Credit risk analysis of cash ow CDO structures

Transcription

1 Credi risk analysis of cash ow CDO srucures Philippos Papadopoulos and Caroline I.M.L. Tan y November 30, 2007 Absrac Raing cash ow CDO srucures is a challenge. We develop a mehod ha o ers consisen and compuaionally e cien credi risk analysis of cash ow CDO srucures. The proposal makes use of simple porfolio models ha admi semi-analyic represenaions of he loss disribuion, combined wih deailed and fas calculaions of realisic ineres and principal cash ow waerfalls. We de ne in his conex and sudy credi ranche risk measures such as he probabiliy of loss and expeced loss-given-defaul and he variance of he laer. We benchmark our approach agains he sress-scenario based analysis favored by cash ow CDO marke praciioners. ABN AMRO Bank, Group Risk Managemen y Disclaimer: The resuls, opinions and conclusions presened in his aricle re ec he personal opinion of he auhors and no ha of ABN AMRO Bank. 1

2 2 1 Inroducion The credi risk analysis of CDOs is a challenging and imporan aciviy. A good example is he "credi crunch" over he summer of 2007, which sared wih a crisis in US sub-prime morgages and ended wih an illiquid srucured credi marke. The illiquidiy was amongs ohers caused by he inransparency of srucured credi producs. Despie some similariies beween he synheic and cash ow CDO marke segmens, here are signi can srucural di erences in he respecive conracs, wih he synheic insrumens being signi canly more ransparen o describe and model. In conras o he sandard synheic ranche conrac (implemened wih a credi defaul swap) a ypical cash- ow CDO srucure is a complex rue sale securiizaion, whereby a diversi ed pool of credi asses is sold ino a Special Purpose Vehicle which in urn issues muliple classes of noes. The CDO liabiliies are serviced exclusively by he cash ows generaed by he collaeral asses. Cash ow CDOs are characerized by a large number of srucural elemens and covenans which govern very precisely he disribuion of income and principal o he issued noes (see [6]). In common wih oher credi risky bonds, CDO invesors are ineresed in he credi risk pro le of a given ranche. Commonly used risk measures for credi risky bonds are he probabiliy of defaul and recovery (or loss given defaul). Modelling defaul raes and recoveries for corporae bonds and loans has received enormous aenion in recen years (see [2]). However, CDO cash ows and heir risk characerisics di er signi canly from he cash ows of a corporae bond. In widely used marke pracise, he credi risk analysis of cash ow CDOs is based on he following sraegy: rs, esimae a defaul rae disribuion of he collaeral porfolio using (possibly) a porfolio model (see [7]). Nex, using his disribuion, esablish represenaive sress scenarios (each associaed wih a scenario realizaion probabiliy). Then, calculae cash ow paerns for each sress scenario. Finally, he severes sress scenarios "susained" by a given ranche wihou moneary loss deermines he raing of ha ranche. For insance, if under he assumpions of a defaul scenario corresponding o a AAA sress (and expeced o occur wih AAA frequency), he ranche sill receives all he scheduled cash ows, he ranche is raed AAA. Noe, ha radiional cash ow calculaions ransform a single defaul scenario ino a single cash ow scenario. While reasonable as a saring poin, he above procedure su ers from a key shorcoming: The ranche risk pro le is probed only parially (on a single cash ow scenario basis) insead of considering he enire sochasic disribuion of cash ows. We invesigae a more comprehensive approach where we calculae and weigh cash ow scenarios wih he probabiliies assigned by he underlying collaeral model, hereby creaing he full probabiliy disribuion of cash ow scenarios. Using his disribuion we de ne risk measures for he various ranches: e.g., probabiliy of ranche loss, expeced loss or LGD volailiy. Combining a porfolio credi risk model wih deailed cash ow calculaions is eminenly feasible in a Mone Carlo conex, bu no wihou subsanial echnical overhead. We illusrae here ha a simple semi-analyical framework based on Vasicek s large pool model permis fas and insighful analysis of srucures while sill capuring realisic levels of complexiy.

3 3 We focus on a generic CLO srucure (CDOs where he underlying pool is a pool of leveraged loans) o avoid he addiional complexiy of a CDO of ABS srucure. Since mos acual CLO ransacions are managed CDO s, any acual cash ows are complicaed by he uncerain (and very pah dependen) porfolio pro le. For concreeness, in his implemenaion we adop a saic porfolio view. Consequenly, we incorporae a key subse of srucural CLO feaures (he OC/IC riggers), bu leave ou elemens ha are linked o porfolio composiion. As wih any analysis of a nancial insrumen wih he amoun of complexiy of a cash ow CDO, our resuls cover only a subse of he relevan risk facors. The conribuion of his paper is wofold: Firs, we illusrae ha he modelling of complex cash ow CDO waerfalls can be achieved wih simple (low dimensional) porfolio models ha can be compued semi-analyically (in he simples case reducing he analysis o onedimensional inegraion). This resul follows because, wihin he assumpions of he curren sudy, ineres and principal cash ows are deerminisic funcions of he porfolio loss rae. Secondly, we propose ha consisen measures of ranche risk can be de ned using he enire disribuion of cash ows, insead of a parial, scenario based, approach. The laer mehod can run ino di culies in he presence of cash ow mechanisms ha violae sandard senioriy prioriizaion. 1 The srucure of he remaining paper is as follows: Secion 2 oulines a simple muli-period exension of he analyic large pool model (Vasicek model), which is a well known approximae descripion of he credi risk in a pool of homogeneous asses. We illusrae how his model can be used o derive an aggregae descripion of collaeral performance for a saic pool of asses. Nex, in Secion 3 we discuss a mahemaical speci caion of a cash ow CDO srucure. The deailed waerfall mechanics is presened in an appendix B.2. In Secion 4 we inroduce various risk measures for capuring he performance of CDO ranches and in Secion 5 we apply our mehod o a ypical srucure and compare he resuls wih a sylized scenario based analysis. 2 Using he Large Pool Model as he cash ow CDO Collaeral model We adop he large pool (Vasicek) credi risk model as he basis for modeling collaeral performance. This implies a homogeneous porfolio wih in niely many asses, all wih he idenical defaul curves q and idenical expeced recoveries g per asse and uniform asse correlaion 2. More deails on he large pool model are given in Vasicek s paper (2002, see [5]). This model has been widely used o provide an approximae descripion of porfolio loss, no leas for esimaing regulaory capial for nancial insiuions. We noe ha his model can be (and has been) improved in many ways. Besides removing he homogeneiy (as 1 This is likely o be he case each and every ime he following wo saemens are no sricly idenical: "The ranche will experience no loss in a fracion p of all possible cash ow realisaions" versus "The ranche will experience no loss in a fracion p of all possible porfolio loss realisaions". Clearly, we only really care abou he former measure.

