Sense and Sensitivity: An Input Space Odyssey for Asset-Backed Security Ratings

Transcription

1 Sense and Senstvty: An Input Space Odyssey for Asset-Backed Securty Ratngs Francesca D Grolamo (Correspondng author) European Commsson, JRC, Va E. Ferm 2749, Ispra (VA), Italy & Department of Mathematcs, K.U.Leuven, Celestjnenlaan 200 B, B-3001, Leuven, Belgum Tel: E-mal: francesca.d-grolamo@jrc.ec.europa.eu Henrk Jönsson European Commsson, JRC, Va E. Ferm 2749, Ispra (VA), Italy Tel: E-mal: henrk.joensson@jrc.ec.europa.eu Francesca Campolongo European Commsson, JRC, Va E. Ferm 2749, Ispra (VA), Italy Tel: E-mal: francesca.campolongo@jrc.ec.europa.eu Wm Schoutens Department of Mathematcs, K.U.Leuven, Celestjnenlaan 200 B, B-3001, Leuven, Belgum Tel: E-mal: Wm.Schoutens@ws.kuleuven.be Receved: September 13, 2012 Accepted: September 24, 2012 Onlne Publshed: October 11, 2012 do: /jfr.v3n4p46 URL: The presented study was ntated n the framework of the research project Quanttatve analyss and analytcal methods to prce securtsaton deals, sponsored by the European Investment Bank va the unversty research sponsorshp programme EIBURS, and realsed at the nternatonal research nsttute EURANDOM ( The authors acknowledge the ntellectual support from the partcpants of the project. Abstract The ratng of asset-backed securtes s partly based on quanttatve models for the defaults and prepayments of the assets n the pool. Ths quanttatve approach contans a number of assumptons and estmatons of nput varables whose values are affected by uncertanty. The uncertanty n these varables propagates through the model and produces uncertanty n the ratngs. The objectves of ths paper are twofold. Frstly, we advocate the use of uncertanty and senstvty analyss technques to enhance the understandng of the varablty of the ratngs due to the uncertanty n the nputs used n the model. Secondly, we propose a novel ratng approach called global ratng, that takes ths uncertanty n the output nto account when assgnng ratngs to tranches. Keywords: Structured fnance, Asset-backed securty, Ratng, Senstvty analyss, Default model 1. Introducton Asset-backed securtes (ABSs) are securtes created through a securtzaton process whose value and ncome payments are backed by a specfc pool of underlyng assets. Illqud assets that cannot be sold ndvdually are pooled together by the orgnator (Issuer) and transferred to a shell entty specally created to be bankruptcy remote, (a so called Specal Purpose Vehcle (SPV)). The SPV ssues notes (labltes) to nvestors wth dstnct rsk return profles and dfferent maturtes: senor, mezzanne, and junor notes. Ths technque s called tranchng of the Publshed by Scedu Press 46 ISSN E-ISSN

2 lablty. Cashflows generated by the underlyng assets are used to servce the notes; the rsk of the underlyng assets results to be dversfed because each securty now s representng a fracton of the total pool value. Fgure 1 shows a general ABS structure. A securtsaton credt ratng s an assessment of the credt rsk of a securtsaton transacton, addressng how well the credt rsk of the assets s mtgated by the structure. The ratng process s based on both quanttatve assessment and a qualtatve analyss of how the transacton mtgates losses due to defaults. For the quanttatve assessment, dfferent default scenaros, combned wth other assumptons, for example, prepayments, are generated usng more or less sophstcated models. Typcally the nput parameters to ths assessment are unknown and estmated from hstorcal data or gven by expert opnons. In any way, the values used for the parameters are uncertan and these uncertantes propagates through the model and generates uncertanty n the ratng output. Ths ntroduces uncertanty nto the assessment and t therefore becomes mportant to understand the ratngs parameter senstvty. For an ntroducton to ABS and the rsks and the ratng methodology see (Jönsson and Schoutens, 2010), (Jönsson et al., 2009), and (Campolongo et al., 2013). There has been an ncreased attenton to the ratng of asset-backed securtes durng the credt crss due to the enormous losses antcpated by nvestors and the large number of downgrades among structured fnance products. Ratng agences have been encouraged to sharpen ther methodologes and to provde more clarty to the lmtatons of ther ratngs and the senstvty of those ratngs to the rsk factors accounted for n ther ratng methodologes (see, for example, Global Stablty Report, Aprl 2008, (IMF, 2008) p. 81). Moody s, for example, ntroduced n (Moody's, 2009) two concepts, V Scores and Parameter Senstvty. Moody s V Scores provde a relatve assessment of the qualty of avalable credt nformaton and the potental varablty around the varous nputs to ratng determnaton. The ntenton wth the V Scores s to rank transactons by the potental for ratng changes owng to uncertanty around the assumptons. Moody s Parameter Senstvty provdes a quanttatve calculaton of the number of ratng notches that a rated structured fnance securty may vary, f certan nput parameters used dffered. Moody s analyss s performed by varyng one nput at a tme, whle holdng all the others fxed at predetermned values. Typcally, t s done by stressng just the two key nput parameters that have the greatest mpact wthn the sector, for example, the mean portfolo default rate and the mean recovery rate. Ths s a local approach whch does not analyse all the nput parameters and s not able to detect the mportance of nteractons among nputs. In contrast, n our paper we want to explore the whole nput space. The objectves are twofold. Frstly, we advocate the use of uncertanty and global senstvty analyss technques to enhance the understandng of the varablty of the ratngs due to the uncertanty n all the nput parameters. Uncertanty analyss quantfes the varablty n the output of nterest due to the varablty n the nputs. Global senstvty analyss assesses how the uncertanty n the output can be allocated to ts dfferent sources. Through global senstvty analyss, we quantfy the percentage of output varance that each nput or combnaton of nputs accounts for. Furthermore, we nvestgate the mportance of nteractons among dfferent nputs. Secondly, we propose a novel ratng approach called global ratng, that takes ths uncertanty n the output nto account when assgnng ratngs to tranches. The global ratngs should therefore become more stable and reduce the rsk of clff effects, that s, that a small change n one or several of the nput assumptons generates a dramatc change n the ratng. The global ratng methodology proposed gves one answer of a way forward for the ratng of structure fnance products. The rest of the paper s outlned as follows. In the next secton, we ntroduce the ABS structure we are gong to use as example: we descrbe the basc steps of modellng the cashflows produced by the asset pool, we pont out how these cashflows are dstrbuted to the labltes and we outlne the procedure to get ratngs. A descrpton of the general elements of global senstvty analyss s provded n Secton 3, wth a partcular attenton to the technques used n ths paper. In Secton 4, we apply uncertanty and global senstvty analyss technques to the ratngs exercse of the example structure. The global ratng, ntroduced n Secton 5, s an attempt to take nto account the uncertanty n the ratngs process when assgnng credt ratngs to ABSs. The paper ends wth conclusons. 2. Asset-Backed Securtes The assessment of the ABS s related wth the rsks nherent n the structure. The ratngs are ndcators of the credt rsk embedded n these nstruments. To derve a fnal ratng of asset-backed securtes, ratng agences combne both a qualtatve assessment and quanttatve analyss. The qualtatve assessment assesses the orgnator s and the servcer s operatons and legal ssues concernng the transfer of the assets from the orgnator to the ssuer ((Moody's, 2001), (Moody's, 2007a), and (Standard and Poor's, 2007)). Publshed by Scedu Press 47 ISSN E-ISSN

