credit models and the crisis
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The Journal of Credit Risk (3981) Volume 6/Number 4, Winter 2010/11
Credit models and the crisis:
default cluster dynamics and thegeneralized Poisson loss model
Damiano Brigo
Department of Mathematics, Kings College London, Strand,
London WC2R 2LS, UK; email: [email protected]
Andrea Pallavicini
Banca Leonardo, Via Broletto 46, Milan 20121, Italy;
email: [email protected]
RobertoTorresettiQuaestio Capital Management, Via del Lauro 14, 20100 Milan, Italy;
email: [email protected]
We consider collateralized debt obligations (CDOs), analyzing their valuation
(both pre-crisis and in-crisis) with the generalized Poisson loss model. General-
izedPoisson lossis an arbitrage-free dynamic lossmodel capable of calibratingall
tranches for all maturities simultaneously. Alternative tranche analysis using the
implied copula framework or using historical estimation techniques highlights a
multimodal tail in the loss distribution underlying the CDO. An effective descrip-
tion of loss dynamics in terms of the generalized Poisson loss model explains
how the multimodal loss-probability distribution can be considered to arise fromexplicitly modeling the default of subsets of names within the CDOs pool of
names. Such default clusters may in turn be interpreted as sectors of the economy.
Our discussion is supported by abundant market examples through history, both
pre-crisis and in-crisis.
1 INTRODUCTION: PRE-CRISIS AND IN-CRISIS CREDIT MODELING
1.1 Bottom-up models
A common way of introducing dependence in credit derivatives modeling is by means
of copula functions. Typically a Gaussian copula is postulated on the exponentialrandom variables driving the default times of the names of the pool. In general, if we
The opinions expressed here are solely those of the authors and do not represent in any way those
of their employers.
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40 D. Brigo et al
try to model dependence by specifying dependence across single default times, we are
using the so called bottom-up framework, and the copula approach typically falls
within this framework. Such a general procedure cannot be extended in a simple way
to a fully dynamical model. We cannot do justice to the huge amount of literature on
copulas in credit derivatives here. We only mention that there have been attempts to
go beyond the Gaussian copula introduced to the CDO world by Li (2000) that have
led to the implied (base and compound) correlation framework, some important limits
of which have been pointed out in Torresetti et al (2006b). Li and Hong Liang (2005)
also proposed a mixture approach in connection with CDO-squareds. For results on
sensitivities computed with the Gaussian copula models, see, for example, Meng and
Sengupta (2008).
An alternative to copulas in the bottom-up context is to insert dependence among
the default intensities of single names (see, for example, Chapovsky et al (2007)).
Joshi and Stacey (2006) resort to modeling business time to create a default corre-
lation in otherwise independent single-name defaults using an intensity gamma
framework. Similarly, but in a firm-value-inspired context, Baxter (2007) introduces
Lvy firm-value processes in a bottom-up framework for CDO calibration. Lopatin
(2008) introduces a bottom-up framework effective in the CDO context as well, first
using single-name default intensities as deterministic functions of time and of the pool
default counting process, then focusing on hedge ratios and analyzing the framework
from a numerical performance point of view, showing that this model is interest-
ing despite the fact that it lacks an explicit modeling of single-name credit-spread
volatilities.
Albanese et al (2006) introduce a bottom-up approach based on structural model
ideas that can be made consistent with several inputs under both the historical and
the pricing measures that manages to calibrate CDO tranches.
At this point, it is useful to recall the concept of the implied copula (introduced
by Hull and White (2006) as a perfect copula): this is a non-parametric model
that can be used to deduce the shape of the risk-neutral pool loss distribution from
a set of market CDO spreads spanning the entire capital structure. The general use
of flexible systemic factors has been generalized and vastly improved by Rosen and
Saunders (2009), who also discuss the dynamic implications of the systemic factor
framework. In the context of one-factor models, though with a more econometric
flavor, Berd et al (2007) interpret the latent factor as the market return governed
by an asymmetric generalized autoregressive conditional heteroscedasticity model
and introduce the implied correlation surface as a mapping between the given loss-
generating model and a Gaussian benchmark.Factors and dynamics are also discussed
in Inglis et al (2008).
Our calibration results based on the implied copula already seen in Torresetti et al
(2006c) and published later in Torresetti et al (2009) and Brigo et al (2010) point out
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Credit models and the crisis: default cluster dynamics and the GPL model 41
that a consistent loss distribution across tranches for a single maturity features modes
in the tail of the loss distribution. These probability masses on the far-right tail imply
default possibilities for large clusters (possibly sectors of the economy) of names of
the economy. These results were originally published in 2006 on ssrn.com (see Brigo
etal (2006a,b)). We find the same features here using a completely different approach.
The implied copula can calibrate consistently across the capital structure but not
across maturities since it is a model that is inherently static. The next step is therefore
to introduce the dynamic loss model. This moves us into the so called top-down
framework (although dynamic approaches are also possible in the bottom-up context,
as we have seen in some of the above references).
1.2 Top-down framework
We could completely avoid single-name default modeling and instead focus on thepool loss and default counting processes, thereby considering a dynamical model at
the aggregate loss level, associated to the loss itself or to some suitably defined loss
rates. This is the top-down approach (see, for example, Bennani (2005); Giesecke
et al (2009); Sidenius et al (2008); Schnbucher (2005); Di Graziano and Rogers
(2005); Brigo et al (2006a,b); Errais et al (2009); Lopatin and Misirpashaev (2007)).
The first joint calibration results of a dynamic loss model across indexes, tranche
attachments and maturities (Brigo et al (2006a)) show that even a relatively simple
loss dynamics such as a capped generalized Poisson process suffices to account for
the loss-distribution dynamical features embedded in market quotes. This work also
confirms the implied copula findings of Torresetti et al (2006c), showing that the loss-
distribution tail exhibits a structured multimodal behavior implying non-negligibledefault probabilities for large fractions of the pool of credit references, which leaves
the potential for high losses implied by CDO quotes before the beginning of the
crisis. Cont and Minca (2008) use a non-parametric algorithm for the calibration
of top models, constructing a risk-neutral default intensity process for the portfolio
underlying the CDO to look for the risk-neutral loss process closest to a prior loss
process using relative entropy techniques (see also Cont and Savescu (2008)).
However, in general, to justify the down in top-down one needs to show that,
from the aggregate loss model, one can recover a posteriori consistency with single-
name default processes when they are not modeled explicitly. Errais et al (2009)
advocate the use of random thinning techniques for their approach (see also Halperin
and Tomecek (2008), who delve into more practical issues related to random thinning
of general loss models, and Bielecki et al (2008), who also build semi-static hedging
examples and consider cases where the portfolio loss process may not yield sufficient
statistics).
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42 D. Brigo et al
It is often notclear whether or not a fully consistent single-name default formulation
is possible given an aggregate model as the starting point.
There is a special bottom-up approach that can lead to a distinct and rich loss
dynamics. This approach is based on the common Poisson shock (CPS) framework
reviewed in Lindskog and McNeil (2003). This approach allows for more than one
defaulting name in small time intervals, which is in contrast with some of the above-
mentioned top-down approaches. In bottom-up language we see that this approach
leads to a MarshallOlkin copula linking the first jump (default) times of single names.
In top-down language, this model looks very similar to the generalized Poisson loss
(GPL) model of Brigo et al (2006a), where the number of defaults is not capped.
The problem with the CPS framework is that it allows for repeated defaults, which
is clearly wrong since one name could default more than once.
The CPS framework has been used in the credit derivatives literature by Elouer-
khaoui (2006), for example (see also the references therein). Balakrishna (2006)
introduces a semi-analytical approach, again allowing for more than one default in
small time intervals and hinting at its relationship with the CPS framework, that shows
some interesting calibration results. Balakrishna (2007) then generalizes this earlier
paper to include delayed default dependence and contagion.
1.3 Generalized Poisson loss and generalized Poisson
cluster loss models
Brigo et al (2007) address the repeated default issue in the CPS framework by con-
trolling the cluster default dynamics in order to avoid repetitions. They calibrate the
obtained model satisfactorily to CDO quotes across attachments and maturities, but
the combinatorial complexity of a non-homogeneous version of this model is forbid-
ding, and the resulting generalized Poisson cluster loss (GPCL) approach is hard to
use successfully in practice when taking single names into account.
In the context of the present paper, however, the GPL and GPCL models will still
be useful for showing how a loss-distribution dynamics consistent with CDO market
quotes should evolve.
As explained above, the GPL and GPCL models are dynamical models for loss that
have the ability to reprice all tranches and all maturities simultaneously. They mainly
differ in the way that they avoid repeated defaults, and by their stressing of the whole
pool or of individual clusters as fundamental objects. Here we employ a variant that
models theloss directly rather than by using thedefault counting process plus recovery.
