SIMULATION METHODS FOR RISK ANALYSIS OF COLLATERALIZED DEBT OBLIGATIONS. William J. Morokoff

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1 Proceedngs of the 2003 Wnter Smulaton Conference S. Chck, P. J. Sánchez, D. Ferrn, and D. J. Morrce, eds. SIMULATION METHODS FOR RISK ANALYSIS OF COLLATERALIZED DEBT OBLIGATIONS Wllam J. Morokoff New Product Research Moody s KMV 1620 Montgomery Street San Francsco, CA 94111, U.S.A. ABSTRACT Collateralzed Debt Oblgatons (CDOs are sophstcated fnancal products that offer a range of nvestments, known as tranches, at varyng rsk levels backed by a collateral pool typcally consstng of corporate debt (bonds, loans, default swaps, etc.. The analyss of the rsk-return propertes of CDO tranches s complcated by the hghly nonlnear and tme dependent relatonshp between the cash flows to the tranche and the underlyng collateral performance. Ths paper descrbes a multple tme step smulaton approach that tracks cash flows over the lfe of a CDO deal to determne the rsk characterstcs of CDO tranches. 1 INTRODUCTION The term Collateralzed Debt Oblgaton (CDO covers a broad range of structured fnance products. They may be supported by a varety of underlyng collateral, from bonds, loans, credt default swaps, asset-backed securtes, and soveregn debt, to more exotc securtes such as equty default swaps or CDO tranches from other deals. The CDO structures, whch descrbe the sze and number of tranches and the rules for how to dstrbute the collateral proceeds to the tranches, also vary wdely. The structures may be smple pass-throughs, whereby nterest payments are made n order of tranche senorty. However, there may also be complcated rules to redrect cash flows to the senor tranches based on the qualty and performance of the underlyng collateral. Each deal has a unque structure determned by market condtons, collateral propertes and nvestor demand, among other factors, at the tme of ssuance. Many features of a structure are ntended to nsure that the most senor tranche wll be rated at the hghest credt qualty level by the debt ratng agences (usually Moody s, S&P, and Ftch. See Goodman and Fabozz (2002 for a detaled dscusson of the CDO market; Duffe and Sngleton (2003 ncludes a chapter on credt modelng methods appled to CDOs. A further dscusson of credt modelng can be found n Arvants and Gregory (2001. From a legal perspectve, a CDO deal s generally set up as a Specal Purpose Entty (SPE that functons as an ndependent company, often ncorporated n Bermuda. The captal structure of ths company s very smple: the assets owned by the SPE are the collateral (e.g., 100 corporate bonds, whle the labltes are the tranches ssued by the SPE. Investors purchase the tranches, and the SPE uses the proceeds from the sale of the tranches to purchase the collateral assets. Perodcally (typcally quarterly or semannually, the nterest and prncpal cash flows generated by the collateral assets over the perod are collected together nto accounts that are then used to make nterest and prncpal payments to the tranches. The set of rules for how the funds are dstrbuted at a gven payment date s known as the cash flow waterfall for the CDO. In a typcal waterfall, taxes and management fees are pad frst, followed by the nterest due to the senor tranche. Ths senor tranche generally accounts for the largest percentage of nvested prncpal (75% - 90% but gets pad the smallest coupon. Ths s consstent wth the senor tranche beng the least rsky due to ts poston of gettng pad frst. The tranches are pad n order of senorty wth coupons and rsk ncreasng as payments move down the structure. The most unor tranche at the bottom of the waterfall s known as the equty tranche. The equty of a CDO deal usually does not receve a predetermned nterest payment on ts ntal nvestment; nstead the equty receves all the remanng collateral nterest payments that were not requred to make the nterest payments on the more senor tranches. In deals wth substantal excess spread (the amount of nterest generated by the collateral portfolo beyond what s due to the CDO tranches, the equty tranche performs well. As defaults occur n the collateral portfolo, the amount of excess spread decreases and the equty tranche suffers the frst losses. Many deals have collateral qualty trggers that dvert all cash flows away from the equty f the collateral qualty deterorates too much.

