Tag Archives: defaults

Correlations, dimension, and risk measure

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.

 stop.on.error =  FALSE)$value}

a=m*(1-r)/r ;
 for(i in 0:n){
 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).

for(i in 1:length(A)){

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,


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…

de Finetti’s theorem and exchangeability

This week, we will start to work on multivariate models, and non-independence. The first idea to discuss non-independence will be to use the concept ofexchangeability. A sequence of random variable http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif is said to be exchangeable if for all http://freakonometrics.blog.free.fr/public/perso5/exch-05.gif

http://freakonometrics.blog.free.fr/public/perso5/exch-01.giffor any permutation http://freakonometrics.blog.free.fr/public/perso5/exch-02.gif of http://freakonometrics.blog.free.fr/public/perso5/exch-03.gif. A standard example is the case wherehttp://freakonometrics.blog.free.fr/public/perso5/exch-07.gif, with


http://freakonometrics.blog.free.fr/public/perso5/exch-09.gifSince http://freakonometrics.blog.free.fr/public/perso5/exch-19.gif, a necessary condition is that


Since this inequality should hold for all http://freakonometrics.blog.free.fr/public/perso5/exch-05.gif it comes that necessarily http://freakonometrics.blog.free.fr/public/perso5/exch-13.gif.
de Finetti (1931): Let http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif be a sequence of random variables with values in http://freakonometrics.blog.free.fr/public/perso5/exch-14.gifhttp://freakonometrics.blog.free.fr/public/perso5/exch-06.gif is exchangeable if and only if there exists a distribution function http://freakonometrics.blog.free.fr/public/perso5/exch-15.gif on http://freakonometrics.blog.free.fr/public/perso5/exch-16.gif such that

http://freakonometrics.blog.free.fr/public/perso5/credit-04.gifwhere http://freakonometrics.blog.free.fr/public/perso5/exch-20.gif. Note that http://freakonometrics.blog.free.fr/public/perso5/exch-15.gif is the distribution function of random variable

http://freakonometrics.blog.free.fr/public/perso5/exch-22.gifA nice proof of that result can be found in Heath & Sudderth (1995) – see alsoSchervish (1995)Chow & Teicher (1997) or Durrett (2010) and also probably in several bayesian books because that result has a strong interpretation in bayesian inference (as far as I understood, see e.g. Jaynes (1982)).
From the exchangeability condition, for any permutation http://freakonometrics.blog.free.fr/public/perso5/defi02.gif of http://freakonometrics.blog.free.fr/public/perso5/defi03.gif,

http://freakonometrics.blog.free.fr/public/perso5/defi01b.gifthat can be inverted in

http://freakonometrics.blog.free.fr/public/perso5/defi05.gifThe idea is then to extend the size of the vector http://freakonometrics.blog.free.fr/public/perso5/defi09.gif, i.e. for all http://freakonometrics.blog.free.fr/public/perso5/defi07.gif, define

http://freakonometrics.blog.free.fr/public/perso5/defi10.gifso that, if we condition on http://freakonometrics.blog.free.fr/public/perso5/defi11.gif,

http://freakonometrics.blog.free.fr/public/perso5/defi08.gifbut since given the sum of components of http://freakonometrics.blog.free.fr/public/perso5/defi11.gif, all possible rearrangements of the ones among the http://freakonometrics.blog.free.fr/public/perso5/GPD11.gif elements are equally likely, we can write

http://freakonometrics.blog.free.fr/public/perso5/defi15.gifThe first idea is to work on the blue term, and to invocate a theorem of approximation of the hypergeometric distribution http://freakonometrics.blog.free.fr/public/perso5/defi17.gif to a binomial distribution http://freakonometrics.blog.free.fr/public/perso5/defi19.gif, when http://freakonometrics.blog.free.fr/public/perso5/defi50.gif becomes large. Then

http://freakonometrics.blog.free.fr/public/perso5/defi16.gifLet http://freakonometrics.blog.free.fr/public/perso5/defi20.gif and let http://freakonometrics.blog.free.fr/public/perso5/defi21.gif denote the cumulative distribution function of http://freakonometrics.blog.free.fr/public/perso5/defi28.gif.

