Yesterday, while I was attending the IFM2 conference, at HEC Montreal, I heard a nice talk about credit risk, and a comparison between contagion (or at least default correlation), for corporate and retail companies (in the US). And it was mentioned that default correlation was much lower for retail companies than it could be for corporate risk. In a discussion that followed those slides, it was mentioned that banks in the US should actually have been working more with those small firms, since contagion risk was much lower.
A problem here is that the link between correlation, risk and dimension is rather complicated:
- corporate means a small number of firms, high correlation (and possible large individual losses)
- retail means a large number of firms (even perhaps extremely large), lower correlation (and small individual losses)
A simple model for default models is based on the assumption that we deal with an exchangeable portfolio (as in a previous post). With the following code, given an (individual) default probability, a default correlation, and a number of firms, it is possible to calculate the probability to have more than a given number of defaults.
proba=function(s,a,m,n){ b=a/m-a choose(n,s)*integrate(function(t){t^s*(1-t)^(n-s)* dbeta(t,a,b)},lower=0,upper=1,subdivisions=1000, stop.on.error = FALSE)$value} CDF=function(x=10,r=.4,m=.1,n=50){ a=m*(1-r)/r ; V=rep(NA,n+1) for(i in 0:n){ V[i+1]=proba(i,a,m,n)} V=V/sum(V); return(sum(V[1:(x+1)])) }
It is possible to calculate, for a large range of correlations, the probability to have – at least – 20% of default in the portfolio (in order to compare things that are comparable).
R=seq(.01,.99,by=.01) VQ=matrix(NA,length(A),2) for(i in 1:length(A)){ VQ[i,1]=1-CDF(r=A[i],x=4,n=20); VQ[i,2]=1-CDF(r=A[i],x=200,n=1000)}
With 20 firms (corporate) we want to have at least 4 defaults, while with 1000 firms (retail) there should be 200 defaults. As mentioned in the previous post, the relationship between correlation and quantiles of sums is not simple. Hence, it might not be monotone. The dotted line is the probability to have at least 4 defaults when default correlation is 50% (around 10%). The plain line is the probability to have at least 200 defaults, as a function of the correlation,
plot(A,1-VQ[,2],type="l",col="red",ylim=c(0,.22)) abline(h=1-VQ[50,1],lty=2,col="red")

In that case, with only a correlation of 10% among retail firms, the probability of having 20% defaults is the same as the same probability for corporate, but with 50% correlation… One should remember that in portfolio analysis, the links between correlation, dimension and risk measure is a sensitive issue…
is said to be exchangeable if for all 
for any permutation
of
. A standard example is the case where
, with
and
Since
, a necessary condition is that
i.e.
.
.
on
such that
where
. Note that
A nice proof of that result can be found in
of
,
that can be inverted in
The idea is then to extend the size of the vector
, i.e. for all
, define
so that, if we condition on
,
but since given the sum of components of
elements are equally likely, we can write
The first idea is to work on the blue term, and to invocate a theorem of approximation of the hypergeometric distribution
to a binomial distribution
, when
becomes large. Then
Let
and let
denote the cumulative distribution function of
.
The idea is then to write the sum as an integral, with respect to that distribution,
The theorem is then obtained since
, i.e.
In the case of non-binary sequences, there is an extension of the previous result,
.
on
such that
where
and
For instance, in the Gaussian case mentioned earlier, if
then
where
i.e. conditionally on
, the
. The proof can be found in
where
are i.i.d. random variables. To go further on exchangeability and related topics, see 
Since
if we assume that – given the latent factor –
(either the company defaults, or not),
i.e.
Thus, we can derive the (unconditional) distribution of the sum
i.e.
. In the upper corner is plotted the density of the Beta distribution.
of
random variables is exchangeable if and only if
‘s are conditionnally independent, conditionnally on some random variable
.