2 The collateral manager of a CDO deal s responsble for managng the collateral assets as ther credt qualty changes. Ths nvolves buyng and sellng assets, as well as renvestng fund that have been recovered from defaultng or maturng names. The manager s skll and strategy around managng the collateral portfolo can have a large effect on the performance of the CDO deal. From a rsk management perspectve, the most mportant factor affectng the performance of a CDO deal s the total loss n collateral portfolo value over the lfe of the deal due to correlated defaults among the collateral. Each tranche can wthstand a characterstc level of loss on the collateral pool before t does not receve ts promsed nterest and prncpal payments. Performance s also greatly affected by the tmng of the defaults, partcularly for the equty tranche. Other rsk factors are nterest rates, maturng and prepayment rates of collateral, and recovery rates on defaulted collateral. There s also prce rsk,.e. the change n value of the collateral due to changes n credt qualty or nterest rates, and the related ssue of renvestment rsk. 2 MODELING DEFAULT PROBABILITIES A key component of accurately capturng the rsks assocated wth correlated collateral defaults s the determnaton of current default probabltes over the lfe of the deal. There are many methods currently n use, and frequently components of varous approaches are used together. Broadly, there are four dstnct approaches. The frst s qualtatve analyst revew of a companes fnancals, management, busness plan, etc. to determne credt worthness. Ths approach s used nternally by banks makng decsons to lend as well as by ratng agences, whch provde the market wth an ndependent qualtatve revew. The second approach s statstcal, n whch mult-dmensonal regressons are carred out on large data sets of company fnancal nformaton to determne default ndcators and assocated probabltes of default. Ths s partcularly effectve when market data on prces of the corporate debt or equty are not avalable, as s often the case wth prvate frms; see Falkensten (2000 for a dscusson of the mplementaton of ths method. A thrd method uses an approach frst put forth by Merton (1974 that derves from the dea that the equty of a frm s actually a knd of call opton on the underlyng asset value of the frm, and that the frm s n default when that asset value drops below labltes owed by the frm. Merton models takes as nput the observable stock prce for publcly traded frms to back out the unobservable frm asset value, whch when combned wth frm lablty nformaton, leads to default probabltes for publc companes. Fnally, default probabltes may be nferred from prces of corporate debt (bonds, loans, default swaps snce prce s strongly nfluenced by the markets percepton of default probablty. However, accurate prcng nformaton s only avalable on a relatvely small number of names. Also, there are numerous factors other than default probablty that determne the prce of debt, and these factors, such as lqudty, may be dffcult to quantfy. Prces are often used to calbrate parameters for stochastc models for the evoluton of default ntenstes. The models used at Moody s KMV are based on the largest database of corporate defaults n the world, whch s mportant for accurate parameter and model estmaton. For publcly traded companes, MKMV uses the Vascek- Kealhofer model, descrbed n Kealhofer (2003, whch s an extenson of the Merton approach that ncorporates extensve asset volatlty modelng and hstorcal default data, to produce the Expect Default Frequency, or EDF, credt measure. Most of the largest 100 fnancal nsttutons n the world use the EDF credt measure to montor the credt qualty of ther loans and nvestments. The Vascek-Kealhofer model s an example of a structural model, named so because there are explct economc drvers that are quantfed n the model and that determne the default probablty of the company. These drvers are frm asset value, frm asset volatlty and frm lablty structure. There are two mportant further benefts of usng ths structural model. Frst, based on the model, MKMV has produced weekly tme seres of asset returns and asset volatltes for over 28,000 publcly traded companes world-wde. From these tme seres, a factor model for