i
C edi Risk Modelling using Mul i-s a e Ma ko Models
João Paulo Noguei a San os
A Disse a ion as a pa ial equi emen o ob ain he deg ee o Mas e in
S a is ics and In o ma ion Managemen , specializa ion in Risk Analysis and
Managemen
1
NOVA In o ma ion Managemen School
Ins i u o Supe io de Es a ís ica e Ges ão de In o mação
Uni e sidade No a de Lisboa
CREDIT RISK MODELLING USING MULTI-STATE MARKOV MODELS
by
João Paulo Noguei a San os
A Disse a ion as a pa ial equi emen o ob ain he deg ee o Mas e in S a is ics and In o ma ion
Managemen , specializa ion in Risk Analysis and Managemen
Men o : P o . Doc o Jo ge Miguel Ven u a B a o
Augus 2018
2
ACKNOWLEDGMENTS
I would like o hank my ad iso o he oppo uni y o wo k wi h mul i-s a e models and hei
implemen a ion in o he esea ch a eas, mo e speci ically he inancial a ea.
To my amily, o he suppo and lo e du ing my 27 yea s o li e. Especially o my mom, a woman
who I am e y p oud o and who did e e y hing in he powe o allow me o lead he li e I ha e
oday.
3
SUMMARY
This pape is de o ed o c edi isk modelling issues conce ning mo gage comme cial loans.
Mo gage loans a e one o he mos popula ype o loans p o ided by c edi ins i u ions. Like in he
case o o he loans, he main conce n o ins i u ions p o iding his ype o p oduc is a po en ial
inabili y o eco e he amoun assigned o hei clien s (c edi isk). In o de o p e en possible
losses o c edi ins i u ions esul ing om clien s en e ing in de aul , i is he e o e c ucial o s udy
he beha iou o isky clien s. This issue can be add essed h ough se e al models, namely h ough
he mul i-s a e Ma ko model, despi e i cons i u ing a mo e unusual app oach in he con ex o
dealing wi h c edi isk modelling. The mul i-s a e Ma ko model is a use ul way o desc ibing a
p ocess in which an indi idual mo es h ough a se ies o s a es ( ini e numbe ) in con inuous ime. By
i ing his model o he loans o isky clien s, i is possible o es ima e he mean sojou n ime in each
s a e be o e a ansi ion occu s, as well as he ansi ion p obabili ies be ween he di e en s a es
assumed by he con ac s, he e o e p o iding a ele an modelling amewo k o e en his o y
da a. The p esen wo k elies upon 2008-13 da abases om one o he bigges Ame ican companies
ha ac in he seconda y mo gage ma ke , he Fannie Mae. Resul s show ha wi h he applica ion
o he mul i-s a e Ma ko model, con ac s signed du ing 2013 a e mo e p opi ious o a scena io o
eco e y when compa ed o hose e e ing o he yea 2008.
KEY WORDS
P obabili y o de aul ; Mo gage loans; Mul i-s a e Ma ko Model; C edi Risk; Msm
4
INDEX
1. In oduc ion .................................................................................................................. 6
2. Li e a u e e iew .......................................................................................................... 9
3. Me hodology .............................................................................................................. 13
4. Resul s ........................................................................................................................ 18
5. Conclusion .................................................................................................................. 25
6. Bibliog aphy ................................................................................................................ 26

5
LIST OF TABLES
Figu e 1 – Mul i-s a e model o loans p og ession ................................................................ 14
Figu e 2 – T ansi ion ma ix o he p oposed model .............................................................. 15
LIST OF TABLES
Table 1 – De ini ion o he s a es ............................................................................................. 15
Table 2 – Numbe o con ac s o he pe iods o 2008-13 ..................................................... 17
Table 3 – Loan’s a e age ime in he po olio o he pe iods o 2008-13 ............................. 17
Table 4 – Pe cen age o con ac s o he pe iods o 2008 ..................................................... 18
Table 5 – Pe cen age o con ac s o he pe iods o 2013 ..................................................... 18
Table 6 – T ansi ion in ensi ies ma ices o he pe iods o 2008 ........................................... 19
Table 7 – T ansi ion in ensi ies ma ices o he pe iods o 2013 ........................................... 19
Table 8 – Mean sojou n imes o he pe iods o 2008 ........................................................... 20
Table 9 – Mean sojou n imes o he pe iods o 2013 ........................................................... 20
Table 10 – 1 yea es ima ed ansi ion p obabili ies o he pe iods o 2008 ......................... 21
Table 11 – 1 yea es ima ed ansi ion p obabili ies o he pe iods o 2013 ......................... 21
Table 12 – 2 yea es ima ed ansi ion p obabili ies o he pe iods o 2008 ......................... 22
Table 13 – 2 yea es ima ed ansi ion p obabili ies o he pe iods o 2013 ......................... 22
Table 14 – Numbe o ac i e con ac s obse ed in June 2015 ............................................... 23
Table 15 – Es ima ed numbe o con ac s in each s a e in June 2016 and June 2017 o he
po olios o 2008-13 ........................................................................................................ 24
LIST OF ABBREVIATIONS
FMNA Fede al Na ional Mo gage Associa ion
GSE Go e nmen Sponso ed En e p ise
MBS Mo gage Backed Secu i ies
FHFA Fede al Housing Finance Agency
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1. INTRODUCTION
C edi isk is he la ges isk aced by comme cial banks and is o conce n o a a ie y o s akeholde s:
ins i u ions, consume s and egula o s. C edi isk may be de ined as he isk o losses due o c edi
e en s, i.e. de aul (an obligo being unwilling o unable o epay i s deb ) o a change in he quali y
o he c edi ( a ing change). Examples o de aul e en s include bond de aul s, co po a e
bank up cy, c edi ca d cha ge-o and mo gage o eclosu e. O he o ms o c edi isk include
epaymen delinquency in e ail loans, loss se e i y upon he de aul e en , as well as he
unexpec ed change o c edi a ing. C edi isk modelling assis s banks in es ima ing he expec ed
loss (EL) on a c edi exposu e o e a gi en ime ho izon, enabling ins i u ions o p ice c edi isks
mo e e ec i ely and o calcula e how much capi al has o be se aside as a p o ision. C edi expec ed
losses depend in a mul iplica i e way on he P obabili y o De aul (PD), on he Exposu e a De aul
(EAD) and on he Loss Gi en De aul (LGD).
