Risk Neutral versus Objective Loss distribution and CDO tranches valuation

Transcription

1 Rsk Neutral versus Objectve Loss dstrbuton and CDO tranches valuaton Roberto orresett, Damano Brgo, and Andrea Pallavcn Credt Models, Banca IMI Corso Matteott 6 22 Mlano {roberto.torresett,damano.brgo,andrea.pallavcn}@bancam.t I verson: 5th May 26; ths verson: 3 Aprl 27 Abstract We consder the rsk neutral loss dstrbuton as mpled by ndex CDO tranche quotes through a scenaro default rate model as opposed to the objectve measure loss dstrbuton based on hstorcal analyss. he rsk neutral loss dstrbuton turns out to prvlege large realzatons of the loss wth respect to the objectve dstrbuton, thus mplyng the well known presence of a rsk premum. We quantfy ths dfference numercally by prcng CDO tranches and ndces under the two dstrbutons. En passant we analyze the mpled rsk neutral default rate dstrbutons calbrated from Aprl-24 throughout Aprl-26, pontng out ts dstnctve bump feature n the tal. Key Words: ault Rate dstrbuton, CDO, CDO tranches, Perfect Copula, Impled Copula, ranston Matrces, Ratng Classes, Rsk Premum, Recovery Rate. JEL classfcaton code: G3.. Introducton. In ths paper we consder the loss dstrbuton as mpled ether by ndex CDO tranche quotes (Rsk neutral mpled dstrbuton) or by objectve measure statstcs such as ratng mgraton probabltes and hstorcal analyss of default correlaton. he rsk neutral Loss dstrbuton s derved through a modfed scenaro default rate approach, nspred by the perfect copula model of Hull and Whte (25), to whch we refer as mpled copula. Instead, the objectve measure loss dstrbuton s derved through a model takng nto account ratng class default probabltes and transton matrces. We fnd, n agreement wth known market stylzed facts, that the rsk neutral loss dstrbuton prvleges large realzatons of the loss wth respect to the objectve dstrbuton, thus mplyng the well known presence of a rsk premum. We quantfy ths premum by re-prcng the market ndex and tranches under the objectve loss dstrbuton and comparng the resultng net present value (NPV) wth the rsk neutral one. 2. Impled Copula and Impled ault Rate Dstrbuton We take as reference the raxx Europe (25 consttuents) ndex. We fx a set of scenaros for the systemc factor M nfluencng the default of all the consttuent names, and assume that condtonal on M defaults of dfferent names are ndependent. hs way we are relyng on M as the only varable buldng the dependence

2 across the pool. he market standard Gaussan factor copula makes partcular assumptons on the dependence structure across defaults and on M, and we revew them n Appendx A. In the mpled copula setup one mplctly makes a homogeneous pool assumpton, and drectly postulates default ntenstes condtonal on a fnte set of scenaros for the systemc factor M, say 24 M { m, m,, m }. hs amounts to assume that the stochastc default ntensty λ of any name n the pool, defned as Probablt yrskneutral further satsfes, condtonally on M, ( Name defaults n [ t; t + dt) Name has not defaulted by t) = λ dt ystemc cenaro M = m M = m 24 M = m cenaro Probablty p p p 24 Condtonal ntensty λ λ λ 24 hs amounts to assume, for a preferred maturty and for each name =,2,, 25, the followng default probabltes, condtonal on M : ProbRskNeutral Name defaults by tme M = m = e. Agan for a preferred maturty, the pool default rate, condtonal on the systemc scenaro defned as (n our partcular case N = 25) N DRN ( ) = I N = where { condton} { Name defaults before M = m } M = m I s equal to f the condton s satsfed and otherwse, and where, gven our ntal assumpton, the terms n the summaton are ndependent of each other gven M. Now the nfnte (or large ) pool assumpton comes nto play. We may assume the number of names N n the pool to be very large. In ths case the law of large numbers mples that the sample average of the..d random varables I {Namedefaults before M = m }, I{Name 2 defaults before M = m },, s whch s our ) ( DR N, converges n law (under some mld assumptons) to the true mean of each sngle random varable when N tends to nfnty,.e. 2