asset returns has been derved that provdes the most accurate means avalable for descrbng the correlated behavor of corporate asset values. When combned wth default probabltes, the correlaton model of asset returns provdes an estmate of ont default probabltes necessary to descrbe the correlated default behavor of a portfolo of CDO collateral. The second mportant consequence of the structural model s the concept of the Dstance to Default ( DD. Ths s essentally the number of standard devatons that the asset value s above the default level (a functon of the lablty structure; t s a scalar measure that captures the key relatonshp among the three structural drvers (Dstance to Default also depends on the tme horzon of nterest; for the purposes here we wll consder one year DD. It has been shown emprcally to be a stable measure of credt qualty over tme and economc condtons (credt cycles as well as over geographc regons. MKMV has an extensve database of tme seres of Dstance to Default for publc companes. From ths database, emprcal transton probablty dstrbutons have been determned that provde the probablty of a frm wth DD0 at tme T0 mgratng to DD1 at tme T1. These emprcal transton denstes provde a much more realstc descrpton of credt qualty mgraton than the standard approach of assumng geometrc Brownan moton for the asset value process. Although ths model for asset value s reasonable, n order to adequately capture changes n DD (and the correspondng EDF, t would be necessary to also model a correlated process for

3 how frms change ther lablty structure. From the data t s clear ths s not a contnuous stochastc process; whle a ump process would be a possble model, the calbraton of the parameters would be dffcult because the decson to add labltes s drven by the frm s management style and the economc opportuntes avalable. The emprcal DD dstrbutons capture these effects as well as changes n asset volatlty wthout requrng explct modelng. The factor model for asset returns and the emprcal DD dstrbutons, both derved from the structural model for publc frms, can also be used wth the statstcal models for prvate frms or any other default probablty model, as long as an estmate for the R-squared for the frm n queston s asset return regressed aganst the factors (.e., the percentage of varance of the asset return explaned by the factors s avalable, together wth ndustry and country nformaton about the frm. 3 MODELING DEFAULT TIMES Ths secton descrbes the methodology most commonly employed today for smulatng correlated defaults. It s known as the default tme or copula approach and s descrbed by L (2000 and Schmdt & Ward (2002. For an exposure n a CDO collateral pool, the default probablty to maturty (ether of the CDO deal or the exposure, whch ever s sooner gves the probablty of that name defaultng as some pont durng the lfe of a CDO deal. The tmng of the default, however, can also play an crucal role n determnng the performance of the deal. Default tmng s determned from a default probablty term structure whch may be represented as a vector of cumulatve default probabltes specfed at tmes ( CEDF1 CEDF2,,, CEDFN ( T1 T2 T N,,,. The quantty CEDF s nterpreted to mean the probablty of default n the nterval ( 0, T. Thus the CEDF are ncreasng. Ths may be generalzed to a tme contnuous default probablty functon CEDF ( t ; however, default probabltes are usually reported at dscrete tmes, and a contnuous functon s obtaned from nterpolaton. The default tme/copula method of randomly samplng default tmes works as follows. The frst step s to randomly sample a unform (0,1 varate u. Assumng that T s the maturty, f u > CEDF then the exposure does N not default. If CEDF 1 < u CEDF then the exposure defaults n perod. Ths procedure s closely related to N samplng a stoppng tme for a random process crossng a default boundary. A key feature of ths approach s the process for determnng correlated default tmes. Ths requres samplng a set of correlated unform varates ( u 1,, um, where M s the number of exposures n the portfolo. Ths s done by specfyng a copula functon C( u 1, um, whch s a probablty dstrbuton functon defned on the M - dmensonal unt cube. The copula functon s often related to the asset return dstrbuton functon at