The loss is modeled as the sum of independent Poisson processes, each associated
with the default of a different number of entities and capped at the pool size to avoid
infinite defaults. A possible interpretation of these driving Poisson processes is that
of defaults of sectors, although the sectors amplitudes vary in our formulation of the
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Credit models and the crisis: default cluster dynamics and the GPL model 43
model pre-crisis and in-crisis. In the new in-crisis model implementation we fix the
amplitude of the loss triggered by each cluster of defaults a priori without calibrating
it, as we did in our earlier GPL and GPCL works. This makes the calibration more
transparent and the calibrated intensities of the sectors defaults easier to interpret.
We highlight how these models are able to reproduce the tail multimodal feature,
which the implied copula has proved is indispensable for accurately repricing market
spreads of CDO tranches on a single maturity.
We also refer to the related results of Longstaff and Rajan (2008), which point
in the same direction but add a principal component analysis on a panel of credit
default swap (CDS) spread changes and also give further comments on the economic
interpretation of the default clusters as sectors.An econometric investigation of cluster
defaults starting from the Poisson framework can be found in Duan (2009).
Incidentally, we draw the readers attention to the history of defaults, which points
to default clusters being concentrated in a relatively short time period (a few months):
the thrifts in the early 1990s at the height of the loan and deposit crisis, airlines after
2001 and, more recently, autos and financials give evidence of this. In particular,
from September 7, 2008 to October 8, 2008 a time window of one month we wit-
nessedseven credit events affectingmajor financial entities: Fannie Mae, Freddie Mac,
Lehman Brothers, Washington Mutual, Landsbanki, Glitnir and Kaupthing. Fannie
Mae and Freddie Mac conservatorships were announced on the same date (September
7, 2008) and the appointment of a receivership committee for the three Icelandic
banks (Landsbanki, Glitnir and Kaupthing) was announced between October 7 and 8.
Moreover, Standard & Poors (2009) issued a request for comments related to
changes in the rating criteria of corporate CDOs. Thus far, agencies had been adopting
a multifactor Gaussian copula approach to simulate the portfolio loss in the objective
measure. Standard & Poors proposed changing the criteria so that tranches rated
AAA would be able to withstand the default of the largest single industry in the
asset pool with zero recoveries. We believe that this is a move in the direction of
modeling the loss in the risk-neutral measure via GPL-like processes, given that the
proposed changes to Standard & Poors rating criteria imply the admittance of the
possibility that a cluster of defaults in the objective measure could exist as a stressed
but plausible scenario (see also Torresetti and Pallavicini (2007) for the specific case
of constant-proportion debt obligations).
Finally, we comment more generally on dynamical aggregate models and on the
difficulties encountered in deriving single-name hedge ratios when trying to avoid
complex combinatorial analysis. The framework therefore currently remains incom-
plete, because obtaining jointly tractable dynamics and consistent single-name hedges
that can be realistically applied on the trading floor remains a problem. We have pro-
vided some references for the latest research in this field above. We highlight, though,
that even simple dynamical models such as our GPL model or the single-maturity
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implied copula model are enough to appreciate that the market quotes were imply-
ing the presence of large default clusters with non-negligible probabilities well in
advance of the credit crisis, as documented in 2006 and early 2007 (Brigo et al
(2006a,b, 2007)).
Finally, it is important to point out that most of the above discussion and references
(with very few exceptions) center on corporate CDOs mostly synthetic ones and
little literature is available for the valuation of CDOs on other asset classes, with
literature on possibly complex waterfalls and prepayment risk (including collateral-
ized loan obligations, residential mortgage-backed securities and CDOs of residential
mortgage-backed securities) generally relating to the asset class that triggered the cri-
sis. For manysuchdeals the main problem is oftenthe data. The scarce literature in this
area includes Jaeckel (2008), Papadopoulos and Tan (2007) and Fermanian (2009).
2 MARKET QUOTESFor single names, our reference products will be CDSs, but the most liquid multiname
credit instruments available in the market are credit indexes and CDO tranches (eg,
Dow JonesiTraxx (DJiTraxx) and CDX). We discuss these instruments below.
The procedure for selecting the standardized pool of names is the same for the
two indexes. Every six months a new series is rolled at the end of a polling process
managed by MarkIt where a selected list of dealers contributes the ranking of the
most liquid CDS.1
2.1 Credit indexes
The index is given by a pool of names 1 ; 2 ; : : : ; M , typically M D 125, each withnotional 1=M so that the total pool has unit notional. The index default leg consists
of protection payments corresponding to the defaulted names of the pool. Each time
one or more names default, the corresponding loss increment is paid to the protection
buyer until final maturity T D Tb arrives or until all the names in the pool have
defaulted.
In exchange for loss increase payments, a periodic premium with rate S is paid
from the protection buyer to the protection seller until final maturity Tb. This premium
is computed on a notional that decreases each time a name in the pool defaults,
decreasing by an amount corresponding to the notional of that name (without taking
out the recovery).
1 All credit references that are not investment grade are discarded. Each surviving credit reference
underlying the CDS is assigned to a sector. Each sector is contributing a predetermined number of
credit references to the final pool of names. The rankings of the various dealers for the investment
grade names are put together to rank the most liquid credit references within each sector.
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We denote by NLt the portfolio cumulated loss and by NCt the number of defaulted
names up to time t divided by M. Since, at each default, part of the defaulted notional
is recovered, we have 0 6 NLt 6 NCt 6 1. The discounted payout of the two legs of
the index is given as follows:
defleg.0/ WD
ZT0
D.0;t/ d NLt
prmleg.0/ WD S0 DV01.0/
DV01.0/ WD
bXiD1
iD.0; Ti /.1 NCTi /
where D.s;t/ is the discount factor (often assumed to be deterministic) between
times s and t and i D Ti Ti1 is the year fraction. In the second equation, DV01.0/
denotes the sum over all premium payment dates of the outstanding notional, weighted
by the relevant year fraction, and discounted back at time 0. The risk-neutral expec-
tation of such a quantity will provide the sensitivity of the premium leg to a unit
movement in the spread S0, all other things being equal (hence the name DV01,
although usually the spread is moved by 1 basis point (bp)). In the same equation,
the actual outstanding notional in each period would be an average over Ti1; Ti ,
but we have replaced it with 1 NCTi (the value of the outstanding notional at Ti ) for
simplicity, which is commonly done.
Note that, in contrast with the tranches (see Section 2.2), here the recovery is not
considered when computing the outstanding notional because it is only the number
of defaults that matters.
The market quotes the values of S0 that, for different maturities, balance the two
legs. If we have a model for the loss and the number of defaults, we may require that
the loss and the number of defaults in the model, when plugged into the two legs, lead
to the same risk-neutral expectation (and hence price):
S0 DE0
RT0 D.0;t/ d
NLt
E0Pb
iD1 iD.0; Ti /.1 NCTi /
(1)
where E denotes the expectation under the pricing risk-neutral measure.
Assuming deterministic default-free interest rates, we can rewrite:
S0 DRT0 D.0;t / dE0 NLt Pb
iD1 iD.0; Ti /.1 E0NCTi /
(2)
The assumption regarding deterministic default-free interest rates is a practical one
and allows us to obtain analytical or semi-analytical pricing formulas, as is usually
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46 D. Brigo et al
done in market practice. It can be relaxed to an assumption of independence between
defaults and default-free interest rates, with the consequence that the D.t;T/ terms of
the deterministic case become P . t ; T / D Et D.t; T /, the zero-coupon bond prices.
2.2 Collateralized debt obligation tranches
Synthetic CDOs with maturity T are contracts involving a protection buyer, a protec-
tion seller and an underlying pool of names. They are obtained by putting together a
collection of CDSs with the same maturity on different names, 1 ; 2 ; : : : ; M , typically
M D 125, each with notional 1=M, then tranching the loss of the resulting pool
between points A and B , with 0 6 A < B 6 1:
NLt WD1
B A. NLt A/1fA< NLt6Bg C .B A/1f NLt>Bg
A useful alternative expression is:
NLt WD1
B AB NL
0;Bt A NL
0;At (3)
Once enough names have defaulted and the loss has reached A, the count starts.
Each time the loss increases, the corresponding loss change rescaled by the tranche
thickness B A is paid to the protection buyer until maturity arrives or until the total
pool loss exceeds B , in which case the payments stop.
The discounted default leg payout can then be written as:
deflegA;B.0/ WD ZT
0D.0;t/ d NLt
As usual, in exchange for the protection payments, a premium rate SA;B0 , fixed at
time T0 D 0, is paid periodically at times T1; T2; : : : ; T b D T. Part of the premium
can be paid at time T0 D 0 as an upfront UA;B0 . The rate is paid on the surviving
average tranche notional. If we further assume that payments are made on the notional
remaining at each payment date Ti , rather than on the average in Ti1; Ti , the
premium leg can be written as:
prmlegA;B.0/ WD UA;B0 C S
A;B0 DV01A;B.0/
DV01A;B.0/ WD
bXiD1
iD.0; Ti /.1 NLTi /
When pricing CDO tranches, we are interested in the premium rate SA;B0 that sets
to zero the risk-neutral price of the tranche. The tranche value is computed taking the
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TABLE 1 Standardized attachment and detachment for the DJiTraxx Europe main and
CDX.NA.IG tranches.