B oadly speaking, c edi isk models can be classi ied in o wo ypes based on he de ini ion o c edi
loss. Fi s , De aul Mode (DM) models (also called as “ wo s a e” models) ecognize c edi loss only i
a bo owe de aul s wi hin he planning ho izon. This means ha in such models only wo ou comes
a e ele an – non-de aul and de aul . I no de aul occu s, he c edi loss is ob iously ze o. I de aul
occu s, EAD and LGD mus be es ima ed. In con as , “mul i-s a e” (o “ma k- o-ma ke ”, MTM)
models ecognize ha ‘de aul ’ is he only one o he se e al possible c edi a ing g ades o which
he ins umen could mig a e o e he planning ho izon. As such, hese models es ima e he
p obabili y ha he bo owe 's c edi quali y de e io a es (c edi mig a ion), including a change o
de aul s a us. The MTM pa adigm ecognizes ha he e can be an economic impac e en i he
bo owe does no de aul . A "pu e" MTM app oach would ake ma ke -implied alues in di e en
non-de aul ing s a es. In eal ma ke applica ions, because o da a and liquidi y issues banks end o
use in e nal p ices based on c edi loss expe iences.
The wo main app oaches in he MTM pa adigm a e he discoun ed con ac ual cash low (DCCF)
app oach and he isk-neu al alua ion (RNV) app oach. In he DCCF app oach, he cu en alue o a
non-de aul ed loan is measu ed as he p esen alue o i s u u e cash lows, compu ed using
ma ke -de e mined o in e nal c edi sp eads o obliga ions o he same g ade. The u u e alue o a
non-de aul ed loan is dependen on he isk a ing a he end o he ime ho izon and he c edi
sp eads o ha a ing. The RNV app oach is de i ed om op ion p icing heo y. P ices a e an
expec a ion o he discoun ed u u e cash lows in a isk-neu al ma ke .
Fede al Na ional Mo gage Associa ion (FNMA), commonly known as Fannie Mae, is a go e nmen -
sponso ed en e p ise (GSE)1 ounded in 1983 whose mainly pu pose is o p o ide liquidi y in he
seconda y mo gage ma ke 2 in o de o make possible o banks, insu ance companies and
mo gage banking companies o issue mo e mo gages han hose would be able o issue wi h hei
own unds. I s ac i i y encompasses ou p ima y a eas in he Single-Family C edi Gua an y business:
(i) Mo gage Acquisi ions, acili a ing he pu chase o single- amily mo gage loans, gene ally o he
1 GSEs a e “p i a ely owned inancial ins i u ions es ablished by he go e nmen o ul ill a public mission” – see
Cong essional Budge O ice, Decembe 2010.
2 A e a lende o igina e a mo gage loan, he can sell i in he seconda y ma ke o he in es o s in o de o ge a ailable
unds o make new mo gages in he p ima y ma ke .
7
pu pose o secu i izing hem; (ii) Mo gage Secu i iza ion o single- amily mo gage loans deli e ed o
Fannie Mae MBS in lende swap ansac ions; (iii) C edi Risk Managemen , se ing and main aining
s anda ds o o igina ion and se icing; and (i ) C edi Loss Managemen , p e en ing o eclosu es
and educing cos s o de aul ed loans h ough o eclosu e al e na i es, managemen o o eclosu es
and eal es a e owned (REO) p ope ies, and by pu suing con ac ual emedies om lende s,
se ice s, and p o ide s o c edi enhancemen .
Like comme cial banks, Fannie Mae can gene a e p o i s by lending money. In i s ac i i y, Fannie Mae
“buys” mo gage loans, consequen ly assuming he c edi isk associa ed wi h hese loans. C edi isk
is p esen h oughou he en i e li e cycle o con ac s, s a ing wi h he g an ing o c edi , ollowed
by hei moni o ing and po en ially ending up in si ua ions o non-compliance, wi h he p ocess o
c edi eco e y. In his way, Fannie Mae, like o he inancial ins i u ions, is exposed o de aul isk,
de ined as he ailu e o a bo owe o mee i s con ac ual obliga ions o epay a deb in acco dance
wi h he ag eed e ms.
