Credit Portfolio Risk
Overview
correlation
Credit portfolio risk is a subset of
portfolio risk
that focuses on the uncertainty in portfolio returns due to losses from
loan default. While most the principles of risk share some characteristics, there are unique
aspects of credit default that require specialized tools. In many assets, such as stocks or commidities,
one can model the asset value using standard Gaussian tools, credit loass cannot easily be measured this way.
Portfolio Variance
For financial institutions, correlations between defaults is a critical parameter to measure. The reason for this is simple. The
variance of a sum of independent (uncorrelated) variables goes to zero as the number of variables goes to infinity. This means that in
theory, a bank could diversify away its default risk by having a large enough portfolio, in which case the default loss for any time period
would be roughly equal the expected loss.
If you assume that EAD and LGD are fixed (not random) you can derive a formula for the Unexpected Loss of portfolio as a function
of the standard stats above and the loan default correlations {% \rho_{ij} %}.
{% UL^2 = \sum_{i,j} EAD_i \times EAD_j \times LGD_i \times LGD_j \times \sqrt{PD_i(1-PD_i)PD_j(1-PD_j)} \rho_{ij} %}
(
Bluhm pg. 25)
The manner in which a model accounts for correlations is generally model specific.
Measuring Default Correlations
Portfolio Variance Sensitivity
Using the above formula for portfolio variance, we can do a rough back of the envelope calculation for the effect of
chnages to default correlation and total portfolio variance. First we assume a constant correlation among assets,
which we label {% \rho %}.
{% UL^2 =\rho \times \sum_{i,j} EAD_i \times EAD_j \times LGD_i \times LGD_j \times \sqrt{PD_i(1-PD_i)PD_j(1-PD_j) } %}
Default Correlations and Variance
For financial institutions, correlations between defaults is a critical parameter to measure. The reason for this is simple. The
variance of a sum of independent (uncorrelated) variables goes to zero as the number of variables goes to infinity. This means that in
theory, a bank could diversify away its default risk by having a large enough portfolio, in which case the default loss for any time period
would be roughly equal the expected loss.
If you assume that EAD and LGD are fixed (not random) you can derive a formula for the Unexpected Loss of portfolio as a function
of the standard stats above and the loan default correlations {% \rho_{ij} %}.
{% UL^2 = \sum_{i,j} EAD_i \times EAD_j \times LGD_i \times LGD_j \times \sqrt{PD_i(1-PD_i)PD_j(1-PD_j)} \rho_{ij} %}
(
Bluhm pg. 25)
Portfolio Variance Monte Carlo
The above formula for portfolio variance is exact, given the assumptions. However, the assumptions are unrealistic.
In general, we would expect to see correlations among the LGD's as well as probability of default for instance.
Making the model more realistic by adding such things as additional correlations makes the math more
complex, however, this can be tackled in a simple way by using
monte carlo simulations.