DJiTraxxEurope CDX.NA.IGmain (%) (%)
03 03
36 37
69 710
912 1015
1222 1530
22100 30100
(risk-neutral) expectation (in t D 0) of the discounted payout, which is the difference
between the default and premium legs above. We obtain:
SA;B0 D
E0RT0 D.0;t / d
NLt UA;B0
E0Pb
iD1 iD.0; Ti /.1 NLTi /
(4)
Assuming deterministic default-free interest rates, we can rewrite:
SA;B0 D
RT0 D.0;t/ dE0
NLt UA;B0Pb
iD1 iD.0; Ti /.1 E0NLTi /
(5)
The above expression can be easily recast in terms of the upfront premium UA;B0 for
tranches that are quoted in terms of upfront fees.
The tranches that are quoted on the market refer to standardized pools, standardizedattachmentdetachment points A and B and standardized maturities.
The standardized attachment and detachment differ slightly for the CDX.NA.IG
and the DJiTraxx Europe main tranches.2
For the DJiTraxx and CDX pools, the equity tranche .A D 0; B D 3%/ is quoted
by means of the fair UA;B0 , while assuming S
A;B0 D 500bps. Thereasonthat the equity
tranche is quoted upfront is to reduce the counterparty credit risk that the protection
seller is facing. All other tranches are generally quoted by means of the fair running
spread SA;B0 , assuming no upfront fee (U
A;B0 D 0). Following recent market turmoil,
the 36% and the 37% tranches have been quoted in terms of an upfront amount and
a running SA;B0 D 500bps given the exceptional risk premium priced by the market
for this tranche.
2 The attachment points of the CDX tranches are slightly higher, reflecting the average higher
perceived riskiness (as measured, for example, by the CDS spread or by balance-sheet ratios) of the
liquid investment grade North American names.
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3 A FULLY CONSISTENT DYNAMICAL MODEL:
THE GENERALIZED-POISSON LOSS MODEL
The GPL model can be formulated as follows. Consider a probability space supportinga number n of independent Poisson processes Z1; : : : ; Zn with time-varying and
possibly stochastic intensities 1; : : : ; n under the risk-neutral measure Q. The risk-
neutral expectation conditional on the market information up to time t , including the
pool loss evolution up to t , is denoted by Et . Intensities, if stochastic, are assumed to
be adapted to such information.
Define the stochastic process:
Zt WD
nXjD1
jZj.t / (6)
for positive integers 1; : : : ; n. In the following, we refer to the Zt process simply as
the GPL process. We will use this process as a driving process for the cumulated port-
folio loss NLt , which is the relevant quantity for our payouts. In Brigo et al (2006a,b),
the possible use of the GPL process as a driving tool for the default counting processNCt is illustrated instead.
The characteristic function of the Zt process is:
'Zt .u/ D E0eiuZt D E0E0e
iuZt j 1. t / ; : : : ; n.t/
where:
j.t / WD
Zt0
j.s/ ds; i D 1 ; : : : ; n
are the cumulated intensities of each Poisson process. Now, we substitute Zt,
obtaining:
'Zt .u/ D E0
nYjD1
E0eiu jZj .t/ j 1. t / ; : : : ; n.t/
D E0
nYjD1
'Zj .t/jj .t/.u j/
which can be directly calculated since the characteristic function 'Zj .t/jj .t/ of each
Poisson process, given its intensity, is known in closed form, leading to:
'Zt .u/ D E0 exp nX
jD1
j.t/.eiu j 1/ (7)
The marginal distribution pZt of the process Zt can be directly computed at any
time via an inverse Fourier transformation of the characteristic function of the process.
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The characteristic function 'Zt .u/ can be explicitly calculated for some relevant
choices of Poisson cumulated-intensity distributions (see, for example, Brigo et al
(2006a,b)).
3.1 Loss dynamics
The underlying GPL process Zt is non-decreasing and, given sufficiently large times,
it takes arbitrarily large values. The portfolio cumulated loss and the rescaled number
of default processes is non-decreasing, but limited to the interval 0; 1. Thus, we con-
sider the deterministic non-decreasing function W N[ f0g ! 0; 1 and we define
the cumulated portfolio loss process NLt as:
Lt WD L.Zt / WD min.Zt ; M0/ and NLt WD
Lt
M0(8)
where 1=M0
, M0
> M > 0, is the minimum jump for the loss process. M0
is clearlyrelated to the loss granularity.
Remark 3.1 Note that the loss is bounded within the interval 0; 1 by construction,
but that the possibility remains that the loss jumps more than M times, where M is
the number of names in the portfolio.If this is the case, we may checka posteriori that
the probability of such events is negligible. This is the case for all of our examples.
The marginal distribution of the cumulated portfolio loss process Lt can be easily
calculated. We obtain:
Lt D min.Zt ; M0/ D Zt1fZtM0g
Since Zt has a known distribution, the distribution of Lt can be easily derived as a
by-product. The related density (defined on integer values since the law is discrete)
is:
pLt .x/ D pZt .x/1fx M0g1fxDM0g
The density of NLt follows directly. Also, the intensity of Lt , ie, the density of the
absolutely continuous compensator of Lt (see, for example, Giesecke et al (2009)),
can be computed directly and is given by:
hL.t / D
nXjD1
min. j; .M0 Zt/
C/j.t / (9)
(seeBrigo etal (2006a,b) for details). The intensity h goes to zero when Z exceeds M0,
which corresponds to total loss, as expected. Furthermore, if all the possible integer
jump sizes between 1 and M0 are allowed, ie, if j D j and n D M0, the intensity
hL jumps whenever the cumulated portfolio loss process L jumps. The intensity
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50 D. Brigo et al
jumps downward, and this would seem to go in the opposite direction with respect
to self-excitedness, which is considered a desirable feature of loss models in general.
However, self-exciting features are embedded in our model, and embedded in the
possibility of having several defaults in small intervals, contrary to most approaches
to loss modeling. Consider, for example, just two names: instead of having the loss
of one name increase the likelihood of default (intensity) of a second name, we have
both names defaulting together immediately. This embeds self-excitedness, although
in an extreme way. Finally, to view the effect of single names on each other, we need
a more sophisticated formulation based on the common Poisson shocks framework,
leading to the GPCL model, which is analyzed in Brigo et al (2007) and is presented
below with a few updates.
Example 3.2 It is worth presenting an example to further clarify the GPL for-
mulation. Let us take an example considering a GPL model where there are three
independent Poisson processes Z1, Z2 and Z3 correspondingly multiplied by sizes1, 10 and 40, leading to:
Zt D 1Z1.t / C 10Z2.t/ C 40Z3.t/
with constant deterministic intensities 1 D 4%, 2 D 1:5% and 3 D 0:75%.
Assume a 40% deterministic recovery, and take one scenario where the three processes
have eachjumped once before the current time. Theloss of the standardized DJiTraxx
pool of 125 names associated to this scenario with one jump of Z1, Z2 and Z3,
respectively, would be 0.48%, 4.8% and 19.2%, respectively. The expected loss in
one year for the uncapped loss process is:
0:2352% D .0:0048 0:04/ C .0:048 0:015/ C .0:192 0:0075/
In Figure 1 on the facing page we plot the resulting default counting distribution on
four increasing maturities: 3, 5, 7 and 10 years. We note that the modes on the right
tail of the distribution become more evident as the maturity increases. These bumps
are the corresponding probabilities of a jump of a higher amplitude occurring.
From the distribution we also note that the probability is not simply assigned to
the higher-amplitude jumps (10 and 40 defaults in our example) but also to a number
of defaults immediately above: this would be the probability of having jumps of both
a high amplitude (either 10 or 40) and of a lower amplitude (1 in our case), where
jumps with lower amplitudes have higher probability.
Furthermore, we note that some second-order modes start to appear as maturity
increases. These are the bumps corresponding to multiple jumps in the higher-size
(10 and 40), smaller probability and GPL components.
Let us now assume we are to price two tranches: 24.8% and 1019.2%. The
expected tranche loss of these tranches, and, ultimately, their fair spread, will depend
primarily on the probability mass lying above the tranche detachment.
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FIGURE 1 Default counting M NCT probability distribution associated to a GPL model
featuring three independent Poisson processes with jump amplitudes 1 D 1, 2 D 10 and
3 D 40, with constant deterministic intensities 1 D 4%, 2 D 1:5% and 3 D 0:75%.
10
8
6
4
2
00 10 20 30 40 50 60
3 year
%
(a)
10
8
6
4
2
00 10 20 30 40 50 60
%
5 year
(b)
10
8
6
4
2
00 10 20 30 40 50 60
%
7 year
(c)
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52 D. Brigo et al
FIGURE 1 Continued.
