Multi-Period Structural Model of a Mortgage Portfolio with Cointegrated Factors *

Transcription

1 JEL classfcaon: G32 Keywords: cred rsk morgage loan porfolo dynamc model esmaon Mul-Perod Srucural Model of a Morgage Porfolo wh Conegraed Facors * Per GAPKO correspondng auhor (per.gapko@seznam.cz) Marn ŠMÍD boh auhors: Economerc Deparmen Insue of Informaon Theory and Auomaon Czech Academy of Scences Prague Absrac We propose a new dynamc wo-facor model of a loan porfolo. Followng he common approach we quanfy he cred rsk assocaed wh he porfolo by he probably of defaul and he loss gven defaul each of whch s drven by a facor common for all debs n he porfolo and a facor ndvdual o each deb. In lne wh he emprcal evdence he ndvdual facors are assumed o be AR() processes. The common facors on he oher hand may be dependen on he exernal (macroeconomc) envronmen. We apply our model o he US naonwde morgage porfolo fng he dynamcs of he facors wh a VECM model wh several macroeconomc ndcaors as exogenous varables.. Inroducon In 2009 when he fnancal crss fully h he US economy losses from real esae loans n he US ncreased by a facor of en compared wh he relavely que perod endng n The am of our paper s o conrbue o he dscusson on he mechancs causng evens of hs ype. One of possble ways of sudyng hese mechancs s o denfy he key rsk facors drvng losses and use an approprae model of he facors o forecas fuure loss behavor. We propose a srucural facor model of he Meron-Vascek ype (for smlar models see Vascek 987; Frye 2000; Pykhn 2003; Jmenez and Menca 2009; or Wzany 20). In hese srucural models defauls and losses gven defaul of ndvdual loans depend on he wealh of credors and on he prce of collaerals wh whch he loans are secured respecvely. In our approach he wealh and he collaeral prce are each assumed o be drven by wo facors one ndvdual o each debor and he oher common o all debors. Generally he common facors may be undersood as a quanfed nfluence of he exernal envronmen e.g. he macroeconomc suaon and he ndvdual facors mgh be nerpreed as an ndvdual s ably o manan wealh.e. feaures specfc o a collaeral. The dea of connecng srucural cred rsk models wh macroeconomc models s que common n he leraure; see for example e.g. Pesaran e al. (2003) or Vrolanen (2004). From he recen works s worh nong for example Hamerle e al. (20) who showed on a bond porfolo ha changes n he macroeconomc envronmen play a sgnfcan role by comparng a pon-n-me mulfacor cred rsk model wh a hrough-he-cycle one or Sommar and Shahnazaran (2009) who used he vecor error correcon model o esmae he dependency of he expeced defaul frequency of a porfolo of nonfnancal lsed companes * Ths work was suppored by he Czech Scence Foundaon under Gran No. GA5-033S. Fnance a úvěr-czech Journal of Economcs and Fnance no

2 on several macroeconomc facors from whch hey found he mos nfluenal neres rae. Compared o he aforemenoned models he orgnal conrbuon of our approach s hreefold. Frs we ake he dynamcs of he debors wealh whn he porfolo no accoun; n parcular we assume he ndvdual facors o be an AR() process whch corresponds o he emprcal evdence (see Hochguerel & Ohlsson 20). Second we nroduce a mul-generaon approach.e. we le new debors ener he porfolo n each perod. Fnally we use conegraon analyss wh exogenous macroeconomc varables o f he jon dynamcs of he common facors whch o he bes of our knowledge has never been done before. In he emprcal par of our paper we apply our model o he 30+ delnquency and charge-off 2 raes of he US naonwde resdenal morgage porfolo. As a resul we fnd ha here exss a se of sascally sgnfcan macroeconomc nfluencers of he rsk facors namely GDP CPI nflaon and he FED base neres rae. The paper s organzed as follows: In Secon 2 we nroduce he model. In Secon 3 we descrbe he emprcal analyss daa and resuls. Secon 4 concludes he paper. 2. The Model As was already premsed we model he dynamcs of a porfolo of loans. For smplcy we assume ha he fnanced amoun s he same for all loans n he porfolo a un whou loss of generaly. New loans ener he porfolo a each me N. The rao of he newly comng loans ( N ) o he overall sze of he porfolo s assumed o be /mn m for each where m s he duraon of he loans laer assumed o be equal o 30. Smlarly o Vascek (987) we assume ha he dsposable wealh of he -h debor from he -h generaon (.e. wh he frs A repaymen a me ) follows a rended Geomercal Brownan Moon wh a common facor as a rend.e. where Y (common facor) s a general sochasc process Z A exp Y Z () (ndvudual facor) s a sochasc process such ha Z ~ N0 for some 0 U ~ N0 Z Z U for some consans R 0 (noe ha he dsrbuon of he nal wealh dffers among he generaons as s dependen on he common facor). Analogously o he wealh and smlarly o Fronczak and Rosek (205) we assume ha he prce of he collaeral of he -h loan from he -h generaon s Loans more han 30 days pas due. 2 Ne charge-offs of loans from books. 566 Fnance a úvěr-czech Journal of Economcs and Fnance no. 6