3 DR N ( ) ExpectatonRskNeutral law ExpectatonRskNeutral = ProbabltyRskNeutral [ I{Name defaults before M = m }] as N [ I{Name defaults before M = m }] { Name defaults before M = m } = e : = DR ( ) hs set of scenaros DR ( ) =,,, 24, represents the set of all the possble default rates of the raxx portfolo: =,,, 24 out of 25 consttuents of the portfolo default before maturty. In turn, gven ts ntutve meanng, the default rate s also the fracton of names that have defaulted for a gven maturty. As such, ts natural values would be: 24 { DR, DR,, } 24 DR,,, = DR We may use these natural default rate scenaros backwards to deduce sensble scenaros for the ntenstes assocated wth them for the maturty. hs amounts to solve n λ lambda λ 24 e = DR ( ),,,, leadng to 24 ln ln ln( ) λ,,, = = 24 { λ, λ,, λ } o sum up, from our set of default rates above, gven the partcular maturty of the tranches and the ndex we want to prce, we have determned a correspondng set of scenaros for the default ntenstes consttutng the parametrzaton from whch we started, thus fnally closng the loop.. 3. he Instruments Payoff he tranches and the ndex pay ther spread on the dates t, t 2,, t N, expressed as year fractons. We call the start date t =. We call R the recovery rate assocated to the scenaro and we call r the rsk free zero-coupon rate on date t. Gven a generc scenaro the NPV of the premum and default leg of the ndex wll be: he orgnal perfect copula approach of Hull and Whte (25), to whch our methodology s nspred, consders a more complex spacng n the set of default rates. ault rate scenaros are spaced n such a way that the sums of net present values of tranches and ndex under each scenaro are equally spaced. 3

4 Prem OutNotl = spr = N = = ( R OutNotl ( t ) t N = ) exp( r t ) exp( ( t + t ) / 2) exp( r t ) [ exp( t ) exp( t ] ) where the notaton for the ndex outstandng notonal, the ndex premum leg, the ndex spread and the ndex default leg s self evdent. Gven a generc scenaro the NPV of the premum and default leg of the tranche wth attachment A and detachment B wll be: 2 Prem A, B OutNotl = spr A, B = A, B N = ( t OutNotl t A, B ) exp( r t t + t ) rancheloss, 2 [ rancheloss(, t ) rancheloss(, t ] N A, B = exp( r t ) ) = max( PortfLoss(, t) A,) max( PortfLoss(, t) B,) rancheloss(, t) = B A ( exp( )) PortfLoss(, t) = ( R ) λ t where agan the notaton for the A-B tranche outstandng notonal, premum leg, spread, default leg and loss s self evdent. 4. Impled ault Rate Dstrbuton Consstently wth the perfect copula sprt by Hull and Whte (25), our numercal problem s fndng the weghts (.e. the scenaro probabltes p, postve and addng up to ) to assgn to each scenaro so as to reprce the ndex and the tranches consstently wth market quotes. hese weghts p, p,, p 24 wll correspond to the rsk neutral dstrbuton of the default rates. We have 25 scenaros and only 6 nstruments (5 tranches plus the ndex), so that the system s under-determned. eed, we have too many unknowns (up to 25: the scenaro weghts) and too few equatons (down to 6: the nstruments to prce). We call PR and DEF the matrces wth the NPV of the premum and default leg. he rows correspond to the scenaros (remember that scenaro s the scenaro where names out of 25 default before maturty) whereas the columns correspond to the nstruments (the ndex n the frst column). 2 Notce the approxmaton we have ntroduced n the computaton of the ntegrals nvolved n determned the average Outstandng Notonal and ault Leg NPV n each perod. For a thorough exposton of CD prcng and the accuracy of dfferent approxmatons to the relevant ntegrals see O Kane and urnbull (23). 4