10
8
6
4
2
00 10 20 30 40 50 60
%
10 year
(d)
3.2 Model limits
The GPL model that we have introduced can currently be viewed as a particularly
simple parametrization of the market-implied loss-distribution dynamics. A positive
feature is that the loss changes only by positive jumps, which should be the case
in any sensible loss model. Furthermore, this choice allows us to achieve a good
calibration to market data, as we will see in the following section. However, we
are not making explicit assumptions on two important issues: firstly, we have not
addressed possible ways of making our model consistent with single-name dynamics,
and secondly, we have not explained how to choose a full-featured pool spread and
recovery dynamics. One possibility would be to make the intensities in the Poissonprocesses driving Z stochastic and to consider more general transformations of Z to
obtain the loss process.
Since we are focusing only on the calibration of CDO tranches, which depend
only on the loss marginal distribution, we will avoid discussing such problems here
(we direct the interested reader to Brigo et al (2006a,b) for an extensive analysis
of candidate spread and recovery dynamics, and to Brigo et al (2007) for further
discussion, including consistency with single-name data).We pointout that significant
progress and testing in loss modeling will only be possible when more liquid market
quotes for tranche options and forward-start tranches are available.
3.3 Model calibration
We work with the basic GPL model specification as given by the driving GPL process
Z in (6), which we use to model the pool loss through (8). In this basic formulation,
each Poisson mode Zj has a deterministic piecewise-constant intensity j.t/.
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Given that we have modeled the pool loss NLt directly, we do not completely charac-
terize the rescaled default counting process NCt , but instead give only its expectations.
Indeed, such expectationsare the only information on default counting that are implicit
in index market quotes (Equation (1)), whereas tranche quotes (Equation (4)) depend
only on the loss and not on default counting explicitly. We therefore assume:
E0 NCt WD1
1 RE0 NLt ; 0 6 R < 1 E0 NLTb (10)
where the range of definition of the constantR is taken in order to ensure that, at each
time t , the expected value of the rescaled number of defaults is greater than or equal
to the cumulated portfolio loss, and that both are smaller than or equal to one. Note
that we avoid introducing an explicit dynamics for the recovery rate (see Brigo et al
(2006a,b, 2007) for an initial discussion on recovery dynamics). Our R here can be
interpreted as a kind of average recovery rate.
3.4 Detailed calibration procedure
The model parameters found using the calibration procedure are the amplitudes j 2
fm 2 N W m 6 M0g, j D 1 ; : : : ; n, and the cumulated intensities j.T /, which are
real non-decreasing piecewise linear functions in the tranche maturity.
The optimal values for the amplitudes are selected in the following way.
1) Fix the minimum jump size to 1=M0 by choosing the integer M0 > M > 0.
2) Find the integer value for 1 by calibrating the cumulated intensity 1 for each
value of 1 in the range 1;M0, all other modes being set to zero. Calibration isobtained by minimizing the objective function (11) below. Finally, choose the 1for which the calibration error is at its minimum when using the corresponding
1. We call this chosen 1 the best integer value for 1.
3) Add the amplitude 2 and find its best integer value by calibrating the cumulated
intensities 1 and 2, starting from the previous value for 1 as a guess, for
each value of2 in the range 1;M0.
4) Repeat the previous step for i with i D 3 and so on by calibrating the cumu-
lated intensities 1; : : : ; i , starting from the previously found 1; : : : ; i1
as an initial guess, until the calibration error is under a given threshold or untilthe intensity i can be considered negligible.
5) Checka posteriori that the probability of having more than M jumps is negli-
gible and that the value ofR is within the arbitrage-free range given in (10).
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54 D. Brigo et al
TABLE 2 DJiTraxx index and tranche quotes in basis points on May 13, 2005, with
bidask spreads in parentheses.
Maturities Attachment 3 5 7 10detachment years years years years
Index 38 54 65 77(4) (1) (3) (2)
Tranche
03 2,060 4,262 5,421 6,489(100) (118) (384) (124)
36 72 173 398 590(10) (68) (40) (20)
69 28 57 141 188(6) (6) (17) (15)
912 13 31 72 87(2) (5) (20) (15)
1222 3 21 42 60(1) (3) (13) (10)
Index and tranches are quoted through the periodic premium, while the equity tranche is quoted as an upfront
premium (see Section 2).
The objective function f to be minimized in the calibration is the squared sum of
the errors shown by the model to recover the tranche and the index market quotes
weighted by market bidask spreads:
f.;/ DXi
"2i ; "i Dxi . ; / xmidi
xbidi xaski
(11)
where the xi (with i running over the market quote set) are the index values S0for DJiTraxx index quotes and either the periodic premiums S
A;B0 or the upfront
premium rates UA;B for the DJiTraxx tranche quotes.
3.5 Calibration results
The GPL model is calibrated to the market quotes observed weekly from May 6,
2005 to October 18, 2005. Following Albanese et al (2006), we take R D 30% as
our reference value for the recovery rate in the DJiTraxx Europe market for spot
and forward contracts. The quality of our calibration below is not altered if we select
a value R D 40% resembling the recovery typically used in simplified quoting
mechanisms in the market (see Brigo et al (2006a,b, 2007) for examples of this). We
start with M0 D 200, corresponding to a minimum loss jump size of 50bps.
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TABLE 3 (a) Calibration error "i in (11) (calculated with respect to the bidask spread)
for tranches quoted on May 13, 2005; (b) cumulated intensities (integrated up to tranche
maturities) of the GPL model with M0 D 200.
(a)
Maturities Attachment 3 5 7 10detachment years years years years
Index 0.0 0.1 0.3 0.0
Tranche
03 0.0 0.1 0.2 0.2
36 0.0 0.0 0.2 0.0
69 0.0 0.0 0.3 0.1912 0.1 0.1 0.1 0.4
1222 0.0 0.0 0.2 0.3
(b)
.T / 3 5 7 10
years years years years
1 1.955 3.726 4.464 7.694
3 0.000 0.062 0.305 0.305
8 0.016 0.033 0.109 0.10912 0.004 0.013 0.026 0.026
19 0.006 0.006 0.017 0.017
72 0.000 0.009 0.026 0.049
185 0.000 0.002 0.002 0.008
Each row corresponds to a different Poisson component with jump amplitude . Recovery rate is 30%.
As a first example, consider the calibration date May 13, 2005 (see Table 2 on the
facing page). In Table 3 we list the calibration result and the values of the calibrated
parameters. The calibration errors " are very low for all maturities. Note that a cal-
ibration error smaller than one means that the difference between the market quote
and the model price is smaller than the bidask spread.
Consider, as a second example, the calibration date October 11, 2005 (see Table 4
on the next page). In Table 5 on page 57, we list the calibration results and the values of
the calibrated parameters. The calibration errors show that the 10-year equity tranche
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56 D. Brigo et al
TABLE 4 DJiTraxx index and tranche quotes in basis points on October 11, 2005, with
bidask spreads in parentheses.
Maturities Attachment 3 5 7 10detachment years years years years
Index 23 38 47 58(2) (1) (1) (1)
Tranche
03 762 3,137 4,862 5,862(26) (26) (76) (74)
36 20 95 200 515(10) (1) (3) (10)
69 7 28 43 100(6) (1) (2) (4)
912 12 27 54(2) (4) (5)
1222 7 13 23(1) (2) (3)
is not correctly priced. We find such mispricing in many calibration examples, in
particular after October 2005.
For the minimum loss jump size 1=M0, besides 50bps, we try the values 2bps and
10bps, corresponding, respectively, to M0 equal to 5,000 and 1,000. As can be seen
from Table 6 on the facing page, the 10-year maturity tranches (in our experience themost difficult to calibrate) are stable through the three different loss sizes, suggesting
that going below 50bps does not add much flexibility to the model. This is confirmed
by further tests. In particular, the difference between the M0 D 1;000 calibration
and the M0 D 5;000 calibration is always small. Furthermore, the behavior of the
mean calibration error, ie, of the mean of the absolute values of the "i across time and
quotes, for the three different choices of M0 is quite similar and within one bidask
spread.
We also note that as the minimum jump size decreases (granularity increases),
the loss distribution becomes noisier, due to the presence of small amplitudes. Fur-
thermore, very small modes, appearing when the minimum jump size is as small as
a few basis points, may violate the requirement that the loss process jumps fewer
than M times (see Remark 3.1). We also try calibrations with M0 less than 200, ie,
with a minimum loss jump greater than 50bps. In this case the calibration error grows
quickly. Indeed, the minimum jump size, in this case, becomes greater than the typical
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Credit models and the crisis: default cluster dynamics and the GPL model 57
TABLE 5 (a) Calibration error "i in (11) (calculated with respect to the bidask spread) for
tranches quoted on October 11, 2005; (b) cumulated intensities (integrated up to tranche
maturities) of the GPL model with M0 D 200.
