The high-tech method of analyzing single-name risk in CDOs relies on Monte Carlo simulation. Rather than ratios and averages, this approach is based on default probability, default recovery, default correlation, and presents results in the form of probability distributions. This high-tech ideal may at first seem to be something of a straw man, proposed only to be criticized, because it seems unrealistic. However, a small army of CDO software engineers and vendors are currently vying to make this approach practical and, indeed, obligatory. So this chapter not only looks at two different ways to assess single-name risk, it also compares two radically different approaches to CDO analysis.
To implement this second approach, we need a default model of cred- its in the CDO portfolios, cash flow models of the CDO tranches, and aggregation of the results. In Exhibit 23.8 we walk through the process.
In the first row of Exhibit 23.8, the far right cell shows that the out- put of the default model in equally likely default scenarios. Each default scenario details whether credits did or did not default. If the credit defaulted in a particular scenario, the timing of its default and its default recovery are specified. The default model creates thousands of equally- likely default scenarios. We would need a default model that covers all of the credits in all of the CDOs in which there is an investment.
The inputs into the default model that creates these default scenar- ios are default probability, default recovery, and the default correlation between each pair of credits throughout all the CDO portfolios, as also shown in the first row, second column of Exhibit 23.8. The default model uses random draws based on these inputs to create the default scenarios.
c23-QuantSingleName Page 446 Monday, March 6, 2006 11:28 AM
447 EXHIBIT 23.8 Inputs and Outputs in a Monte Carlo Simulation InputsProcessOutput Default modelDefault probabilities and default recovery distributions for every credit in the CDO portfolios. Default correlation among each pair of credits in all the CDOs.
Random draws taking into account default probabilities and default correlations determine whether and when credits default. Ran- dom draws also determine how much credits recover in default.
Thousands of equally likely default scenarios. Each default scenario details whether each credit in the CDO portfolios defaulted, when it defaulted, and how much it recovered. CDO cash flow modelsThousands of default scenarios from the default model.CDO cash flow models generate collateral cash flows and CDO tranche cash flows according to the rules of each CDO’s cash flow waterfall.
In each of the thousands of default scenarios, net present value, e.g., of each CDO tranche. AggregationThe net present value of each CDO tranche in each of the thousands of default scenarios.
Aggregate the portfolio’s net present value across all tranches in each default scenario. Calculate the distribution of portfolio’s net present value across all the default scenarios.
Measures such as the mean and standard deviation of the portfo- lio’s net present value distribu- tion.
448 OTHER CDO TOPICS
EXHIBIT 23.9 Illustrative Distribution of CDO Portfolio NPV: With and Without Nextel
In the second row of the exhibit, each default scenario simulated by the default model is input into cash flow models of the investor’s CDO tranches. These CDO cash flow models translate defaults and recoveries of the names in its portfolio into collateral cash flow and, ultimately, into tranche cash flow according to the CDO’s cash flow waterfall. The sum of tranche cash flow can be quantified in a convenient measure, such as that tranche’s net present value. Each default scenario implies a net present value for each of the CDO tranches the investor owns.
In the third row of Exhibit 23.8, the net present values of each tranche in each default scenario are aggregated into the portfolio’s net present value. Each default scenario, then, ultimately leads to a related portfolio net present value, which is an equally likely occurrence. The average and the distribution of aggregate portfolio net present value over many simulations can be measured. To assess the sensitivity of a portfolio of tranches from different CDOs to the risk of, for example, Nextel defaulting, one would simulate aggregate portfolio net present values, assuming Nextel is default risky and default proof. The result of such a simulation is shown in Exhibit 23.9.
In Exhibit 23.9, the grey bars (“With Nextel”) show the distribution of aggregate portfolio NPV given the default model of collateral and the cash flow models of tranche net present values. The black bars (“With- out Nextel”) also show the distribution of aggregate portfolio NPV, but this time under the assumption that Nextel never defaults. Assuming
c23-QuantSingleName Page 448 Monday, March 6, 2006 11:28 AM
that Nextel never defaults naturally shifts the distribution of portfolio NPV a little to the right. The difference between the “with” and “with- out” distributions is based on all the Nextel-specific factors addressed in the modeling: the amount of Nextel in the different CDO portfolios, Nextel’s default probability and default recovery, and the default corre- lations between Nextel and every other name in all the CDOs. The dif- ference in the two distributions is also the result of non-Nextel factors:
the subordination and credit protections of each of the tranches the investor owns, the size of the investor’s investment in each particular tranche, and the maturity of the tranches.
One can summarize the distinction between the two distributions of Exhibit 23.9, with and without Nextel, by taking the difference of their two means and the difference of their two standard deviations. This directly measures the impact Nextel has on the investor’s CDO portfo- lio. In Exhibit 23.9, Nextel reduces the investor’s mean return by $1.1 million. Nextel’s contribution to risk can then be compared to the con- tribution of other single names calculated the same way. As with excess OC delta, before buying a new CDO, one would look to see how it affects the contributions of single-name concentrations already owned.
An investor would perhaps sell CDO tranches to reduce the single-name contribution of the largest risks.
We would expect, however, that an investor would want to know why a single name had its effect upon portfolio NPV. Accordingly, one would:
■ Look at the model inputs associated with the single name, its default probability, default severity, and default correlation.
■ Test the sensitivity of results to these assumptions by varying these inputs.
■ Look at the tranches in which he owns the particular single name, again to figure out why that name contributes to his portfolio’s risk.
■ Look at the tranche’s maturity, its remaining subordination, and per- haps even its excess OC delta with respect to the name.