4 4 in Lucas, Klaassen, Spreij and Sraemans (2000), see [4]) and large-pool assumpion (as in Anderson, Basu and Sidenius (2003), see [1], and as in Lauren and Gregory (2003), see [3]). Any of hose advances in semi-analyic echniques (which are primarily spurred by sudies of pricing in he synheic CDO marke) can be subsiued here for a collaeral model. The simpliciy of he chosen collaeral model means ha we need o absrac from several feaures of he acual cash srucure in he following areas: he porfolio is assumed fully ramped-up a closing ime. There is no scheduled amorizaion (all exposures are bulles wih same mauriy). There are no subsiuions or reinvesmens (saic srucure) of principal proceeds, which in urn requires ha principal recoveries from workous are paid down o he various ranches before mauriy. Modeling he reinvesmen of principal repaymens, recoveries and excess spread would require a more dynamic porfolio model, where new asses can be inroduced a forward ime poins, coningen on curren marke evens (raing migraion of he porfolio, spread moves), ogeher wih some prede ned porfolio managemen sraegy. A presen, hose addiional complexiies would only hinder our aim o undersand and de ne he basic cash ow CDO risk properies. We inroduce a ime grid (discree ime approximaion), wih he ime of closing denoed as T 0 = 0 and N subsequen periods a imes T and saed mauriy a T N = T. If a variable X is indexed by, we mean ha he variable is evaluaed a T. X = X (T ) : To simplify accouning he cash ows we assume ha defauls happen a he sar of each period, hence coupon from defauled asses is los for ha enire period (and all subsequen ones). We assume recoveries are delayed by one period. Coupon paymens o he ranche invesors, equiy disribuions, any principal adjusmens and any allocaions o auxiliary accouns are done insanly afer he curren period collecions. Recalling he basic feaures of he Large Homogeneous Pool model, he credi sanding of each asse j is driven by an associaed normal variable W j = Z + p 1 2 j, where boh Z and j have a sandard normal disribuion. Defaul before ime happens wih frequency P (W j < ) = q which implies ha = N 1 (q ). Calibraion is o he hisorical defaul performance of similarly raed pools. Condiioning on a realizaion Z = z, he cumulaive defaul rae by period is denoed by q (z), and he condiional defaul rae 2 (condiional on boh he realizaion of Z and survival ill he previous period). is (Z) q (z) = N( z p 1 2 ) (1) (z) = q (z) q 1 (z) 1 q 1 (z) (2) In our analysis we calibrae o hisorical defaul performance (we use widely available S&P daa). The cumulaive defaul probabiliies for he raings we have used are presened in 2 We noe ha using he Vasicek model in a muli-period conex implies a very concree iner-emporal disribuion of defaul raes which may or may no be provide an adequae o acual daa.

5 Hazard raes Cumulaive defaul raes 5 Appendix A.1. A summary of his able (excluding raing modi ers "-" and "+") is PD 1 year 2 years 3 years 4 years 5 years 6 years 7 years AAA 0:02% 0:06% 0:12% 0:19% 0:28% 0:39% 0:52% AA 0:11% 0:24% 0:39% 0:57% 0:76% 0:97% 1:20% A 0:14% 0:32% 0:54% 0:81% 1:11% 1:45% 1:81% BBB 0:22% 0:64% 1:18% 1:81% 2:50% 3:21% 3:94% BB 2:77% 5:26% 7:50% 9:49% 11:25% 12:82% 14:20% B 8:59% 14:51% 18:59% 21:45% 23:49% 25:00% 26:15% CCC 19:82% 30:18% 35:83% 39:09% 41:08% 42:39% 43:32% D 100:00% 100:00% 100:00% 100:00% 100:00% 100:00% 100:00% : As an example, for a B+ raed porfolio we show he quaniles and cumulaive defaul raes q and hazard raes in gure 1. The average defaul raes depend on he choice of a B+ Figure 1: Quaniles of he Vasicek disribuion for B+ raed porfolio Quaniles (e.g. 10%) of he cumulaive defaul raes and hazard raes of a B+ raed porfolio in a Vasicek model. Noe ha he hazard rae curve is downward sloping. We also show he average (cumulaive) defaul raes and he corresponding hazard raes Quaniles Vasicek model for B+ 10% 20% 30% 40% Year 50% Quaniles Vasicek model for B+ 60% 70% % Year 90% Expecaion raed porfolio (and he corresponding hisorical gures). For some conex, we also presen