5 Prem Prem PR = Prem Prem,3 Prem 2,22 Prem,3 Prem2,22 24 Prem 24,3 Prem 24 2,22 DEF = 24,3,3 24,3 2,22 2, ,22 If we call [ ] P = p,, p24 the scenaro weghts vector then the NPV of the nstruments wll be: NPV NPV NPV =,3 = NPV 2,22 ( Prem ) P o solve the under-determned feature problem Hull and Whte (25) look for the set of weghts assgned to the scenaros that best re-prces the nstruments (mnmzes NPV NPV ) and that s also as regular as possble. he objectve functon they select s 23 NPV NPV + c = [( p p ) ( p p )] + /25 2 where the summaton s a regularzaton term. In the objectve functon of Hull and Whte there s thus a trade-off between mnmzaton of the msprcng and regularzaton of the scenaros dstrbuton. 5. Regularzaton through bd-ask nformaton We prefer not to choose a trade-off between msprcng and regularzaton. We am at fndng the smoothest dstrbuton that prces all nstruments exactly (.e. wthn ther Bd Ask spreads). o do ths we splt the problem n two steps. In the frst step we mnmze the msprcng wthout consderng smoothness of the dstrbuton. We consder the frst mnmzaton to be successful f all nstruments are reprced wthn the bd ask spread. In ths case we proceed wth a second optmzaton that takes as startng pont n the numercal algorthm the optmum found n the frst step. In the second step we maxmze the regularty of the dstrbuton gven the constrant that the nstruments are prced wthn the bd ask spread. Followng these two steps we avod choosng any tradeoff between msprcng and smoothness. he nstruments NPV can be rewrtten as: 5

6 NPV spr OutNotl p p NPV = = NPV,3 =,3 = spr r OutNotl,3 p,3 p = = NPV 2, ,22 sprr OutNotl2,22 p 2,22 p = = hus the varaton n the spreads requred to set to the NPV of the nstruments gven a scenaro dstrbuton P s: 24 - NPV / OutNotl p spr = 24 spr spr =,3 = - NPV,3 / OutNotl,3 p = spr 2, NPV2,22 / OutNotl2,22 p = If the absolute value of the components of ths vector s larger than half the bd ask spread then the scenaro dstrbuton P s not able to prce the nstruments wth an error wthn the bd ask spread. 6. A convenent relatonshp between ault Rates and Recovery Rates In ther artcle Hull and Whte (25) have used frst n ther estmaton a flat recovery rate at 4% for all scenaros: R = 4 % =,,,24 he authors report they could ft market data n the frst half of 24 usng a flat recovery of 4%. hey also report that n order to ft more recent data (as of November 25) they found t necessary to ncorporate the followng best ft relatonshp as n Hamlton et al. (25), expressng recovery as a functon of the default rate: R = max %, 52% 6.9 exp( ) =,,, [ ( )] 24 hs means that n correspondence of default rates above 7.53% the recovery s null. Our approach s nstead to ft a nonlnear relatonshp between recovery rates and annualzed default rates as n Altman et al. (22). 3, = [ ] R =.563 ln( exp( λ ), =,2,,24 3 he relatonshp we mplement does not have an ntercept so that we obtan a recovery of n case the entre portfolo defaults. 6

7 7. Impled ault Rate Dstrbuton through me We now apply the methodology outlned n the prevous paragraphs to the market CD ndex and standardzed CDO tranches spreads from Aprl 24 to Aprl 25. We mply a rsk neutral default rate densty. In the top two graphs of fgure we note that the reference swap rate (rght hand scale) and the average default rate (left hand scale) are strongly correlated. In the bottom two graphs of fgure we note nstead that the mpled default rate densty had a much larger dsperson, as measured by the dfference between the 75th and 25th percentle, durng 24 compared to the second half of 25 and the begnnng of 26. hs feature s especally pronounced for the years maturty. Also we note that untl Aprl 25 the dsperson of the dstrbuton was postvely correlated wth the reference spread movements, as proxed by the average default rate: the dsperson decreases (ncreases) when the ndex decreases (ncreases). Also from the mddle two graphs of fgure we note what happened durng the Ford-GMAC crss. Around Aprl 25 the average spread ncreased and smultaneously the dstrbuton narrowed. he dsperson narrowed and at one pont n tme, 6 May 25, on the year maturty the mpled average default rate was above the 75th percentle. Fgure y - Mean ault Rate & Reference wap y - Mean ault Rate & Reference wap Apr-4 Jul-4 Oct-4 Feb-5 mean May-5 Aug-5 ref wap Dec-5 Mar-6 Apr-4 Jul-4 Oct-4 Feb-5 mean May-5 Aug-5 ref wap Dec-5 Mar-6 4 5y Rsk Neutral ault Rate Dstrbuton Percentles 35 y Rsk Neutral ault Rate Dstrbuton Percentles Apr-4 Jul-4 Oct-4 Jan-5 May-5 Aug-5 Nov-5 Mar-6 th perc 25th perc mean 75th perc 9th perc 5 Apr-4 Jul-4 Oct-4 Jan-5 May-5 Aug-5 Nov-5 Mar-6 th perc 25th perc mean 75th perc 9th perc 8. ault Rate mulaton under the Objectve Measure We are gven a set of N reference oblgatons assocated to a set of K ratng classes. We have hstorcally measured the frequency accordng to whch a name belongng to a certan ratng class wll default before a maturty expressed as a year fracton. In order to value CDO tranches we need also a measure of the dependency between the defaults of the N reference oblgatons n the portfolo. 7

