Tailoring the logistic distribution to fit the empirical distribution of financial asset returns

Risk measures, including Value-at-Risk (VaR) and Conditional VaR (Expected Shortfall), turn out to be quite sensitive to the degree to which distributions are thick tailed and asymmetric. Lack of encoding information about asymmetry, leptokurtosis and non-linear dependence is a well-known drawback of the Gaussian law. This, on the one hand, has led to a search for alternative distributions (Student t, Pearson type VII, normal inverse Gaussian, several stable distributions, see, e.g., Mills ibid., Rachev et al. 2010 ), and, on the other hand, has accelerated the development of approaches based on copula theory and related techniques (Nelsen1999, Szego 2004). In this paper, we will tackle the issue of accounting for asymmetry, (possibly severe) excess kurtosis and dependence by following the alternative approach of adjusting bell-shaped distributions using orthogonal polynomials as shape adapters.