6 Hazard raes Cumulaive defaul raes 6 q and for he oher raings in gure 2. Any oher funcional speci caion q (z) could also be used insead of he Vasicek disribuion. E.g. he q (z) could be based on a -disribuion insead of a Gaussian disribuion, in order o inroduce faer ails. Figure 2: Cumulaive defaul raes and hazard raes for di eren raing classes For each raing class (e.g. AAA), he cumulaive defaul raes and hazard raes. Noe ha he hazard rae curve is downward sloping for he lower raing classes and upward sloping for he higher raing classes. 0.8 Expeced cumulaive defaul raes 0.6 AAA Year Expeced hazard raes AA A BBB BB 0.1 B 0.05 CCC Year We denoe by N he ousanding noional a any period. By convenion his is iniially scaled o uniy. The reducion of performing collaeral due o defauls is governed by he equaion N = (1 )N 1 : (3) In equaion (3) and in he sequel we drop he z dependence on all equaions in order o simplify he noaion. All relaionships are assumed o hold condiionally on he facor realizaion, unless explicily noed. We assume all asses and liabiliies are oaing rae noes. We de ne as c he weighed average ineres paymen (coupon rae) of he performing collaeral. The oaing rae collaeral is assumed o have an average coupon of c = r + s, where s is he average spread over some reference rae r wih he same mauriy T. I follows from he homogeneous and saic porfolio assumpion ha he coupon rae c remains he same for he lifeime of he srucure. We de ne for each period he ineres

7 7 cash ow based on he coupon and performing capial w = c N : (4) The recovery cash ow equals he defaul rae imes he recovery rae (g) r = g N 1 : (5) When handling recovery cash ow, in principle we need o disinguish beween recovered principal and recovered coupons, as hose may be processed di erenly in he cash ow waerfall. We will rea recovered amouns as principal recoveries. 3 The cash ow CDO srucure In his secion we describe in mahemaical erms he cash ow CDO srucure. The srucure involves M ranches (alernaively noes or bonds), indexed by i = 1; : : : ; M in order of decreasing senioriy, plus an equiy posiion. We denoe T0 i he iniial size for he i-h ranche. The ranche size may be reduced by conracual repaymens or be increased by he addiion of deferred ineres. T i denoes he running noional for he i-h ranche during he -h period. Similarly, E 0 is he iniial equiy size while E denoes he curren "book" equiy, E = N M X j=1 T j : (6) The subordinaion available o he i-h ranche a each period is de ned as: MX i = E + T j = N i ; (7) j=i+1 where in he above equaion i is useful o de ne he complemenary noion of he suppored deb level for a ranche, i.e., he sum of oal ranche sizes down o and including he i-h ranche: ix i = T j : (8) j=1 In order o keep rack of he collaeral cash ows we will inroduce he following accouns: The ineres proceeds accoun i and he principal proceeds accoun p. The iniializaion and updae of hose accouns is given in he appendix. Moving on o he liabiliy side, for each ranche, he paid coupon c i is a sum of a risk-free rae and a ranche spread (c i = r + s i ), which a any given period leads o scheduled ineres paymens, S i = c i T i. Subjec o he paymen waerfall each noe will receive an acual ranche paymen b i. The residual cash ow, afer all deb servicing and coss, is paid ou as dividend d o he equiy ineres. Scheduled ineres ha is no paid during a paymen period (i.e., when b i < S) i is added o he ousanding noional of he ranche as deferred ineres 3 : T i := T i + S i b i : (9) 3 The implicaion of his reamen of missed coupons is ha in subsequen periods deferred ineres is accruing a he ranche coupon rae. For simpliciy he above srucure applies o all noes. In acual ransacions i may be he case ha some senior noes do no defer ineres bu defaul immediaely.

8 8 In equaion (9) we inroduced he ":=" symbol o denoe he "updae" of a variable during he disribuions occurring on a CDO paymen dae. This noaion implies ha he acual nal values of any assigned variable depend on he ordering of hose disribuions and adjusmens, as per he speci c waerfall which we will de ne laer in his secion. An imporan noe here is ha due o deferred ineres and possible unscheduled amorizaion (in order o comply wih deb covenans), he scheduled paymens are pah dependen. E.g., we only know of he required nal repaymens one period before mauriy. In conras o sandard synheic CDO srucures, he noional of a cash ow CDO ranche is no conracually reduced due o defauls during he life of he ransacion (excep for he adjusmens menioned above). Such reducion is obviously implici in he accumulaing noional losses in he collaeral, bu is normally only revealed as an explici noional wriedown (loss) when, as per he erms of he indenure, he ranche mus be redeemed and here are no available funds. Ulimaely in analyzing he credi risk of he i-h ranche, we are concerned wih he probabiliy disribuion of he realized reurns b i. Since missed coupons accrue a he ranche rae, wih lile loss of generaliy we can focus on he nal period paymens b i T. If he realizaion of hose variables is lower han he scheduled nal redempions ST i here has been some loss. More on de ning suiable risk measures in Secion 4. We de ne running measures of he available overcollaeralizaion (OC) for he various ranches: L i N = ~ i : (10) In analogy wih curren pracice, we used in he numeraor an adjused ousanding noional ~N = h N + r ; (11) in order o properly re ec he curren leverage. Hence any risk-free cash (available in he form of curren recoveries r ) is added o he adjused noional. We inroduced h as a haircu applied o he ousanding noional. The purpose of his haircu is o provide proecion agains a marke value decline of deerioraing credis. presened here we will assume ha h = 1. 4 For simpliciy in he calculaions In addiion o he leverage consideraions above, we de ne ineres coverage raios as running measures of he deb service paymen abiliy. This is de ned as he raio of he ineres coverage amoun (IC), ~w = w f s (de ned as he curren ineres proceeds minus senior fees), over he cumulaive scheduled ineres paymens down o he ranche under consideraion: l i = ~w i ; (12) 4 In pracice his haircu is applied only o he mos deerioraed credis in collaeral pool, i.e., he par ha would be expeced o sell a a discoun o par value due o downward migraion. In he curren aggregae porfolio model haircus canno be applied consisenly. In addiion, i is he curren collaeral raing ha deermines he haircu, which requires a dynamic porfolio model.

