Deconstructing the Gaussian copula, part II and a half

August 11, 2009 in Finance,Math,Risk

(Parts I, II and III of this series are also available.)

One of the comments to part II of this series asked about recovery assumptions, observing that a 40% static recovery assumption would cause a 60-100% tranche to always be worth par, since it would never take damages. That's correct and raises an issue I will dance around in part three: recoveries constitute another unobserved input to the copula model.

When issuers default, they do not drop completely out of the picture - there is a some recovery level assumed, on the premise that in bankruptcy there will be enough assets to cover some portion of the debt owed. It could be anywhere from 0% to 100%. If the issuer recovers 100%, the CDO takes no damage (as there is no need to pay out insurance).

The most simple way to address recovery assumptions is to take them as exogenous static inputs. They don't necessarily have to be 40% (widely accepted as the long-run average recovery rate). Recovery assumptions could be based on an analyst's estimate, implied from CDS, taken from a market quote (recoveries are traded quantities), spit out of a random number generator, etc. It doesn't matter in an exogenously-specified static model.

The second way of dealing with them is to model them as stochastic quantities, which may or may not be correlated with default intensities. A simple (and hardly stochastic) method would be to lower recoveries as more issuers default, on the assumption that in such a scenario the business environment must be very bad. This would have the effect of compounding losses - not only are issuers defaulting, but the surviving issuer recoveries are falling over time. Alternatively, you could get very fancy and let the recoveries dance as random variables - incorporating a gamma process, for example. However, that's a little beyond the scope of my series (for the moment).

There is generally enough information about recoveries available through research and market information that an exogenous model provides satisfactory results. One question we must ask is whether it is "bad" that 40% recoveries would cause a senior tranche to never take damages. Honestly, I think it isn't because that's a scenario that could easily be reflected in reality: if it really happened, then the tranche wouldn't take damages. It might trade down, however, as people fear recoveries would drop - and that is where the model specification comes into play. It is the role of the model caretaker, not the model itself, to make sure its input parameters and dynamics are reflective of reality.

There's nothing wrong with creating a risk free bond - and to take this example to extreme, the 99-100% tranche is most likely risk-free under any recovery assumptions. The only problem is to expect to get paid for holding it. Moreover, even a 60-100% tranche with 40% recoveries could trade under par depending on interest rates and (in some cases) prepayment probabilities. But that's another chapter.

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