In my article, "Is One Rating Enough?," in the Summer 2005 issue of The Journal of Structured Finance, I emphasized the need for a second measure of risk for structured credit products that could be used in conjunction with existing agency ratings. I argued that it is not sufficient to quantify the probability of default as high or low, or the expected loss as X or Y (as current ratings do) because these measures fail to identify significant elements of risk. For example, if one were given the probability of default (a rating from S & P or Fitch, for instance), it would be natural to ask what the loss given default might be. Two bonds may have vastly different recovery profiles even though the probabilities of default (and ratings) are identical. Similarly, if one were given a measure of expected loss (a Moody's rating, for instance), it would be natural to ask what the variation around that expectation might be.
In fact, help may be on the way. Fitch Ratings recently issued an exposure draft to solicit feedback on a proposal to quantify expected recoveries given default for corporate and structured finance securities that Fitch rates single B or below. (1) This is a good first step, but it is just that; this approach should be extended up and down the rating spectrum, at least where structured credit products are concerned.
Furthermore, what are investors to do until rating agencies supply such ratings? In the February 15 paper, I suggested investors develop their own measures. These measures could be created using existing rating agency methodologies with very little additional computational effort. I still hold out that possibility for consideration, though its implementation requires a CDO model, which is often neither available nor expedient. Fortunately, there are alternative approaches, albeit less precise ones, and I present one here to estimate recovery rates for CDO tranches. It is the Two-Tranche Method (TTM), and it is most easily used with ratings from S & P or Fitch (which reflect the probability of default of a security) but an adaptation of this method can be used with Moody's ratings at a slight cost of complexity. Both are presented below.
THE TWO-TRANCHE METHOD EXPLAINED
The expected recovery rate can be estimated for an individual tranche by inspection of that tranche relative to the rest of the capital structure. Consider the expected recovery rates for the second- and third-priority tranches of a CDO with four tranches (see Exhibit 1, left side). What do we know about this CDO? If it is rated by S & P or Fitch, we can estimate the probability of default for each tranche by mapping the rating to a default number. Suppose that we have Fitch ratings and our transaction has a legal maturity of 10 years; then the probabilities of default for the "AAA" rated tranche, the "BBB" rated tranche, and the "BB" rated tranche are 0.19%, 3.74%, and 13.53%, respectively (Exhibit 1, right side).
Given these default probabilities, three scenarios for each tranche may be considered: 1) the possibility that the tranche will not default, 2) the possibility that the tranche defaults but the tranche above it does not default, and 3) the possibility that the tranche and the tranche above it both default, in which case the likely recovery for the tranche is zero or nearly zero (see Exhibit 2).
Case 1: The Selected Tranche Does Not Default
Though this case occurs quite frequently (96% of the time for BBB tranches and 86% of the time for BB tranches with 10 years until maturity), this case has no bearing on the expected recovery of a tranche given the fact that it does not default. So for the purpose of this analysis, it can be ignored.
Case 2: The Selected Tranche Defaults, but No Tranches Above the Selected Tranche Default
In this case, the selected tranche defaults but the tranche above the selected tranche does not default. Therefore, some recovery is expected and, if the distribution of recoveries is uniform, one would expect a 50% recovery ([R.sub.case 2]) on average for case 2 (see Exhibit 2). In general, however, one would expect an average recovery rate greater than 50% as the probability of "minor" defaults is larger than the probability of "large" defaults. In other words, the probability of incremental collateral loss decreases as losses surpass the amount of expected losses as illustrated in Exhibit 3. (2) That said, since the exact number is not known, I suggest investors adopt a conservative tranche recovery value of 50% for this case. (3) The probability of a case 2 event occurring is also simple to estimate: It is the probability of the tranche itself defaulting minus the probability of the tranche above it defaulting. This is 3.55% and 9.79% for the "BBB" rated tranche and the "BB" rated tranche, respectively, and I designate this probability as [P.sub.case 2].