3 nally equal o he (un) sze of he loan and laer follows dynamcs smlar o he wealh: P P exp I I E (2) where I (he collaeral common facor) s a general sochasc process and E s a sochasc process fulfllng E E V V N R ( E 0 by defnon). As s usual n srucural facor models we assume ha 2 2 N N 2 2 N 2 T U V U V U V U V V U ~ 0 and are muually ndependen and ndependen of Y I. The -h loan from he -h generaon defauls a me f he dsposable wealh of he correspondng debor does no suffce for repayng he morgage specfcally f A B (3) B where denoes he lables of he debor. Generally may nclude varous lables; n he presen paper however we ake only accumulaed nsalmens no B b where b he one-perod nsallmen (noe ha s accoun.e. deermnsc hen). The correspondng percenage loss assocaed wh he loan a equals Q max 0 h P G (4) h where Q s he ndcaor of nequaly (3).e. he zero-one varable ndcang he defaul and h s he prncpal ousandng a compued n he sandard way (see Šmíd 205). The overall defaul rae and he charge-off rae on he porfolo are hen defned as Q N Q (5) N G N N G B B (6) respecvely where for each whch dd no defaul unl. N s he number of loans from he -h generaon Fnance a úvěr-czech Journal of Economcs and Fnance no

4 2 Once he nal szes N N grow o nfny he nfluence of he ndvdual facors on quanes Q and G s canceled ou by he Law of Large Numbers and boh of he quanes become unquely deermned by he common facors; o be precse here exss one-o-one mappng Φ for each fulfllng Q G Q G Q G Y I Y I Y I (7) (see Theorem 3 of Šmíd 205 where a rgorous descrpon of he whole model may also be found). Thanks o he one-o-one propery he values of he common facors may be unquely rereved from he values of he loss raes n he lm case. Consequenly havng a me seres of he loss raes a hand s fuure values may be predced by ransformng no he facors predcng he facors and ransformng he predcons back no he loss raes. 3. Daa and Emprcal Esmaon As prevously menoned we appled our model o he US naonwde porfolo of morgages. 3. The Daase The daase was provded by he Uned Saes Federal Reserve Sysem. The me range covered was he perod beween 99 and 204 n quarerly granulary so he number of observaons was 96. The daase consss of me seres of wo quanes namely he morgage delnquency rae whch s a proporon of loans more han 30 days pas due (30+) on he oal morgage balance and he morgage charge-off rae whch s an annualzed proporon of loans charged off ne of recoveres on he average oal balance. In pracce here exs accouns whch afer becomng delnquen are no charged off bu are cured.e. he amoun pas due s pad and subsequen nsallmens connue o be pad on me. As cured accoun would brng addonal complexy o he model we om he cure rae and assume ha he cure amoun s repad wholly n he 30+ delnquency sae. Table and Fgure summarze he descrpve sascs and show he developmen of he npu daa. The 30+ delnquency rae was used as a proxy for he defaul rae and he charge-off rae represened he real loss from he unpad balance. From Fgure s obvous ha here exss a vsual suspcon ha he wo me seres are srongly correlaed. Also he recen economc crss whch sared n he US n lae 2007 and mpaced he US morgage and real esae markes excessvely s clearly vsble as boh me seres rockeed upward n he perod beween 2007 and 200 o mulples of her prevous values. 3.2 Underlyng Facors Exracon We appled he nverse ransformaon (v) o our npu daa wh he followng parameers: 568 Fnance a úvěr-czech Journal of Economcs and Fnance no. 6

5 Fgure Developmen of he 30+ Delnquency Rae (rgh axs) and Charge-Off Rae (lef axs) Table Descrpve Sascs of he Inpu Daa Sasc 30+ delnquency rae Charge-off rae Mean value Medan Mnmum Maxmum Sandard devaon Varance Skewness Kuross % percenle % percenle he sandard devaon of he nal wealh was se o 0.5 whch roughly corresponds o a long-erm sandard devaon of famly ncome n he US. - he sandard devaon of he loss gven defaul was se o 0.2 accordng o Gapko & Šmíd (202b). - he auocorrelaon coeffcen of he defaul rae ndvdual facor was se o 0.8 whch corresponds o fndngs n Hochguerel & Ohlsson (20). - he auocorrelaon coeffcen of he loss gven defaul ndvdual facor was se o The morgage neres rae was se o zero for smplcy. The resulng me seres of he exraced common facors Y (defaul rae) and I (loss gven defaul) are depced n Fgure 2. 3 Accordng o Guren (204) he annual AR coeffcen of house prces n he US ranges beween 0.4 and 0.7. We expec ha mos of he auocorrelaon s caused by common facors such as cyclcal developmen on he real esae marke. Thus we se he AR coeffcen of he LGD ndvdual facor o 0.. Fnance a úvěr-czech Journal of Economcs and Fnance no

