Exchangeability is an extremely concept, since (most of the time) analytical expressions can be derived. But it can also be used to observe some unexpected behaviors, that we will discuss later on with a more general setting. For instance, in a old post, I discussed connexions between correlation and risk measures (using simulations to illustrate, but in the context of exchangeable risk, calculations can be performed more accurately). Consider again the standard credit risk problem, where the quantity of interest is the number of defaults in a portfolio. Consider an homogeneous portfolio of exchangeable risk. The quantity of interest is here

or perhaps the quantile function of the sum (since the Value-at-Risk is the standard risk measure). We have seen yesterday that – given the latent factor – (either the company defaults, or not), so that

i.e. we can derive the (unconditional) distribution of the sum

so that the probability function of the sum is, assuming that

Thus,  the following code can be used to calculate the quantile function

> 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
+ }
> QUANTILE=function(p=.99,a=2,m=.1,n=500){
+ V=rep(NA,n+1)
+ for(i in 0:n){
+ V[i+1]=proba(i,a,m,n)}
+ V=V/sum(V)
+ return(min(which(cumsum(V)>p))) }

Now observe that since variates are exchangeable, it is possible to calculate explicitly correlations of defaults. Here

i.e.

Thus, the correlation between two default indicators is then

Under the assumption that the latent factor is beta distributed

we get

Thus, as a function of the parameter of the beta distribution (we consider beta distributions with the same mean, i.e. the same margin distributions, so we have only one parameter left, with is simply the correlation of default indicators), it is possible to plot the quantile function,

> PICTURE=function(P){
+ A=seq(.01,2,by=.01)
+ VQ=matrix(NA,length(A),5)
+ for(i in 1:length(A)){
+ VQ[i,1]=QUANTILE(a=A[i],p=.9,m=P)
+ VQ[i,2]=QUANTILE(a=A[i],p=.95,m=P)
+ VQ[i,3]=QUANTILE(a=A[i],p=.975,m=P)
+ VQ[i,4]=QUANTILE(a=A[i],p=.99,m=P)
+ VQ[i,5]=QUANTILE(a=A[i],p=.995,m=P)
+ }
+ plot(A,VQ[,5],type="s",col="red",ylim=
+ c(0,max(VQ)),xlab="",ylab="")
+ lines(A,VQ[,4],type="s",col="blue")
+ lines(A,VQ[,3],type="s",col="black")
+ lines(A,VQ[,2],type="s",col="blue",lty=2)
+ lines(A,VQ[,1],type="s",col="red",lty=2)
+ lines(A,rep(500*P,length(A)),col="grey")
+ legend(3,max(VQ),c("quantile 99.5%","quantile 99%",
+ "quantile 97.5%","quantile 95%","quantile 90%","mean"),
+ col=c("red","blue","black",
+"blue","red","grey"),
+ lty=c(1,1,1,2,2,1),border=n)
+}

e.g. with a (marginal) default probability of 15%,

> PICTURE(.15)

On this graph, we observe that the stronger the correlation (the more on the left), the higher the quantile… Note that the same graph can be plotted with on the X-axis the correlation,

Which is quite intuitive, somehow. But if the marginal probability of default decreases, increasing the correlation might decrease the risk (i.e. the quantile function),

> PICTURE(.05)

(with the modified code to visualize the quantile as a function of the underlying default correlation) or even worse,

> PICTURE(.0075)

And it because all the more counterintuitive that the default probability decreases ! So in the case of a portfolio of non-very risky bond issuers (with high ratings), assuming a very strong correlation will lower risk based capital !