(a)
Maturities Attachment 3 5 7 10detachment years years years years
Index 0.0 0.0 0.1 0.1
Tranche
03 0.1 0.1 1.2 2.1
36 0.1 0.1 0.3 1.0
69 0.0 0.1 0.3 0.9
912 0.4 0.8 0.81222 0.0 0.0 0.0
(b)
.T / 3 5 7 10
years years years years
1 0.441 2.498 4.466 7.555
2 0.435 0.435 0.435 0.671
11 0.004 0.023 0.023 0.023
22 0.000 0.001 0.006 0.030
29 0.000 0.000 0.001 0.00132 0.001 0.004 0.004 0.004
192 0.000 0.001 0.005 0.011
The three-year maturity quotes lack two tranches.
TABLE 6 GPL calibration error for different minimum loss sizes 1=M0 with respect to the
bidask spread for 10-year tranches on October 11, 2005.
Attachmentdetachment 50bps 10bps 2bps
03 2.1 1.8 1.8
36 1.0 1.0 1.0
69 0.9 0.9 0.9
912 0.8 0.9 0.8
1222 0.0 0.2 0.0
Recovery rate is 30%.
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58 D. Brigo et al
TABLE 7 Values of the Poisson amplitudes =M0 for different values of the minimum loss
jump 1=M0.
(a) 50bps
Poisson amplitudes 1 2 3 4 5 6 7
Date (%) (%) (%) (%) (%) (%) (%)
May 6, 2005 0.50 1.50 4.00 6.00 9.50 39.50 92.50
September 2, 2005 0.50 1.00 4.00 5.50 12.50 39.00 100.00
October 11, 2005 0.50 1.00 5.50 11.00 14.50 16.00 96.00
(b) 10bps
Poisson amplitudes 1 2 3 4 5 6 7
Date (%) (%) (%) (%) (%) (%) (%)
May 6, 2005 0.10 1.50 4.60 5.90 9.60 39.60 53.00
August 5, 2005 0.20 1.10 1.40 8.10 11.30 49.00 62.40
October 11, 2005 0.10 0.70 1.00 6.30 11.50 14.50 93.70
(c) 2bps
Poisson amplitudes
1 2 3 4 5 6 7Date (%) (%) (%) (%) (%) (%) (%)
May 6, 2005 0.02 1.50 5.26 9.64 17.58 39.64 99.78
August 12, 2005 0.38 1.06 1.14 7.38 12.24 41.34 99.80
October 3, 2005 0.02 0.98 1.16 7.52 9.74 43.34 65.16
October 11, 2005 0.16 0.68 1.00 6.30 10.98 14.46 94.90
Only the calibration dates between May 6, 2005 and October 18, 2005 where the =M0 values change are listed.
portfolio loss given when one name defaults. 50bps seems, then, to be a reasonable
reference value.
Also, the values of the Poisson amplitudes are quite stable across the calibration
dates. Indeed, in six months we observe at most four changes in their values, as shown
in Table 7.
The loss distribution implied by the GPL model is multimodal and the probabil-
ity mass moves toward larger loss values as the maturity increases. These features
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Credit models and the crisis: default cluster dynamics and the GPL model 59
FIGURE 2 Loss-distribution evolution of the GPL model with a minimum jump size of
50bps at all the quoted maturities up to 10 years, drawn as a continuous line.
0 0.02 0.04 0.06 0.08 0.100
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Loss
0.10 0.15 0.20 0.25 0.300
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0x 10
3
Loss
(a)
3 years5 years7 years10 years
(b)
3 years5 years7 years10 years
are shared by different approaches. For instance, static models, such as the implied
default-rate distribution in Torresetti et al (2006c), suggest multimodal loss distribu-
tions, as mentioned in the introduction regarding the implied copula. The evolution
of the implied loss distribution is shown in Figure 2.
The dynamic credit correlation model of Albanese et al (2006) shows implied
loss distributions whose modes tend to group as the maturity increases, leading to a
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60 D. Brigo et al
distribution approaching normality. The GPL model reproduces this behavior (see,
for example, Brigo et al (2006a,b)).
4 APPLICATIONS TO MORE RECENT DATA ANDTHE CRISIS
In this section we check whether the critical features that we discussed regarding
implied correlation and the subsequent elements coming from more advanced models
are still present in-crisis after mid 2007. We will observe that the features are still
present and are often amplified in the market after the beginning of the crisis.
4.1 Customizing the generalized Poisson loss model
to deal with sectors
We now consider the GPL model to assess how well it performs in-crisis. We change
the model formulation slightly in order to have a model that is more in line with the
current market, while maintaining all the essential features of the modeling approach.
Compared with the model described in Section 3, we introduce the following modifi-
cations. We fix the jump amplitudes a priori rather than calibrating them through the
detailed calibration procedure seen in Section 3.4, and we associate different recov-
eries to different . Fixing the before calibration will make the model less flexible
but quicker to calibrate and possibly more stable. To fix the a priori, we reason as
follows. Fix the independent Poisson jump amplitudes to the levels just above each
tranche detachment, when considering a 40% recovery.
For the DJiTraxx index, for example, this would be realized through jump ampli-
tudes ai D i=125, where:
5 D roundup125 0:03
1 R
; 6 D roundup
125 0:061 R
7 D roundup
125 0:09
1 R
; 8 D roundup
125 0:12
1 R
9 D roundup
125 0:22
1 R
; 10 D 125
and, in order to have more granularity, we add the sizes 1, 2, 3 and 4:
1 D 1; 2 D 2; 3 D 3; 4 D 4
In total we have n D 10 jump amplitudes. We then modify the obtained sizes slightlyin order to account for CDX attachments that are slightly different. Eventually, we
obtain the set of amplitudes:
125 ai i 2 f1;2;3;4;7;13;19;25;46;125g
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We want to associate a recovery of 40% to all of these amplitudes , except 125, to
which we want to associate a recovery of 0% in order to size the premium quoted
by the market for super-senior tranches. We start by considering the default counting
process which, by introducing the non-Armageddon indicator It , can be cast in the
following form:
NCt D 1 It .1 Nct/ (12)
with:
Nct WD min
n1XiD1
aiZi .t/;1
; It WD 1fZn.t/D0g
We refer to the component associated with n D 10 D 125 as the Armageddon
mode, since whenever Zn D Z10 jumps, the whole pool defaults. We refer to the
jump Zn D Z10 as an Armageddon event. Indeed, whenever theArmageddon com-
ponent Zn jumps for the first time, the default counting processN
Ct jumps to the entirepool size and every name in the pool has defaulted, with no more defaults being
possible.
We then introduce the stopping time O as the minimum time between the Armaged-
don jump event and the time when the reduced pool without the Armageddon compo-
nent has completely defaulted. This is also the time when the full pool has defaulted:
O D inf
t W
nXiD1
aiZi .t/ > 1
We note that I is 0 after the Armageddon event, and otherwise it is equal to 1, while
Nc is equal to 1 after the reduced pool is over, so, after some algebra, we obtain:
NCt D 1fO6tg C Nct1fO>t g (13)
Note that, as expected, NC depends on Nc only at terminal time t .
To derive the loss process with a zero recovery for the Armageddon event, we first
calculate the evolution of the counting process:
d NCt D It d Nct dIt.1 Nct/
Here, notation such as d NCt denotes the left increment d NCt D NCt NCt, since we are
considering right-continuous jump processes. Intuitively, this quantity is zero if NC
does not jump at t , whereas it is one if NC jumps at t .We then apply the recovery only to the terms coming from jumps that are not due
to the Armageddon event:
d NLt D .1 R/It d Nct dIt.1 Nct/
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62 D. Brigo et al
We cannow integrate the previous equation to obtain theloss process. To manipulate
the above equation we observe three things. Firstly, I is zero after the Armageddon
event, otherwise it is equal to one. Secondly, dI is non-zero only at the Armageddon
event. Finally, Nc is equal to one and d Nc is zero after the reduced pool is over:
NLt D .1 R NcO/1fO6tg C .1 R/ Nct1fO>t g (14)
Note that NL now depends on Nc both at full-pool exhaustion time O and at terminal
time t .
Remark 4.1 (Interpretation of the model) Whenever the Armageddon component
Zn jumps the first time, we will assume that the recovery rate associated to the
remaining names defaulting in that instant will be zero. The pool loss, however, will
not always jump to one, since there is the possibility that one or more names already
defaulted before the Armageddon component Zn jumped, and that they defaulted
with recovery rate R. If, at a given instant t , the whole pool defaults, ie, NCt D 1, this
may happen in one of two ways.
1) Zn jumped by t . In this case, the portfolio has been wiped out with the help of
an Armageddon event. Note that, in this case, d NCt D d NLt ifZn jumps at t . In
fact, the recovery associated to the pool fraction defaulting in that instant will
be equal to zero.
2) Zn has not jumped by t . In this case the portfolio has been wiped out, not with
the help of anArmageddon event, but because of defaults of more small or large
sectors that do not comprise the whole pool. Note that, in this case, the loss is
less than the whole notional, as all these defaults had recovery R > 0.