6 Fgure 2 Developmen of he Exraced Common Facors Y (defaul rae lef axs) and I (loss gven defaul rgh axs) 3.3 Esmaon of he Facors Dynamcs In he nex sep we suded he nerconnecedness of he wo common facors Y and I. Accordng o he naure of he cred rsk he defaul rae and he charge-off rae 4 should be nerconneced.e. durng mes of economc expansons when he fnancal ncome of households and real esae prces ncrease he expeced decrease of he defaul rae should be accompaned by he expeced decrease of he charge-off rae and vce versa for he perods of economc downurns. The hypohess of nerconnecedness s suppored by he fac ha he muual correlaon beween he facors s 57%. To examne he muual relaonshp beween he facors Y and I n more deal we used conegraon analyss. Exsence of he un roo n boh me seres whch s one of he necessary condons for conegraon wasn rejeced. We furher ran he Johansen conegraon es whch confrmed ha conegraon of rank one s presen. 5 As he nex sep we esmaed he dynamcs of Y and I and he dependence on he exernal envronmen usng he VECM model wh Y and I beng endogenous varables and a se of macroeconomc ndcaors represenng exogenous varables. We consdered GDP he house prce ndex (HPI) consumer prces (nflaon) he FED base neres rae ndusral producon and personal ncome as represenaves of he exernal envronmen. The daase of macroeconomc varables was obaned from he US Bureau of Economc Analyss. The esmaon resuls are summarzed n Table 2. The predcve power of he esmae measured by R-square s hgh n he case of Y (over 90%) bu que low n he case of I (jus below 23%). The resuls show how srong he muual relaonshp of Y and I s and hey also prove he dependence of boh Y and I on he exernal envronmen. The mean squared errors of he equaons for Y and I are and respecvely. The dscovered relaonshp beween he facors and he macroeconomc envronmen s largely n lne wh our expecaons and s easly nerpreable cred rsk grows wh a rsng neres rae 4 LGD can be easly obaned by dvdng he charge-off rae by he defaul rae. 5 The deals of he Johansen conegraon es are provded n he Appendx. 570 Fnance a úvěr-czech Journal of Economcs and Fnance no. 6

7 Table 2 VECM Esmaon Resuls Varable Facor Dependen Y I Consan 0.32 *** - Y *** ** I *** ** GDP YoY *** *** HPI - - CPI nflaon YoY *** FED n. rae *** - Ind. producon - - Personal ncome - - Correcon erm *** ** Fgure 3 Predcon of Y Gven an Unchanged Macroeconomc Envronmen (lef) and Real Macroeconomc Developmen n Q and Q2 205 (rgh) and growng nflaon and decreases wh ncreasng GDP. Noe ha here s a negave relaonshp beween he facors and cred rsk.e. he hgher he value of a facor he lower he cred rsk. On he oher hand he negave sgn a he coeffcen of I - s counernuve as suggess negave auocorrelaon of he I me seres and negave correlaon beween Y and I. The reason for hs mgh be he naure of he charge-offs: as he charge-off s a purely accounng operaon here exss hgher proporonal varance n he charge-off rae compared o he delnquency rae whch s smooher. Thus he hsorcal varaon n he charge-off rae mos probably nroduced a negave auocorrelaon n he I me seres as well as he negave correlaon beween Y and I. From our esmaon we consruced wo ses of predcons of Y and I. In he frs se he predcon horzon was four quarers and he exogenous varables were se so ha he exernal envronmen was unchanged. The second predcon horzon was wo quarers and was based on he values of he macroeconomc ndcaors (GDP nflaon and neres rae) from Q and Q The former predcon allows us o see he developmen of Y and I ceers parbus whereas he laer can be compared o he real developmen and hus s able o backes he model. The predcons of Y and I are shown n Fgures 3 and 4. Fnance a úvěr-czech Journal of Economcs and Fnance no. 6 57

