Pricing CDOs with state-dependent stochastic recovery rates

Abstract

Up to the 2007 crisis, research within bottom-up CDO models mainly concentrated on the dependence between defaults. Since then, due to substantial increases in market prices of systemic credit risk protection, more attention has been paid to recovery rate assumptions. In this paper, we use stochastic orders theory to assess the impact of recovery on CDOs and show that, in a factor copula framework, a decrease of recovery rates leads to an increase of the expected loss on senior tranches, even though the expected loss on the portfolio is kept fixed. This result applies to a wide range of latent factor models and is not specific to the Gaussian copula model. We then suggest introducing stochastic recovery rates in such a way that the conditional on the factor expected loss (or, equivalently, the large portfolio approximation) is the same as in the recovery markdown case. However, granular portfolios behave differently. We show that a markdown is associated with riskier portfolios than when using the stochastic recovery rate framework. As a consequence, the expected loss on a senior tranche is larger in the former case, whatever the attachment point. We also deal with implementation and numerical issues related to the pricing of CDOs within the stochastic recovery rate framework. Due to differences across names regarding the conditional (on the factor) losses given default, the standard recursion approach becomes problematic. We suggest approximating the conditional on the factor loss distributions, through expansions around some base distribution. Finally, we show that the independence and comonotonic cases provide some easy to compute bounds on expected losses of senior or equity tranches.

Keywords:

• Applications to credit risk
• Copulas
• Correlation structures
• Credit derivatives
• Credit models
• Insurance mathematics

JEL Classification::

• G1
• G3
• G13
• G32

Acknowledgements

The authors thank the two referees, Xavier Burtschell, Laurent Carlier, Pierre Miralles, and Thierry Rehmann for numerous and helpful comments. Additional feedback from Fakher Ben Atig, Areski Cousin, Michel Crouhy, Ersnt Eberlein, Jean-David Fermanian, Steven Hutt, Benjamin Jacquard, Marek Musiela, Olivier Vigneron and participants at the Large Portfolio Concentration and Granularity conference and at the third Financial Risks International Forum on Risk Dependencies have also been welcome. Jean-Paul Laurent acknowledges support from the BNP Paribas Cardiff chair ‘Management de la Modélisation’. All errors are ours. The views expressed are the authors’ own and not necessarily those of BNP Paribas.

Notes

†Note that the notion of recovery rate in a CDO pricing context depends upon the precise definition of a default event and of the settlement procedures. This especially concerns the notion of the restructuring and auction mechanism. Thus, one should use historical data with caution, as emphasized by Guo et al. (2008) and Verde et al. (2009). Let us also stress that as far as CDO tranche pricing is involved, we need to consider the joint distribution of default times and recovery rates across all names, which also includes the cross-sectional dependence between recovery rates, which is not usually addressed in the econometrics literature. Eventually, one needs to consider risk-neutral recovery rates as in Pan and Singleton (2008).

‡We refer to Müller and Stoyan (2002) and Shaked and Shanthikumar (2007) for textbooks that survey the topic.

§For simplicity, we omit the dependence in t in the default probability P_i .

†We refer the reader to Li (2009) for a discussion of the differences between the two approaches. Just as the Gaussian correlation observed on equity tranches varies with maturity, which is compatible with the copula of the default indicator approach, but not the copula of default time, the stochastic recovery model proposed here aims to be compatible with the copula of default indicators only. The CDO price can be obtained as a linear combination of options on the portfolio loss maturing at different times t . When practitioners are asked to price a linear combination of options, on different underlyings, the option model corresponding to each underlying is used rather than trying to come up with a model consistent with all the underlyings at once.

‡Part of this result is obvious. If the recovery rate goes down, say from 40% to 15%, then all senior tranches with will have a zero premium with the 40% recovery assumption. With positive default probabilities, they obviously have a positive premium with the latter recovery rate assumption. The point that we make here is that this results remains true for all .

§We recall that, given two random variables , we say that X is smaller than Y with respect to the convex order, and we denote if for all convex functions f such that the expectations are well-defined. Convex order is a standard tool in actuarial studies and reliability theory. Since and are convex, implies that . If we think of X and Y as losses, they can be compared with respect to the convex order only if they share the same expectation. Moreover, since is convex, we readily have . It can be shown that is equivalent to and for all increasing and concave functions u . The latter condition means that X is less risky than Y with respect to second-order stochastic dominance, commonly used in microeconomics. When are Gaussian, that is equivalent to and .

¶See appendix A for details about comparing the two distribution functions.

†For any convex function , is directionally convex. For instance, is directionally convex, which we will use for the analysis of senior tranches.

