Dynamic selection of Gram–Charlier expansions with risk targets: an application to cryptocurrencies

This paper implements a procedure for dynamically selecting the Gram–Charlier approximation that best fits the empirical distribution of cryptocurrency returns at any point in time. The endogenous selection of the Gram–Charlier expansion length exploits its property for approximating frequency distributions through a flexible number of parameters that allows capturing changes at the tails provoked by new extreme events. The procedure is based on the differences between the cumulative distribution function of Gram–Charlier distributions with a particular focus on the fitting of the distribution left tail for risk assessment purposes. The method is tested through backtesting techniques for a group of major cryptocurrencies. The results show that the selection of the Gram–Charlier expansion order on the basis of cumulative distribution function dynamics, provides, in most cases, a significant improvement for conditional coverage compared to the use of fixed-order Gram–Charlier expansions. The method seems to be a useful tool for risk management purposes, especially for highly volatile assets such as cryptocurrencies.