In this way, whenever Zn jumps at a time when the pool has not yet been wiped
out, we can rest assured that the pool loss will be above 1 R. We do this because
the market in 2008 quoted CDOs with prices assuming that the super-senior tranche
would be impacted to a level that was impossible to reach with recoveries fixed at
40%. For example, there was a market for the DJiTraxx five-year 60100% tranche
on March 25, 2008 quoting a running (bid) spread of 24bps bid.
In the dynamic loss model, recovery can be made a function of the default rate NC
(or other solutions are possible; see Brigo et al (2006b) for more discussion). Here
we use the above simple methodology to allow losses of the pool to penetrate beyond
1 R and hence severely affect even the most senior tranches, in line with market
quotations.
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4.2 Numerical calculation of loss distribution
We know how to calculate the distribution of both NCt and NLt . The distribution of the
counting process may be directly calculated from Equation (12) as the distribution ofa reduced GPL, ie, a GPL where the jump Zn is excluded, whose counting process
will be Nct .
The distribution of the loss process can be calculated starting from Equation (14).
A simple approach involves explicitly calculating the forward Kolmogorov equation
satisfied by the probability distribution:
p NLt .x/ D QfNLt D xg;
d
dtp NLt .x/ D
Xy
At.x;y/p NLt .y/ (15)
where x and y are generic states of the loss process and where the generator matrix
At is given by:
At.x;y/ WD limt!0
Qf NLtCt D x j NLt D yg 1fxDyg
t
This procedure is allowed since, at each point in time, starting only from the value
of the loss process, it is possible to say whether the Armageddon event happened, so
we are able to reduce the loss process, which depends on the stopping time O, to a
Markov process. In order to do so, we consider the following states for the pool loss
process divided by M:NLt 2 A[B
whereA is the set of states where the Armageddon event has not happened, andB is
the set of states where the Armageddon event has happened. The two sets are defined
as follows:
A WD
0; .1 R/
1
M; .1 R/
2
M; : : : ; . 1 R/
M 1
M; 1 R
B WD
.1 R/
M 1
MC
1
M; : : : ; . 1 R/
1
MC
M 1
M; 1
Note that the two sets are disjoint if R > 0, so if we know the value of NL only at time
t , we can state whether the Armageddon event has happened.
Example 4.2 Consider a pool of six names with the following GPL modes with
constant deterministic default intensities:
Zt D Z1.t/ C 3Z2.t / C 6Z.t /
1 D 0:020; 3 D 0:010; D 0:003
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TABLE 8 Generator matrix for a simple GPL process of three amplitudes for a pool of six
names with 40% recovery for each mode, and the Armageddon mode, whose recovery is
zero.
y x 0 10 20 30 40 50 60 100
0 3.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00
10 1.00 3.30 0.00 0.00 0.00 0.00 0.00 0.00
20 0.00 1.00 3.30 0.00 0.00 0.00 0.00 0.00
30 2.00 0.00 1.00 3.30 0.00 0.00 0.00 0.00
40 0.00 2.00 0.00 1.00 3.30 0.00 0.00 0.00
50 0.00 0.00 2.00 0.00 1.00 3.30 0.00 0.00
60 0.00 0.00 0.00 2.00 2.00 3.00 0.00 0.00
67 0.00 0.00 0.00 0.00 0.00 0.30 0.00 0.00
73 0.00 0.00 0.00 0.00 0.30 0.00 0.00 0.0080 0.00 0.00 0.00 0.30 0.00 0.00 0.00 0.00
87 0.00 0.00 0.30 0.00 0.00 0.00 0.00 0.00
93 0.00 0.30 0.00 0.00 0.00 0.00 0.00 0.00
100 0.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Starting states (y) on columns, arrival states (x) on rows (see Equation (15)). See text for intensity and amplitude
data. All values in the table are in percent.
where we have called Z.t / the mode corresponding to the Armageddon event. We
consider a recovery of 40% for the first two modes and a zero recovery for the last
mode. The loss states are NLt 2 A[B with:
AWD f0%; 10%; 20%; 30%; 40%; 50%; 60%g
B WD f67%; 73%; 80%; 87%; 93%;100%g
We can directly calculate the generator matrix since the pool loss may only jump
via the three GPL modes, and each mode only corresponds to one loss transition. We
display the matrix entries in Table 8, with starting states on columns and arrival states
on rows. Note that the main diagonal is calculated to ensure probability conservation
through time.
If no default has happened at time t D 0, the boundary condition for our Kol-
mogorov equation is p NL0.x/ D 1fxD0g, so we can integrate Equation (15) by means
of matrix exponentiation:
p NLt .x/ D exp Zt
0
Xy
Au.x;y/ du
p NL0.y/ D exp
tXy
A.x;y/
p NL0.y/
where the last step is due to the fact that, in this case, the generator matrix does
not depend on time. Matrix exponentiation can be quickly computed with the Pad
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FIGURE 3 Loss distribution evolution up to 10 years predicted by the GPL model for a
pool of six names.
0
2040
6080
100
2
4
6
8
10
0
Years to maturity
Attachment points (%)
Loss
(%)
10
5
0
40% recovery for each mode except the Armageddon mode, whose recovery is zero. See text for intensity and
amplitude data.
approximation (Golub andVan Loan (1983)), leading to a closed-form solution for the
probability distribution p NLt .x/. This distribution can then be used in the calibration
procedure. In Table 9 on the next page and in Figure 3, we show the loss distribution
for our example.
4.3 Calibration results
Bearing this interpretation of the modes in mind, we decided to select the GPL ampli-
tudes by choosing the independent Poisson jump amplitudes to be at the level just
above each tranche detachment when using a 40% recovery. This led to the specific
values of5; : : : ; 10 that we introduced earlier in the paper. With an eye on the vari-
ety of shapes of the implied copula historical calibrated distributions, as summarized
in Brigo et al (2010) and in Torresetti et al (2006c), we added four amplitudes (from
one to four), corresponding to a small number of defaults.
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TABLE 9 Loss distribution numerical values for 5-year and 10-year maturity dates, pre-
dicted by the GPL model for a pool of six names.
0 5 10years years years(%) (%) (%)
0% 100.00 84.79 71.89
10% 0.00 4.24 7.19
20% 0.00 0.11 0.36
30% 0.00 8.48 14.39
40% 0.00 0.42 1.44
50% 0.00 0.01 0.07
60% 0.00 0.46 1.72
67% 0.00 0.00 0.00
73% 0.00 0.00 0.02
80% 0.00 0.07 0.2587% 0.00 0.00 0.00
93% 0.00 0.04 0.12
100% 0.00 1.38 2.55
40% recovery for each mode, except the Armageddon mode whose recovery is zero.
We now present the goodness of fit of this GPL through history. We measure the
goodness of fit by calculating, for each date, the relative mispricing:
MisprRel
A;B
D
8
SA;B;ask0
0 otherwise
where SA;B;theor0 is the tranche theoretical spread as in Equation (4), where, for the
calculation of the expectations of both numerator and denominator, we take the loss
distribution resulting from the calibrated GPL.
In Figure 4 on the facing page and Figure 5 on page 68 we present the relative
mispricing for all tranches, for all maturities and for both indexes throughout the
sample (March 2005 to June 2009).
We note that, while the 5-year tranches could be repriced fairly well in the cur-
rent credit crisis for both DJiTraxx and CDX, the 10-year tranche calibrations for
both indexes have sensibly been made less precise following the collapse of Lehman
Brothers. Recently, with the stabilization of credit markets we again see fairly precise
calibration results (ie, a relative mispricing in the 24% range) for all tranches and
for both indexes and both maturities.
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Credit models and the crisis: default cluster dynamics and the GPL model 67
FIGURE 4 Relative mispricing resulting from the GPL calibration: DJiTraxx.
20
15
10
5
0
5
10
15
20
03/2005 01/2006 11/2006 09/2007 08/2008 06/2009
03%36%
69%912%
1222%22100%
Index
%
(a)
20
15
10
5
0
5
10
15
20
03/2005 01/2006 11/2006 09/2007 08/2008 06/2009
%
03%36%
69%912%
1222%22100%
Index
(b)
To highlight where the problems in calibration come from, we have grouped mis-
pricings across three different categories.
Instrument: we have grouped all tranches, independent of maturity and seniority, into
one group and we have put the remaining calibrated instruments, ie, the 5-year and10-year indexes, in the other group.
Maturity: we have grouped all tranches and indexes in two groups according to their
maturity (5 and 10 years).
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68 D. Brigo et al
FIGURE 5 Relative mispricing resulting from the GPL calibration: CDX.
20
15
10
5
0
5
10
15
20
%
03/2005 01/2006 11/2006 09/2007 08/2008 06/2009
03%36%69%912%1222%22100%Index
(a)
20
15
10
0
5
10
15
20
%
03%36%
69%
912%1222%
22100%
Index
5
01/2006 11/2006 09/2007 08/2008 06/200903/2005
(b)
Seniority: we have grouped all tranches (leaving out the indexes) into three categories
depending on the seniority of the tranche in the capital structure.