8 We model ths dependency va a one factor Gaussan copula: ) we randomly draw a vector X of N jontly standardzed normal random varables wth correlaton matrx Σ; 2) we compute the cumulatve normal U = φ( X) for all components X of X ; 3) we fnally compute the smulated default tme for each name as the nverse of the hstorcally measured survval functon = τ = ( U ), ( ) exp λ ( s) ds where λ s here the determnstc tme dependent default ntensty of name. If the smulated default tme τ s less than then the assocated reference oblgaton n the current smulaton wll default before the maturty of the CDO. he loss ncurred by the portfolo wll be the notonal nvested n the name (for the raxx t wll be.8%=/25) tmes one mnus the recovery assocated to that name. In the rsk neutral framework the way the correlaton s mpled from market quotes assumes the correlaton matrx Σ to be flat n the sense that the off-dagonal terms are all equal to a sngle parameter ρ. Gven the market spread of a tranche the correspondng compound correlaton would be the ρ value to be put nto a Gaussan copula lnkng the default of dfferent names that sets to the NPV of the tranche gven the rsk neutral default ntenstes λ (s) strpped by the CD term structure of the ndex consttuents. Clearly rather than a model ths s a quotng mechansm, and to acheve consstency the related quotes of dfferent tranches need to be calbrated wth a sngle model, such as for example the perfect copula model above, where the noton of correlaton matrx s somehow lost apart from ts quotng mechansm nterpretaton. he correlaton matrx we use to obtan the default rate and loss rate dstrbutons under the objectve measure s nstead block dagonal, as n the CDO Evaluator of &P. he correlaton between any two names wll be 5% f both names belong to the same global or regonal sector and 5% otherwse. Remark : ( &P CDO Evaluator 3 correlaton assumptons). In determnng the block dagonal structure of the pool of oblgors the &P CDO Evaluator 3 dfferentates between local, regonal and global sectors. A local sector s only affected by the macroeconomc forces wthn the country where the asset resdes (for example buldng and development ). A regonal sector s affected by the macroeconomc forces of the regon (for example: ral ndustres ). A global sector assumes that the same economc forces affect all companes n that sector, regardless of locaton (for example: ol and gas ). 9. Rsk Premum Evdence n the ex wap and the IRAXX ranches Gven the dollar value of the default leg of an nstrument (ndex CD or CDO tranche) we can ask ourselves the queston of how much of ths dollar value s justfed n terms of default rsk. In other words we want to know the dfference between the NPV of the default leg under the rsk neutral and objectve measures. A dfferent queston we mght consder s the followng: how much more the premum leg would have pad n excess of the default leg f the default dstrbuton were the hstorcal one? hs corresponds to the dfference between the NPV of the premum and default leg under the objectve measure. Both dfferences can be thought as related to a compensaton for the rsk of default n the nstrument notonal. Gven the market quotes of a set of nstruments as of January 27 th 26 ( st row, left column of table ), n order to be remunerated for the rsk of default of the notonal amount, we would expect both ths dfferences 8

