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 !
, a Generalized Pareto Distribution will fit nicely, then we can use it to derive an estimator of the quantile function (for percentages such that the quantile is larger than the threshold)
, i.e. we keep the
largest observations to fit a GPD, then this estimator can be written








, where
is a slowly varying function. Then, for all
,






. Then (under standard suitable conditions)







, is defined as



, so that
, where
is some slowly varying function. Equivalently, the exists a slowly varying function
such that
. Then
is the order statistic
, the slope of the straight line is the opposite of tail index
. The estimator of the slope is (considering only the
largest observations)
, as
), i.e.
, but not two fast, i.e.
(one can even get
with stronger convergence assumptions). Further

, then
. And similarly
as
with additional assumptions on the rate of convergence), and












is the total number of balls, and if
is the number of white
balls then
, or to be more precise, in 


admits a pair of solutions, then
. Further, the difference between
and
is precisely
. Thus, recursively, it is extremely simple to get all possible answers. Below, we have
and the difference between
, consider an urn with
balls. We draw two balls at the same time. It is equally likely that the two will be the same color as different colors. Then the number of colors within the bag are respectively


, for some
, if
. An this concept can be related to sums and maxima (see
: let
and
. Then it can be shown easily that
if and only if
for some
if and only if the exists a non-degenerate variable
such that
if and only if



, define
and
. Then analogous results can be derived,
if and only if
if and only if the exists a non-degenerate variable
such that









Via 











. For bounded distribution, consider e.g. the uniform distribution on the unit interval, i.e.
on the unit interval. Let
and
. Then, for all
and
,

. Let
and
, then

. Let
and
, then

sample. We can use the following approximation of the cumulative distribution function (based on l’Hopital’s rule)
. Let
and
. Then we can get
. I.e. the limiting distribution of the maximum of a Gaussian sample is Gumbel’s. But what we do not see here is that for a Gaussian sample, the convergence is extremely slow, i.e., with 100 observations, we are still far away from Gumbel distribution,

, if


















