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Calculating CDOs with the Recursion Method

A typical CDS on CDO tranches structure is as follows:

• A number of entities, bonds or names, are collectively referred to as the pool. Their individual notional amounts add up to the entire pool's notional.
• After the inception of the pool some of these bonds or names might undergo changes in their credit quality - either as downgrades or even defaults.
• Every default event results in the reduction of the pool's notional.
• A tranche is a predetermined loss-range over the notional - and is described by its attachment (lower bound) and detachment (upper bound) points.
• A tranche (say, 3-9% tranche) will decrease in notional only after the entire pool's losses (via credit events leading to notional reductions) exceed 3%.
• A CDS on a CDO tranche is a standard CDS, where one is long or short the underlying loss-range, which in turn is contingent on the behavior of entities that make up the lower tranches.

In FINCAD XL v8.0, the default times of all entities were simulated via Monte Carlo methods; and thereafter the default times were sorted and cumulative losses were calculated. This method, while intuitive, is computationally resource consuming - particularly for bespoke CDOs.

To download the latest trial version of FINCAD Analytics, contact a FINCAD Representative.

In the subsequent release of FINCAD XL v9.0, FINCAD implemented a one-factor Gaussian Copula to model the correlation structure of the underlying entity defaults. A one-factor Gaussian Copula implies that the credit indices are conditionally independent for some fixed level. In the valuation of a CDO, the key is to calculate the loss distribution of the underlying pool. In order to calculate the loss distribution, the Fast Fourier Transform was used to determine the characteristic function of the loss distribution. While an improvement over the Monte Carlo, the performance of the FFT led valuation was not satisfactory especially when the underlying pool was not homogenous.

In the latest release of FINCAD XLv10.0, a more sophisticated counting mechanism (that counts the number of defaulted entities) is employed. The new method employed is called a grouped recursion method which iteratively calculates the probability that the reference pool has n defaults at time, t. Using convolution, conditional loss distribution is calculated. Further, in case of a heterogeneous subgroup - the underlying pool is approximated by homogenous subgroups. The improvements in computational efficiency by the grouped recursion method are displayed below:

A comparison of the two methods (FFT & Recursion) is provided below:

As one can see from the above, the recursion method is faster in the calculation. In tests that have been done, the outputs are the same or extremely close between the recursion and the Fast Fourier Transform.

References:

Luo L., (May 2006), CDOs and Search for Simplicity, Wilmott Magazine

Disclaimer

Your use of the information in this article is at your own risk. The information in this article is provided on an "as is" basis and without any representation, obligation, or warranty from FINCAD of any kind, whether express or implied. We hope that such information will assist you, but it should not be used or relied upon as a substitute for your own independent research.

For more information or a customized demonstration of the software, contact a FINCAD Representative.