3 The quanttatve analyss reles on modellng of the cashflows produced by the assets (based on default and prepayment models of dfferent level of sophstcaton), the collecton of these cashflows, and the dstrbuton of the cashflows to the labltes accordng to a payment prorty (waterfall) descrbed n the deal s prospectus. In ths secton, the ABS structure used n the numercal experment s ntroduced; the basc steps of modellng the cashflows produced by the assets n the pool (default models) are descrbed; the collecton of the cashflows, and the dstrbuton of these cashflows to the labltes are ponted out; fnally, the procedure to get ratngs of asset-backed securtes s explaned. 2.1 The ABS Structure for the Experment Throughout the paper we assume that the collateral pool s homogeneous,.e., that all the consttuents of the pool are dentcal wth respect to ntal amount, maturty, coupon, amortsaton, and payment frequency, (see Table 1), and wth respect to rsk profle (.e. probablty of default). Ths mples that all the assets n the pool are assumed to behave as the average of the assets n the pool. We also assume the pool to be statc,.e. no replenshment s done. Ths collateral pool s backng three classes of notes: A (senor), B (mezzanne), and C (junor). The detals of the notes are gven n Table 2 together wth other structural characterstcs. To ths basc lablty structure we have added a cash reserve account. The reserve account balance s ntally zero and s founded by excess spread. The prorty of payments of the structure, the waterfall, s presented n Table 3. The waterfall s a so called combned waterfall where the avalable funds at each payment date consttutes of both nterest and prncpal collectons. 2.2 Cashflow Modellng We denote by t m, m=0,1,, mt, the payment date at the end of month m, wth t 0 =0 beng the closng date of the deal and tm T = T beng the fnal legal maturty date Cashflow Collecton The cash collecton each month from the asset pool conssts of nterest payments and prncpal collecton (scheduled repayments) whch together wth the prncpal balance of the reserve account consttute avalable funds. We begn by modellng the asset behavour for the current month, say m. The number of performng loans n the pool at the end of month m wll be denoted by Nm. ( ) We denote by nd ( m ) the number of defaulted loans n month m. The followng relaton holds true for all m : Nm ( ) = Nm ( 1) nd ( m). The outstandng prncpal amount of an ndvdual loan at the end of month m, after any amortsaton, s denoted by B( m ). Ths amount s carred forward to the next month and s, therefore, the current outstandng prncpal balance at the begnnng of (and durng) month m 1. Denote by BA( m ) the scheduled prncpal amount repad ( A stands for amortsed) n month m. The outstandng prncpal amount of an ndvdual loan at the end of month m: B( m) = B( m1) BA( m). The total outstandng prncpal amount of the pool at the end of month m s: P BC ( m)= N( m) B( m). Defaulted prncpal s based on prevous months endng prncpal balance tmes number of defaulted loans n current month: PD( m)= B( m1) nd( m). Interest collected n month m s calculated on performng loans,.e., prevous months endng number of loans less defaulted loans n current month: I( m)=( N( m1) nd ( m)) B( m) rl, where r L s the loan nterest rate. It s assumed that defaulted loans pay nether nterest nor prncpal. Scheduled repayments are based on the performng loans from the end of prevous month less defaulted loans: PSR ( m)= ( N( m1) nd ( m)) BA( m). We wll recover a fracton of the defaulted prncpal after a tme lag, T RL, the recovery lag: PRec ( m)= PD ( mtrl ) RR( m TRL ), where RR s the Recovery Rate. Publshed by Scedu Press 48 ISSN E-ISSN

4 The avalable funds n each month, assumng that total prncpal balance of the cash reserve account ( B CR ) s added, are: AF( m) = I( m) PSR( m) PRec( m) BCR( m). (1) The total outstandng prncpal amount on the asset pool has decreased wth: PRed ( m)= PD ( m) PSR ( m), (2) and to make sure that the notes reman fully collateralsed we have to reduce the outstandng prncpal amount of the notes wth the same amount Payment Waterfall The senor expenses represent payments to transacton partes, e.g. the ssuer and the servcer, that are necessary for the structure to functon properly. The senor expenses due to be pad durng the month m are based on the outstandng pool balance durng the month, m 1, (plus any unpad fees from prevous month): ( Sr) ( P) ( Sr) ( Sr) ( Sr) IDue ( m)= BC ( m1) f ISF ( m1) (1 rsf ), ( ) where Sr ( Sr f s the senor fee (expressed as a per cent of the outstandng pool balance), I ) SF ( m 1) s the shortfall ( Sr) (.e. unpad fees) n prevous month and r SF s the nterest rate on any shortfall (see Table 2). The actual amount pad to the ssuer s: ( Sr) ( Sr) IP ( m)= mn IDue( m), AF ( m ). After payment of the senor expenses, the avalable funds are updated: (1) ( Sr) AF ( m)= max0, AF ( m) IP ( m). We use the superscrpt (1) n A (1) ( m ) to ndcate that t s the avalable funds after tem 1 n the waterfall. F The nterest due to be pad to the class A notes s based on the current outstandng prncpal balance of the A ( A notes at the begnnng of month m,.e. before any prncpal redempton. Denote by B ) C ( m 1) the outstandng balance at the end of month m 1, after any prncpal redempton. Ths amount s carred forward and s, therefore, the current outstandng balance at the begnnng of (and durng) month m. To ths amount, we add any shortfall from the prevous month. The nterest due to be pad s: ( A) ( A) ( A) ( A) ( A) IDue ( m) = BC ( m1) r ISF ( m1) (1 r ), ( A where I ) SF ( m 1) s any nterest shortfall from month m 1 and r ( A) s the fxed nterest rate for the A notes. We assume the nterest rate on shortfalls s the same as the note nterest rate. The nterest pad to the A notes depends of course on the amount of avalable funds: ( A) ( A) (1) IP ( m)= mn IDue( m), AF ( m ). (3) If the avalable funds are not enough to cover the nterest due we get a shortfall that s carred forward to the next month: ( A) ( A) ( A) ISF ( m)= max IDue ( m) IP ( m),0. After class A nterest payments, the avalable funds are updated: (2) (1) ( A) AF ( m)= max0, AF ( m) IP ( m). The nterest payments to the B notes are calculated dentcally. The prncpal payments to the notes are based on the total prncpal reducton of the collateral pool PRed ( m ) calculated n equaton (2). The allocaton of prncpal due to be pad to the notes s supposed to be done sequentally, whch means that prncpal due s allocated n order of senorty. In the begnnng, prncpal due s allocated to the class A notes. Untl the class A notes has been fully redeemed no prncpal s pad out to the other classes of notes. After the class A notes are fully redeemed, the class B notes are started to be redeemed, and so on. Note that we here are dscussng the calculaton of prncpal due to be pad. The actual amount of prncpal pad to the dfferent notes depends on the avalable funds at the relevant level of the waterfall. That s: ( A) ( A) ( A) P ( m)= mn B ( m1), P ( m) P ( m 1), (4) Due C Red SF Publshed by Scedu Press 49 ISSN E-ISSN