9 9 where we have de ned he suppored deb service level, i ix ix i = c j T j = S j : (13) j=1 j=1 The cash ow srucure is disinguished by he fac ha he running leverage and deb service raios (L i ; l) i are checked agains pre-speci ed (and consan) riggers or barriers (L i B ; li B ). Depending on he oucome of hose comparisons, he indenure may modify he manner in which ineres and principal paymens are disribued o he various noes and equiy. For he sake of simpliciy we assume here ha each ranche has an associaed OC/IC pair of ess, whereas in pracice several ranches may share a pair of ess. We also assume, as is usual in pracice, ha he wo ypes of ess are always examined in pairs and only he join oucome a ecs he waerfall. A each period, each ranche s OC es saus is given by he indicaor variable O i = 1 fl i >L i B g : (14) Similarly, each ranche s IC es saus is given by he indicaor variable I i = 1 fl i >lb i g : (15) Finally, we are ineresed in he join indicaor J i = 1 fl i >L i B ;li >li B g : (16) If he i-h OC es is failing during a period, he paymen waerfall may require ha a ha poin principal is (re)paid sequenially o all noes senior o or equal he senioriy of he es level, unil he failing i-h es is me. 5 For each OC es we de ne he "supporable" noional size Q i : Q i = N ~ : (17) This is he (reduced) size of he various ranches ha will resore he OC indicaor for his period). The scheduled noional reducion of he k-h ranche for he curing of he i-h OC es will be given (recursively, saring from he senior-mos ranche) by he expression L i B T ki = max(min( i kx 1 m=1 T mi Q i ; T k ); 0): (18) The meaning of he above expression is ha he currenly suppored deb level i mus be reduced ieraively by pursuing all possible ranche reducions, saring from he senior-mos (i = 1) ranche. Similarly, if he i-h IC es is failing, he indenure requires ha principal is repaid sequenially o all senior noes unil he es is me. We de ne he supporable ranche noional for each IC es: R i = ~w lb i : (19) 5 E.g., if he second (Class B) OC/IC es fails, we need o repay noional o he class A and Class B noes

10 10 The scheduled deb service reducion of he k-h ranche, for he curing of he i-h IC es will be given (recursively) by he expression P ki = max(min( i kx 1 m=1 P mi R i ; S k ); 0): (20) Reducing he deb service can only be achieved via a corresponding noional reducion of he noes. Using he ranche coupon, we calculae he required noional reducion as of he k-h ranche, for he curing of he i-h IC es: T ki = P ki c k : (21) Due o he cusomary join applicaion of he OC/IC ess, we combine he reducion requiremens T ki n and T ki ki = max(t ki ; T ki ): (22) Obviously he join required ki can only be achieved up o he available funds as per he cash ow waerfall. I will be useful here o inroduce he quaniy M i = ix mi (23) which denoes he oal required deb reducion per es. As proceeds are applied sequenially o he various ranches o achieve he reducion, he updaed variables M i conrol wheher he curing of he ess was successful or no. 6. Formally his is achieved wih he indicaor funcion C i C i = 1 fm i =0g: (24) Wih he de niions given in he previous subsecion we are ready o provide a precise, even if high level de niion of he cash ow waerfall. For conciseness we do use he indicaor funcions we de ned above. A more deailed descripion is given in Appendix B.2. For all paymen periods before mauriy: Collec curren period paymens Pay senior fees Pay senior ranche ineres Loop over mezzanine ranches (i < M) If J i is rue, pay scheduled ineres o he i + 1 ranche, proceed o he nex ranche If J i is false, aemp o amorize senior ranches o he required level If C i is rue, pay scheduled ineres, proceed o nex ranche 6 In he evaluaion of he required deb reducion for junior ess we always incorporae previous deb reducions for more senior ess

11 11 If C i is false, defer ineres on junior ranches, proceed o nex period If J M If J M is rue, pay mezzanine fees, pay equiy dividend, proceed o nex period. is false, aemp o amorize senior ranches o he required level If C M is rue, pay mezzanine fees, pay equiy dividend, proceed o nex period. If C M is false, proceed o nex period 4 Consisen CDO Risk Measures In credi risk modelling of corporae bonds or loans, i is marke pracise o esimae defaul probabiliy (P D) and loss given defaul (LGD). For a credi risk analysis of cash CDOs we are ineresed primarily in he probabiliy of experiencing a loss, raher han he probabiliy of defaul. The probabiliy of loss for each ranche can be derived once all nal cash disribuions have been compued (inermediae losses may be remedied). This is given by P D i = Z fb i T (z)<s i T (z)g n(z)dz (25) where b i T (z) is he nal coupon and principal paymen. Once we know he nal scheduled paymen ST i (z) (which, depending on he defaul pah, includes noional and any deferred ineres) we simply check wheher b i T (z) is enough o make his paymen, else he ranche experiences a loss even. Apar from he probabiliy of no receiving promised cash ows, we are ineresed in he expeced amoun of loss (i.e., loss given loss). We discoun cash ows received a di eren imes wih risky discoun facors DF including he credi spread on he ranche. Before de ning he LGD, he loss rae LR i (z) is de ned as: LR i (z) = 1 P T =1 bi (z)df P T =1 Si (z)df ; (26) where The LGD i can hen be de ned as DF = 1 (1 + s i ). LGD i = whereas he Tranche Expeced Loss is de ned as EL i = E[LR i ] = R +1 1 LRi (z) n(z)dz P D i ; Z +1 1 = LGD i P D i LR i (z) n(z)dz