tme T N, F R R, by the formula ( 1,, M where 1 M 1 1 ( M M 1 1 C( u,, u = F F ( u,, F ( u 1 F ( dstrbuton for the s the nverse of the margnal probablty th exposure. However, any copula functon may be used for ths purpose. The most commonly used are Gaussan and T-copulas, although a varety of other methods, ncludng Archmedean copulas, have been consdered. For the Gaussan copula, the samplng procedure s partcularly smple. Based on the correlaton matrx for the asset returns, a correlated sample of standard Normal varates ( ε 1,, εm s sampled, ether from a Cholesky decomposton of the correlaton matrx or from a factor model decomposton. The unform varates are then obtaned from the formula u =Φ( ε. Here Φ s the one dmensonal standard cumulatve Normal dstrbuton functon. If the factor model underlyng the correlaton structure has more than a few dmensons, t s necessary to use Monte Carlo smulaton to sample correlated defaults and default tmes that are then used to evaluate expectaton ntegrals such as the probablty of havng more than k defaults or the expected value of the cash flows to a tranche. Under more restrctve assumptons on the correlaton structure, sem-analytcal solutons can be derved. For example, the latent varable approach, proposed by Vascek (1987 for credt portfolo rsk problems, has been extended to CDOs by Gregory and Laurent (2003. The dea s that there exsts a low dmensonal underlyng latent varable x such that condtonal on x the default probabltes and tmes for the exposures are ndependent. The law of condtonal expectatons then allows the portfolo propertes of nterest to be expressed as an expectaton over x of the portfolo propertes of an ndependent portfolo. Often x s taken to be one dmensonal, so the problem reduces to a one dmensonal quadrature.

4 4 MULTI-STEP SIMULATION An alternatve to the default tme approach based on smulatng the frm asset value as a stochastc random varable has been descrbed by Hull and Whte (2001, Arvants and Gregory (2001 and Fnger (2000. In ths secton we descrbe an mplementaton of ths approach and descrbe a new mult-step approach based on the emprcally derved Dstance to Default dstrbutons. Whle the default tme approach captures the margnal default probabltes of each ndvdual exposure correctly over the lfe of the smulaton, substantal error may be ntroduced nto the correlated default structure, dependng on how the correlaton structure and the underlyng stochastc default process are vewed. Tme seres of asset, equty or debt prce returns are usually based on daly or weekly tme ntervals. Gven the relatvely hgh default probablty of most assets over tme horzons of fve years or longer, usng a correlaton structure based on weekly returns as a proxy for mult-year horzon correlatons can lead to skewed results. In partcular, the sngle step approach may not adequately capture the absorbng nature of the default state (.e., the stochastc process has an absorbng boundary. Thus t s better to consder a smulaton based on a sequence of shorter tme steps that one sngle step to maturty. It s possble to model the credt mgraton of a sngle asset as a contnuous tme stochastc process, such as geometrc Brownan moton or an Ohrnsten-Uhlenbeck process, wth an absorbng boundary mpled by the cumulatve default probablty functon CEDF( t. In ths formulaton a free boundary problem PDE can be derved as descrbed by Avellaneda and Zhu (2001. However, the exstence of CEDF( t as a tme contnuous functon usually arses from mposed model or nterpolaton assumptons; there s generally not enough market data or fnancal nformaton avalable to mply forward default probabltes over short tme wndows. Thus the contnuous approach does not add accuracy relatve to a dscrete approach as long as the correlated behavor of asset over the tme step s consstent wth the correlaton modelng. In any case, unless a low-dmensonal latent varable approach s appled, computaton of the propertes of a portfolo of many exposures wll requre a Monte Carlo smulaton based on dscrete tme steps. For analyzng a sngle CDO deal, t s most convenent to use smulaton tme steps based on the CDO payment dates. For one smulaton step, the names defaultng durng that perod are dentfed, recoveres on defaulted names are determned, nterest