VaR and Expected Shortfall

ferro-risk ships three families of Value-at-Risk estimators. All three share the same trait surface so they can be swapped in a portfolio config without touching call sites.

Historical VaR

The empirical quantile of a P&L sample. Robust to fat tails because it makes no distributional assumption, but slow to react to regime changes because every observation is weighted equally.

Inputs

• A sorted slice of historical P&L observations.
• A confidence level $\alpha \in (0, 1)$, typically 0.99.

Output

The negative of the $\alpha$-quantile of the P&L distribution, returned as a positive number representing a loss.

Parametric VaR

Assumes Gaussian P&L with mean $\mu$ and standard deviation $\sigma$. The $\alpha$-VaR is then

$$\mathrm{VaR}_\alpha = -\mu - \sigma\,\Phi^{-1}(1 - \alpha)$$

Fast and analytically tractable, but understates tail risk on real return distributions.

Monte Carlo VaR

Simulate $N$ P&L paths from a calibrated model and take the empirical $\alpha$-quantile. Slower than parametric but lets you plug in arbitrary distributions, jumps, or copulas.

Expected Shortfall

For any of the three estimators, expected_shortfall(samples, alpha) returns the conditional mean loss beyond the VaR threshold — the coherent risk measure preferred by regulators.