12 12 V ar[lr i ] is hen de ned o be he Tranche Loss Volailiy and V ar LR i jlr i > 0 he LGD Volailiy. V ar[lr i ] = V ar LR i jlr i > 0 = Z +1 1 Z +1 1 LR i (z) 2 n(z)dz E[LR i ] 2 (27) LR i (z) 2 n(z) P D i dz LGDi 2 : (28) When we refer o he LGD disribuion, we mean he condiional disribuion LR i condiioned on LR i > 0. 5 Analysis of represenaive cash ow srucure We illusrae here he proposed approach on a realisic example. 5.1 Descripion srucure The asse porfolio consiss of B+ raed asses wih a spread of 300 bps and a recovery rae of 40%. The senior fees are 20 bps. The mezzanine fees are 45 bps. There are four ranches. The class A noe is considered o be a senior ranche in he waerfall srucure. The srucure looks as follows Lower Bound Upper Bound Spread (bps) OC Triggers IC Triggers Class A 22:08% 100% % 120% Class B 18:05% 22:08% :4% 110% Class C 12:2% 18:05% % 105% Class D 7:42% 12:2% :4% 100% The ransacion mauriy is 7 years. 5.2 Credi risk measures We illusrae here he main credi risk measures derived for his srucure: Raing PD LGD LGD Volailiy EL Class A AAA 0:39% 4:17% 3:52% 0:02% Class B AA 1:01% 63:67% 36:92% 0:64% Class C BBB 4:00% 53:89% 36:72% 2:16% Class D BB 12:71% 61:21% 32:92% 7:78% As expeced, he mos senior ranche, has a high raing, low PD and a low LGD. We observe in addiion ha he LGD volailiy of he mos senior ranche is much lower han he LGD volailiy of lower ranches. We observe also ha he LGD Volailiy of all mezzanine ranches is of similar order of magniude. 7 7 This is an ineresing conras wih he corporae bond marke, where he recovery amoun is an independen risk facor. raher direcly For a ranche, all he momens of he LGD disribuion can (in heory) be modeled

13 Comparison wih he sress scenario approach The adverised di erence in our analysis compared o sandard sress scenarios is ha we inpu he full defaul disribuion ino he cash ow model. This resuls in a full cash ow disribuion (insead of a se of sressed cash ow scenarios per ranche). While a judicious choice of waerfall mechanics can maximize he observed di erences, we benchmark our resuls using a ypical srucure. In order o perform a meaningful comparison we reinerpre he sress scenario mehodology wihin he quaniaive framework esablished here. This is done as follows: Using he Vasicek defaul model we deermine a se of sressed defaul scenarios. More speci cally, for each period we use he quaniles of he Vasicek defaul model which correspond o he defaul probabiliies of he various raing classes. Dependen on he iniial raing of he porfolio his leads o sressed defaul scenarios. We inpu hese scenarios in our cash ow model. The severes sressed defaul scenario a ranche can handle wihou losing scheduled cash ows deermines he raing of his ranche. The sressed defaul scenarios (for he complee able including raing modi ers "+" and "-", we refer o Appendix A.2) correspond o he following defaul probabiliies: Quaniles 1 year 2 years 3 years 4 years 5 years 6 years 7 years AAA 48:25% 58:09% 63:41% 66:54% 68:41% 69:50% 70:08% AA 38:12% 48:68% 54:89% 58:76% 61:21% 62:75% 63:67% A 36:79% 46:68% 52:38% 55:87% 58:02% 59:32% 60:06% BBB 33:45% 41:23% 45:82% 48:71% 50:56% 51:72% 52:43% BB 16:80% 23:45% 28:09% 31:42% 33:83% 35:58% 36:87% Noe ha he AAA quanile scenario is he severes sress scenario. The sress scenarios could be compared wih he expeced defaul scenario 8, which is: Expeced 1 year 2 years 3 years 4 years 5 years 6 years 7 years B+ 3:67% 7:53% 11:08% 14:12% 16:66% 18:74% 20:44% For concreeness, we compare he derived probabiliy of loss (P D i ) wih he hisorical defaul probabiliies and infer a "raing". E.g., if he probabiliy of loss of he Class A noes maches he corresponding cumulaive defaul probabiliy of he AAA raing class, we assign a AAA raing. We obain he following resuls: CDO noes Full cash ow analysis Sress scenario analysis Class A AAA AA+ Class B AA AA- Class C BBB BBB- Class D BB BB We observe ha he full cash ow and sress scenario resuls are broadly similar for a ypical srucure. This re ecs he underlying senioriy based prioriizaion of cash ows in such srucures. Neverheless, here are measurable di erences, wih he full cash ows framework producing slighly higher raings (one noch di erence) compared o he radiional approach. 8 The sress scenarios are quanile scenarios corresponding o he cumulaive defaul raes of he di eren raings. The quaniles are subsanially a eced by he skew of he disribuion compared o he mean.

