(p.77) Appendix B: Residual Gaussian Copula Correlation
(p.77) Appendix B: Residual Gaussian Copula Correlation
(p.77) Appendix B:
Residual Gaussian Copula Correlation
Here, we will define the notion of a copula, provide a brief critique of the manner in which copula models have been applied in the financial industry, and describe the empirical fit of a “residual copula model” that estimates the extent to which the default correlation in our public‐firm default data is not explained by the default intensities estimated in Chapter 4.
Consider default times τ _{1}, …, τ _{n} whose cumulative distribution functions (CDFs) F _{1},…,F _{n} are, for simplicity, assumed to be strictly increasing and continuous. The copula for τ _{1}, …, τ _{n} is defined as the joint probability distribution of the (uniformly distributed) random variables F _{1} (τ _{1}), …, F _{n} (τ _{n}). If there are joint Gaussian variables X _{1}, …, X _{n} with CDFs G _{1}, …, G _{n} such that C is also the joint probability distribution of G _{1}(X _{1}), …, G _{n}(X _{n}), then we say that the copula of τ _{1}, …, τ _{n} is Gaussian. A Gaussian copula, although not necessarily realistic in default‐time applications, is easy to apply. For example, the default times can be simulated with the copula‐implied joint distribution by first simulating X _{1}, …, X _{n} and then letting ${\tau}_{i}={F}_{i}^{1}\left({G}_{i}\left({X}_{i}\right)\right)$. In financial‐industry practice, it has been common to address portfolio default risk with a Gaussian copula. In fact, for many practical applications, it is assumed that the default times τ _{1},…,τ _{n} have the same CDFs, and moreover that the correlation of any pair of the underlying Gaussian variables is the same as the correlation of any other pair. The single correlation parameter is said to be the “fat Gaussian‐copula correlation.” This surprisingly simple copula is undoubtedly too simple for most default‐risk applications.
Putting aside the disadvantages associated with the lack of realism of commonly used copula models, the copula approach to modeling correlated default times is hampered by the lack of any tractable accompanying framework for computing conditional default risk given information about the borrowers that can arrive over time, such as the market prices of credit default swaps. There is no role in the definition of the copula for auxiliary new information, beyond the information revealed by the arrival of the defaults themselves.
Salmon (2009) provides a popular account of the problems created by the dependence of the financial industry on the Gaussian copula model of correlated default risk, leading up to the financial crisis of 2007–2009.
Schönbucher and Schubert (2001) suggested a model that allows one to augment the effect of correlation induced through doubly‐stochastic default intensity processes with additional correlation parameterized through a copula model. In order to gauge the degree to which default correlation in our data on U.S. public corporations is not captured by the default intensity processes estimated in Chapter 4, Das, Duffie, Kapadia, and Saita (2007) calibrated the intensity‐conditional copula model of Schönbucher and Schubert (2001) to these intensity processes and the associated observed default times. Specifically, they estimated the amount of copula correlation that must be added, after conditioning on the intensities, to match the upper‐quartile moments (p.78) of the empirical distribution of defaults per time bin. This measure of residual default correlation depends on the specific copula model. Here, we employ the industry‐ standard flat (single‐parameter) Gaussian copula. The resulting calibrated Gaussian copula correlation is a measure of the degree of correlation in default times that is not captured by co‐movement in default intensities. This “residual” Gaussian copula correlation is estimated by the following algorithm.

1. We fix a particular correlation parameter r and cumulative‐intensity bin size c.

2. For each name i and each bin number k, we calculate the increase in cumulative intensity ${C}_{i}^{c,k}$ for name i that occurs in this bin. (The intensity for this name stays at zero until name i appears, and the cumulative intensity stops growing after name i disappears, whether by default or otherwise.)

3. For each scenario j of 5,000 independent scenarios, we draw one of the bins, say k, at random (equally likely), and draw joint standard normal X _{1},…,X _{n} with corr(X _{i},X _{m}) = r whenever i and m differ.

4. For each i, we let U _{i} = G(X _{i}), the standard normal cumulative distribution function G(∙) evaluated at X _{i}, and draw “default” for name i in bin k if ${U}_{i}\u3009\mathrm{exp}\left({C}_{i}^{c,k}\right)$.

5. A correlation parameter r is “calibrated” to the data for bin size c, to the nearest 0.01, if the associated upper‐quartile mean across simulated samples best approximates the upper‐quartile mean of the actual data reported in Table 5.3.
The results are reported in Table B.1. As anticipated by the tests reported in Chapter 5, the calibrated residual Gaussian copula correlation r is non‐negative for all time bins, and ranges from 0.01 to 0.04. The largest estimate is for bin size 10; the smallest is for bin size 2.
Table B.1: Residual Gaussian copula correlation. Using a Gaussian copula for intensity‐conditional default times and equal pair wise correlation r for the underlying normal variables, Monte Carlo means are shown for the upper quartile of the empirical distribution of the number of defaults per bin. Set in boldface is the correlation parameter r at which the Monte Carlo estimated mean best approximates the empirical counterpart. (Under the null hypothesis of correctly measured intensities, the theoretical residual Gaussian copulation r is approximately zero.)
Bin size 
Mean of upper copula correlation quartile (data) 
Mean of simulated upper quartile copula correlation 


r = 0.00 
r = 0.01 
r = 0.02 
r = 0.03 
r = 0.04 

2 
4.00 
3.87 
4.01 
4.18 
4.28 
4.48 
4 
7.39 
6.42 
6.82 
7.15 
7.35 
7.61 
6 
9.96 
8.84 
9.30 
9.74 
10.13 
10.55 
8 
12.27 
11.05 
11.73 
12.29 
12.85 
13.37 
10 
16.08 
13.14 
14.01 
14.79 
15.38 
16.05 
Source: Das, Duffie, Kapadia, and Saita (2007).
(p.79) We can place these “residual” copula correlation estimates in perspective by referring to Akhavein, Kocagil, and Neugebauer (2005), who estimate an unconditional Gaussian copula correlation parameter of approximately 19.7% within sectors and 14.4% across sectors by calibrating with empirical default correlations (that is, before “removing,” as we do, the correlation associated with covariance in default intensities).^{1} Although only a rough comparison, this indicates that correlation of default intensities accounts for a large fraction, but not all of the default correlation.