In 2008-09, he Uni ed S a es en e ed in o he deepes ecession since he G ea Dep ession o 1929-
33. Al hough no iceably simila , he dep h o in eg a ion be ween mode n global inancial cen e s
exace ba ed he e ec s o his ecession, he e o e igge ing a global c isis. Al hough he e is s ill
much esea ch o be done, he esiden ial mo gage lending p ac ices in he Uni ed S a es we e in
he oo o he shock (Demyanyk and O o, 2011). Many o he inancial p oduc s c ea ed du ing he
pe iod leading up o he c isis, enabled by he p ocess o secu i iza ion, played a cen al ole in he
de elopmen o he sub-p ime mo gage indus y. The p oduc s we e ambiguous; he unde lying isk
was di icul o pic u e and hence, p ice. The p ima y p oduc s ha enabled he inancing o he sub-
p ime ma ke segmen we e Mo gage Backed Bonds (MBB's) and Colla e alized Mo gage
Obliga ions (CMO's). These p oduc s e ec i ely package pools o mo gage deb in o he o m o a
bond which could hen be sold on he in es men ma ke . These bonds o e ed in es o s highe han
a e age yields when compa ed o exis ing s aigh bonds such as T easu y bonds and co po a es and
p o ided he liquidi y necessa y o keep inancing he U.S. eal es a e ma ke . On Sep embe 7, 2008,
he ede al go e nmen ook con ol o Fannie Mae. The company was pu in conse a o ship (a
s a u o y p ocess wi h he objec i e o es o ing he inancial s eng h o he company in o de o
ensu e i s sol ency), unde he e ms and supe ision o FHFA3, and emains in his si ua ion o he
p esen day4.
Gi en he impo ance o de aul isk o inancial ins i u ions, which in mo e se e e cases can esul
in bank up cy, i is undamen al o ensu e an e icien managemen o c edi po olios based on
mode n me hodologies o quan i ica ion o isk, as well as a mo e adi ional c edi isk analysis,
h ough which cus ome s c edi p o ile a e e alua ed.
This pape aims o con ibu e o he cu en li e a u e on his opic by de eloping a modelling
app oach speci ically o discuss loan pe o mance du ing 2008-13, using da a on mo gages acqui ed
by Fannie Mae. As a whole, i seeks o answe he ollowing 2 main ques ions:
1. Wha is he main con ibu ion o mul i-s a e Ma ko models in c edi isk modelling;
3 ”FHFA is a membe agency o he Financial S abili y O e sigh Council. The Council is cha ged wi h iden i ying isks o he
inancial s abili y o he Uni ed S a es; p omo ing ma ke discipline; and esponding o eme ging isks o he s abili y o he
Uni ed S a es' inancial sys em” – see h p://www. h a.go /Abou Us.
4 See h ps://www. h a.go /Conse a o ship.
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2. A e he e signi ican di e ences in he loan pe o mance o Fannie Mae po olios o 2008-
13.
Taking in o accoun how clien s can beha e h oughou he li e cycle o hei loans, his s udy aims o
analyse he ansi ion p obabili ies be ween he di e en s a es ha a con ac can assume, by
eso ing o a model less adi ionally adop ed o c edi isk modelling, namely he mul i-s a e
Ma ko model. No only a e he e ew e e ences in exis ing li e a u e ega ding he applicabili y o
his model in he inancial sec o , bu also i can be conside ed ha his me hod ep esen s an
impo an app oach o de ine con ac s’ mo emen , gi en ha i ende s i possible o es ima e he
p obabili y o a con ac mo ing o a ce ain s a e based on i s p esen posi ion. The scope o his
wo k also includes an es ima e o he mean sojou n ime in each ansi ion s a e, he 1 and 2 yea
ansi ion p obabili ies, as well as a compa ison o he esul s o 2008-13. Addi ionally, he e is also
an analysis o he applica ion o he mul i-s a e Ma ko model in hese wo pe iods, gi en ha in
2008 we a e aced wi h Fannie Mae c isis, while in 2013 ep esen s a ela i ely mo e ecen pe iod o
s udy, based on which i is possible o unde s and i he c edi po olios o Fannie Mae ep esen
compa a i ely mo e o less isk o he ins i u ion. Al hough he a iables included in he da a se a e
disc e e, his s udy elies on he applica ion o mul i-s a e Ma ko model in con inuous ime once
ha his model p o ide an impo an con ibu e in e ms o ime p edic ion o he possible
ansi ions. Mul i-s a e models a e also used by conside ing co a ia es, in o de o s udy he ela ion
o cons an o ime- a ying cha ac e is ics o indi iduals wi h hei ansi ion a es. Al hough his is a
possible app oach o he issue a hand, his wo k does no ocus on a simula ion o models wi h
hese ypes o cha ac e is ics.
Taking in o accoun all o he abo e men ioned analysis, his pape is o ganized as ollows. In sec ion
2 we illus a e he li e a u e e iew abou he company unde s udy, a sample o models used in
c edi isk modelling and he u ili y o he mul i-s a e model in he obse a ion o an indi idual. In
sec ion 3, he me hodology o he mul i-s a e Ma ko model, as well as he da a used in he
es ima ion o he model a e p esen ed. Finally, in sec ion 4, he esul s om he implemen a ion o
he model a e add essed, while in sec ion 5 he o e all conclusions o his wo k a e p esen ed.
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Table 1 – De ini ion o he s a es
Sou ce: he au ho .
