Fitting a Pareto-Normal-Pareto distribution to the residuals of financial data

The Pareto-Normal-Pareto (PNP) distribution assumes that, for log returns of financial series, the innovations are normally distributed between two threshold values with Pareto tails below and above the respective thresholds. These threshold values can be estimated by maximum likelihood estimation (MLE). Monte Carlo simulations of normal, as well as heavy tailed error distributions, are used to compare the methods using this distribution with other methods to calculate Value-at-Risk (VaR) and Expected Shortfall (ESf). It is also applied to South African stock exchange data.