{
  "slug": "var-historical",
  "name": "Value at Risk — Historical Method",
  "category": "Risk Management",
  "description": "Maximum expected loss at a given confidence level over a horizon, computed from the empirical distribution of historical returns.",
  "formula": "VaR_α = -Quantile(returns, α)",
  "formula_latex": "\\text{VaR}_\\alpha = -\\text{Quantile}(R, \\alpha)",
  "parameters": [
    {
      "name": "Returns",
      "symbol": "R",
      "type": "vector",
      "description": "Historical daily return series",
      "default": []
    },
    {
      "name": "Confidence level",
      "symbol": "α",
      "type": "percent",
      "default": 0.95,
      "min": 0.5,
      "max": 0.999
    },
    {
      "name": "Holding period (days)",
      "symbol": "T",
      "type": "integer",
      "default": 1
    }
  ],
  "outputs": [
    {
      "name": "VaR",
      "symbol": "VaR",
      "type": "percent",
      "description": "Loss threshold (positive number) at confidence α"
    }
  ],
  "worked_example": {
    "narrative": "1000-day return series, 95% VaR at 1-day horizon.",
    "inputs": {
      "alpha": 0.95,
      "T": 1
    },
    "expected_outputs": {
      "var": 0.018
    }
  },
  "limitations": [
    "Backward-looking; misses regime shifts",
    "No information about losses beyond VaR (use CVaR/Expected Shortfall)",
    "Assumes future losses look like historical sample"
  ],
  "alternatives": [
    {
      "slug": "var-parametric",
      "when_to_prefer": "Returns approximately normal"
    },
    {
      "slug": "expected-shortfall",
      "when_to_prefer": "Need conditional tail expectation"
    }
  ],
  "related_terms": [
    "var",
    "expected-shortfall",
    "cvar",
    "tail-risk"
  ]
}