cash flows from nondefaulted collateral are aggregated, scheduled and unscheduled prncpal payments from the collateral are collected, etc. The resultng pools of nterest and prncpal cash flows are then passed to the cash flow waterfall engne to be dstrbuted to the CDO tranches. If desred, the exact default tme of an exposure can be sampled usng the default tme methodology descrbed above wthn one smulaton perod; n practce, however, the default on a partcular exposure wll occur on a coupon date, not at a random tme. The key queston for the smulaton s thus whether the default occurs n a gven perod. There are numerous approaches that can lead to multstep smulatons for correlated defaults dependng on how the default process s modeled. We focus here on two methods related to structural models for whch correlated default behavor s derved from the underlyng frm asset value correlatons. Both methods take as nput the cumulatve default functon CEDT ( t specfed at dscrete tmes ( T, 1, TN for each oblgor n the collateral portfolo, ndexed by. In addton, the frm asset value correlaton matrx for all oblgors must be specfed. The frst approach assumes that the asset value process for each oblgor follows correlated geometrc Brownan moton. The assocated asset value (log return process therefore follows a standard Brownan moton process. An oblgor defaults durng a perod ( T T ] f the asset return whle > α for all k < (.e., there was no prevous default. α t (, 1 R at tme T s less than some threshold level α, R k k In a contnuous tme formulaton, the functon s the default boundary such that the default tme s the stoppng tme of the Brownan moton process assocated wth crossng the boundary. Obvously the default thresholds must be related to the default probablty. Specfcally the relatonshp s > > = ( PR ( α,, R α CEDF T As ths equaton suggests, the determnaton of the default thresholds requres a non trval calculaton as t relates to nvertng an varate cumulatve Normal dstrbuton (n the contnuous case, the default boundary s the soluton to a free boundary PDE. One approach that gets around the need to nvert a mult-dmensonal dstrbuton 1 s to determne the dstrbuton of R, condtonal on no defaults up to tme T 1. Assumng we know ths dstrbuton and usng the fact that R = R + ϕ 1 1 where ϕ s an ncrement ndependent of R (snce the return process s Brownan moton wth a Normal dstrbuton, we can obtan by convoluton the dstrbuton of condtonal on no defaults up to T 1. R,, from the condtonal

5 dstrbuton for default threshold Once 1 R α and ϕ. We can then solve for the P( R α no defaul ts up to T 1 ( 1 CEDF( T 1 = CEDF T CEDF from the equaton ( ( T 1 α has been determned, the dstrbuton of. R condtonal on no defaults up to T can be determned by truncat- ng the dstrbuton of R condtonal on no defaults up to tme T 1. By repeated applcaton of ths procedure, the entre set of default thresholds can be determned. The man computatonal cost s assocated wth the convoluton. Ths can be handled easly wth the fast Fourer transform algorthm, whch s effectve snce the condtonal dstrbuton s always convolved wth a Normal dstrbuton. Once the default thresholds are determned, the smulaton proceeds by samplng correlated Brownan moton paths for the asset returns at the specfed tmes. Default occurs for a gven oblgor durng the frst perod for whch ts return falls below the assocated threshold. For names that don t default, condtonal default probabltes at each tme step can be used as nput n valuaton algorthms to provde consstent, correlated mark-to-model prcng for the collateral. As mentoned above, the assumpton of geometrc Brownan moton for the asset value process often does not adequately capture how a frm s credt qualty changes over tme because t does not take nto account the assocated changes n lablty structure. It s known that as frms do well (e.g. as the asset value of the frm ncreases, they tend to take on more debt, thereby keepng ther credt qualty more stable over tme. For example, a Baa rated frm wll tend to mantan that ratng by borrowng more when opportuntes arse. It would be unusual for such a frm to grow wthout addng leverage to become a Aaa rated. However, ths tends to be the consequence of the geometrc Brownan moton model: over longer