5 ( A where P ) SF ( m 1) s prncpal shortfall from prevous month. The amount pad s: ( A) ( A) (3) PP ( m) = mn PDue( m 1), AF ( m). (5) Fnally, we have to update the outstandng balance after the prncpal redempton: ( A) ( A) ( A) BC ( m) = BC ( m1) PP ( m), (6) and avalable funds: (4) (3) ( A) AF ( m)= max0, AF ( m) P ( m). Snce we apply a sequental allocaton of prncpal due, no prncpal wll be pad to the B and C notes untl the A notes are fully redeemed. Note that the total prncpal reducton amount n month m allocated to the notes s the sum of the prncpal reducton allocated to the A notes, ( A P ) Red ( m ), and the prncpal reducton allocated to the B and C notes, ( BC, P ) ( m ): Red ( A) ( B, C) Red Red Red P ( m) = P ( m) P ( m). From the above, t s clear that the porton of the total prncpal reducton allocated to class A s: ( A) ( A) ( A) PRed ( m)= mn PRed ( m), BC ( m1) PSF ( m 1). ( A) ( A) Note that BC ( m1) PSF ( m1) 0. From the above expresson we see that there are two cases to take nto account: ( A 1) If ) ( BC, PRed ( m)= PRed ( m ), then P ) Red ( m )= 0. ( A) ( A) ( A) ( BC, ) ( A) 2) If PRed ( m)= BC ( m1) PSF ( m 1), then PRed ( m)= PRed ( m) PRed ( m). The class B prncpal due s: ( B) ( B) ( B, C) ( B) Due C Red SF ( B P ) P ( ) P ( m)= mn B ( m1), P ( m) P ( m 1). Prncpal pad, m, and the update of outstandng amount, ( B B ) ( m ), are dentcal to the A notes. Next tem n the waterfall s the rembursement of the reserve account. The reserve account balance after rembursement s: ( CR ) ( CR) (5) B ( m)= max BTarg ( m), AF ( m ), ( CR) where the target balance on the reserve account s gven as a fracton ( q Targ ) of the outstandng pool balance (see Table 2): ( CR) ( P) ( CR) BTarg ( m) = BC ( m) qtarg. After the reserve account rembursement, the avalable funds are updated: (6) (5) ( CR) AF ( m)= max0, AF ( m) BP ( m). The nterests to the C notes are calculated as for the nterest payments to the A and B notes. The payment of prncpal to the C notes s dentcal to the B notes, wth the small change that one has to make sure that no prncpal s pad untl the B notes are fully redeemed. Any resdual amount left s pad out n the fnal tem n the waterfall, class C addtonal returns, ( C) (8) PAdd ( m) = AF ( m ). 2.3 Default Modellng Dfferent default scenaros are generated by frst samplng a cumulatve portfolo default rate from a default dstrbuton and then dstrbute ths default rate over tme wth the help of a default curve. The default dstrbuton of the pool s assumed to follow a Normal Inverse dstrbuton and the default curve s modelled by the Logstc functon. These dstrbutons are charactersed by nput parameters whch have to be gven by expert opnons or estmated from hstorcal data on the performance of asset pools wth smlar characterstcs as the asset pool under consderaton. Thus, the quanttatve analyss ntroduces an exposure to parameter uncertanty. Under the assumpton to set up the default modellng by usng these two dstrbutons we are not focusng on model uncertanty. The mpact of model choce has been presented n (Jönsson and Schoutens, 2010), and (Jönsson et al., 2009). Recovery rate and recovery tmng assumptons are added to each default scenaro. C Publshed by Scedu Press 50 ISSN E-ISSN

6 The default curve - Logstc Functon The default curve represents the cumulatve portfolo default rate s evoluton over tme. It provdes the percentage of the total cumulatve default rate that wll be applcable n each month. The curve should therefore be monotoncally ncreasng and the slope of the curve should be always non negatve. The functon used to model the default tmng wll be a common sgmod curve: the Logstc functon. We are usng a four parameters verson: ()= a Ft,0 t T, (7) ct ( t 0 ) 1 be where abc,,, t 0 are postve constants and t [0, T ]. Parameter a s the asymptotc cumulatve default rate; b s a curve adjustment or offset factor; c s a tme constant (spreadng factor); and t 0 s the tme pont of maxmum margnal credt loss. The shape of the Logstc functon and the nfluence of the parameters are llustrated n Fgure 2. We can observe that the tmng of the peak of the monthly default rate to a large extent s controlled by t 0 and that the sharpness of the peak s controlled by the spreadng factor c. The curve adjustment factor b shfts the peak around t 0. If b =1, the curve becomes symmetrc around t 0. Note that the Logstc default curve has to be normalsed such that t starts at zero (ntally no defaults n the pool) and FT ( ) equals the expected cumulatve default rate. One default scenaros s thus generated by samplng a value for a from the default dstrbuton. The default dstrbuton - Normal Inverse Let PDR( T ) denote the portfolo default rate at tme T of our large homogeneous portfolo. The dstrbuton of PDR( T ) s gven by the Normal Inverse dstrbuton: ( y) ( p( T)) FPDR( T )( y) = P[ PDR( T)< y]=, (8) where 0% y 100%, s the oblgor correlaton, and pt ( ) s the probablty of default by T of a sngle oblgor n the pool. The Normal Inverse dstrbuton s derved as an approxmaton to the dstrbuton of the portfolo default rate at maturty T when the Gaussan one-factor model s used to model the defaults n a large homogeneous portfolo where the number of assets n the pool s assumed to grow to nfnty. The default dstrbuton n equaton (8) s a functon of the oblgor correlaton,, and the default probablty, pt ( ), whch are unknown and unobservable. Instead of usng these parameters as nputs t s common to ft the mean and standard devaton of the dstrbuton to the mean and standard devaton, respectvely, estmated from hstorcal data (see, for example, (Moody's, 2007b) and (Raynes and Rutledge, 2003)). Let us denote by cd and cd the estmated mean and standard devaton, respectvely. The mean of the dstrbuton s equal to the probablty of default for a sngle oblgor, pt ( ), so pt ( )= cd. As a result there s only one free parameter, the correlaton, left to adjust to ft the dstrbuton s standard devaton to cd. 2.4 Ratngs of ABSs Credt ratngs are based on assessments of ether expected loss or probablty of default. The expected loss assessment ncorporates assessments of both the lkelhood of default and the loss severty, gven default. The probablty of default approach assesses the lkelhood of full and tmely payment of nterest and the ultmate payment of prncpal no later than the legal fnal maturty. In ths paper, the expected loss ratng approach under the assumpton of a large, granular portfolo, s beng used, followng (Moody's, 2006). The ratngs are based on cumulatve expected loss (EL) and expected weghted average lfe (EWAL). Expected loss s based on the relatve net present value loss (RPVL) whch s calculated by dscountng the cashflows (both nterest and prncpal) receved on that note and by comparng t to the ntal outstandng amount of the notes. The present value of the cashflows under the A notes, for a gven scenaro j, s: Publshed by Scedu Press 51 ISSN E-ISSN

