Tail risk of Bitcoin and major cryptocurrencies: a distributional and extreme-value analysis

We study the distributional and tail-risk properties of Bitcoin and the major cryptocurrencies using daily returns from June 2014 to May 2026. For a panel of 10 assets we fit a family of heavy-tailed distributions by maximum likelihood, rank them by the Akaike and Bayesian information criteria, and estimate the risk quantities that matter for management: Value-at-Risk, Expected Shortfall, and extreme-value tail indices. Returns are decisively non-Gaussian. Normality is rejected for every asset, excess kurtosis ranges from single digits to several dozen, and on the Bayesian information criterion the parsimonious Student-t and skewed generalized-t are preferred for almost every asset (4 and 5 of 10, respectively), while the heavily parameterised generalised hyperbolic, despite the highest raw likelihood, is selected for at most one; the lighter Akaike penalty moves more assets to the generalised hyperbolic (4 of 10) but never to the Gaussian. The Gaussian model understates the 99 % daily Value-at-Risk of Bitcoin by roughly a fifth relative to both the empirical and the extreme-value estimates, and Expected Shortfall is materially larger still. A provider-controlled refit on the 2014-2016 window changes the fitted degrees of freedom by a median of only 9 %, confirming that these results are not an artefact of the data source. We translate the statistics into concrete guidance for risk measurement, product design, and investor communication, and we set out a companion agenda on volatility dynamics and cross-asset dependence.