Suppose one is offered the choice of selling credit protection via two dif- ferent 5-year credit derivatives. Both reference BBB rated credits, but one is based on exposure to a single credit while the other is based on tranched exposure to 100 credits. The two choices are:
■ Sell credit protection for five years on a single BBB rated corporate for 82 basis points via a single-name CDS.
■ Sell credit protection for five years on 100 underlying BBB corporates for 325 basis points and be responsible for default losses that occur on the 5% to 7% synthetic CDO tranche.1
Below we describe the available data to assess these two trades.
Historical Default Rates
What historical data do we have at our disposal to see how these two choices would have fared in the past? Exhibit 14.1 shows the results of studies from Moody’s,2 S&P,3 and Fitch4 with respect to the average 5- year historical default rate of BBB/Baa corporates.
1 All market prices herein were realistic at the time this chapter was written.
2 Exhibits 17 and 18 in David T. Hamilton, et al., Default and Recovery Rates of Corporate Issuers, 1920–2004, Moody’s Investors Service (January 2005).
3 Table 11 in Diane Vazza, Devi Aurora, and Ryan Schneck, Annual Global Corpo- rate Default Study: Corporate Defaults Poised to Rise in 2005, Standard and Poor’s (January 2005).
4 Charlotte L. Needham, Mariarosa Verde, and Stephanie K. Mah, Fitch Ratings Corporate Finance 2004 Transition and Default Study, Fitch Ratings (May 17, 2005), p. 8.
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EXHIBIT 14.1 Average BBB/Baa 5-Year Default Rates
Source: Fitch Ratings, Moody’s Investors Servic, and Standard & Poor’s.
The Fitch study averages results over one decade, the S&P study over two decades, and the Moody’s studies over three and eight decades, respectively. In the bottom row of the Exhibit 14.1, we calculated the average of these average 5-year BBB/Baa default rates as 3.05%.
These average default rates are not very useful for analyzing credit derivatives, especially CDO tranches. According to any of the default rates in Exhibit 14.1, default losses would never rise to a level that would affect a 5% to 7% CDO tranche, even if there were zero recover- ies and loss given default was therefore 100%.
For the CDO tranche, the fluctuation around the average default rate matters a lot. The wider the fluctuation, the more often the default rate will rise to a level where it affects the 5% to 7% CDO tranche.
Default rate fluctuation has two sources:
1. Default rates vary from time to time, even among credits with the same rating.
2. Default rates differ among portfolios created at the same time, again even if they contain credits with the same ratings.
Rating Cohort and Portfolio Defaults
Exhibit 14.2 shows how default rates have varied over time. The exhibit shows the 5-year default rate for BBB/Baa rating cohorts. A “rating cohort” comprises all corporates with the same senior unsecured rating at a particular time. BBB and Baa rating cohorts were formed at the beginning of every year from 1970 through 2000, and their defaults were tracked over the next five years.
For example, the 1.43% default rate for 1970 in the Moody’s time series indicates that of all corporates with senior unsecured ratings of Baa at January 1, 1970, 1.43% of them defaulted within five years up to January 1, 1975. At the opposite end of the exhibit, the 4.02% default rate at 2000 in the S&P time series indicates that of all corporates with senior unsecured ratings of BBB at January 1, 2000, 4.02% defaulted within five years up to January 1, 2005.
Fitch 1990–2004 3.45%
Moody’s 1920–2004 3.40%
Moody’s 1970–2004 2.08%
S&P 1981–2004 3.25%
Average of studies 3.05%
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EXHIBIT 14.2 Five-Year BBB/Baa Rating Cohort Default Rates
Source: Moody’s Investors Service and Standard & Poor’s.
EXHIBIT 14.3 Number of Defaults in 100-Name Portfolio, Default Probability = 1%
The 5-year default rates of Moody’s Baa rating cohorts have varied from 0.00% to 5.26%, while the S&P BBB rating cohorts have varied from 0.25% to 4.59%. This gives us perspective on the fluctuation of default rates over time. However, default rates also fluctuate among portfolios cre- ated at the same time because of the simple workings of chance.
When 100-name portfolios are created in an environment where there is, for example, a 1% default rate, each portfolio will not experience exactly one default. Using the binomial distribution, Exhibit 14.3 shows the probability
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that zero, one, two, or more defaults will occur out of a 100-name portfolio if the general population-wide default rate is 1%. As one would expect, the most likely possibility is one default out of 100 names. But there also are decreasing probabilities for two, three, four, and even more defaults.5 Default Timing
The time-fluctuation default rates shown in Exhibit 14.2 and the portfo- lio-fluctuating default rates in Exhibit 14.3 provide a lot more detail than the average default rates of Exhibit 14.1. With Exhibits 14.2 and 14.3, we can begin to imagine how the 5% to 7% CDO tranche could be touched by defaults. This could happen because the default rate of a particular rating cohort is high or because the default rate of a particu- lar portfolio of credits within a rating cohort is high.
