Jun 03

The problem with copulas

by David Harper, CFA, FRM, CIPM

FRM |

The Chains

Arturo Cifuentes illustrates a problem with copulas in Modeling Credit Risk in Fixed-Income Portfolios. A copula is a function that links marginal (unconditional) distributions to a joint distribution. They are useful and popular for credit portfolios: if a marginal (CDF) distribution M1(X1) characterizes the probability of default of an individual credit and M2(X2) characterizes the same for another credit, then J(X1,X2) = C[M1(x1),M2(x2)] is the joint CDF distribution that links the marginals by way of the copula C() function.

My favorite copula is the one I best understand, the profoundly simple independence copula: if u1 = probability that X1 defaults and u2 = probability that X2 defaults, under independence we know the joint probability of default = [X1 = default, X2 = default] = (u1)(u2). The independence copula is simply C(u1,u2) = (u1)(u2). This copula function is so simple we don't really need it; various copula functions can be selected that characterize various dependence structures among the credits.

Copula (asset) correlation doesn't equal default correlation

In the EditGrid below, I illustrated Mr. Cifuentes example of two correlated normal variables. This underlies the classic Gaussian copula transformation. The spreadsheet is simple. The only input is the copula (asset) correlation. Then three random standard normal variables are created (one trial per row). Then, based on the correlation, the independent random variables are transformed into two random variables that are correlated. To generate a standard random normal variable, you only need = NORMSINV(RAND()). Under normality here, default occurs when the random normal variable is less than some value; e.g., 5% PD matches = NORMSINV(5%) = -1.645.

Mr. Cifuentes makes the point that these are asset correlations not default correlations. That's fair enough, but he goes further and says the translation from the one to the other is non-trivial and asset class dependent (i.e., the function varies with PD). Did somebody say model risk?

His recommendation meets the simplicity test, and certainly you've got to agree, what is the point in tweaking the correlation matrix if the probability of defaults aren't rightly calibrated:

In my opinion, correlation should be ignored in credit risk models. To be more precise, the emphasis should be placed on estimating the default probabilities more accurately and then using the assumption that the correlation is zero…In other words, a much better way to assess the credit risk associated with a CDO portfolio is to make good estimates of the default probabilities of each asset in the portfolio on an individual basis. This approach is simpler and more direct and thus offers a better view of the true credit risk of a fixed income portfolio

…A second approach, for those not comfortable with eliminating correlation, is to throw away the correlation matrix and adopt a single number to describe the correlation of all the assets in the portfolio. Based on my experience, a single number very closely proxies the multiple numbers that populate a correlation matrix. And the single-number approach involves a lot less pain than generating a correlation matrix. Arturo Cifuentes, Modeling Credit Risk in Fixed-Income Portfolios - MD, R.W. Pressprich & Company

 

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