14 14 6 Conclusions The coninuous expansion of he srucured credi marke requires ever more incisive analyical ools. Much of he moivaion for furher modelling developmen comes from he credi derivaives area, where conracs are usually resriced o relaively simple cash ow paerns. The adven of CDS on non-corporae securiies and he need o beer undersand he credi risk of cash CDO moivaes a closer look a he quaniaive analysis of cash srucures. In his direcion, we inroduced an approach ha signi canly faciliaes his analysis. A he compuaional level, he mehod illusraes ha one can combine he widely used "condiional independence" (or facor models) for porfolio credi risk wih very deailed cash ow calculaions (in paricular cash ow redirecions riggered by breaching leverage hresholds). In essence we showed by consrucion ha he pah-dependency of he basic CLO srucure can be capured in a few sae variables (he various indicaors) linked o he cumulaive loss rae. The enire calculaion is hen reduced o simple one-dimensional inegraion, which is execued almos insananeously in curren compuer hardware. We de ned and compued key risk measures applying o CDO ranches uilizing he full relevan cash ow disribuion. Those measures are he probabiliy of loss, he LGD bu also he LGD volailiy. For a ypical srucure, he rs measure is in broad agreemen wih a scenario based approach, bu here are neverheless measurable (one noch level) di erences. The laer wo measures canno be compued consisenly in a scenario framework. We presened resuls using a widely adoped single facor model, and a sylized cash ow scheme represenaive of a saic CLO ransacion. In he curren paper we do no pursue an exhausive analysis of he impac and ineracion of srucural elemens wih he porfolio model bu leave his for fuure work. A key advanage of a scenario based approach ha is no ye incorporaed here is ha i allows exibiliy in specifying he iming (iner-emporal disribuion) of a given number of defauls. In fuure research we aim o incorporae his aspec in a consisen manner. Relaxing he oher simplifying assumpions of his sudy (e.g., he saic porfolio assumpion) may require porfolio models ha are oo complex o be capured in semi-analyic form.

15 15 References [1] L. Andersen, S. Basu, and J. Sidenius. All your hedges in one baske. Risk, November [2] D. Du e and K.J. Singleon. Credi Risk: Pricing Measuremen and Managemen. Princeon Universiy Press, [3] J. Lauren and J. Gregory. Baske defaul swaps, CDO s and facor copulas. ISFA Acuarial School, Universiy of Lyon and BNP Paribas working paper, [4] A. Lucas, P. Klaassen, P. Spreij, and S. Sraemans. An analyic approach o credi risk of large corporae bond and loan porfolios. The Journal of Banking and Finance, [5] O.Vasicek. Loan porfolio value. Risk, December [6] D. Picone. Srucuring and raing cash- ow CDOs wih raing ransiion marices. Case Business School London working paper, [7] Sandard and Poor s. General cash ow analyics for CDO securiizaions. Srucured Finance CDO Spoligh: Crieria, 2004.

16 16 A Appendix: Vasicek model A.1 Cumulaive defaul probabiliies Cum. PD 1 year 2 years 3 years 4 years 5 years 6 years 7 years AAA 0:02% 0:06% 0:12% 0:19% 0:28% 0:39% 0:52% AA+ 0:02% 0:07% 0:14% 0:24% 0:36% 0:50% 0:66% AA 0:11% 0:24% 0:39% 0:57% 0:76% 0:97% 1:20% AA- 0:14% 0:29% 0:46% 0:66% 0:88% 1:11% 1:37% A+ 0:14% 0:30% 0:50% 0:73% 0:98% 1:26% 1:57% A 0:14% 0:32% 0:54% 0:81% 1:11% 1:45% 1:81% A- 0:14% 0:36% 0:63% 0:96% 1:33% 1:74% 2:17% BBB+ 0:22% 0:53% 0:91% 1:35% 1:84% 2:37% 2:92% BBB 0:22% 0:64% 1:18% 1:81% 2:50% 3:21% 3:94% BBB- 0:54% 1:36% 2:32% 3:34% 4:39% 5:42% 6:41% BB+ 1:67% 3:32% 4:92% 6:44% 7:87% 9:19% 10:41% BB 2:77% 5:26% 7:50% 9:49% 11:25% 12:82% 14:20% BB- 2:79% 5:67% 8:38% 10:83% 12:97% 14:83% 16:44% B+ 3:67% 7:53% 11:08% 14:12% 16:66% 18:74% 20:44% B 8:59% 14:51% 18:59% 21:45% 23:49% 25:00% 26:15% B- 9:56% 16:63% 21:56% 24:96% 27:32% 28:99% 30:21% CCC+ 14:69% 23:40% 28:70% 32:02% 34:20% 35:69% 36:76% CCC 19:82% 30:18% 35:83% 39:09% 41:08% 42:39% 43:32% CCC- 46:55% 53:45% 57:22% 59:39% 60:72% 61:60% 62:21% D 100:00% 100:00% 100:00% 100:00% 100:00% 100:00% 100:00% A.2 Sressed defaul scenarios corresponding o B+ porfolio Sressed Quanile Cum. PDs 1 year 2 years 3 years 4 years 5 years 6 years 7 years AAA 48:25% 58:09% 63:41% 66:54% 68:41% 69:50% 70:08% AA+ 48:22% 57:25% 62:17% 65:09% 66:84% 67:85% 68:37% AA 38:12% 48:68% 54:89% 58:76% 61:21% 62:75% 63:67% AA- 36:80% 47:35% 53:61% 57:54% 60:03% 61:59% 62:53% A+ 36:79% 47:01% 53:00% 56:72% 59:06% 60:50% 61:36% A 36:79% 46:68% 52:38% 55:87% 58:02% 59:32% 60:06% A- 36:38% 45:75% 51:13% 54:42% 56:46% 57:69% 58:39% BBB+ 33:46% 42:68% 48:07% 51:41% 53:50% 54:77% 55:52% BBB 33:45% 41:23% 45:82% 48:71% 50:56% 51:72% 52:43% BBB- 27:51% 35:06% 39:70% 42:73% 44:76% 46:10% 46:99% BB+ 20:07% 27:45% 32:38% 35:79% 38:17% 39:84% 41:02% BB 16:80% 23:45% 28:09% 31:42% 33:83% 35:58% 36:87%