tme horzons, frms that do not default undergo systematc mprovement n ther credt qualty. To capture the effects of changes to both asset value and lablty structure on credt qualty n long horzon mult-step smulatons, at MKMV we have developed a mult-step smulaton based on the Dstance to Default transton denstes. We now consder the mplementaton of ths second, emprcally-based method. A key pont to consder when workng wth hstorcally observed data s the need to bucket the data n order to buld a sutable sample sze. For example, the frst step n determnng the probablty of transtonng from a DD value of 3 over a one year horzon to a DD value of 4 s to dentfy all names n the hstorcal sample that have at some tme pont a DD value of 3. However, snce DD s determned as a contnuous varable, t s unlkely that any of the sample wll have a DD value of exactly 3. Thus t s necessary to repose the queston as to the probablty of transton from a bucket, or nterval, contanng the DD value 3 to a DD value less than 4. The dstrbuton of arrval DD s after one year does not necessarly have to be bucketed a parametrc dstrbuton for the cumulatve transton probablty dstrbuton can be selected and the actual data used to estmate the dstrbuton s parameters. However, for use n a mult-step smulaton, t s convenent to work wth the transton probabltes from one bucket to another bucket n the form of a transton matrx. The mult-step smulaton s then carred out as a dscrete Markov chan by repeated applcaton of the transton matrx to an ntal state vector. The sze of the transton matrx, whch s determned by the sze of the DD buckets, s chosen to balance the desre for hgh resoluton n DD space wth the need to mnmze the statstcal errors arsng from small sample szes. Ultmately ths s a queston of the sze of the orgnal data set. The MKMV smulaton s based on 9 years of monthly data on over frms. There are a number of mportant observatons to be made about the DD transton matrx. Frst, the default state, convenently labeled as DD = 0, s an absorbng state. The total probablty of transtonng to ths default state over a gven tme perod s the forward EDF. Ths EDF s dfferent for each frm; however, the transton matrx was determned by poolng data on many frms. Thus the transton matrx must be vewed as frm aggregate behavor. In order to capture the frm-specfc behavor dctated by the nput EDF term structure for each frm, t s necessary to make a frm-specfc calbraton of the transton matrx. The calbraton conssts of satsfyng the constrant that over a gven tme perod, the probablty of transtonng from a non-default state to the default state must be the uncondtonal (or more precsely, condtonal only on data specfed at T forward default probablty: FWD EDF( T, T 1 0 CEDF( T CED F ( T 1 ( =. 1 CEDF T There are numerous ways ths constrant could be enforced. One smple approach s to rescale all the orgnal, frm aggregate transton probabltes to default by a sngle factor such that ther sum, weghted by the uncondtonal probabltes of beng n each non-default state at tme T 1, matches the forward EDF. Once the transton probabltes are adusted by ths scalng, the uncondtonal probabltes for each state at tme T can be determned, thereby allowng the calbraton for the next tme step. Ths s equvalent to the convoluton and truncaton steps employed for the geometrc Brownan moton model. A second consderaton for the transton matrx s whether the underlyng data supports the model of a 1

6 Markov process. Not surprsngly, the frm-aggregate transton matrces for tme horzons of 6 months, 1 year, 2 years, 5 years, etc., derved from the data do not ft perfectly n a Markov framework. In other words, the one year matrx s not exactly the convoluton of the 6 month matrx wth tself; nor s the fve year transton matrx exactly the fve-fold convoluton of the one year transton matrx. The agreement of these transton matrces s however suffcent, partcularly gven the complexty of the underlyng factors whch drve credt mgraton of frms as well as the frm-aggregate nature of the transtons themselves, to warrant the approxmaton by a sngle, Markov transton matrx, whch s determned by optmally fttng, n a least-squares sense, one matrx (and ts convolutons to the emprcal transton matrces. Ths avods the exceptonally