Risk

Module: Shoals.Risk.

This module computes value-at-risk and conditional value-at-risk (expected shortfall) on a tensor of losses, both parametrically under a Gaussian assumption and empirically from the loss sample. The convention is that the input tensor holds losses, so a higher confidence selects a deeper point in the loss tail.

Parametric measures

def parametric_var[n](losses: tensor[n, f32], confidence: f32) -> f32
def parametric_cvar[n](losses: tensor[n, f32], confidence: f32) -> f32

parametric_var fits a Gaussian to the sample mean and standard deviation of the losses and returns mu + sigma * z, where z is the inverse normal CDF at the confidence level (drawn from Nautilus.Distributions). The standard deviation uses one degree of freedom. parametric_cvar returns the Gaussian conditional value-at-risk, mu + sigma * phi(z) / (1 - confidence), the expected loss conditional on exceeding the VaR level.

Historical measures

def historical_var[n](losses: tensor[n, f32], confidence: f32) -> f32
def historical_cvar[n](losses: tensor[n, f32], confidence: f32) -> f32
def empirical_loss_quantile[n](losses: tensor[n, f32], q: f32) -> f32

historical_var is the empirical quantile of the loss sample at the confidence level, computed by Nautilus.Stats.quantile_vec. historical_cvar is the mean of all losses at or above that quantile (the tail mean). empirical_loss_quantile is the raw empirical quantile at an arbitrary level q, the same primitive historical_var uses.

The empirical measures are exercised through the extended risk module, whose mc_var and mc_expected_shortfall delegate to historical_var and historical_cvar. On the integer losses 0..100, the 95% historical VaR is 95 and the 95% historical CVaR is the tail mean 97.5.