However, this default data is still inadequate for our purposes. We need to know more about the timing of defaults within the 5-year period because this greatly influences the valuation of credit derivatives.
If defaults happen early in the life of a credit derivative, the seller of protection receives fewer or smaller premium payments and makes default loss payments sooner. That reduces the present value of the credit protection seller’s position relative what its present value would be if the same defaults occurred later in the life of the credit derivative.
Exhibits 14.4 and 14.5 show the timing of default rates by focusing on Baa marginal default rates using Moody’s default experience. “Mar- ginal default rates” are the default rates that occur within the first year, within the second year, within the third year, and so on. The exhibits show the wide variability of marginal default rates. First-year marginal default rates for Baa credits range from 0.00% to 1.33% and fifth-year marginal defaults range from 0.00% to 1.65%.
5 Those familiar with techniques for looking at default risk in credit portfolios will notice that we are doing things unusually. The common practice is to assume a single unvarying default rate and some level of positive default correlation (actually posi- tive correlation of changes in asset values) to create default rate fluctuations in a portfolio. But we vary default probability and accept the zero default correlation as- sumption of the binomial distribution. Regardless of how one feels about the domi- nant credit modeling approach, we have an advantage in performing this historical analysis, because we know the default rate for each rating cohort in each year from 1970 through 2000. Looking back in time at each rating cohort, we are not so much dealing with default probabilities as with default rates. Given that a particular rating cohort of hundreds of names experienced a particular default rate, the default rate experienced by a randomly drawn subset of 100 names from the rating cohort is an independent or uncorrelated process and the binomial distribution is appropriate.
We assess the assumption that synthetic CDO portfolios are randomly drawn from the credit population near the end of this chapter.
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EXHIBIT 14.4 Baa Marginal Default Rates
Source: Moody’s Investors Service.
1st 2nd 3rd 4th 5th
1970 0.27% 0.00% 0.00% 0.87% 0.30%
1971 0.00% 0.00% 0.80% 0.27% 0.00%
1972 0.00% 0.73% 0.25% 0.26% 0.27%
1973 0.46% 0.23% 0.48% 0.24% 0.52%
1974 0.00% 0.47% 0.25% 0.51% 0.53%
1975 0.00% 0.00% 0.26% 0.53% 0.00%
1976 0.00% 0.27% 0.29% 0.00% 0.31%
1977 0.28% 0.29% 0.00% 0.00% 0.00%
1978 0.00% 0.00% 0.00% 0.00% 1.32%
1979 0.00% 0.30% 0.00% 1.30% 0.34%
1980 0.00% 0.00% 0.96% 0.33% 0.34%
1981 0.00% 0.61% 1.27% 0.67% 1.03%
1982 0.31% 0.00% 1.01% 1.05% 0.37%
1983 0.00% 1.16% 0.41% 1.71% 0.47%
1984 0.36% 0.00% 0.41% 0.46% 0.51%
1985 0.00% 1.20% 0.00% 0.51% 1.08%
1986 1.33% 0.00% 1.68% 0.88% 1.38%
1987 0.00% 1.05% 0.37% 1.56% 1.65%
1988 0.00% 0.33% 0.70% 1.45% 1.16%
1989 0.59% 0.63% 0.65% 1.04% 0.00%
1990 0.00% 0.62% 0.00% 0.00% 0.00%
1991 0.27% 0.00% 0.00% 0.00% 0.00%
1992 0.00% 0.00% 0.00% 0.00% 0.00%
1993 0.00% 0.00% 0.25% 0.00% 0.00%
1994 0.00% 0.20% 0.00% 0.22% 0.22%
1995 0.00% 0.00% 0.00% 0.42% 0.87%
1996 0.00% 0.00% 0.17% 0.54% 0.56%
1997 0.00% 0.14% 0.60% 0.61% 1.13%
1998 0.12% 0.36% 0.50% 1.03% 1.21%
1999 0.10% 0.53% 0.78% 1.27% 0.37%
2000 0.38% 0.49% 1.35% 0.45% 0.24%
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EXHIBIT 14.5 Distribution of 5-Year Baa Defaults
Source: Moody’s Investors Service.
