A Fitting Return to Fitting Returns: Cryptocurrency Distributions Revisited

This study fits 22 theoretical distribution functions, four of them originally derived, onto 772 cryptocurrency daily returns with goodness-of-fit evaluated using Cramer-von Mises, Anderson-Darling, Kuiper, Kolmogorov-Smirnov, and Chi-squared tests, as well as a harmonic mean p-value synthetic criterion. Most cryptocurrency return distributions can be sufficiently approximated with a Johnson SU function or an asymmetric power function. Johnson SU, asymmetric Student, and asymmetric Laplace distributions have better fit for larger cryptocurrencies, while error, generalised Cauchy, and Hampel (a Gaussian-Cauchy mixture) distributions are more characteristic of smaller cryptocurrencies, with larger coins demonstrating better overall fit. Less than 8% of sample coins and less than 4% of the top quartile by size do not fit into any of the investigated distributions, three largest “misbehaving” cryptocurrencies being Litecoin, Dogecoin, and Decred. Bitcoin and Ethereum are best modelled with error and asymmetric power law distributions, respectively, with asymmetric power law distributions stable through time. More than 30% of sample cryptocurrencies, and 26% from the top quartile, have infinite theoretical variance, severely limiting the diversification potential with such cryptoassets. Three most prominent infinite-variance coins are Bitcoin SV, Tezos, and ZCash. This study has substantial implications for risk management, portfolio management, and cryptocurrency derivative pricing.