On the distribution of the sum of cograduated discrete random variables with applications to credit risk analysis

This paper focuses on the notion of cograduation which was first introduced in 1939 by the Italian statistician Tommaso Salvemini. In few words, a certain number of random variables are cograduated if they are associated with the maximum positive dependence. Here, it is shown how to derive the probability distribution of the sum of cograduated discrete random variables. This theoretical result is applied to the CreditMetrics and CreditRisk+ models in order to study the probability distribution of the value/loss of a credits portfolio under the assumption of cograduation of single credits. These applications are particularly meaningful since cograduation represents the ‘‘worst scenario’’ which is useful in order to obtain a prudential evaluation of the risk implicit in the credits portfolio.