TradingCalcEVT Tail Risk Calculator
RISK/STATS · EVT TAIL RISK · ADVANCED
Peaks-Over-Threshold Extreme Value Theory: fits a Generalized Pareto Distribution to the loss tail and extrapolates VaR/Expected Shortfall beyond it, without assuming a normal distribution.
Comma, space, or newline separated. No thousand separators (write 43250.50, not 43,250.50).
Fits a Generalized Pareto Distribution to losses beyond the threshold percentile via Grimshaw's profile-likelihood MLE, then extrapolates VaR/Expected Shortfall - no normal-distribution assumption.
Why not just use parametric VaR?
Parametric VaR/CVaR assumes returns are normally distributed. Crypto returns aren't: excess kurtosis of 12-16 is common for daily BTC/ETH returns, meaning extreme moves happen far more often than a normal distribution predicts. Extreme Value Theory sidesteps the assumption entirely by fitting a distribution only to the losses that actually happen beyond a high threshold, then extrapolating from that fit, not from a bell curve.
Peaks-Over-Threshold (POT)
Pick a high threshold u (this calculator: a percentile of the loss distribution). The Pickands-Balkema-de Haan theorem says the excesses above any sufficiently high threshold converge to a Generalized Pareto Distribution (GPD), regardless of the parent distribution:
- P(loss − u > y | loss > u) = (1 + ξy/β)−1/ξ
- ξ > 0: heavy/fat tail · ξ = 0: exponential tail · ξ < 0: bounded tail
Fitted here via Grimshaw's (1993) profile-likelihood MLE. VaR and Expected Shortfall beyond the threshold follow in closed form once (ξ, β) are known (McNeil, Frey & Embrechts, Quantitative Risk Management, ch. 7).
Reading ξ asymptotically normal
For a strongly negative shape parameter, the maximum-likelihood estimator loses its usual statistical guarantees: standard confidence-interval theory only applies when ξ > −0.5. Between −1 and −0.5 the fit is still a valid point estimate, just a less certain one. Values at or below −1 are excluded from the fit entirely - that region has a genuinely unbounded likelihood (Smith, 1985), not just a wide one.
Where to go next
Compare this against the normal-distribution assumption directly with the VaR / CVaR calculator, or see the full risk picture including drawdown ratios with the portfolio tearsheet.
Use via API or MCP
This calculation is available as a deterministic API call for bots and AI agents.
How is this different from the VaR / CVaR calculator?
The VaR/CVaR calculator assumes returns are normally distributed (plus an optional skew/kurtosis correction). This calculator makes no distributional assumption at all: it fits a Generalized Pareto Distribution only to the actual losses beyond a high threshold, the standard approach for fat-tailed series like crypto returns.
What is the threshold percentile?
The percentile of the loss distribution (losses = -returns) used as the cutoff u: only losses beyond this point are used to fit the tail model. A higher percentile (e.g. 95th) uses fewer, more extreme observations; a lower one (e.g. 85th) uses more data but risks including observations that aren't truly in the tail. 90th is a common default.
What do xi and beta mean?
xi (shape) controls how heavy the tail is: positive means heavier-than-exponential (fat tail, common for crypto), zero means exponential, negative means a bounded tail with a finite worst case. beta (scale) sets the magnitude of typical excesses beyond the threshold.
Why can Expected Shortfall be undefined?
When xi >= 1, the Generalized Pareto Distribution's mean doesn't exist (the tail is too heavy for a finite expectation), so Expected Shortfall has no closed form. VaR is still well-defined in that case; only ES is affected.
How much data do I need?
At least 100 return values, and enough that the chosen threshold percentile leaves at least 10 exceedances to fit the two-parameter GPD. Longer series and a lower threshold percentile both help.