EXHIBIT 14.6 Senior Unsecured Loss Given Default
Source: Moody’s Investors Service and Standard & Poor’s.
Loss Given Default
Of course, default rate is only half of the credit loss equation. Exhibit 14.6 shows the results of rating agency studies of loss given default (default loss severity, or one minus recovery as a percent of par) for senior unsecured bonds.6 Over various periods and subject to different calculation methods, all of the studies come close to the 50.50% average.
Yet, just as with default rates, there is a great deal of variability in loss given default from year to year. This is illustrated in Exhibit 14.7, Moody’s Issuer Weighted 51.56%
Moody’s Par Weighted 49.70%
S&P Ultimate 50.25%
Average of studies 50.50%
6 The Moody’s data are from David T. Hamilton and Richard Cantor, Rating Tran- sitions and Defaults Conditional on Watchlist, Outlook and Rating History, Moody’s Investors Service (February 2004). The S&P data are from David Keisman,
“Recovery Trends and Analysis,” PowerPoint presentation, Standard and Poor’s (March 24, 2004).
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which shows average loss given default year-by-year.7 It is also impor- tant to understand that within any particular year, loss given default varies from one defaulted credit to another. On average, the standard deviation of loss given default among defaulted corporates within the same year has been 25%.
The Historical Credit Model
Assume in our original problem a $100 notional for both trades. We now have the ingredients for our historical exploration of these two credit derivative opportunities:
■ Five years of marginal default rates for Baa corporates beginning each year from 1970 through 2000, as shown in Exhibit 14.4.
■ Loss given default rates for each year 1970 through 2004, as shown in Exhibit 14.7, and the additional information that the standard devia- tion of loss given default within any year is around 25%.
For the 1970 Baa rating cohort, we used marginal default rates of 0.27%, 0.00%, 0.00%, 0.87%, and 0.30% as shown in Exhibit 14.4. We
7 Loss given default for 1982 through 2003 are actual senior unsecured bond prices one month after default. Loss given defaults from 1970 through 1981 were estimated via a regression using Moody’s all-corporate default rate. This regression was Loss given default = 41.7% + 5.8 × All corporate default rate and had an adjusted R- square of 26%.
EXHIBIT 14.7 Yearly Loss Given Default
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further divide each yearly marginal default rate into two, so that we can determine defaults in the first six months and last six months of each year.
Given the default rate, the binomial distribution determines how many defaults occur in each 6-month period. For each default, loss given default is determined first by looking at the average loss given default for that par- ticular year. These are 57%, 43%, 44%, 44%, and 43% for the years 1970 through 1974. Each particular loss given default is simulated using these average yearly loss given defaults and a 25% standard deviation and normal distribution. Premium payments and credit loss payments were tracked and cash flows were present valued at a 5% discount rate.
Exhibit 14.8 shows the results of our modeling. There is no contest.
The expected present value of the 5% to 7% CDO tranche across all 31 rat- ing cohorts from 1970 through 2000 is $14.04, while the expected present value of the single-name CDS is only $2.67. The $11.37 difference is caused by both the pricing of the two alternatives and by their expected losses. The CDO’s premium is 325 basis points, which creates a $14.22 present value if there are no default loss payouts. Meanwhile, the premium for the CDS is only 82 basis points, which creates a $3.59 present value if there are no default loss payouts. So right away, the CDO starts with a $10.63 advan- tage. Moreover, expected default loss payouts are less for the CDO than for the CDS: specifically, $0.18 for the CDO, and $0.92 for the CDS.
What if one simply puts more notional at risk in the CDS trade? In Exhibit 14.9, we show our model’s results after increasing the notional amount of the single-name CDS trade to $527. The average present value of each alternative becomes $14.04.
EXHIBIT 14.8 Present Values of Two BBB-Underlying Trades
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EXHIBIT 14.9 Present Value of Two BBB-Underlying Trades
Obviously, the extreme downside of the two trades is now very much different. It is only possible to lose $100 in the CDO trade, but it is theoretically possible to lose $527 in the CDS trade. More practically, the volatility of CDS trade becomes obvious. Year-by-year, the average present value of the CDO ranges from $11.94 to $14.22; meanwhile, the average present value of the CDS ranges from $5.76 to $18.90.
Moreover, within each rating cohort there is greater variability of returns in the CDS trade than in the CDO trade. In Exhibit 14.10 we also show the average present value of the two trades minus one stan- dard deviation.8 Under this stress, the $527 single-name CDS usually has a negative present value. Clearly, using historical defaults and recov- ery and the assumed pricing, the 5% to 7% CDO tranche is a much less risky than the single-name CDS.