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Value at Risk (VaR)

Worst expected loss over a horizon at a confidence level.

A 1-day 95% VaR of $1M means there is a 5% chance of losing more than $1M tomorrow. The historical method reads it off the return distribution; the parametric method assumes normality. VaR ignores how bad the tail gets — see Expected Shortfall.

VaR_α = −quantile(returns, 1 − α)

import numpy as np
from scipy.stats import norm
# Historical 95% VaR
var_hist = -np.percentile(returns, 5)
# Parametric (normal) 95% VaR
var_param = -(returns.mean() + norm.ppf(0.05) * returns.std())