1) Equity: equity tranche for both the DJiTraxx and CDX indexes.
2) Mezzanine: comprising the two most junior tranches after the equity tranche.
For the DJiTraxx index this means the 36% and 69% tranches, whereas
for the CDX this means the 37% and 710% tranches.
3) Senior: comprising the remaining most senior tranches. For the DJiTraxx
this means the 912%, 1222% and 22100% tranches, whereas for the CDX
this means the 1015%, 1530% and 30100% tranches.
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Credit models and the crisis: default cluster dynamics and the GPL model 69
FromFigure 6 onthe nextpage and Figure 7 onpage 71we notethat the GPL model
produced calibrations that resulted in a relative mispricing that was larger (though of
relatively small magnitude) in the period from June 2005 to October 2006. For both
the DJiTraxx and the CDX pools the mispricing could be ascribed to the 10-year
tranches, in particular to the equity and mezzanine tranches.
We also note that from October 2008 to June 2009 (ie, after the default of Lehman
Brothers), the GPL calibration again resulted in a non-zero but still fairly contained
relative mispricing that, in this case, could be ascribed to both the 5-year and 10-year
tranches independent of their positions in the capital structure (equity, mezzanine or
senior).
5 ADDING SINGLE-NAME AND CLUSTER DYNAMICS:
THE GENERALIZED-POISSON CLUSTER LOSS MODEL
The GPL model introduces a simple mechanism to avoid repeated defaults by cap-
ping the default counting process to the number of names in the pool. However, this
approach prevents us from extending the model to incorporate single names, since
we are limiting the number of defaults irrespective of which name is defaulting.
In Brigo et al (2007), to overcome this issue we introduced the GPCL model,
starting from the standard CPS framework with repeated defaults and using it as an
engine to build a new model for (correlated) single-name defaults, clusters defaults
as well as for the default counting process and the portfolio loss process. Two possible
strategies (aside from the GPL capping function) were introduced to deal with single
names.
We now review the GPCL strategy based on an adjustment at the single-name levelto avoid repeated defaults, leading to (correlated) single-name default processes. This
approach leads to a less clear cluster dynamics in terms of the original cluster repeated
default processes, but it shows interesting properties at the single-name level. We refer
the reader to Brigo et al (2007) for the strategy based directly on cluster dynamics.
5.1 Common Poisson shock basic framework
In the CPS framework we consider the observation that a single names default can
be originated by different events or factors. Occurrence of the event/factor number
e is modeled as a jump of independent Poisson processes Ne, e D 1 ; : : : ; m. Each
event can be triggered many times r D 1 ; 2 ; : : : as jumps go on. The r th jump ofNe
triggers a default event for name k with probability per;k
. Note that a defaulted name
k may default again. We address this limitation later.
The CPS framework may accommodate systemic and idiosyncratic factors in a
natural way. Consider, for instance, the following two cases:
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70 D. Brigo et al
FIGURE 6 DJiTraxx relative mispricing resulting from the GPL calibration grouped by
(a) maturity (5 and 10 years), (b) instrument type (index and tranches) and (c) seniority
(equity, mezzanine and senior).
12
10
8
6
4
2
003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009
10 year all5 year all
(a)
12
10
8
6
4
2
003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009
TrancheIndex
%
%
(b)
12
10
8
6
4
2
003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009
MezzanineEquity
%
14
(c)
Senior
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Credit models and the crisis: default cluster dynamics and the GPL model 71
FIGURE 7 CDX relative mispricing resulting from the GPL calibration grouped by (a)
maturity (5 and 10 years), (b) instrument type (index and tranches)and (c)seniority (equity,
mezzanine and senior)
12
10
8
6
4
2
0
03/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009
10 year all5 year all
(a)
12
10
8
6
4
2
003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009
TrancheIndex
%
%
(b)
12
10
8
6
4
2
003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009
MezzanineEquity
%
14
(c)
Senior
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72 D. Brigo et al
1) e is a totally systemic factor ifper;1 D per;2 D D p
er;n D 1;
2) e is a totally idiosyncratic factor if there exists a k such that, for all j k, we
obtain per;k D 1 and p
er;j D 0.
We define the dynamics for the single-name default process Nk, jumping each time
name k defaults, as:
Nk.t/ WD
mXeD1
Ne.t/XrD1
Ier;k
where Ier;k
is a Bernoulli variable with probabilityQfIer;k
D 1g D per;k
. Note that the
process Nk itself turns out to be Poisson. Furthermore, by differentiating the above
relationship, we obtain:
dNk.t/ D
m
XeD1 Ie1CNe.t/;k dN
e.t /
so that, if we consider two names k and h, we have that the respective default counting
processes Nk and Nh are not independent, since their dynamics is explained by the
same driving events Ne.t /.
Thecore of the CPS framework involves mapping thesingle-name default dynamics
(which consist of the dependent Poisson processes Nk) into a multiname dynamics
explained in terms of independent Poisson processes QNs , where s is a subset (or
cluster) of names of the pool, defined as follows:
QNs.t / D
m
XeD1Ne.t/
XrD1 Xs0s.1/js
0jjsj
Yk02s0Ier;k0
where jsj is the number of names in the cluster s. In a summation, s 3 k means
that we are adding up across all clusters s containing k, k 2 s means that we are
adding across all elements k of cluster s, jsj D j means that we are adding across all
clusters of size j , and, finally, s0 s means that we are adding up across all clusters
s0 containing cluster s as a subset.
Theproof of the independence of QNs for different subsets s canbe found in Lindskog
and McNeil (2003). Note that a jump in an QNs process means that all the names in
the subset s, and only those names, have defaulted at the jump time. We denote by Qsthe intensity of the Poisson process QNs.t /, and we assume it to be deterministic for
the time being. We refer to Brigo et al (2007) for model extensions.
5.2 Cluster processes as common Poisson shock building blocks
We do not need to remember the above construction. All that matters for the following
developments are the independent clusters default Poisson processes QNs.t /. These
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Credit models and the crisis: default cluster dynamics and the GPL model 73
can be taken as fundamental variables from which (correlated) single-name defaults
and default counting processes follow. The single-name dynamics can be derived
based on these independent QNs processes in the so-called fatal shock representation
of the CPS framework:
Nk.t/ DXs3k
QNs.t / or dNk.t / DXs3k
d QNs.t / (16)
where the second equation is the same as the first one but in instantaneous jump
form. We now introduce the process Zj.t/, which describes the occurrence of the
simultaneous default of any j names whenever it jumps (with jump size 1):
Zj.t/ WDXjsjDj
QNs.t/ (17)
Note that each Zj.t /, being the sum of independent Poisson processes, is itself Pois-
son. Furthermore, since the clusters corresponding to the different Z1; Z2; : : : ; ZMdo not intersect, the Zj.t / are independent Poisson processes.
The multiname dynamics, that is, the default counting process Zt for the whole
pool, can be easily derived by carefully adding up all the single-name contributions:
Zt WD
MXkD1
Nk.t / D
MXkD1
Xs3k
QNs.t/ D
MXkD1
MXjD1
Xs3k;jsjDj
QNs.t / D
MXjD1
jXjsjDj
QNs.t/
leadingto the relationship that links the setof dependentsingle-name default processes
Nk with the set of independent and Poisson distributed counting processes Zj:
MXkD1
Nk.t / DMX
jD1
jZj.t/ DW Zt (18)
Hence, the CPS framework offers us a way of consistently modeling the single-
name processes along with the pool-counting process, taking into account the cor-
relation structure of the pool, which remains specified within the definition of each
cluster process QNs . Note, however, that the Zt=M process is not, properly speaking,
the rescaled number of defaults NCt , since the former can increase without limit, while
the latter is bounded in the 0; 1 interval. We address this issue in the next section,
along with the issue of avoiding repeated single-name defaults.
5.3 The single-name adjusted approach
In order to avoid repeated defaults in single-name dynamics, we can introduce con-
straints on the single-name dynamics ensuring that each single name makes only one
default. Such constraints can be implemented by modifying Equation (16) in order to
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74 D. Brigo et al
allow for only one default. Given the same repeated cluster processes QNs as before,
we define the new single-name default processes N1k
replacing Nk as solutions of the
following modification of Equation (16) for the original Nk :
dN1k .t / WD .1 N1k .t
//Xs3k
d QNs.t/
DXs3k
d QNs.t /Ys3k
1f QNs.t/D0g (19)
Remark 5.1 (Interpretation) This equation amounts to saying that name k jumps
at a given time if some cluster s containing k jumps (ie, QNs jumps) and if no cluster
containing name k has ever jumped in the past.
We can compute the new cluster defaults QN1s consistent with the single names
N1k
as:
d QN1s .t/ D Y
j2sdN
1j .t/ Y
j2sc.1 dN
1j .t// (20)
where sc is the set of all names that do not belong in s.