The nex s a e o which he indi idual mo es, and he ime o he change, a e go e ned by a se o
ansi ion in ensi ies 𝑞𝑞𝑟𝑟𝑟𝑟(𝑡𝑡,𝑧𝑧(𝑡𝑡)) o each pai o s a es and s. The in ensi ies may also depend on
he ime o he p ocess 𝑡𝑡, o mo e gene ally, on a se o indi idual-speci ic o ime- a ying
explana o y a iables 𝑧𝑧(𝑡𝑡). The in ensi y ep esen s he ins an aneous isk o mo ing om s a e o
s a e s and is gi en by:
𝑞𝑞𝑟𝑟𝑟𝑟�𝑡𝑡,𝑧𝑧(𝑡𝑡)�=lim
𝛿𝛿𝑡𝑡→0𝑃𝑃(𝑆𝑆(𝑡𝑡+𝛿𝛿𝑡𝑡)=𝑠𝑠|𝑆𝑆(𝑡𝑡)=𝑟𝑟/𝛿𝛿𝑡𝑡
The in ensi ies o m a ma ix Q whose ows sum o ze o, so ha he diagonal en ies a e de ined by
𝑞𝑞𝑟𝑟𝑟𝑟 =−�𝑞𝑞𝑟𝑟𝑟𝑟
𝑟𝑟≠𝑟𝑟
To i a mul i-s a e model o he a ailable da a, a ansi ion in ensi y ma ix was es ima ed. The
Ma ko assump ion was ha u u e e olu ion only depends on he cu en s a e. Tha is,
𝑞𝑞𝑟𝑟𝑟𝑟(𝑡𝑡,𝑧𝑧(𝑡𝑡)𝐹𝐹𝑡𝑡) is independen o 𝐹𝐹𝑡𝑡, he obse a ion his o y 𝐹𝐹𝑡𝑡 o he p ocess up o he ime
p eceding (Cox and Mille ).
As a esul , and acco ding wi h he model de ini ion o his wo k, he s a es may be modelled as a
homogeneous con inuous- ime Ma ko p ocess, wi h a ansi ion ma ix Q, pic u ed below:
Figu e 2 – T ansi ion ma ix o he p oposed model
Sou ce: he au ho .
Kalb leisch and Lawless, and la e Kay, desc ibed a gene al me hod o e alua ing he likelihood o a
gene al mul i-s a e model in con inuous ime, applicable o any o m o ansi ion ma ix. The
likelihood is calcula ed om he ansi ion p obabili y ma ix 𝑃𝑃(𝑡𝑡). Fo a ime-homogeneous p ocess,
he (𝑟𝑟,𝑠𝑠)) en y o 𝑃𝑃(𝑡𝑡), 𝑃𝑃𝑟𝑟𝑟𝑟(𝑡𝑡), is he p obabili y o being in s a e s a a ime 𝑡𝑡+𝑢𝑢 in he u u e,
gi en he s a e a ime u is . I does no say any hing abou he ime o ansi ion om o s, indeed
he p ocess may ha e en e ed o he s a es be ween imes u and 𝑡𝑡+𝑢𝑢. 𝑃𝑃(𝑡𝑡) can be calcula ed by
aking he ma ix exponen ial o he scaled ansi ion in ensi y ma ix (see, o example, Cox and
Mille ):
𝑃𝑃(𝑡𝑡)=exp (𝑡𝑡𝑡𝑡)
De ini ion
1Pe o ming Cu en o less han 30 days pas due
2Non pe o ming 30 o +270 days pas due
3Modi ied Modi ied loan
4De aul De aul
S a e

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3.3. Es ima ion P ocedu e
As men ioned be o e, he mo gage loans p o ided by Fannie May o he i s wo qua e s o 2008-
13 we e used in o de o s udy he ansi ion p obabili ies be ween s a es, h ough a mul i-s a e
Ma ko model. Since he e a e a ia ions in hese kind o models (as illus a ed in he e iew
li e a u e chap e ), an analysis o he a iables p o ided by Fannie May was necessa y, as well as
some ea men o he in o ma ion p io o he implemen a ion o he model, al eady desc ibed
p e iously.
The ollowing a iables o he da ase we e used:
1) LOAN_ID: iden i ie o he con ac ;
2) LOAN_AGE: li e pe iod, in mon hs, o he c edi mo gages;
3) DELQ_STAT: s a e o a con ac ;
4) ZB_CODE (ze o balanced code): a code indica ing he eason he loan's balance was educed o
ze o o expe ienced a c edi e en . The possible codes7 a e:
• 01 = P epaid o ma u ed;
• 03 = Sho sale, hi d pa y sales and o he o eclosu e al e na i es;
• 06 = Repu chased;
• 09 = REO disposi ion o deed-in-lieu.
Th ough he analysis o he a iable “LOAN_AGE” i was iden i ied ha some con ac s p esen ed
incomple e his o ical in o ma ion, and will as a esul no be eligible o en e he model. Also, he e
was a conce n ha some con ac s could no ha e enough his o ical in o ma ion. Based on he
me hodology used in c edi sco ing models, whe e he classi ica ion o an indi idual as “good” o
“bad” is obse ed du ing a ce ain pe iod o ime, o he s udy a hand i was de e mined ha
con ac s p esen ing less han 128 mon hs o his o ical in o ma ion would be excluded. Addi ionally,
he e we e con ac s whe e a iable “LOAN AGE” was se o minus one. Since he e was no e idence
abou he eal ini ial da e o hese con ac s, hey we e excluded om he inal da a se in o de o
ensu e mo e eliable esul s.
Wi h espec o he a iable “DELQ_STAT”, we concluded ha some con ac s ha e ansi ions
be ween s a es, namely om one mon h o he nex , highe han 30 days ( o ins ance, a con ac in
he p esen mon h has 30 days pas due and in he ollowing mon h p esen s 120 days pas due).
