AGGREGATION OF DEPENDENT RISKS WITH MARGINALS IN JOHNSON SYSTEM AND GIVEN CORRELATION MATRIX

In modern risk management the notion of correlation is central. Essentially, correlation is used to measure dependence between risks assuming multivariate normally distributed returns, but the inclusion of non-derivative products invalidates many of the distributional assumptions underlying the use of correlation. Therefore, the need for measuring correlation and dependence in risk management must account both for the new sources of problems and for new technologies for solving arising problems. In this framework, it is important to construct multivariate distributions that are consistent with given marginal distributions and correlations, since this is a question that anyone wanting to simulate dependent random vectors is likely to encounter. Given fixed marginal distributions and a correlation matrix, it is possible to construct infinitely many joint distributions. Ideally, models for dependency structures should be easy to implement and require relatively few input parameters. As well, the choice of the dependency model and its parameter values should reflect the underlying correlation-generating mechanism.In this paper we present a multivariate distribution with marginals in the Johnson system and given correlation matrix that adequately interprets returns from a portfolio of assets when the assumption of normality is inadequate. We show the statistical characteristics of the distribution for risk analysis and introduce methods for estimating the parameters. The method is applied to two portfolios of Italian and English assets.