7 m ( A) ( A) T IP m j PP m j A j m/12 m=1 (1 ra/12) ( ; ) ( ; ) PVCF ( )=, (9) ( A where I ) P ( m; j ) and ( A P ) P ( m; j ) s the nterest and prncpal payment receved, respectvely, n month m under scenaro j (see Secton 2.2). We have ncluded j n the expressons to emphasze that these quanttes depend on the scenaro. Thus, for the A notes the relatve present value loss under scenaro j s gven by: ( A) B0 A j ( A) B0 PVCFA ( j ) RPVL ( )=, ( A) where B 0 s the ntal nomnal amount of the A tranche. The expected loss estmate usng M number of scenaros s: EL M A A j M j1 The weghted average lfe WAL ( ) for the A notes (n years) s defned as: 1 m T ( A) ( A) WALA( )= j P ( ; ) ( ; ), ( ) P m A j m BC mt j m T 12 B 0 m=1 A j 1 RPVL ( ). (10) ( A where B ) C ( mt; j) s the current outstandng amount of the A notes at maturty (month m T ) after any amortsaton. Thus, we assume that f the notes are not fully amortsed after the legal maturty, any outstandng balance s amortsed at maturty. Snce we assume monthly payments, the factor 1/12 s used to express WAL n years. Exactly as n the case wth the expected loss we apply Monte Carlo smulatons to estmate the EWAL: EWAL M 1 = WAL ( ). (11) A A j M j=1 The ratng of the note s found from Moody s Idealsed Cumulatve Expected Loss Table whch maps the Expected Average Lfe and Expected Loss combnaton to a specfc quanttatve ratng. Each run of the ratng algorthm s rather tme consumng as the expected loss and the expected average lfe of the notes are themselves the results of Monte Carlo smulatons. Thus, n order to speed up the senstvty analyss experment, we make use of Quas-Monte Carlo smulatons based on Sobol sequences n order to sample a value for the cumulatve default rate from the Normal Inverse dstrbuton. (See (Kucherenko, 2007), (Kucherenko, 2008), (Kucherenko et al., 2009), (Kucherenko et al., 2011) for more nformaton on Sobol sequences and ther applcatons.) 3. Global Senstvty Analyss Senstvty analyss (SA) s the study of how the varaton (uncertanty) n the output of a statstcal model can be attrbuted to dfferent varatons n the nput of the model. In other words, t s a technque for systematcally changng nput varables n a model to determne the effects of such changes. Very often, senstvty analyss s performed by varyng one nput at a tme, whle holdng all the others fxed at predetermned values. In most nstances the senstvty measure s chosen to be a partal dervatve and nputs are allowed only small varatons around a nomnal value (local senstvty analyss). However, when the addtvty of the model s not known a pror and nteractons among the nputs cannot be excluded, an analyss of ths knd s unrelable. In contrast to the local approach, global senstvty analyss does not focus on the model senstvty around a sngle pont but ams at Publshed by Scedu Press 52 ISSN E-ISSN

8 explorng the senstvty across the whole nput space. Usually global analyss s performed by allowng smultaneous varatons of the nputs thus allowng to capturng also potental nteractons effects among the varous nputs. For a general ntroducton to global senstvty analyss, see (Saltell et al., 2004) and (Saltell et al., 2008). In ths secton we ntroduce the two senstvty methods that we are gong to apply to our ratng exercse: the elementary effect method and the varance based method. The elementary effect method belongs to the class of screenng methods. Screenng methods are employed when the goal s to dentfyng the subsets of nfluental nputs among the many contaned n a model, relyng on a small number of model evaluatons. The varance based method s more accurate but computatonally more costly and therefore not always affordable. Through the varance based method t s possble to dentfy the parameters that contrbute the most to the total varance n the output. In our analyss we follow a two-steps approach. Frst we apply the elementary effect method to dentfy the subset of nput parameters that can be vewed as non-nfluental. The non-nfluental ones wll be gven fxed values. Then, we apply the varance based technque to quantfy and dstrbute the uncertanty of our model output among the nfluental nput parameters. In the present secton, we gve a general descrpton of the elementary effect method and the varance based technque. The notaton adopted s the followng. We assume that there are k uncertan nput parameters X 1, X 2,, Xk (assumed to be ndependent) n our model, and denote by Y the output of our generc model. Y s a functon of the nput parameters, whch we wrte Y ( X ) = f ( X1, X2,, X k ). Examples of nput parameters n our model are the mean and standard devaton of the default dstrbuton. Example of outputs are the expected loss or expected weghted average lfe of a tranche. To each nput parameter we assgn a range of varaton and a statstcal dstrbuton. For example, we could assume that X 1 s the mean of the default dstrbuton and that t takes values n the range [5%,30%] unformly, that s, each of the values n the range s equally lkely to be chosen. We could of course use non-unform dstrbutons as well, for example, an emprcal dstrbuton. These nput parameters and ther ranges create an nput space of all possble combnatons of values for the nput parameters. 3.1 Elementary Effects A very effcent method wthn the screenng methods n dentfyng mportant nputs wth few smulatons s the elementary effects method (EE method). It s very smple, easy to mplement and the results are clear to be nterpreted. It was ntroduced n (Morrs, 1991) and has been refned by (Campolongo et al., 2007). Because of the complexty of ts structure, the ABS s computatonally expensve to evaluate and the EE method s very well suted for the senstvty analyss of the ABS model s output. The method starts wth a one-at-a-tme senstvty analyss. It computes for each nput parameter a local senstvty measure, the so-called Elementary Effect (EE), whch s defned as the rato between the varaton n the model output and the varaton n the nput tself, whle the rest of the nput parameters are kept fxed. Then, n order to obtan a global senstvty measure, the one-at-a-tme analyss s repeated several tmes for each nput, each tme under dfferent settngs of the other nput parameters, and the senstvty measures are calculated from the emprcal dstrbuton of the elementary effects. To apply the EE method we map each of the nput parameter ranges to the unt nterval [0,1] such that the nput space s completely descrbed by a k -dmensonal unt cube. In order to estmate the senstvty measures, a number of elementary effects must be calculated for each nput parameter. Morrs suggested an effcent desgn that bulds r trajectores n order to compute r elementary effects. Each trajectory s composed by ( k 1) ponts n the nput space such that each nput changes value only once and two consecutve ponts dffer only n one component of a step equal to. Once a trajectory has been generated, the model s evaluated at each pont of the trajectory and one elementary effect for each nput can be computed. Let () l X and ( l1) X, wth l n the set {1, 2,, k}, denote two ponts on the th j trajectory. These ponts dffer Publshed by Scedu Press 53 ISSN E-ISSN