9 to be postve. In both cases we subtract the NPV of the default leg under the objectve measure ( st row, rght column of table ). In the former defnton we subtract t from the NPV of the default leg under the rsk neutral measure (2 nd row, rght column of table ) and n the latter defnton from the NPV of the premum leg under the objectve measure (2 nd row, left column of table ). In fact we see that the two dfferences are ndeed postve for all nstruments and maturtes (3 rd row of table ). able Market Quotes (27 Jan 26) ndex.875%.355%.4725%.575% % 27.75% 48.25% % %.79%.875% 5.3% 6-9.3%.27%.48%.5% %.27%.445% %.2%.225% Npv ault Leg under the Objectve Measure ndex.393%.592%.899%.3776% % % % % %.949%.9632% % 6-9.%.27%.32%.2979% 9-2.%.%.2%.224% 2-22.%.%.%.% Npv Premum Leg under the Objectve Measure ndex.579%.5996% % % % % % % % 3.587%.5529% % %.2242% % % 9-2.%.5668%.6667% % 2-22.%.255%.747%.899% Npv ault Leg under the Rsk Neutral Measure ndex.558%.5784% 2.823% % % % % % % 3.532%.25% % %.244% 2.98% 8.438% 9-2.%.5636%.6477% % 2-22.%.2538%.7335%.866% Npv Premum Leg under the Objectve Measure MINU Npv ault Leg under the Objectve Measure ndex.286%.94%.9898% 3.382% % % % % % %.5897% 4.458% %.225% 2.936% 8.443% 9-2.%.5668%.6647% 3.773% 2-22.%.255%.747%.8898% Npv ault Leg under the Rsk Neutral Measure MINU Npv Premum Leg under the Objectve Measure ndex.265%.9882%.922% 3.28% % % % % % %.69% 34.99% %.28% 2.885% % 9-2.%.5636%.6457% % 2-22.%.2538%.7334%.865% We note that for all maturtes roughly half (between 4% and 6%) of the NPV of the equty tranche s justfed n terms of default rsk. We also note that only a small porton of the mezzanne (3-6) year tranche s justfed n terms of default rsk. Of course the quantfcaton of the remuneraton for rsk for each nstrument (ndex and tranche) depends heavly on the smulaton engne outlned n secton 8. If for example we were to use non homogeneous ratng transton matrces or a dfferent copula we would get dfferent NPVs of the default leg under the objectve measure and thus dfferent numbers n the thrd row of table. 9

10 . Comparson between the ault Rate Densty under the Objectve and Rsk Neutral Measure In fgure 2 we plot the default rate densty under the objectve and rsk neutral measure (the abscssas correspond to the number of defaulted oblgor n the IRAXX CDO pool: 25 names). he rsk neutral densty s obtaned from the market quotes n the bottom left part of table (27-Jan-26) usng the methodology descrbed n sectons 2 to 6. he objectve measure densty s obtaned nstead from a smulaton followng the methodology outlned n secton 8. Fgure 2 45% 4% 35% 3% 25% 2% 5% % 5% % ault Rate Densty under the Objectve Measure &P 5 &P 7 &P &P 45% 4% 35% 3% 25% 2% 5% % 5% ault Rate Densty under the Rsk Neutral Measure % PC 5 PC 7 PC PC We notce mmedately the dfferent centers of the denstes correspondng to the same maturty between the objectve and rsk neutral measure. he rsk neutral denstes are shfted to the rght, correspondng thus to a rsk premum beng prced n the traded assets underlyng the rsk neutral measure. In fgure 3 we fnally zoom on two features, correspondng to dfferent areas of the abscssa (number of defaulted oblgors at maturty) of the densty under the rsk neutral measure. In the left plot we notce a seres of bumps of ncreasng sze and shftng further to the rght, as maturty ncreases, n the range of to 4 defaults: a scenaro of extremely severe loss n the CDO oblgors pool. In the rght plot we notce the far end tal, referrng to 8 to 25 defaults out of 25 oblgors n the IRAXX pool. We notce a small bump ncreasng wth the maturty also for ths catastrophc scenaro. he top part of fgure 4 shows the mpled default rate dstrbuton calbrated at dfferent tmes for the 5 and years maturty. he bottom part of fgure 4 zooms on the tals of the mpled dstrbuton, pontng out the overall persstence of the above mentoned bumps n terms of sze (probablty mass) and locaton (range of default numbers). nce ths feature perssts, t may be approprated to look for more complex dynamcal loss models that can produce a bump feature n the tal. A related dynamcal loss models that can be consstently calbrated to tranche and ndex data for dfferent maturtes s the Generalzed Posson Loss model of Brgo, Pallavcn and orresett (26). Brgo, Pallavcn and orresett (27) further address consstency wth sngle name data and default clusters, leadng to a top-down approach known as GPCL model. A dfferent model free approach to extract market nformaton from standardzed CDO tranches that s also consstent across maturtes can be found n Walker (26) and n orresett, Brgo and Pallavcn (26).

