un _ r = 0.0403353    (s ta nd ard d e via tion)




58
Figure A12 - Unemployment simulations – 1000 paths for 12 month period




Un emplo ymen t p a ths (DN B Stress Scen ar io ):

                                           unt +1 = unt * (1 + un _ rt )

where     un _ r ~ N ( μun _ r , σ un _ r ) is a ra ndom r ela tive cha ng e fo llow in g a no rma l-
d is tr ib u ti on w i th th e fol l ow i ng p ar a me t er s ( es t im ated on da ta f or th e l as t 9 ye ars ) :


μun _ r = 0.0405    ( mea n)


σ un _ r = 0.0403353    (s ta nd ard d e via tion)




                                                                                                                      59
     Figure A13 – Expected PD Distribution of mortgage contract           l : X DTI ,l = 30%

     1000 simulations with initial   X UN ,l = 4.81%   and initial   X IR ,l = 1.5%




60
Figure A14 – Expected LGD Distribution of mortgage contract           l : X DTI ,l = 30%

1000 simulations with initial   X UN ,l = 4.81%   and initial   X IR ,l = 1.5%




                                                                                           61
     Figure A15 – Expected Loss Distribution of mortgage contract                 l : X DTI ,l = 30%

     1000 simulations with initial     X UN ,l = 4.81%    and initial   X IR ,l = 1.5%




     N o t e t ha t e v en tho ug h PD is n o t t ha t l ow the E xpec t ed Los s es ar e v er y s ma ll

     b ecaus e on ly ≈ 5 0 % o f t he d e fau l ts r es u l t in ac tua l los s




62
Figure A16 - STORM 2007-I B.V. Collateral pool characteristics.




Figure A17 - STORM 2007-I B.V. Notes Structure




                                                                  63
     F i gu re A 18   - Expected Loss Distribution of Collateral pool (DNB Base Scenario with MVD
     20%) – 2 year simulation period - 5000 simulations




     Figure A19 - Expected Loss Distribution of the Notes (DNB Base Scenario w ith MVD 20%) –

     2 year simulation period - 5000 simulations




64
Figure A20 - Expected Loss Distribution of Collateral pool (DNB Base Scenario with MVD
20%) – 3 year simulation period - 5000 simulations




Figure A21 - Expected Loss Distribution of the Notes (DNB Base Scenario w ith MVD 20%) –

3 year simulation period - 5000 simulations




                                                                                           65
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