9 n the th component such that The EE of nput s: ( l1) X =( X, X,, X,, X ). 1 2 ( l1) ( l) X YX k Y j () l EE ( X ) =. (12) Practcally, accordng to the recent work of (Campolongo et al., 2011), a large number of dfferent trajectores (e.g ) s constructed and then r of them are selected n order to get the maxmum spread n the nput space. The number of trajectores ( r ) depends on the number of nputs and on the computatonal cost of the model; typcal values of r are between 4 and 10 (see (Morrs, 1991) and (Campolongo et al., 2007)). See (Campolongo et al., 2011) for the all detals about the desgn that bulds the r optmzed trajectores of ( k 1) ponts n the nput space. For each nput r elementary effects are then estmated, one per trajectory. Note that elementary effects obtaned from dfferent trajectores are ndependent snce the startng ponts of the trajectores are ndependent and thus the trajectory ponts. Startng from the absolute values of the elementary effects, the followng senstvty measure s used to assess the mportance of each parameter n the model. EE * j=1 =. (13) r Secton 4 presents the results obtaned by applyng the EE methodology to the ABS model. Results do not depend on the choce of the strategy employed to compute the elementary effects. 3.2 Varance Based Method We begn our dscusson on varance based method by notng that the varance of our generc output, VY ( ), can be decomposed nto a man effect and a resdual effect: r j V( Y) = V ( E ( Y X )) E ( V ( Y X )). X X : X X : (14) Here EX ( Y X ) s the condtonal expectaton gven X : calculated over all nput parameters and V X denotes the varance calculated wth respect to X. Equvalently, VX ( Y X ) s the varance wth respect to all : parameters condtonal on X. The frst term n equaton (14) s of most nterest to us. It tells how much the mean of the output vares when one of the nput parameters ( X ) s fxed. A large value of V( E( Y X )) ndcates that X s an mportant parameter contrbutng to the output varance. When we dvde ths varance wth the uncondtonal varance VY ( ) we obtan the frst order senstvty ndex wth respect to X : S VX ( E ( )) X Y X = :. (15) V ( Y ) These frst order senstvty ndces represent the man effect contrbuton of each nput. When nputs do not nteract and the model s purely addtve. k S =1 =1, the However, when k S =1 <1, the nteractons among the nputs play an mportant role n explanng the output varance. For nstance, the second order senstvty ndex, S, j, quantfes the extra amount of the varance correspondng to the nteracton between nputs and j that s not explaned by the sum of ther ndvdual effects. The second order senstvty ndex s: Publshed by Scedu Press 54 ISSN E-ISSN

10 S, j VX, X ( E (, )) ( ( )) ( ( )) j X Y X, X j V j X E Y X VX E Y X : X : j X : j j =. VY ( ) (16) In general, for a model output dependng on k ndependent nputs, the followng relaton has been shown to hold: k S S S =1, (17), j 1,2,, k =1 j> where S are the frst order senstvty ndces, S, j are the second order senstvty ndces, and so on untl S, whch s the k th order senstvty ndex. 1,2,,k The sum of all the terms n expresson (17) that contan descrbes Ths s called the total effect term, S and s expressed as follows: T S T V EX Y =1 X X : :. VY ( ) Xs ' total contrbuton to the output varance. For detals, see (Saltell, 2002), (Saltell et al., 2004), (Saltell et al., 2008), and (Sobol', 1993). For the techncal computaton of the senstvty ndces see (Saltell et al., 2010) and (Ratto and Pagano, 2010). 4. The SA Experment In order to apply the elementary effect method we frst have to dentfy the outputs we want to study and the nputs whch are controllable (.e. known) and those ones whch are uncontrollable (.e. unknown). We also have to dentfy sutable ranges for the uncontrollable nputs. The senstvty analyss (SA) s performed on the structure presented n Secton and the default model presented n Secton 2.3. The fundamental output n our study s the ratng of the ABSs. These ratngs are derved from the Expected Average Lfe and the Expected Loss of the notes as calculated n Expresson (10) and (11), respectvely. The SA thus nvestgates how the uncertanty n each nput parameter contrbutes to the uncertanty of the Expected Average Lfe and Expected Loss and hence the ratngs. To get one ratng, 2 14 scenaros are used under each parameter settng of the nputs. Ths guarantees the convergence of the ABS model. Wthout loss of generalty, the nvestor s assumed to be nformed about the collateral pool s characterstcs and the structural characterstcs gven n Table 1 and Table 2, respectvely, and the waterfall n Table 3. These are treated as controllable nputs. Assumng the default dstrbuton of the pool to follow a Normal Inverse dstrbuton and the default curve to be modelled by the Logstc functon, the uncertanty n the SA s not related to the model choce but to the parameters of the cumulatve default rate dstrbuton, the default tmng (the Logstc functon), and the recoveres: the mean ( cd ) and the standard devaton ( cd ) of the Normal Inverse dstrbuton; b, c, and t 0 of the Logstc functon; the recovery rate ( RR ) and the recovery lag ( T RL ). The nput ranges are summarsed n Table 4 and n the subsequent sectons we wll gve some motvaton to our choce of ranges. Ranges Assocated wth cd and cd The mean and standard devaton of the default dstrbuton are typcally estmated usng hstorcal data provded by the orgnator of the assets (see (Moody's, 2005) and (Raynes and Rutledge, 2003)). In our SA we wll assume that the mean cumulatve default rate at maturty T ( cd ) takes values n the nterval [5%,30%]. Ths s equvalent to assumng that the probablty of default before T for a sngle asset n the pool ranges from 5% to 30%. (Recall that the mean of the Normal Inverse dstrbuton s equal to the probablty of default of an ndvdual asset). We make the range of the standard devaton ( cd ) a functon of cd by usng a range for the coeffcent of varaton, / ([0.25,1]). Ths gves us the opportunty to assume hgher standard devaton for hgh values of cd cd (18) Publshed by Scedu Press 55 ISSN E-ISSN