17 17 B Appendix: Generic Cash ow Paymens In his appendix we de ne wih he necessary precision he paymens waerfall. This waerfall re ecs he main sylized feaures of conemporary cash CLO srucures. Some adapaions/simpli caions were necessary. B.1 Paymen Funcion In order o specify he waerfall concisely, i is helpful o inroduce a funcion Paymen which akes hree argumens: Source: The source (accoun) from which he scheduled amoun is o be paid. SA: Scheduled paymen amoun. P A: Toal paid amoun. The argumens are updaed wihin he funcion as follows: If Source > SA hen he oal scheduled amoun is paid: P A := P A + SA (Toal paid amoun is updaed.) Source := Source SA (The source is updaed for he paymen and here migh be sill somehing available for oher paymens). SA := 0 (This indicaes ha he scheduled amoun has been successfully made). If Source < SA hen: P A := P A + Source (Only he oal available source is paid.) SA := SA Source (Par of he scheduled amoun is paid, he remainder is sill scheduled for oher available sources). Source := 0 (The source is fully exhaused. No oher paymens can be made anymore from his source). B.2 Waerfall The following is a descripion of a generic ineres/principal proceeds waerfall. For speci c ransacions here can be subsanial variaions in srucure. 1. Iniializaion of accouns: For each period before mauriy, we rack he ow of he curren ineres proceeds i and he curren principal proceeds p. The size of he ineres proceeds accoun a he end of period and before disribuions is given by any lefover ineres proceeds from he previous cycle (normally zero) accruing a he risk free rae r, plus any new ineres proceeds w from he collaeral. Similarly, he value of he principal proceeds accoun a he end of period and before disribuions is given

18 18 by any lefover principal proceeds from he previous cycle (normally zero) accruing a he risk free rae, plus any new principal proceeds r from he collaeral. I.e., i = (1 + r)i 1 + w ; (29) p = (1 + r)p 1 + r : (30) Nex, he paymen and reserve accouns are iniialized a 0: f s = 0 f m = 0 b i = 0 a = 0: 2. Calculae scheduled paymens SF s, SF m ; S i : SF s SF m = f s N = f m N S i = (r + s i ) T i : 3. Paymen of senior fees: Use he available ineres proceeds accoun i o pay he senior fees SF s. If here is a shorfall in senior fees paymen, use he principal proceeds accoun p o compensae Paymen SF s ; i ; fs (31) Paymen SF s ; p ; f s : (32) 4. Paymen of senior bonds: Use any remaining ineres proceeds i o pay scheduled ineres o senior noes. A his poin he ineres waerfall receives funds from he principal waerfall o make up (if possible) for any shorfall in paying he senior noes coupon. The size of he acual paymen a ecs he relevan ranche sizes and accouns as follows: Incremen he senior ranche noional T 1 wih any ineres paymen shorfall. This means ha while we allow senior ranches o defaul only a mauriy, we do keep rack of missed coupons and heir accrued ineres. Paymen S 1 ; i ; b 1 Paymen S 1 ; p ; b 1 (33) (34) T 1 := T 1 + S 1 (35) 5. Mezzanine paymens loop: In his porion of he waerfall, a loop of idenical operaions is performed for every OC/IC es i = 1; : : : ; M indicaor J i For each es, check he i-h join 9 The las OC/IC es (i=m) will lead o equiy insead of bond paymens and is hence reaed separaely

19 19 (a) If J i = 1, pay scheduled ineres o he i + 1 noes from i. (So he Class B noes will receive paymen if he Class A OC/IC es is PASS). Paymen S i+1 ; i ; b i+1 (36) T i+1 := T i+1 + S i+1 (37) Once he above paymens have been made, he loop proceeds o he nex lower OC/IC es. (b) If J i = 0, we ener ino OC/IC "cure mode". In he rs insance we use ineres proceeds i, o sequenially amorize principal on noes down o i-h noe, and hen aemp o mee any shorfall using noional reducion from principal proceeds p or he reserve accoun a. In sequence for each of he noes j = 1; : : : ; i his leads o he updaes. Inroduce also a emporary variable e b j : e b j = 0 (38) Paymen Paymen ji ; i ; e b j ; p ; e b j ; a ; e b j (39) (40) (41) b j := b j + e b j (42) T j := T j e b j (43) We nex check he indicaor C i, which indicaes wheher here were enough funds o achieve he necessary reducions. ix i. If = 0 and C i = 1, he required noional reducion has been achieved. ji Now he waerfall revers back o Poin 5a of he Mezzanine porion above. ix ii. If = 0 and C i = 0, he required reducion was no successful. We now ji have exhaused all funds and simply defer ineres on noes from (i + 1) onwards, while equiy receives no dividend his period. Hence for j = i+1; : : : ; M b j = 0 (44) T j = T j + S j (45) d = 0 (46) Wih hose adjusmens, we proceed o he nex period (Poin 1). 6. Paymen of mezzanine fees: Use he available ineres proceeds accoun i o pay he mezzanine fees fm. Paymen SF m ; i ; fm (47)