‡Note that a directionally convex function is supermodular. As a consequence, , where stands for the supermodular order.

§Clearly the notion of conditional increase is law-invariant. Thus, we can compare distribution functions instead of the corresponding random vectors.

¶Stated slightly differently, while the expected loss is kept unchanged, all convex risk measures increase after a markdown.

†A well-managed trading book of CDS is likely to behave as a portfolio of long positions in premium legs of CDS, since it corresponds to the outcome of profitable CDS trades after hedging the default leg exposure. This is likely to change after the big bang CDS protocol since undoing a CDS trade will only result in an upfront premium.

‡Krekel (2008) is another example of a suitable stochastic recovery rate model for the pricing of CDO tranches. While our approach is associated with dichotomous individual losses, the Krekel model can be viewed as a multivariate polytomous item response probit model, using the statistical terminology, which extends the standard multivariate dichotomous item response probit model associated with the Gaussian copula and fixed recovery. The use of factor models in that framework can be traced back to Bock and Lieberman (1970). In the credit field, one may also note that the Krekel approach is quite similar to that used by Gupton et al. (1997) in Creditmetrics. The only difference is that the former considers different levels of default severity while the latter concentrate on pre-default quality, by looking at rating migrations.

§Since marginal default probabilities also remain unchanged, using the stochastic recovery rate model will have no effect on the value of a book of credit default swaps.

†Actually, we do not need call–put parity since is convex.

‡This obviously assumes that the markdown is done appropriately, i.e. expected losses are the same in the two models.

†This subsection does not aim at providing a full account of the relevant literature. It intends to show that existing expansion techniques can be well suited for the pricing of CDO tranches in our stochastic recovery framework.

‡Prampolini and Dinnis (2009) suggest some bucketing approach to deal with the curse of dimensionality. Assessing rigorously the discretization errors related to the choice of the loss unit is not a standard issue. For simplicity, we did not tolerate any approximation in the losses given default when computing the optimal loss unit in table 1 . Other numerical schemes do not rely on such approximations of the loss unit. The Fourier transform inversion method of Gregory and Laurent (2003) and the saddle point approximation scheme described by Martin et al. (2001) are among them.

§Note that matching higher moments is sometimes achieved at the expense of the positivity of the measure in these constructions. Also, increasing the number of matched moments does not necessarily lead to more accurate approximations. We refer to Kolassa (2006) for an extensive discussion of such techniques.

†It was noted by El Karoui et al. (2008) in the case of the Gaussian distribution.

†Note that V does not need to be scalar, although we do not need such an extension here.

‡See the proof of lemma 2.1 where the notion of ‘less dangerous’ is detailed.

§The conditional convex order implies that which was already stated and proven in subsection 4.2 through a direct computation.

†A real-valued function f defined on is said to be component-wise convex if it is convex in each argument when the others are held fixed.

‡Conditionally on V , the individual losses , , are independent. The same conditional independence result holds for the set of individual losses , .

§Note that the previous proof only applies when expected conditional losses are equal, which was not the case, for instance, in the recovery markdown case studied in section 2. However, we stress that the above comparison result between a markdown and the corresponding stochastic recovery rate model is not specific to the Gaussian copula case. To follow up the Clayton copula case and using the same notation as above, we have since . Due to conditional independence upon V , we also have Thus, portfolio losses when applying a recovery markdown and when using the stochastic recovery rate model are ordered the same way as in the Gaussian copula case.

†For notational simplicity the dependence of the loss given default upon ρ is not stated explicitly.

†If , , then and are defined by

†Note that all individual spreads have been matched to CDS quotes. However, there is some discrepancy between the Index spread and the average CDS spread of the names within the index. This basis effect is reported, for example, by Beinstein (2009). To cope with this, we used some multiplicative adjustment on individual credit spreads. This guarantees consistency of the individual credit spreads with the index quote. In the modelling field, Eckner (2007) and Herbertsson (2008) deal with similar issues.

†For the numerical results, which are displayed subsequently, we made the following choices. First, the scaling factor is taken equal to the average loss given default, , so that takes integer values when all are the same. Second, the shifting factor is chosen in order to prepare the rescaled loss distribution for a Poisson approximation

where denotes the floor function. Indeed, with this specification, satisfies , whereas for a true Poisson distribution, we would have equality of mean and variance. Finally, a last trick consists of approximating, with the above expansion, the ‘mirror’ of the portfolio loss instead of the loss itself, when , to minimize the probability of the approximated loss being greater than its theoretical maximum value.

†The conditional independence can be defined in slightly different ways. Here, we say that are conditionally independent of V if for all measurable functions such that the expectations are well-defined.

†If , , then and are defined by