# Correlations, dimension, and risk measure

May 4, 2012
By

(This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers)

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 been working 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...

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