dffcult task of specfyng and calbratng a non- Markov process for the credt mgraton. Once the transton matrx s specfed for each oblgor at each tme step, the smulaton proceeds by samplng from F ( DD DD 1, the probablty dstrbuton of DD states at tme T determned from the approprate probablty dstrbuton (as gven by the transton matrx condtonal on the DD state at tme T 1. By nterpolaton from the cumulatve probabltes for the dscrete transton matrx DD states, F D can be assumed to be a ( D DD 1 1 F ( u contnuous, non-decreasng functon wth nverse defned on the unt nterval [0,1]. For values of u n the nterval [0, PDD ( 1 0] (.e., between 0 and the condtonal probablty of defaultng, t follows that 1 F u =. We ntroduce correlatons among oblgors by ( 0 assumng mult-varate Brownan moton for the asset return process and samplng the correlated asset return ncrements accordng to the specfed asset return correlaton matrx. The cumulatve Normal dstrbuton functon s then used to map the sampled asset return ncrements to the unt nterval; ths value s then used as the argument for 1 F u. More precsely, the DD sample for oblgor at ( tme s gven by 1 ( ( DD = F Φ ε where the ε are the normalzed, correlated Normal samples of asset returns. If the random asset return sample falls below the default threshold (determned by the DD state at the prevous tme step and the orgnal EDF term structure, the default state of DD = 0 s sampled. In ths case, a random recovery may be drawn from an approprate dstrbuton of recovery rates. If the oblgor does not default, the sampled DD state at T can be used to determne a condtonal EDF term structure lookng forward that can be used to dscount future cash flows accordng to ther credt rsk n order to obtan a prce for the exposure at tme T. (Note that a dscusson of the modelng of a stochastc nterest rate process, mportant for determne both prce and cash flow characterstcs of debt nstruments, has not been ncluded here. 5 COMPARISON OF MODELS In ths secton we use smulaton experments to compare the two model choces dscussed above. The frst concerns the dfferences between the sngle step default tme model and the mult-step smulaton, whle the second concerns the dfference between modelng the asset value process as geometrc Brownan moton and modelng credt mgraton through the Dstance to Default emprcal transton dstrbutons. For the default tme/mult-step comparson, we consder a portfolo of 120 hgh yeld bonds of maturty greater than fve years ssued by 120 correlated frms. A hstogram of the cumulatve fve year EDFs are shown n Fgure 1. Ths shows the portfolo to have a substantal component of dstressed names, wth a portfolo mean 5 year EDF of around 18.7% (correspondng to annualzed default rate of 4%. The expected number of defaults, computed as the average EDF tmes the number of bonds, s However, based on the mult-step smulaton there s a 5.4% chance of havng more than 41 defaults. Fgure 2 plots the cumulatve probablty dstrbuton for the total number of defaults over the fve year perod for each method. Ths shows that the default tme model overestmates the probabltes of the extreme events (very few defaults, say under 10, or very many defaults, say over 40 relatve to the mult-step model. For example, the default tme model puts the probablty of havng more than 60 defaults at around ten tmes greater than the mult-step model (2.6% versus 0.26%. Ths dfference may be put nto the CDO context f we assume all exposures are of equal sze Fgure 1: Dstrbuton of 5 Year EDFs for Bond Portfolo

7 Fgure 2: Comparson of Default Tme and Mult- Step Models for Generatng Correlated Defaults and a recovery of 50%; n ths case 60 defaults corresponds to a loss of 25%. For a senor tranche wth 25% subordnaton, ths gves us sense of the default probablty of the tranches over the 5 year horzon (not accountng for structural effects of the cash flow waterfall. Under the default tme model, the tranche would be consdered a moderate nvestment grade asset, whle under the mult-step model ths would be a trple A nvestment. For the second queston of comparng the Brownan moton model wth the emprcal dstrbuton model, we consder a two horzon case and compare the relatve role of each tme step s asset return draw n determnng