Case 3: The Selected Tranche Defaults and the Tranche Above the Selected Tranche Defaults as Well
In case 3, where even highly rated tranches are impaired, the selected tranche will receive very little if any return of principal. (4) Thus, although rare, this case is clearly a significant event because the recovery rate for this case is likely zero. The probability of being "zeroed-out" is 0.19% for the "BBB" rated tranche and 3.74% for the "BB" rated tranche. For each tranche, I designate the probability of case 3 occurring as [P.sub.case 3] and the recovery for case 3, [R.sub.case 3] (which is equal to zero).
Putting it All Together
The recovery of each tranche may now be estimated as:
E[Recovery] = [[P.sub.case 3] [R.sub.case 3] + [P.sub.case 2] [R.sub.case 2]]/[[P.sub.case 3] + [P.sub.case 2]] (1)
or, because [R.sub.case 3] = 0, more simply,
E[Recovery] = [[P.sub.case 2] [R.sub.case 2]]/[[P.sub.case 3] + [P.sub.case 2]] (2)
Exhibit 4 summarizes our findings for the example. Estimated recovery rates for the "BBB" rated tranche and "BB" rated tranche are 46% and 33%, respectively. However, one should not conclude from this example that recovery rates for senior tranches will always be more than recovery rates for junior tranches. In fact, the recovery rate of a tranche is highly dependent on the likelihood of the next senior tranche defaulting, which is a function of the tranche's thickness. Had we chosen a different example--one where the Class B tranche was relatively thin--the estimated recovery rates could easily have been reversed.
Senior-Most Tranches Should Be Treated Separately
Of course, the Two-Tranche Method for estimating recovery rates is less applicable to the senior-most tranche in a CDO (generally the class A, "AAA" rated tranche); there are no tranches above the senior-most tranche by definition (a required input). In addition, the class A tranche is generally so large relative to the rest of the capital structure that the 50% recovery assumption used for case 2 is simply too arduous. In fact, one would expect rather high recovery rates (more than 90%) for the senior-most tranches based on Monte Carlo simulation analysis. Thus, 90% is a good, but conservative estimate of recovery for senior most tranches. Standard & Poor's and Moody's have published empirical studies that confirm this result. (5)
The Two-Tranche Method for estimating recovery rates can be modified for use with Moody's ratings as well. The chief difficulty is that a Moody's rating is generally interpreted as a measure of expected loss, not a probability of default. However, one can rewrite Equation (2) in terms of the expected loss as follows.
Initially, all of the variables on the right-hand side of Equation (3) are known only for the senior-most tranche ([R.sub.case 2] is assumed to be near 90%, E[L] is deduced from the Moody's rating, and [P.sub.case 3] = 0) and for this special case, the expected recovery rate reduces to the expected recovery rate for case 2, or 90%.
For the other tranches, all variables are known except for the probability of the senior tranche defaulting ([P.sub.case 3]). However, [P.sub.case 3] can be calculated from the estimated recovery rate and expected loss for the tranche immediately senior. Therefore, we can "bootstrap" the Two-Tranche Method if we assume a recovery rate for the senior-most tranche and use that to estimate a default probability, which can then be used to calculate the recovery of the tranche immediately junior. This process can then be repeated for each tranche in the capital structure.
Example
For the "BBB" rated tranche of our CDO with four tranches (see Exhibit 6, left side) and a legal maturity of 10 years, we find from Moody's that the expected loss is 1.98% (see Exhibit 6, center). This expected loss value, coupled with the newly estimated probability of default for the tranche A (0.0055%/(1%-90%) = 0.055%), can then be used to estimate the expected recovery rate for the "BBB" tranche using formula (3):
Readers will note that these estimates of recovery are similar to the recovery values obtained using the Fitch ratings (47% and 36%).
CONCLUSION
Investors need a second measure of risk for structured credit products that can be used in conjunction with existing rating agency ratings. I have argued previously that it is not sufficient simply to quantify the probability of default as high or low. It is important to know what the loss given default might be as well. The Two-Tranche Method enables investors to estimate quickly the expected recovery of a tranche based on the CDO's capital structure. The method is most easily used with ratings from S & P or Fitch (which reflect the probability of default of a security) but an adaptation of the method can be used with Moody's ratings at a slight cost of complexity.
