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Mahalanobis meinte am 3. Mar, 15:02:
The Model
Assume every friend i earns a certain return Ri each year. When this return drops below a given cut-off level ci, he will default on the loan, i.e. the respective default probability is:

As we've said before, we want the return to be a function of the overall state of the economy and something that reflects idiosyncratic risk which is independent of the state of the economy.

The overall state of the economy is denoted by Y (= systemic factor) and the idiosyncratic shock by εi. The squared "beta", ρ which lies between 0 and 1, tells you the importance of the systemic factor relative to the idiosyncratic one.

Usually, we don't have any good empirical data to calibrate this model. So we make the following assumption:
  • Everybody is exposed to systemic risk to the same extent, i.e. the correlation coefficient ρ is the same for everybody. (A kind of average...)
  • Every individual defaults with the same probability pi = p and this unconditional default probability is known to us (i.e. the portfolio has to be somehow homogeneous, like "only BBB corporates")
  • Y and εi follow a standard normal distribution.
Given that Y and εi follow a standard normal distribution, we know that Ri must follow a standard normal distribution as well*. Given that Ri follows a standard normal distribution and we know the unconditional default probability, p, we can infer the cut-off value c which is just a quantile of the standard normal distribution. In case the unconditional default probability is 2%, the cut-off value is qnorm(0.02) = -2.05

And that's all we need! Now we can ask: Given that we know the unconditional default probability, p, and the correlation, ρ, how does the distribution of the portfolio default rate look like for a large portfolio (see blog post)?

Regulators are usually looking at this problem from a stress testing point of view. They ask: How many defaults would you have, in case Y turns out to be really bad, e.g. a 1 in a 100 years event (y = qnorm(0.01)= -2.33)?

What's so neat about the Vasicek model is that given Y, R is normally distributed with a variance that goes to zero as n goes to infinity! So given that you know Y and given your portfolio is very large, you can actually calculate your portfolio default rate given the unconditional default rate:

related items:
The Merton Model (Moody's KMV) : Video : David Harper, Bionic Turtle

Kreditrisikomessung, Henking et al., Springer, 2006
Kreditderivate und Kreditrisikomodelle, Martin et al., vieweg, 2006

* E(Ri) = 0, Var(Ri) = 1, and a linear combination of two independent standard normal distributed random follows a standard normal distributin as well. 



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