default. For each model the total default probablty s the same. However, dependng on the credt mgraton over the frst step (determned by the frst asset return draw, the range of asset return draws requre for default n the second tme perod can be substantally dfferent. Ths s llustrated n Fgure 3, whch plots regons n the two dmen- ( 1 2 sonal space ε, ε, where ε s the scaled asset return for perod ( = 1, 2 normalzed to be a standard Normal varate. In ths example the tme nterval for each perod are equal and the cumulatve default probabltes are 1.8% for the frst perod and 3.57% for the frst and second perods together. Ths plot shows that under the Gaussan process, the condtonal default probablty followng a postve asset return on the frst step s much lower than for the emprcal dstrbuton. By the same token, a negatve asset return on the frst step under the Brownan moton model s much more lkely to lead to default n the second step. Ths model behavor dffers from the Dstance to Default emprcal dstrbuton behavor because t fals to capture the frms response to good asset returns (addng more debt and bad asset returns (takng measures to avod default. Another strkng dfference between the models can be seen by consderng the case for whch the frm behaves accordng to expectaton n the frst perod. Ths corresponds to ε 1 = 0. The example has be chosen such that the default probablty n each perod s 1.8%. Yet for the Gaussan model, f the frm behaves at ts expected level for the frst perod, the condtonal default probablty for the second perod drops to 0.27%. For the DD Dynamcs model, the second perod default probablty condtonal on the expected asset return s around 1.4%. Note that the Gaussan process default boundary s lnear because there s a fxed threshold such that f the two perod cumulatve asset return (the sum of the two one perod asset returns falls below t, the frm s n default. 6 CONCLUSIONS The complexty of CDOs and the default behavor of the underlyng collateral demand sophstcated smulatons to capture the behavor accurately. True mult-step smulatons have been shown here to yeld sgnfcantly dfferent results from sngle step default tme approxmatons. In addton, a substantally more realstc credt mgraton behavor can be captured by usng emprcal dstrbutons n place of the standard mathematcal modelng approach. ACKNOWLEDGMENTS The work n ths paper was bult upon a great deal of research carred out by a number of people n the San Francsco research group at Moody s KMV. In partcular, specal thanks are due to Ym Lee and Jeff Bohn. Fgure 3: Comparson of Gaussan Process and Emprcal Dstance to Default Process for Two Steps of a Mult-Step Smulaton REFERENCES Arvants, A and J Gregory Credt: The Complete Gude to Prcng, Hedgng and Rsk Management. Rsk Publcatons.

8 Avellaneda, M and J Zhu Dstance to default. Rsk (December. Duffe, D and K Sngleton Credt Rsk: Prcng, Management and Measurement (Prnceton Seres n Fnance. Prnceton Unversty Press. Falkensten, E RskCalc for prvate companes: Moody s default model [onlne]. Avalable va < lng.html>[accessed August 5, 2003]. Fnger, C A comparson of stochastc default rate models. RskMetrcs Journal 1 (November, Goodman, L. and F. Fabozz Collateralzed Debt Oblgatons: Structure and Analyss. John Wley & Sons. Gregory, J and Laurent, J I wll survve. Rsk. (June, Hull, J. and A. Whte Valung credt default swaps II: modelng default correlatons. Journal of Dervatves 8 (3, Kealhofer, S Quantfyng credt rsk I: default predcton. Fnancal Analyst Journal (January/February, L, D On default correlaton: a copula functon approach. Journal of Fxed Income 9 (4, Merton, R On the prcng of corporate debt: the rsk structure of nterest rates. Journal of Fnance 29 (2. Schmdt, W. and I. Ward Prcng default baskets. Rsk (January, Vascek, Probablty of loss on loan portfolo [onlne]. Avalable va < research/portfolotheory.html> [accessed August 5, 2003]. AUTHOR BIOGRAPHY WILLIAM MOROKOFF s a drector at Moody s KMV n the San Francsco research department where he heads the New Product Research Group. Hs research focuses on Monte Carlo smulaton appled to rsk management and dervatves prcng n fnance. He holds a Ph.D. from the Courant Insttute at New York Unversty. He can be contacted at <morokoff@mkmv.com>. Morokoff