11 the default mean than for low values of the mean, whch mples that we get hgher correlaton n the pool for hgh values of the mean than for low values, see Fgure 3. Ranges Assocated wth b, c, and t 0 n the Logstc Functon The parameters can be estmated from emprcal loss curve by fttng the Logstc curve to a hstorcal default curve (see (Raynes and Rutledge, 2003)). Because we want to cover a wde range of dfferent default scenaros we have chosen the followng parameter ranges: b [0.5,1.5] ; c [0.1,0.5]; 0 T 2T t [, ]. 3 3 Inspectng the behavour of the Logstc functons n Fgure 2 provdes some nsght to the possble scenaros generated wth these parameter ranges and gves an ntutve understandng of the dfferent parameters nfluence on the shape of the curve. Ranges Assocated wth Recovery Rate and Recovery Lag Recovery rates and recovery lags are very much dependent on the asset type n the underlyng pool and the country where they are orgnated. For SME loans, for example, Standard and Poor s made the assumpton that the recovery lag s between 12 months to 36 months dependng on the country (see (Standard and Poor's, 2004a)). Moody s uses dfferent recovery rate ranges for SME loans ssued n, for example, Germany ( 25% 65% ) and Span ( 30% 50% ) (see (Moody's, 2009)). The range assocated wth recovery lag T RL has been fxed to be equal to [6,36] months and wth the recovery rate to be equal to [5%,50%]. 4.1 Uncertanty Analyss The emprcal dstrbutons of the ratngs of the tranches n Fgure 4 can be used to obtan nformaton on the uncertanty n the model. All three hstograms show evdence of dsperson n the ratng outcomes. The dsperson s most sgnfcant for the mezzanne tranche. The ratngs of the senor and the junor tranches behave n a more stable way: we get ratngs wth low degree of rsk 78% of tmes for the A notes, and the C notes s unrated 51% of tme. Ths s not surprsng because losses are allocated to the notes n reverse order of senorty, t s the junor tranche that absorbs any losses frst. The uncertanty analyss hghlghts an mportant pont: the uncertanty n the ratng of the mezzanne tranche s very hgh. As a measure of the ratngs dsperson we look at the nterquartle range, whch s defned as the dfference between the 75 th percentle and the 25 th percentle. Ratngs percentles are provded n Table 5. It does not come as a surprse that the range s the hghest for the B notes, 9 notches, gven the very dspersed emprcal dstrbuton shown n Fgure 4. From Table 5, we can also conclude that the nterquartle range s equal to fve and three notches for the A notes and the C notes, respectvely. Ths dsperson n the ratng dstrbuton s of course a result of the uncertanty n the expected losses and expected average lves whch are used to derve the ratngs of each note. In the next secton, we apply senstvty analyss methods to assess whch sources of uncertanty among the nput parameters are contrbutng the most to the uncertanty n the outputs. 4.2 Senstvty Analyss Senstvty analyss assesses the contrbuton of each nput parameter to the total uncertanty of the outcome and the mportance of the nteractons among parameters. We analyse sx outputs: the expected loss and the expected weghted average lfe of each of the three classes of notes. Due to the fact that the ABS model s computatonally expensve we wll start our senstvty analyss by usng the elementary effect method to dentfy non-nfluental nput parameters. Each of the non-nfluental nputs wll be fxed to a value wthn ts range. After that, the varance Publshed by Scedu Press 56 ISSN E-ISSN

12 based method wll be appled to quantfy and to dstrbute the uncertanty of our model outputs among the nput parameters dentfed to be nfluental. The startng pont for both methods s the selecton of a number of settngs of the nput parameters. The number of SA evaluatons to get senstvty analyss results depends on the technque used. In the elementary effect method, we select 80 settngs of nput parameters. We apply the method wth usng r =10 trajectores of 4 ponts. Havng k =7 nput parameters the total number of SA model evaluatons s 80 ( N = r( k 1) ). In the varance based method, we select 2 8 settngs of nput parameters. These choces have been demonstrated to produce valuable results n general applcatons of elementary effects and varance based method (see (Ratto and Pagano, 2010)). For 14 each settng of the nput parameters, the ABS model runs 2 tmes to provde the outputs and the ratngs Elementary Effects * For a specfc output, the elementary effect method provdes one senstvty measures,, for each nputs. These senstvty measures are used to rank each nput parameter n order of mportance relatve the other nputs. The nput * parameter wth the hghest value s ranked as the most mportant one for the varaton of the output under consderaton. It s mportant to keep n mnd that the rankng of the nputs are done for each output separately. In Fgure 5, bar plots vsually depct the rank of the nput parameters for each of the sx outputs. The least nfluental parameters across all outputs are the recovery lag and the Logstc functon s b parameter. Hence they could be fxed wthout affectng the varance of the outputs of nterest and therefore the uncertanty n the ratngs to a great extent. Among the other nput parameters, the mean of the default dstrbuton ( cd ) s clearly the most mportant nput * parameter over all for all three notes. It s characterzed by hgh values for both the expected loss and the expected average lfe of all the notes. Ths hghlghts the strong nfluence the mean default rate assumpton has on the assessment of the ABSs. The only excepton from rankng the mean default rate as the most nfluental nput s the Expected Loss of the A notes. Here the coeffcent of varaton s ranked the hghest wth the recovery rate as second and the mean default rate as thrd. Changng the thckness of the junor tranche In the present secton, we nvestgate f the results of elementary effect method are affected by changng the ABS structure. We thus ncrease the ntal prncpal amount of the C notes accordng to the Table 6. We kept the ntal prncpal amount of the B notes, reducng only the ntal prncpal amount of the A notes. All other characterstcs of the structure are kept as prevously. In ths way, we ncrease the credt enhancement or loss cushon protecton of the mezzanne tranche. * The bar plots n Fgure 6 depct the rank of the nputs accordng to the values for the old and the new structure. The rankngs of the nput parameters for the new structure are consstent and coherent wth the result obtaned for the orgnal structure Varance Based Method In the elementary effect analyss performed above, two out of seven nput parameters were dentfed as non-nfluental. These two nputs can therefore be fxed to values wthn ther ranges. We have chosen to let b = 1 and T RL = 18. For the other nput parameters, we are gong to apply varance based method to quantfy ther contrbuton to the outputs varances. 8 We select now 2 settngs of nput parameters, we run our model for each of them and fnally we obtan the frst order senstvty ndces. Fgure 7 shows a decomposton of the output varance hghlghtng the man contrbutons due to the ndvdual nput parameters (frst order effects) and due to nteractons (second and hgher order effects whch are ndcated n whte n Fgure 7). For the B and C notes the mean cumulatve default rate, cd, s clearly contrbutng the most to the varance, accountng for approxmately more than 60% and more than 70%, respectvely. The uncertanty analyss performed earler ponted out that the uncertanty n the ratng of the mezzanne tranche s very hgh. The frst order senstvty ndces ndcate that mprovng the knowledge of cd can help to reduce the varablty of the outputs. In fact, f we could know the value of cd for certan, then the varance n expected loss and expected average lfe of the B notes could be reduced by more than 60%. Publshed by Scedu Press 57 ISSN E-ISSN