http://freakonometrics.blog.free.fr/public/perso5/defi33.gifThe idea is then to write the sum as an integral, with respect to that distribution,

http://freakonometrics.blog.free.fr/public/perso5/defi30.gifThe theorem is then obtained since http://freakonometrics.blog.free.fr/public/perso5/defi31.gif, i.e.

http://freakonometrics.blog.free.fr/public/perso5/defi32.gifIn the case of non-binary sequences, there is an extension of the previous result,
Hewitt & Savage (1955): Let http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif be a sequence of random variables with values in http://freakonometrics.blog.free.fr/public/perso5/exch-24.gif.  http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif is exchangeable if and only if there exists a measure http://freakonometrics.blog.free.fr/public/perso5/exch-25.gif on http://freakonometrics.blog.free.fr/public/perso5/exch-26.gif such that

http://freakonometrics.blog.free.fr/public/perso5/exc99.gifwhere http://freakonometrics.blog.free.fr/public/perso5/exch-25.gif is the measure associated to the empirical measure


http://freakonometrics.blog.free.fr/public/perso5/exc98.gifFor instance, in the Gaussian case mentioned earlier, if



http://freakonometrics.blog.free.fr/public/perso5/exch-31.gifi.e. conditionally on http://freakonometrics.blog.free.fr/public/perso5/exch-32.gif, the http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif are conditionally independent, with distribution http://freakonometrics.blog.free.fr/public/perso5/exch-33.gif. The proof can be found in Kingman (1978) and is based on martingale arguments.
Note that in the Gaussian case, http://freakonometrics.blog.free.fr/public/perso5/excccc02.gif where http://freakonometrics.blog.free.fr/public/perso5/exccc03.gif are i.i.d. random variables. To go further on exchangeability and related topics, see Aldous (1985)  (see also here).
This construction can be used in credit risk, to model defaults in an homogeneous portfolio, see e.g. Frey (2001),


Assuming a Beta distribution for the latent factor, we can derive the probability distribution of the sum


http://freakonometrics.blog.free.fr/public/perso5/exch61.gifif we assume that – given the latent factor – http://freakonometrics.blog.free.fr/public/perso5/exch67.gif (either the company defaults, or not),


http://freakonometrics.blog.free.fr/public/perso5/exch63.gifThus, we can derive the (unconditional) distribution of the sum



> 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)$value
+ }

Based on that function, it is possible to plot the probability distribution over http://freakonometrics.blog.free.fr/public/perso5/credit-5.gif. In the upper corner is plotted the density of the Beta distribution.

> a=2
> m=.2
+ n=10
+ V=rep(NA,n+1)
+ for(i in 0:n){
+ V[i+1]=proba(i,a,m,n)}
> barplot(V,names.arg=0:10)


Those two theorems are extremely close,

De Finetti’s theorem: a random sequence http://freakonometrics.blog.free.fr/public/perso5/dfhs1.gif of http://freakonometrics.blog.free.fr/public/perso5/dfhs4.gif random variables is exchangeable if and only if http://freakonometrics.blog.free.fr/public/perso5/dfhs2.gif‘s are conditionnally independent, conditionnally on some random variable http://freakonometrics.blog.free.fr/public/perso5/dfhs3.gif.

Hewitt-Savage’s theorem: a random sequence http://freakonometrics.blog.free.fr/public/perso5/dfhs1.gif is exchangeable if and only if http://freakonometrics.blog.free.fr/public/perso5/dfhs2.gif‘s are conditionnally independent, conditionnally on some sigma-algebra http://freakonometrics.blog.free.fr/public/perso5/dfhs5.gif

Olshen (1974), proposed an interesting discussion about those theorems, see also in the Encyclopedia of Statistical Science,


The subtle difference between those two theorem is also discussed in Freedman (1965)