This a iable, o iginally, was ep esen ed by 10 s a es, om he s a e 0 (cu en o less han 30 days
pas due), o s a e 9 (30 o +270 days pas due). The o he s a es we e ep esen ed by in e als o 30
days pas due (s a e 1, om 30 o 59, and so on). Fo his wo k we choose o modi y he a iable and
de ine he s a e 1 as ”cu en o less han 30 days pas due” and agg ega e he emaining s a es in o
jus one, namely s a e 2. Also, i was no ed ha in some ins ances he a iable assumed a alue “X”
in he las da e o obse a ion o he con ac s. When his occu ed, he a iable “ZB_CODE” was in
some ins ances (no all) illed wi h one o he 4 codes p esen ed be o e. Based on his, he s a e in
his obse a ion is modi ied: i is conside ed ha when he “ZB_CODE” is “03” o “09” we a e aced
wi h a de aul si ua ion and when is “01” o “06” we a e aced wi h a pe o ming si ua ion (which can
lead o a ansi ion om pe o ming o de aul ). Also, when he e is no code associa ed o he ze o
7 The de ini ion o he p esen ed codes is based on he Fannie Mae guideline.
8 Pe iod commonly used in c edi sco ing models in Po uguese banking.
17
balanced code a iable and he penul ima e posi ion has he non pe o ming s a e, i was assumed
ha he con ac ends i s li e ime cycle in a s a e o de aul .
Table 2 below p esen s he numbe o con ac s o each qua e o he yea s being analysed, as well
as he numbe o con ac s excluded based on he p e iously p esen ed exclusion ules.
Table 2 – Numbe o con ac s o he pe iods o 2008-13
Sou ce: he au ho .
In u n, he loan’s a e age ime in he po olio o he con ac s unde analysis is p esen ed below in
able 3:
Table 3 – Loan’s a e age ime in he po olio o he pe iods o 2008-13
Sou ce: he au ho .
Q1 Q2 Q1 Q2
37 36 14 13
2008
2013
18
4. RESULTS
This sec ion p esen s he esul s om he implemen a ion o he mul i-s a e Ma ko model,
including he analysis o he numbe o ansi ions be ween s a es (on a mon hly basis), he
ansi ion in ensi ies ma ix, ha illus a es he ins an aneous isk o con ac s mo ing om one
s a e o o he (wi h 95% con idence in e als), he es ima ed mean sojou n imes in each ansien
s a e, he one and wo yea s es ima ed ansi ion p obabili ies and he u u e numbe o con ac s in
each s a e, applying hese p obabili ies.
Table 4 and able 5 below p esen he pe cen age o con ac s o each qua e o he yea s unde
analysis:
Table 4 – Pe cen age o con ac s o he pe iods o 2008
Sou ce: he au ho .
Table 5 – Pe cen age o con ac s o he pe iods o 2013
Sou ce: he au ho .
Taking in accoun he o al numbe o obse ed ansi ions o each qua e , i is possible o conclude
ha s a e combina ion (1 – 1) is e i ied a ound 94% o he ime. This means ha 94% o he o al
numbe o ansi ions occu ed when a con ac in he p esen mon h o obse a ion was in s a e 1
and con inued in s a e 1 in he ollowing mon h.
In analogous e ms, looking a con ac s ini ially in s a e 2, i can be concluded ha in bo h qua e s
o 2008 ci ca 83% o he ansi ions occu ed wi h con ac s emaining in s a e 2 in he ollowing
mon h. In 2013 his numbe is conside ably lowe (abou 42%), while he s a e combina ion (2 – 1)
occu s 54% o he imes, i.e., ci ca 54% o he o al numbe o ansi ions occu s when a con ac in
he p esen mon h o obse a ion is in s a e 2 and mo es o s a e 1 in he ollowing mon h. S a e
combina ion (3 – 3) ep esen s ci ca 97% and 80% o ansac ions o 2018 and 2013, espec i ely.
Conside ing his s a e (s a e 3) as he s a e unde obse a ion, o 2013, 14% o ansi ions ep esen
con ac ha mo e o s a e 1 in he ollowing mon h.
The analysis p esen ed abo e allowed o a be e unde s anding o he da a se s used in his wo k,
namely ha hese include con ac s ha emains mos o he imes in he same s a e. Howe e , i
was also possible o conclude ha a conside able numbe o ansi ions occu when a con ac is in
s a e 2 and mo es o s a e 1 in he nex mon h.
19
Table 6 – T ansi ion in ensi ies ma ices o he pe iods o 2008
Sou ce: he au ho .
Table 7 – T ansi ion in ensi ies ma ices o he pe iods o 2013
Sou ce: he au ho .
By looking a he ansi ion in ensi ies ma ices o he pe iod o 2008 p esen ed abo e, i is possible
o conclude ha he e is a low ins an aneous isk o he con ac s o mo e om s a e 1 o s a e 2
which is e en lowe when we analyse he ansi ion o s a e 4. Fo con ac s ini ially in s a e 2, he
ins an aneous isk o mo emen om s a e 3 o 4 is e y simila (app oxima ely 1,8% and 2,2%,
espec i ely), while a highe alue is obse ed in s a e 1 (12% and 13%). Fo con ac s ini ially in s a e
3, we conclude ha he e is a minimum isk o mo emen o s a es 1 and 4 (1,9% and 0,6%
espec i ely). The esul s o 2013 la gely con as wi h hose o 2008, whe e he possibili y o
eco e y om s a e 2 o s a e 1 is no iceable. Al hough his endency exis s in bo h yea s, in 2013 he
possibili y o his ansi ion is highe han 50%. O e all, he e isn’ a signi ican isk o mo emen
om he pe o ming s a e o he s a es o eco e y, es uc u ing o de aul in 2008 and 2013. Also, a
highe possibili y o eco e y ega ding he non pe o ming con ac s o 2013 is obse ed.