11 Fgure 3 ault Rate Densty under the Rsk Neutral Measure.%.9%.8%.7%.6%.5%.4%.3%.2%.%.% PC 5 PC 7 PC PC ault Rate Densty under the Rsk Neutral Measure.%.9%.8%.7%.6%.5%.4%.3%.2%.%.% PC 5 PC 7 PC PC Fgure 4

12 ACKNOWLEDGEMEN We are grateful to Massmo Morn for suggestng techncal mprovements on a frst draft. REFERENCE Altman Edward I., Brady Brooks, Rest Andrea and ron Andrea (22), he Lnk Between ault and Recovery Rates: Implcatons for Credt Rsk Models and Procyclcalty Brgo Damano, Pallavcn Andrea and orresett Roberto (26), Calbraton of CDO ranches wth the Dynamcal Generalzed-Posson Loss Model, Fnancal Engneerng - Credt Models, Banca IMI Workng Paper, avalable at RN.com: Brgo Damano, Pallavcn Andrea and orresett Roberto (27), ault Correlaton, Cluster Dynamcs and ngle Names: he GPCL Dynamcal Loss Model, Workng Paper, avalable at RN.com: CDO Evaluator Handbook (26), tandard & Poor s tructured Fnance Group De ervgny Arnaud and Renault Olver (24), Measurng and Managng Credt Rsk, Mc Graw-Hll. Hamlton Davd., Varma Praveen, Ou haron and Cantor Rchard (25), ault and Recovery Rates of Corporate Bond Issuers, 92-24, pecal Comment, Moodys Investor ervce. Hull John and Whte Alan (25), he Perfect Copula, Workng Paper O Kane Domnc and urnbull tuart (23), Valuaton of Credt ault waps, Fxed Income Quanttatve Credt Research, Lehman Brothers orresett Roberto, Brgo Damano, and Pallavcn Andrea, Impled Expected ranched Loss urface from CDO Data, 26, Workng paper. Avalable at Walker Mcheal, CDO models. owards the next generaton: ncomplete markets and term structure, 26, Workng paper. Avalable at crdrv9.htm APPENDIX A he Gaussan factor copula models assume a Gaussan copula structure drvng the exponental random varables generatng jumps n the related default processes of the names n the pool. hs results n the default probablty for each name, condtonal on the Gaussan systemc factor M, to be gven by [ 2] ProbabltyRskNeutral{ Name defaults before M } N = N ( PD( )) ρ M ρ where PD ( ) s the rsk neutral probablty that any name defaults by tme. 2

13 As before defaults are ndependent gven M, and ths allows to compute the jont default probabltes for the whole pool by smply multplyng the condtonal rsk neutral probabltes n [2] and ntegratng over M under ts Gaussan densty. Under the large pool assumpton the above probablty s also the pool default rate by gven M. In the factor Gaussan Copula the systemc factor s a contnuous random varable. If we dscretse the doman of the systemc factor we can sum up the Gaussan copula as: One factor Gaussan Copula ystemc cenaro M [ m;m + dm) cenaro Probablty N(m + dm) N(m) Condtonal default rate N N ( Pd( )) ρ m ρ It s now possble to compare ths to the mpled copula assumpton we used n ths paper: In our case we replaced the parametrc formula [2] for the default probablty wth the more natural, ntensty based one ProbabltyRskNeutral Name defaults before M = m = e leadng to { } ystemc cenaro M = m Impled Copula M = m 24 M = m cenaro Probablty p p 24 p Condtonal default rate e e 24 e he end result of these approaches as far as prcng s concerned s the default rate dstrbuton. In the Gaussan One Factor Copula case ths dstrbuton has lttle flexblty, n that one can play only wth the sngle copula parameter ρ, scenaro probabltes beng fxed by the Gaussan assumpton. If one s to prce a set of nstruments (e.g. CDO tranches) wth a sngle model specfcaton, havng just one parameter can be unrealstc. In the mpled copula approach nstead we can play wth the scenaro probabltes so as to obtan a rch varety of possble default rate dstrbutons, whch can help n prcng a set of nstruments wth a sngle model specfcaton. 3