13 For the senor tranche, the frst order ndces ndcate that cd s the largest ndvdual contrbutor to the varaton n the expected loss of the A notes (17% ) and that c s the largest ndvdual contrbutor to the varaton n expected average lfe of the A notes ( 24% ). However, large parts of the varaton n expected loss and expected average lfe of the A notes come from nteracton among nput parameters. Ths ndcates that the frst order ndces cannot solely be used to dentfy the most mportant nputs and more sophstcated senstvty measures must be used. When nteractons are nvolved n the model, we are not able to understand whch nput s the most responsble of them by just usng the frst order effect contrbutons. Fgure 8 depcts the decomposton of the varance ncludng explctly the second order effect contrbutons due to the parwse nteractons between nput parameters. From the partton of the expected loss of the A notes varance we can clearly see that, for example, the nteracton between cd and the coeffcent of varaton and the nteracton between cd and RR sgnfcantly contrbute to the total varance wth 15% and 10%, respectvely. For the other outputs the frst order ndces are n most cases larger than the second order effects. Less than 5% for the mezzanne and junor tranche and less than 15% n the senor tranche refer to nteractons among more than two parameters (the whte slce Fgure 8): thus the sum of the frst and second order effects can be consdered an acceptable approxmaton of the total ndex. Fnally n Fgure 9, the approxmaton of the total effect ndces for the nput parameters for the dfferent outputs s presented. For all outputs t s clear that the mean of the default dstrbuton s the most nfluental nput parameter. 5. Global Ratng In the prevous secton we saw that the uncertanty n the nput parameters propagates through the model and generates uncertanty n the outputs. The ratng of the A notes, for example, shown n Fgure 4s rangng from Aaa to Unrated. The queston s how to pck the ratng of the A notes f we have ths varablty. By usng senstvty analyss we have been able to quantfy ths uncertanty and dentfy ts sources. If we knew the true value of the most mportant nputs, we could elmnate the most of the varablty n the model. In practse, these values are unknown to us. Ths mples that we have an ntrnsc problem n the ratng of ABSs. In ths secton, we propose to use a new ratng approach that takes nto account the uncertanty n the outputs when ratng ABSs. Ths new approach should be more stable, reducng the rsk of clff effects when assgnng ratngs to tranches. The clff effect refers to the rsk that a small change n one or several of the nput assumptons generates a dramatc change of the ratng. The dea s to assgn the ratng accordng to the uncertanty/dsperson of the credt rsk. We call ths new approach a global ratng, because t explores the whole nput space when generatng the global scenaros. The global ratng procedure s bascally the same as the one used for the uncertanty analyss and global senstvty analyss: 1) Identfy the uncertan nput parameters, ther ranges and dstrbutons; 2) Generate N global scenaros,.e., N dfferent settngs of the nputs, from the nput space; 3) For each global scenaro generate a ratng of each note; 4) Derve a ratng of each note by a percentle mappng. 5.1 Methodology The global approach derves the ratng of a note from the emprcal dstrbuton of ratngs generated from the global scenaros. An mportant fact s that ths procedure s ndependent of whch ratng methodology s used to derve the ratng of each global scenaro, that s, f t s based on expected loss or probablty of default. We propose a global ratng scale that reflects the dsperson of the credt rsk of a tranche. In other words the global scale should not reflect a sngle ratng but a range of possble credt rsks, thus takng nto account the uncertantes that affect the ratng process. The global ratng scale that we propose s supermposed on a ratng scale used by a ratng agency or by a fnancal nsttuton. The global ratng s based on a percentle mappng of the underlyng ratng scale, that s, a global ratng s Publshed by Scedu Press 58 ISSN E-ISSN

14 assgned to a tranche f a predetermned fracton of the ratngs generated usng the uncertanty scenaros s better than or equal to a gven underlyng ratng. Hence, to set up a global ratng scale we frst have to decde on the underlyng ratng scale. Imagne we use Moody s. A proposal for the global ratng scale A-E s provded n Table 7. The global ratng B n Table 7, for example, ndcates that a substantal fracton of the ratngs generated under dfferent scenaros fall n Moody s ratng scale Aaa-Baa3. Ths nforms the potental nvestor that the tranche shows low credt rsk for certan scenaros but that there are scenaros where the credt rsk s on a medum level. Secondly, we have to choose the fracton of ratng outcomes that should be layng n the credt rsk range. As frst attempt, we have defned the global scale wth respect to the 80 th percentle of the local scale (n ths case Moody s ratngs). The mappng s shown n Fgure 10. From the graph one can see that to assgn a global ratng B, for example, at least 80% of the ratngs must be better than or equal to Baa3. The dea to base ratngs on percentles s related to the Standard and Poor s, that s usng a percentle approach for assgnng ratngs to CDOs. For further nformaton, see (Standard and Poor's, 2001), (Standard and Poor's, 2004b), (Standard and Poor's, 2009), and (Standard and Poor's, 2010). Example Usng the percentles of the ratngs n Table 5 we can derve the global ratngs of the three notes. The global ratngs based on the ratng scale provded n Table 7 for dfferent ratng percentles are shown n Table Conclusons In ths paper, we have shown how global senstvty analyss can be used to analyse the man sources of uncertanty n the ratngs of asset-backed securtes (ABSs). The global senstvty analyss was appled to a test example consstng of a large homogeneous pool of assets backng three classes of notes (senor, mezzanne, and junor). Due to the fact that dervng ratngs for ABSs s computatonally expensve, the elementary effect method was chosen for an ntal analyss amng at dentfy the non-nfluental nput parameters. As a second step, varance based method wasappled to quantfy and to dstrbute the uncertanty of the outputs among the nput parameters dentfed to be nfluental and to analyse ther nteractons. The global senstvty analyss led to the concluson that the least nfluental nputs across all outputs are the recovery lag and the Logstc functon s b parameter. Hence they could be fxed wthout affectng the varance of the outputs of nterest and therefore the ratngs to a great extent. The mean of the default dstrbuton ( cd ) was found to be the most nfluental nput parameter among all nputs for all three notes. For the mezzanne and the junor tranche the mean cumulatve default rate, cd, s clearly contrbutng the most to the varance, accountng for approxmately more than 60% and more than 70%, respectvely, of the total varance of the expected loss and the expected weghted average lfe of the tranches. For the senor tranche, the frst order ndces ndcated that cd s the largest ndvdual contrbutor to the varaton n expected loss (17% ) and that c s the largest ndvdual contrbutor to the varaton n expected weghted average lfe ( 24% ). However, large parts of the varaton n the outputs for the senor tranche came from nteractons among nput parameters. Ths ndcates that the frst order ndces cannot solely be used to dentfy the most mportant nputs and more sophstcated senstvty measures must be used. In the fnal secton, we propose a new ratng approach called global ratng. The global ratng approach takes nto account that the uncertanty n the nput parameters propagates through the model and generates uncertanty n the outputs. The global approach derves the ratng of a note from the emprcal dstrbuton of ratngs generated from uncertanty scenaros. Each scenaro s a unque combnaton of values of the nput parameters. An mportant fact s that ths procedure s ndependent of whch ratng methodology s used to derve the ratng of each global scenaro, that s, f t s based on expected loss or probablty of default. The global ratng scale s chosen to reflect the dsperson of the credt rsk of a tranche. The dea s to let the global ratng reflect a range of possble credt rsks. Ths scale s supermposed on a ratng scale used by a ratng agency or by a fnancal nsttuton. The scale s based on a percentle mappng of the underlyng ratng scale, that s, a global ratng s assgned to a tranche f a Publshed by Scedu Press 59 ISSN E-ISSN