20
Table 8 – Mean sojou n imes o he pe iods o 2008
Sou ce: he au ho .
Table 9 – Mean sojou n imes o he pe iods o 2013
Sou ce: he au ho .
Fo he es ima ed mean sojou n imes in each ansien s a e, we e i ied ha in 2008 ( able 8) he
con ac s s ay in s a e 1 o 19 mon hs be o e mo ing o ano he s a e. When con ac s each s a e
2, which is a non pe o ming si ua ion, i is possible o obse e ha he mean sojou n ime in his
s a e be o e passing o a si ua ion o eco e y, es uc u ing o de aul is abou 6 mon hs. As o
con ac s in s a e 3, hey end o emain in his s a e o 40 mon hs be o e assuming a eco e y o
de aul s a us. This ex ensi e pe iod o ime is due o Fannie Mae conside ing ha om he momen
he con ac s a e subjec o any kind o c edi modi ica ion, hey assume his s a e un il he las
momen o obse a ion, when he e is a e i ica ion as o whe he he con ac passes o a eco e y
si ua ion o emains in a de aul .
In 2013 he obse ed mean sojou n imes we e much lowe o s a es 2 and 3, which can be
explained by he ac ha con ac s o his yea ha e less his o ical in o ma ion when compa ing o
hose in 2008. O e all, con ac s in bo h yea s end o s ay in a pe o ming s a e o a pe iod in-
be ween 17 and 20 mon hs, be o e passing on o wo s case scena ios, namely non pe o ming,
es uc u ing o de aul .
Es ima es SE L U
119,29726 0,05100 19,19757 19,39747
26,24807 0,01652 6,21578 6,28053
340,27077 0,32242 39,64376 40,90769
1s qua e o 2008
Es ima es SE L U
119,90934 0,05473 19,80236 20,01690
25,86376 0,01613 5,83224 5,89545
339,67767 0,33359 39,02920 40,33691
2nd qua e o 2008
Es ima es SE L U
119,10750 0,13543 18,84391 19,37478
21,71209 0,01213 1,68847 1,73604
35,42748 0,47420 4,57329 6,44122
1
s
qua e o 2013
Es ima es SE L U
116,73994 0,12717 16,49254 16,99106
21,76085 0,01357 1,73444 1,78765
34,65476 0,50788 3,75858 5,76463
2
nd
qua e o 2013

21
Table 10 and able 11 below p esen he esul s o he simula ion o one yea es ima ed ansi ions
p obabili ies o he pe iods o 2008 and 2013:
Table 10 – 1 yea es ima ed ansi ion p obabili ies o he pe iods o 2008
Sou ce: he au ho .
Table 11 – 1 yea es ima ed ansi ion p obabili ies o he pe iods o 2013
Sou ce: he au ho .
Based on he esul s p esen ed abo e we can conclude he ollowing:
• Con ac s o he pe iod o 2008 - The 1 yea es ima ed ansi ion p obabili ies can be conside ed
o e lec he s a e o he con ac s in one yea and show ha con ac s s ay in s a e 1 wi h 70%
o p obabili y, and ha e a p obabili y o abou 20% o passing on o s a e 2. In u n, he
p obabili ies o ansi ion om s a e 1 o s a es 3 and 4 a e minimal. Analysing s a e 2, we e i y
ins ead ha con ac s will assume his s a e wi h app oxima ely 25% o p obabili y. The esul s
also show ha i is mo e likely a con ac will change o a eco e y si ua ion (abou 50% o
p obabili y) han mo e o a es uc u ing o de aul si ua ion (9% and 14% o p obabili y,
espec i ely). As o con ac s in s age 3, we conclude ha hese will main ain hei s a us wi h
T ansi ion Es ima es LU
1-1 0,71327 0,71191 0,71450
1-2 0,21478 0,21374 0,21590
1-3 0,02844 0,02802 0,02888
1-4 0,04351 0,04297 0,04412
2-1 0,51059 0,50866 0,51249
2-2 0,26308 0,26142 0,26471
2-3 0,08746 0,08618 0,08879
2-4 0,13887 0,13737 0,14068
3-1 0,15810 0,15555 0,16082
3-2 0,03057 0,03005 0,03113
3-3 0,74472 0,74124 0,74826
3-4 0,06661 0,06468 0,06860
4-1 0,00000 - -
4-2 0,00000 - -
4-3 0,00000 - -
4-4 1,00000 1,00000 1,00000
1
s
qua e o 2008 - 1 yea
T ansi ion Es ima es L U
1-1 0,72896 0,72763 0,73042
1-2 0,20164 0,20045 0,20268
1-3 0,02811 0,02766 0,02854
1-4 0,04129 0,04066 0,04188
2-1 0,53500 0,53308 0,53715
2-2 0,24406 0,24245 0,24555
2-3 0,08759 0,08624 0,08897
2-4 0,13334 0,13157 0,13507
3-1 0,16542 0,16253 0,16812
3-2 0,02996 0,02942 0,03048
3-3 0,74149 0,73803 0,74535
3-4 0,06313 0,06113 0,06503
4-1 0,00000 - -
4-2 0,00000 - -
4-3 0,00000 - -
4-4 1,00000 1,00000 1,00000
2
nd
qua e o 2008 - 1 yea
T ansi ion Es ima es L U
1-1 0,88101 0,87832 0,88327
1-2 0,07692 0,07561 0,07819
1-3 0,00138 0,00113 0,00167
1-4 0,04070 0,03892 0,04296
2-1 0,84295 0,83877 0,84682
2-2 0,07412 0,07277 0,07538
2-3 0,00225 0,00175 0,00291