Now we can use Equation (18), with the N1k
replacing the Nk , to calculate how the
new counting processes Z1j are to be defined in terms of the new single-name default
dynamics:
MXkD1
dN1k .t/ D
MXkD1
.1 N1k .t//
Xs3k
d QNs.t /
D
M
XkD1
.1 N1k .t//
M
XjD1
Xs3k;jsjDj
d QNs.t /
D
MXjD1
XjsjDj
d QNs.t /Xk2s
.1 N1k .t//
D
MXjD1
XjsjDj
d QNs.t /Xk2s
Ys03k
1f QNs0 .t/D0g
This expression should match:
dZ1.t / WD
Xjj dZ1j.t /
so that the counting processes are to be defined as:
dZ1j.t / WD1
j
XjsjDj
d QNs.t/Xk2s
Ys03k
1f QNs0 .t/D0g(21)
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The intensities of the above processes can be directly calculated in terms of the
density of the process compensator. By direct calculation, we obtain:
hN1k
.t/ D Ys3k
1f QNs.t/D0gXs3k
Qs.t/
hZ1j
.t/ D1
j
XjsjDj
Qs.t /Xk2s
Ys03k
1f QNs0 .t/D0g
9>>>>=>>>>;
(22)
where, in general, we denote by hX.t/ the compensator density of process Xat time t ,
referred to as intensity ofX, and where Qs is the intensity of the Poisson process QNs .
If we consider the repeated Poisson cluster default building blocks QNs to be
exogenously given, the model N1k
; QN1s ; Z1j is a consistent way of simulating the
single-name processes, the cluster processes and the pool-counting process from the
point of view of avoiding repeated defaults. In particular, we obtain:
NCt WD1
M
Xk
N1k .t / D1
MZ1t 6 1
Note, however, that the definition ofN1k
in (19), even if it avoids repeated defaults of
single names, is not consistent with the spirit of the original repeated cluster dynamics.
Consider the following example.
Example 5.2 (Single names versus clusters) Consider two clusters s D f1;2;3g
and z D f3;4;5;6g. Assume that no names defaulted up to time t except for cluster z,
in that in a single past instant preceding t names 3, 4, 5 and 6 (and only these names)
defaulted together (ie, QNz jumped at some past instant). Now suppose that, at time t ,
cluster s jumps, ie, names 1, 2 and 3 (and only these names) default, so QNs jumps for
the first time. Does name 2 default at t ?
According to our definition of N12 , the answer is yes, since no cluster containing
name 2 has ever defaulted in the past. However, we have to be careful in interpreting
what is happening at cluster level. Indeed, clusters z and s cannot both default since,
in this way, name 3 (in both clusters) would default twice. So we see that the actual
clusters default dynamics of this approach, implicit in Equation (20), does not have
a clear intuitive link with processes QNs . This is why, in Brigo et al (2007), we present
a second strategy to avoid repeated defaults that is also consistent at the cluster level.
5.4 Homogeneous pool limit
To simplify the parameters and combinatorial complexity, we may assume that the
cluster intensities Qs depend only on the cluster size jsj D j . Then it is possible to
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76 D. Brigo et al
directly calculate the intensity of the pool-counting process C D Z1 as:
hZ1.t/ D 1 Z
1t
M Xj j M
j ! Qjwhere Qj is the common intensity of clusters of size j .
We see that the pool-counting process intensity hZ1 is a linear function of the
counting process C D Z1 itself, as would be expected from general arguments for
a pool of independent names (again with homogeneous intensities). In such a pool,
a default of one name does not affect the intensity of default of other names, and
the pool intensity is the common homogeneous intensity multiplied by the number
of outstanding names. Each new default simply diminishes the pool intensity of one
common intensity value and the pool intensity is always proportional to the number
(fraction) of outstanding names .1 NC /.
5.5 Credit default swap calibration
In this section we tackle the single-name CDS calibration using the above strategy for
avoiding repeated defaults. In such a strategy, the compensator for the single default
event of name k is given by Equation (22). This leads to the following expression for
the default leg of a CDS on name k with deterministic recovery R paying protection
up to maturity T, under deterministic risk-free interest rates, and assuming that the
cluster intensities Qs are deterministic:
E0defleg.0/ D LGD
ZT0
P.0;u/E0dN1k .u/
D LGDZT0
P.0;u/E0hN1k
.u/ du
D LGD
ZT0
P.0;u/Ys3k
E01f QNs.u/D0gXs3k
Qs.u/ du
D LGD
ZT0
P.0;u/Ys3k
eQs.u/
Xs3k
Qs.u/ du
D LGD
ZT0
P.0;u/ exp
Xs3k
Qs.u/
Xs3k
Qs.u/ du
which is the same as the default leg in a standard deterministic intensity single-name
model for the CDS when the intensity of name k at time t is given by:
hCDSk .t / WDXs3k
Qs.t /
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In particular, if cluster intensities Qs are also constant in time, besides being deter-
ministic, and if we consider CDSs with continuous payments in the default leg, we
can see that single-name consistency with CDSs on name k having fair running spread
SkCDS.0;T/ for maturity T is achieved with:
Xs3k
Qs DSkCDS.0;T/
LGD
Imposing this (linear) constraint on the Q0s for all k values in the CDO calibration
ensures single-name consistency with CDSs. Of course, there is no need to assume
constant cluster intensities, so the method also works when one tries to fit the term
structure of both CDSs and CDOs. It is sufficient to go through the obvious general-
ization for piecewise constant cluster intensities.
This highlights a great advantage of this capping strategy for avoiding repeated
defaults: it makes single-name CDS calibration very easy to achieve.
5.6 Customizing the generalized Poisson cluster loss model to
deal with sectors
As was previously done for the GPL model, we can split the default counting process
and the loss process to factor out the Armageddon event in order to assign a zero
recovery to such a mode to allow for a better calibration of super-senior tranches.
We consider the rescaled pool-counting process NCt and the corresponding reduced
process Nc1
t , which can be defined as:
NCt WD1
M
MXjD1
jZ1j.t /
Nc1t WD1
M 1
M1XjD1
j dZ1j.t /
Starting from the counting process, we obtain:
d NCt D
1
M
MXjD1
j dZ1j.t /
D1
M
MXjD1
XjsjDj
d QNs.t/Xk2s
Ys03k
1f QNs0 .t/D0g
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78 D. Brigo et al
D 1f QN .t/D0g1
M
M1XjD1
XjsjDj
d QNs.t /Xk2s
Ys03k;s0
1f QNs0 .t/D0g
C 1f QN .t/D0g dQN .t/ 1
M
Xk2
Ys03k;s0
1f QNs0 .t/D0g
D 1f QN .t/D0g d Nc1t C 1f QN .t/D0g d
QN .t/.1 Nc1t/
and, by introducing the non-Armageddon indicator I1t , we obtain:
d NCt D I1t d Nc
1t C .1 Nc
1t/I
1t dI
1t ; I
1t WD 1fZ1
M.t/D0g D 1f QN .t/D0g
where is the set of all possible names.
For the GPL model we then introduce the stopping time O as the minimum time
between the Armageddon event time and the time when the reduced pool is over:
namely, the time when the full pool is over. Now we can integrate in order to obtain thecounting process by observing the following: firstly, I1 is zero after the Armageddon
event, otherwise it is equal to one; secondly, d I1 is non-zero only at the Armageddon
event; and finally, Nc is equal to one and d Nc is zero after the reduced pool is over.After
some algebra, we obtain:
NCt D 1fO6tg C Nc1t 1fO>t g
Note that NCt depends on the value of Nc1 only at time t and that the equation relating
the two counting processes has the same form as the corresponding Equation (13) of
the GPL model.
The same line of reasoning used for the GPL model may be repeated for the GPCLloss process as well by considering a zero recovery for the Armageddon event. We
obtain:
d NLt D .1 R/I1t d Nc
1t C .1 Nc
1t/I
1t dI
1t
and, by integrating:
NLt D .1 R Nc1O/1fO6tg C .1 R/ Nc
1t 1fO>t g
we obtain an equation relating the loss of the pool to the counting process of a reduced
GPCL model, which has the same form as the corresponding Equation (14) of the GPL
model, allowing for the same numerical techniques to calculate the loss distribution.
6 CONCLUSIONS AND FUTURE DEVELOPMENTS
In this paper we have considered CDOs. We analyzed their valuation (both pre-crisis
and in-crisis) using our pre-crisis GPL model, an arbitrage-free dynamic loss model
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Credit models and the crisis: default cluster dynamics and the GPL model 79
capable of calibrating all the tranches for all the maturities simultaneously. We have
also confirmed the CDO model-implied loss-distribution features that we had already
highlighted in Brigo et al (2006a): we found a multimodal loss probability distribution
that can also be obtained independently within the implied copula framework. This
behavior can be associated to the default of clusters of names within the CDO pool,
which, in turn, may be interpreted as default of sectors of the economy. We repe