20 20 7. Paymen of Equiy dividends: In his porion of he waerfall, passing he junior mos OC/IC es (i = M) will lead o equiy paymens. (a) If J M = 1, pay any remaining ineres proceeds i o equiy and amorize any remaining principal proceeds o noes Paymen (i ; i ; d ) a : = (1 + r) a 1 + p p : = 0. The paymen waerfalls is now complee and we rever o he nex period (Poin 1). (b) If J M = 0, similarly o he mezzanine loop, we use any remaining i o sequenially amorize all he noes, unil he es is cured. Hence for each of he noes j = 1; : : : ; M e b j = 0 (48) jm ; i ; e b j (49) b j := b j + e b j (50) T j := T j e b j : (51) We now check wheher he M-h (junior) OC/IC cure has been successful. We check he indicaor C M i. If C M = 1, he required noional reducion has been achieved. The waerfall revers back o Poin 7a. ii. If C M = 0, he required reducion was no successful. Equiy receives no paymen his period. d := 0 (52) We now have exhaused all funds and simply move on o he nex period (Poin 1). During he nal period ( = T ), all noes receive nal scheduled coupon and principal (if possible). Accumulae he ineres rae and principal proceeds ino one accoun a T a T := (1 + r) a T 1 + N T + i T + p T (53) For all ranches j = 0; : : : ; M, aemp sequenial repaymen of principal and accrued coupons Paymen S j T ; a T ; b j T Any remaining funds accumulae o he equiy ineres. d T = a T : (54)

21 21 Figure 3: Ineres waerfall In he gure we demonsrae how he ineres waerfall ows hrough he cash ow CDO srucure before mauriy. Ineres waerfall: coupon paymens Senior fees Coupons senior bond (Class A) Class A OC/IC Tes Tes passed Coupons Class B Tes cured Tes failed Redeem Class A Noes unil Class A OC/ IC Tes has been cured Class B OC/IC Tes Tes passed Coupons Class C Tes cured Tes failed Redeem Class A and B Noes unil Class B OC/IC Tes has been cured Tes passed Coupons Class D Class C OC/IC Tes Tes cured Tes failed Redeem Class A,B and C Noes unil Class C OC/IC Tes has been cured Mezzanine fees Class D OC/IC Tes Tes passed Equiy paymens Tes cured Tes failed Redeem Class A,B, C and D Noes unil Class D OC/IC Tes has been cured

22 22 Figure 4: Principal waerfall before mauriy In he gure we demonsrae how he principal waerfall ows hrough he cash ow CDO srucure before mauriy. Principal waerfall: recoveries and noional redempion Senior fees Coupons senior bond (Class A) If Class A OC/IC Tes fails Redeem Class A Noes unil Class A OC/IC Tes has been cured If Class B OC/IC Tes fails Redeem Class A and B Noes unil Class B OC/IC Tes has been cured If Class C OC/IC Tes fails Redeem Class A,B and C Noes unil Class C OC/IC Tes has been cured Mezzanine fees If Class D OC/IC Tes fails Redeem Class A,B, C and D Noes unil Class D OC/IC Tes has been cured Add remainder of principal waerfall o reserve accoun

23 23 C Diagram of cash ow waerfalls Before mauriy he ineres and principal waerfall is presened in gures 3 and 4. A mauriy he ineres and principal waerfall and he ousanding noional of he porfolio are all accumulaed in he reserve accoun and disribued according o gure 5. Figure 5: Waerfall a mauriy In he gure we demonsrae how he waerfall a mauriy ows hrough he cash ow CDO srucure. Reserve accoun Senior fees Coupons and redempion Class A Noes Coupons and redempion Class B Noes Coupons and redempion Class C Noes Coupons and redempion Class D Noes Mezzanine fees Equiy paymens

24 24 D Noaion Symbol De niion q Defaul curve g Recovery 2 Correlaion k Evaluaion ime-poins W j Asse credi facors Z Sysemaic credi facor Defaul hresholds Hazard rae N Ousanding noional T Mauriy r Risk free rae c (Weighed) average asse coupon s (Weighed) average asse spread w Ineres proceeds r Principal proceeds T0 i Iniial ranche size T i Running ranche size c i Tranche coupon s i Tranche spread S i Scheduled ranche paymen D i Deferred ranche paymen b i Acual ranche paymen E 0 Iniial equiy size E Running equiy size i Ineres proceeds accoun p Principal proceeds accoun L i Overcollaeralizaion raio (leverage) ~N Adjused Noional L i B l i lb i Overcollaeralizaion raio rigger Ineres cover raio Ineres cover rigger ~w Ineres cover amoun f s f m SF s SF m f s f m d Senior fees and coss Mezzanine fees and coss Scheduled senior fee paymens Scheduled mezzanine fee paymens Acually realised senior fee paymens Acually realised mezzanine fee paymens Equiy paymen (dividend)

25 25 Symbol a I i O i J i C i T ki T ki De niion Reserve accoun Indicaor of IC es Indicaor of OC es Indicaor of join IC/OC es Indicaor of cured IC/OC es Scheduled noional reducion of k-h noe o cure i-h OC es Scheduled noional reducion of k-h noe o cure i-h IC es Scheduled noional reducion of k-h noe o cure join i-h IC/OC es