15 predetermned fracton of the ratngs generated usng the uncertanty scenaros s better than or equal to a gven underlyng ratng. References Campolongo, F., Carbon, J., & Saltell, A. (2007). An effectve screenng desgn for senstvty analyss of large models. Envromental Modellng & Software, 22(10), Campolongo, F., Carbon, J., & Saltell, A. (2011). From screenng to quanttatve senstvty analyss. A unfed approach. Computer Physcs Communcatons, 182(4), Campolongo, F., Jönsson, H., & Schoutens, W. (2013). Quanttatve assessment of securtsaton deals. Sprnger, n press. IMF. (2008, Aprl). Global Fnancal Stablty Report: Contanng systemc rsks and restorng fnancal soundness. Jönsson, H., Schoutens, W., & van Damme, G. (2009). Modelng default and prepayment usng Lévy processes: an applcaton to asset-backed securtes. Radon Seres on Computatonal and Appled Mathematcs, 8, , de Gruyter, Berln. Jönsson, H., & Schoutens, W. (2010). Known and less known rsks n asset-backed securtes. In D. Wgan, (Eds.), Credt Dervatves - The March to Maturty (pp ). IFR Market Intellgence. Kucherenko, S. (2007). Applcaton of global senstvty ndces for measurng the effectveness of Quas-Monte Carlo methods. Proc. of the Ffth Internatonal Conference on Senstvty Analyss of Model Output. Kucherenko, S. (2008). Hgh dmensonal Sobol s sequences and ther applcaton, Techncal Report. Kucherenko, S., Rodrguez-Fernandez, M., Panteldes, C., & Shah, N. (2009). Monte Carlo evaluaton of dervatve-based global senstvty measures. Relablty Engneerng System Safety, 94(7), Kucherenko, S., Fel, B., Shah, N., & Mauntz, W. (2011). The dentfcaton of model effectve dmensons usng global senstvty analyss. Relablty Engneerng System Safety, 96(4), Moody s Investor Servce. (2001). The combned use of qualtatve analyss and statstcal models n the ratng of securtsatons, 11 July Moody s Investor Servce. (2005). Hstorcal default data analyss for ABS transactons n EMEA, 2 December Moody s Investor Servce. (2006). Moody s TM ABSROM v 1.0 user gude, 22 May Moody s Investor Servce. (2007a). Informaton on EMEA SME securtsatons - Moody s vew on granular SME loan recevable transactons and nformaton gudelnes, 12 March Moody s Investor Servce. (2007b). Moody s approach to ratng granular SME transactons n Europe, Mddle East and Afrca, 8 June Moody s Investor Servce. (2009). V scores and parameter senstvtes n the EMEA small-to-medum enterprse ABS sector, 15 June Morrs, M. D. (1991). Factoral samplng plans for prelmnary C. Technometrcs, 33(2), Ratto, M., & Pagano, A. (2010). Usng recursve algorthms for the effcent dentfcaton of smoothng splne ANOVA models. Advances n Statstcal Analyss, 94(4), Raynes, S., & Rutledge, A. (2003). The analyss of structured securtes: precse rsk measurement and captal allocaton. Oxford Unversty Press. Saltell, A. (2002). Makng best use of model evaluatons to compute senstvty ndces. Computer Physcs Communcatons, 145(2), Saltell, A., Tarantola, S., Campolongo, F., & Ratto, M. (2004). Senstvty analyss n practce. Wley. Saltell, A., Ratto, M., Andres, T., Campolongo, F., Carbon, J., Gatell, D.,... Tarantola, S. (2008). Global senstvty analyss: The prmer. Wley. Saltell, A., Annon P., Azzn, I., Campolongo, F., Ratto, M., & Tarantola, S. (2010). Varance based senstvty Publshed by Scedu Press 60 ISSN E-ISSN

16 analyss of model output. Desgn and estmator for the total senstvty ndex. Computer Physcs Communcatons, 181(2), Sobol, I. (1993). Senstvty analyss for non-lnear mathematcal models. Mathematcal Modelng and Computatonal Experment, 1, Translated from Russan: I.M. Sobol. (1990). Senstvty estmates for non-lnear mathematcal models. Matematcheskoe Modelrovane, 2, Standard, & Poor s. (2001). CDO Evaluator apples correlaton and Monte Carlo smulaton to determne portfolo qualty, 13 November Standard, & Poor s. (2004a). Credt rsk tracker strengthens ratng analyss of CLOs of European SME loans, 10 June Standard, & Poor s. (2004b). CDO Spottng: General cashflow analytcs for CDO securtsaton, 25 August Standard, & Poor s. (2007). Prncples-based methodology for global structured fnance securtes, 29 May Standard, & Poor s. (2009). Update to global methodologes and assumptons for corporate cashflow and synthetc CDOs, 17 September Standard, & Poor s. (2010). CDO Evaluator system verson User gude, 22 Aprl Table 1. Collateral characterstcs Collateral Number of loans Intal prncpal amount Weghted average maturty 5 years Weghted average coupon (per annum) 9% Amortsaton Level-Pay Payment frequency Monthly Table 2. Lablty and structural characterstcs Class of Notes Labltes Intal Prncpal Amount Interest Rate (per annum) Credt enhancement (%) A % 20% B % 6% C % 0% Fnal Maturty Payment frequency Prncpal allocaton Shortfall rate (per annum) Sr f General Features Senor expenses 10 years Monthly Sequental Applcable note coupon Senor fees ( ( ) ) 2% of Outstandng Pool Balance Payment frequency Monthly Shortfall rate (per annum) ( ( Sr r ) ) 20% SF Reserve account ( CR) Target amount ( q ) 1.0% of Outstandng Pool Balance Targ Interest rate ( ( ) CR r ) 1.0% Publshed by Scedu Press 61 ISSN E-ISSN

17 Table 3. The waterfall used n the analyss Waterfall Level Basc amortsaton 1) Senor expenses 2) Class A nterest 3) Class B nterest 4) Class A prncpal 5) Class B prncpal 6) Reserve account remburs. 7) Class C nterest 8) Class C prncpal 9) Class C addtonal returns Table 4. Ranges for the uncertan nput parameters Parameter Range cd [5%,30%] cd Coeff. Varaton ( ) [0.25,1] cd b [0.5,1.5] c [0.1, 0.5] t 0 T 2T [, ] 3 3 T RL [6,36] RR [5%,50%] Table 5. Ratng percentles and nterquartle ranges Percentle Interquartle Range Note Number of notches A Aaa Aa1 A2 A3 Baa3 Ba1 5 B A2 Ba1 B2 B3 Caa Caa 9 C B2 Unr. Unr. Unr. Unr. Unr. 3 Table 6. Tranche thckness n the new structure Class of Notes Intal Prncpal Amount A (Senor) B (Mezzanne) C (Junor) Publshed by Scedu Press 62 ISSN E-ISSN

18 Table 7. A proposal of global ratng scale and the correspondng ranges n Moody s ratng scale Global Ratng Moody s A A3 Aaa B Baa3 Aaa C Ba3 Aaa D B3 Aaa E Unr. Aaa Table 8. Global ratngs for dfferent percentles Percentle Note 75% 80% 90% A A A B B D D E C E E E Fgure 1. ABS general structure Publshed by Scedu Press 63 ISSN E-ISSN

19 Fgure 2. The Logstc functon and ts dervatve for dfferent values of b, c and t 0. Parameter values: a =1 and T =60 Fgure 3. Impled correlaton versus coeffcent of varaton Publshed by Scedu Press 64 ISSN E-ISSN

20 Fgure 4. Emprcal dstrbuton of Moody s ratngs obtaned by 256 smulatons Fgure 5. Bar plots of the * values for the A, B, and C notes Publshed by Scedu Press 65 ISSN E-ISSN

21 Fgure 6. Bar plot of the * values for all the notes n the orgnal and the new structure, respectvely Publshed by Scedu Press 66 ISSN E-ISSN

22 Fgure 7. Frst order senstvty ndces for the orgnal structure Fgure 8. Frst and second order senstvty ndces for the orgnal structure Publshed by Scedu Press 67 ISSN E-ISSN