8 Fgure 4 Predcon of I Gven an Unchanged Macroeconomc Envronmen (lef) and Real Macroeconomc Developmen n Q and Q2 205 (rgh) Fgure 5 Comparson of he Model Predcon and Real Developmen of he 30+ Delnquency Rae (Q) Fgure 6 Comparson of he Model Predcon and Real Developmen of he Charge-Off Rae (G) Fnally we ranslaed he predced values of Y and I back no he 30+ delnquency rae (Q) and charge-off rae (G). Fgures 5 and 6 show he npu daase Q and G respecvely snce he end of 20 where he comparson of he modelpredced delnquency rae and charge-off rae n Q and Q2 205 o he acual developmen of delnquency and charge-off raes s added. The resuls show ha our model correcly esmaes he drecon of he ou-of-sample developmen; however 572 Fnance a úvěr-czech Journal of Economcs and Fnance no. 6

9 slghly overreacs o changes n he macroeconomc envronmen and suggess bgger movemens n Q and G han hose observed. 4. Concluson We consruced a mul-perod mulgeneraonal dynamc model of cred losses he dynamcs of whch are dependen on he exernal envronmen. The model nroduces several mporan orgnal enhancemens namely he nernal dynamcs of he ndvdual facors whn he loan porfolo a mul-generaon approach and fnally ncluson of macroeconomc varables n he esmaon process by means of a VECM model. We appled he model o he US naonwde porfolo of morgage loans. The emprcal analyss confrmed he well-known fac ha here exss a clear and esmable relaonshp beween he cred rsk and he macroeconomc envronmen. We found ou ha he mos sgnfcan macroeconomc varables nfluencng cred rsk facors are GDP nflaon and he neres rae. Addonally we proved he nerconnecedness beween defauls and recorded losses (or n oher words loss gven defaul). Fnally we demonsraed he predcve power of our model. A smple backes showed ha he model gves reasonable resuls. On he oher hand he model was found o overreac o changes n macroeconomc varables. Apar from he predcon oher applcaons of our model sugges hemselves. Thanks o he ncluson of macroeconomc varables he model may also be used for cred rsk sress esng. Obvously he model can also be used as a model of economc capal whn a fnancal nsuon. APPENDIX Table A Resuls of he Johansen Conegraon Tes Rank Egenvalue Trace es (p-value) (0.000) (0.095) Lmax es (p-value) (0.000) (0095) Fnance a úvěr-czech Journal of Economcs and Fnance no

10 REFERENCES Fronczak R Rosek S (205): Modelng loss gven defaul wh sochasc collaeral. Economc Modellng Volume 44 January 205: Frye J (2000): Collaeral Damage. Rsk Volume 3 Number 4. Gapko P Šmíd M (202a): Dynamc Mul-Facor Cred Rsk Model wh Fa-Taled Facors. Fnance a úvěr-czech Journal of Economcs and Fnance 62(2): Gapko P Šmíd M (202b): Modelng a Dsrbuon of Morgage Cred Losses. Ekonomcký časops Vol. 60 no. 0 (202) p Guren AM (204): The Causes and Consequences of House Prce Momenum. Harvard Unversy Workng Paper. Hamerle A Darsch A Jobs R Plank K (20): Inegrang macroeconomc rsk facors no cred porfolo models. Journal of Rsk Model Valdaon Volume5/ Number 2 :3 24. Hochguerel S Ohlsson H (20): Wealh Mobly and Dynamcs over Enre Indvdual Workng Lfe Cycles. ECB Workng Paper Seres no. 30. Jmenez G Menca J (2009): Modellng he dsrbuon of cred losses wh observable and laen facors. Journal of Emprcal Fnance Volume 6 Issue Meron RC (974): On he Prcng of Corporae Deb: The Rsk Srucure of Ineres Raes. Journal of Fnance 29 Chaper 2. Pesaran MH Schuermann T Treuler B-J Wener SM (2003): Macroeconomc Dynamcs and Cred Rsk: A Global Perspecve. CESfo Workng Paper no Pykhn MV (2003): Unexpeced Recovery Rsk. Rsk Augus: Šmíd M (205): Model of rsk and losses of a mulgeneraon morgage porfolo. 0h Inernaonal Scenfc Conference Fnancal managemen of frms and fnancal nsuons. Avalable a: hp://ssrn.com. Sommar PA Shahnazaran H (2009): Inerdependences beween Expeced Defaul Frequency and he Macro Economy. Inernaonal Journal of Cenral Bankng Sepember:83 0. Vascek OA (987): Probably of Loss on Loan Porfolo. KMV. Vrolanen K (2004): Macro Sress Tesng wh a Macroeconomc Cred Rsk Model for Fnland. Bank of Fnland Dscusson Paper no. 8/2004. Wzany J (20): A Two-Facor Model for PD and LGD Correlaon. Bullen of he Czech Economerc Socey Volume 8 Issue Fnance a úvěr-czech Journal of Economcs and Fnance no. 6