2-4 0,08068 0,08068 0,08508
3-1 0,59842 0,53060 0,65814
3-2 0,04935 0,04352 0,05459
3-3 0,11029 0,07362 0,15428
3-4 0,24194 0,18522 0,31340
4-1 0,00000 - -
4-2 0,00000 - -
4-3 0,00000 - -
4-4 1,00000 1,00000 1,00000
1
s
qua e o 2013 - 1 yea
T ansi ion Es ima es L U
1-1 0,85239 0,84939 0,85550
1-2 0,08674 0,08502 0,08843
1-3 0,00103 0,00079 0,00134
1-4 0,05984 0,05724 0,06238
2-1 0,80520 0,80059 0,81019
2-2 0,08255 0,08087 0,08423
2-3 0,00150 0,00107 0,00206
2-4 0,11075 0,10585 0,11587
3-1 0,51922 0,42756 0,60248
3-2 0,05027 0,04112 0,05855
3-3 0,07642 0,04079 0,12564
3-4 0,35409 0,27053 0,45172
4-1 0,00000 - -
4-2 0,00000 - -
4-3 0,00000 - -
4-4 1,00000 1,00000 1,00000
2
nd
qua e o 2013 - 1 yea
22
74% o p obabili y, and consequen ly he e is a 16% p obabili y o eco e y. Resul s also allow o
a conclusions ha de aul si ua ions a e minimal (6% o p obabili y).
• Con ac s o he pe iod o 2013 - In 2013 we can see a signi ican change in s a e 2, gi en ha
he e is a high p obabili y o eco e y when a con ac is in a non pe o ming si ua ion (a ound
85%). A change is also obse ed in s a e 3. In his case, highe ansi ion p obabili ies a e
obse ed o he eco e y s a e (be ween 52% and 60%), al hough he p obabili ies o ansi ion
o de aul also inc eased when compa ing o 2008.
The same simula ion was pe o med by conside ing ins ead he s a e o con ac s in 2 yea s, and
esul s a e shown below in able 12 and in able 13:
Table 12 – 2 yea es ima ed ansi ion p obabili ies o he pe iods o 2008
Sou ce: he au ho .
Table 13 – 2 yea es ima ed ansi ion p obabili ies o he pe iods o 2013
Sou ce: he au ho .
T ansi ion Es ima es L U
1-1 0,62292 0,62116 0,62489
1-2 0,21057 0,20941 0,21172
1-3 0,06025 0,05935 0,06120
1-4 0,10626 0,10484 0,10754
2-1 0,51235 0,51014 0,51470
2-2 0,18155 0,18036 0,18277
2-3 0,10267 0,10117 0,10434
2-4 0,20344 0,20090 0,20571
3-1 0,24612 0,24259 0,24968
3-2 0,06477 0,06382 0,06579
3-3 0,56178 0,55641 0,56655
3-4 0,12733 0,12409 0,13082
4-1 0,00000 - -
4-2 0,00000 - -
4-3 0,00000 - -
4-4 1,00000 1,00000 1,00000
1
s
qua e o 2008 - 2 yea s
T ansi ion Es ima es L U
1-1 0,64391 0,64314 0,64584
1-2 0,19704 0,19590 0,18250
1-3 0,05899 0,05807 0,05989
1-4 0,10005 0,09854 0,10134
2-1 0,53506 0,53272 0,53748
2-2 0,17007 0,16889 0,17137
2-3 0,10136 0,09977 0,10301
2-4 0,19351 0,19089 0,19584
3-1 0,25927 0,25539 0,26290
3-2 0,06288 0,06179 0,06386
3-3 0,55708 0,55189 0,56271
3-4 0,12077 0,11735 0,12423
4-1 0,00000 - -
4-2 0,00000 - -
4-3 0,00000 - -
4-4 1,00000 1,00000 1,00000
2
nd
qua e o 2008 - 2 yea s
T ansi ion Es ima es L U
1-1 0,84184 0,83788 0,84559
1-2 0,07354 0,07220 0,07490
1-3 0,00154 0,00122 0,00197
1-4 0,08309 0,07958 0,08695
2-1 0,80647 0,80097 0,81162
2-2 0,07044 0,06913 0,07180
2-3 0,00158 0,00122 0,00207
2-4 0,12151 0,11619 0,12734
3-1 0,63481 0,56524 0,69261
3-2 0,05513 0,04890 0,06423
3-3 0,01310 0,00646 0,02496
3-4 0,29696 0,23564 0,37347
4-1 0,00000 - -
4-2 0,00000 - -
4-3 0,00000 - -
4-4 1,00000 1,00000 1,00000
1
s
qua e o 2013 - 2 yea s
T ansi ion Es ima es L U
1-1 0,79695 0,79205 0,80183
1-2 0,08115 0,07951 0,08266
1-3 0,00109 0,00081 0,00144
1-4 0,12081 0,11582 0,12641
2-1 0,75359 0,07674 0,76022
2-2 0,07674 0,07510 0,07841
2-3 0,00107 0,00078 0,00145
2-4 0,16860 0,16151 0,17609
3-1 0,52274 0,43531 0,59481
3-2 0,05303 0,04414 0,06055
3-3 0,00645 0,00229 0,01567
3-4 0,41778 0,33780 0,51421
4-1 0,00000 - -
4-2 0,00000 - -
4-3 0,00000 - -
4-4 1,00000 1,00000 1,00000
2
nd
qua e o 2013 - 2 yea s
23
Based on he esul s o his analysis we can conclude he ollowing:
• Con ac s o he pe iod o 2008 – The e is a lowe p obabili y o con ac s in s a e 1 o emain
in his s a e (62% and 64%) and he p obabili y o de aul inc eases om 4% o 10%. As o s a e
2, we can see ha con ac s will main ain hei s a us wi h a p obabili y o 18% (lowe when
compa ing wi h 2013 and he p obabili y o de aul ises o 20%. In s a e 3 he e is 56% o
p obabili y ha con ac s will main ain in his s a e and he p obabili y o ansi ion o s a e 1 is
highe when compa ed wi h he de aul s a e (26% and 13%, espec i ely).
