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This post explains the Vasicek/Merton single factor model which is part of the Basel framework (IRB approach) and has been used to evaluate CDOs.

Imagine you loan money to a friend who will default with a probability of 1%. When it comes to paying back the loan you will either receive 100% (plus any interest) or 0%. That's pretty risky. So you figure you can improve your situation by making same size loans to n friends. (NB: You will need a license for doing so). In case they all default individually with a probability of p = 1%, the law of large numbers tells you that the more loans you make (increase n), the closer the average default rate (= portfolio default rate) will be to 1%. "The" central limit theorem states that the portfolio default rate will be normally distributed with a mean of 1% and a variance that goes to zero as n increases.

Now here is the result for n = 5000 loans and 100,000 simulations (portfolios):
vasicek01
Unfortunately, this is only true when default solely happens due to idiosyncratic reason such as illness or a divorce, i.e. when the default of one friend is not related to the default of another friend. But in case you make loans to colleagues from work this assumption won't be correct since bankruptcy of the company would turn most of your loans sour at the same time no matter how large (number of obligors) your loan portfolio actually is. In other words: Certain systemic risk can't be diversified away.

Assuming you really have lots of "similar" friends (p = 1%) and they are all pretty evenly distributed across the sectors the economy has to offer one could argue that defaults are actually only a function of this idiosyncratic risk (as before) and a single systemic risk factor reflecting the overall state of the economy. In case the economy does very well, hardly anybody will default (even an expensive divorce is not an issue) and in case the economy enters into a deep recession, the default rate goes north. The sensitivity of each obligor to this systemic factor and the correlation among the obligors is given by √ρ and ρ, respectively.

Result for n = 5000 loans and 100,000 simulations (portfolios):
vasicek02vasicek03
The average portfolio default rate is not affected by an increase in correlation, but the higher the correlation the more likely extreme portfolio default rates become (good or bad). In case the correlation is one, we are back to a single obligor. Either nobody (0%) or everybody (100%) defaults.

The model (see comment section for details) is a useful starting point but since both systemic and idiosyncratic risk is assumed to be normally distributed and are connected via an uncertain correlation coefficient you could easily be on the wrong end of the trade. NB: That doesn't mean the Basel guys did a bad job. They had two parameters (correlation and systemic shock size) for calibrating the model.
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

Source:
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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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: buytherc.com 
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Mahalanobis meinte am 3. Mar, 18:07:
R Code
n <- 5000
t <- 100000
rho <- 0.3
p <- 0.01

port.def <- array(0,t)
for(i in 0:t)
{
y <- rnorm(1)
e <- rnorm(n)
r <- sqrt(rho)*y + sqrt(1-rho)*e
port.def[i] = length(r[r<qnorm(p)])/n
}

# Plot histogram
library(MASS)
truehist(port.def, xlim=c(0,0.1), xlab="portfolio default rate", main="Vasicek Distribution, p = 0.01 and rho = 0.3") 
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Unfortunately, this is only true when default solely happens due to idiosyncratic reason such as illness or a divorce, i.e. when the default of one friend is not related to the default of another friend.

This is why you quants will always need one or two streetwise neanderthals like me about, to remind you that there are all too many cases where the divorce of one set of friends is intimately related to the divorce of another set of friends! 
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