Aggregation and Risk Measurement for a Portfolio of Exchangeable Bernoulli Risks

Aggregation and Risk Measurement for a Portfolio of Exchangeable Bernoulli Risks

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In this paper, we consider the computation of risk measures, such as the VaR and the TVaR, for a portfolio of n dependent losses assuming that the marginal distributions of the loss random variables are known but that the dependence structure is only known partially. We will consider a portfolio of exchangeable loss random variables for which the dependence relationship is defined through a common factor random variable (rv). We suppose the distribution of the common factor rv to be unknown while its first moments, such as the mean, the variance, and the skewness are assumed to be known. Based on the link between the joint distribution of the vector of n losses and the moments of the common factor rv, we propose an approach to derive upper and lower bounds on risk measures on the aggregate losses of the portfolio. Briefly, using stochastic ordering arguments, it is possible to derive first distributional lower and upper bounds on the distribution of the common factor rv. Then, we obtain lower and upper bounds on risk measures, such as the VaR and the TVaR of the aggregate losses of the portfolio. For example, assuming the probability of occurrence of a default and the covariance between the occurrences of two defaults in a portfolio of credit risks to be known, we are able to find the smallest and the largest value of the VaR and the TVaR of the aggregate losses of the portfolio. Numerical examples are provided to illustrate our proposed approach.

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