{
  "slug": "annualization",
  "name": "Annualize a Periodic Return / Volatility",
  "category": "Performance Math",
  "description": "Convert periodic returns and volatilities to annual figures using √T scaling and compound conversion.",
  "formula": "R_ann = (1 + R_period)^(periods/year) - 1\nσ_ann = σ_period × √(periods/year)",
  "formula_latex": "R_{\\text{ann}} = (1 + R_p)^{N} - 1,\\quad \\sigma_{\\text{ann}} = \\sigma_p \\sqrt{N}",
  "parameters": [
    {
      "name": "Periodic return",
      "symbol": "R_p",
      "type": "percent",
      "default": 0.012
    },
    {
      "name": "Periodic volatility",
      "symbol": "σ_p",
      "type": "percent",
      "default": 0.018
    },
    {
      "name": "Periods per year",
      "symbol": "N",
      "type": "integer",
      "default": 12
    }
  ],
  "outputs": [
    {
      "name": "Annual return",
      "symbol": "R_ann",
      "type": "percent"
    },
    {
      "name": "Annual volatility",
      "symbol": "σ_ann",
      "type": "percent"
    }
  ],
  "worked_example": {
    "narrative": "Monthly return 1.2%, monthly vol 1.8%.",
    "inputs": {
      "R_p": 0.012,
      "sigma_p": 0.018,
      "N": 12
    },
    "expected_outputs": {
      "R_ann": 0.1538,
      "sigma_ann": 0.0624
    }
  },
  "limitations": [
    "√T scaling assumes i.i.d. returns; serial correlation distorts annual vol"
  ],
  "related_terms": [
    "annualization",
    "cagr",
    "volatility"
  ]
}