Expected Credit Loss not dependent on Correlation?

[rajeshtr]

Hi David,
I am not able to understand why the Expected Credit loss is not dependent on Default Correlation?

Eg., If the default events between A and B are correlated then..

E[A and B] = E[A] * E [ B ] + Correlation[A,B] * SD[A] * SD [ B ]

From this formula the Expected Credit Loss for 2 Bonds A and B should depend on the Probability of A and B occuring together --> which should inturn depend on Correlation between A and B.

Kindly Clarify?

 

[David Harper CFA FRM]

Hi @rajeshtr

But I think you mean E[A + B]? This can be tricky. Consider the difference between addition and multiplication:

• In the case of a portfolio's expected credit loss determined by summing individual credits, we are adding E[A] and E such that E[A+B] = E[A] + E. The test is absolutely correct that correlation does not impact "portfolio expected loss" where we typically are referring to the summation of individual credits
• However, in the case of EL = PD * LGD, that's a multiplication and because Cov[PD,LGD] = E[PD*LGD] - E[PD]*E[LGD], we have E[PD*LGD] = E[PD]*E[LGD] + Cov[PD,LGD]. A key relationship is cov(x,y) = E(x*y) - E(x)*E(y), which is zero if E(xy) = E(x)*E(y) and is our classic test of independence; i.e., if E(xy) = E(x)*E(y), the X and Y are independent and therefore covariance/correlation is zero.

The text is also correct that unexpected loss does vary with correlation. Our learning XLS demonstrates this nicely. But this should be intuitive:

• expected loss is a distributional mean,
• but unexpected loss is a standard deviation (variance) or multiple of standard deviation, so it exhibits sensitivity to correlation in a way (similar to) that of variance property var(x+y) = var(x) + var(y) + 2*cov(x,y) even as E[x+y] does not care about the cov(x,y). I hope that helps, i don't mind reviewing it myself but it's actually not easy. Thanks!