• Con ac s o he pe iod o 2013 – These con ac s p esen highe p obabili ies o eco e y when
we analyse he non pe o ming and es uc u ing s a es. In he i s si ua ion, he p obabili ies
inc ease, a leas , 20%. The es uc u ed con ac s p esen be ween 52% and 63% o eco e y,
which is conside ably highe when compa ing o he igu es o 2008. Al hough he e is a good
eco e y pe spec i e o he es uc u ing e en , one also has o conside ha he ansi ion o
he s a e o de aul p esen s a signi ican isk when compa ing wi h he case in 2008, gi en ha
he p obabili ies inc ease om 13% (in 2008) o a maximum o 41% (2013).
To b ie ly summa ize, o he yea o 2008 one can conclude ha con ac s in s a es 1 and 3 will
main ain hei posi ion wi h a high p obabili y. Resul s also show ha non pe o ming and
es uc u ed con ac s a e mo e sui able o a eco e y scena io han he de aul . Rega ding he 2013
da a, we obse e ha con ac s in s a e 1 will main ain hei posi ion wi h an e en highe p obabili y
(when compa ing wi h esul s o 2008). In u n, con ac s in s a e 3 ha e a signi ican ly di e en
beha io , gi e ha hey no only p esen highe p obabili ies o eco e y bu also highe p obabili ies
associa ed o he de aul s a us. The p obabili y o emaining in his s a e is conside ably lowe
(again, when compa ing wi h 2008) bu his can be explained in pa by he sho his o ical pe iod o
hese con ac s. O e all, he beha io o he con ac s o 2013 is less isky when compa ed wi h ha
obse ed o he 2008 da ase .
The la es obse a ion poin o he con ac s used in his wo k is June 2015. Based on he
assump ion ha con ac s ha each his poin a e s ill ac i e in Fannie Mae po olios, i is
impo an o s udy and p edic he beha iou o hese con ac s ega ding hei u u e s a us.
The e o e, he ollowing analysis elies on he obse a ion o he numbe o con ac s by each s a e,
in June 2015, and he applica ion o he es ima ed ansi ion p obabili ies in o de o p edic he
numbe o con ac s in each s a e in June 2016 and June 2017.
Table 14 – Numbe o ac i e con ac s obse ed in June 2015
Sou ce: he au ho .
2008_Q1 2008_Q2 2013_Q1 2013_Q2
S a e 1 19.367 18.936 12.285 10.372
Numbe o ac i e con ac s in June 2015
24
Table 15 – Es ima ed numbe o con ac s in each s a e in June 2016 and June 2017 o he po olios
o 2008-13
Sou ce: he au ho .
Table 14 abo e illus a es he numbe o con ac s in each po olio unde analysis and hei s a us in
June 2015. Since ha a e only obse ed con ac s in s a es 1 and 4 in his da e, i is only possible o
es ima e ansi ions om s a e 1 (s a e 4 is igno ed gi en he ac ha his s a e is he abso ben
s a e). Table 15 shows he esul s ega ding he expec ed numbe o con ac s in each s a e in June
2016 and June 2017, gi en by he p oduc be ween he es ima ed ansi ion p obabili ies wi hin 1
and 2 yea s and he numbe o con ac s by s a e in June 2015. Resul s show ha he p edominan
s a e in e ms o numbe o con ac s is s a e 1, while s a e 3 appea s o be he leas ep esen ed.
Taking in conside a ion po olios o 2008, esul s also show ha he numbe o con ac s in s a e 1
dec eases be ween 2016 and 2017, while he numbe o con ac s in s a es 3 and 4 duplica es. The
same conclusion can be d awn o he po olios o 2013 ega ding s a es 1 and 4. Finally, we also
conclude ha con ac s end o be in highe isk s a es he longe he o ecas ime.
2008_Q1 2008_Q2 2013_Q1 2013_Q2
S a e 1 13.814 13.804 10.823 8.841
S a e 2 4.160 3.818 945 900
S a e 3 551 532 17 11
S a e 4 843 782 500 621
June 2016
2008_Q1 2008_Q2 2013_Q1 2013_Q2
S a e 1 12.064 12.193 10.342 8.266
S a e 2 4.078 3.731 903 842
S a e 3 1.167 1.117 19 11
S a e 4 2.058 1.895 1.021 1.253
June 2017