{
  "slug": "information-ratio",
  "title": "Arithmetic information ratio",
  "topic": "validation",
  "summary": "Relate mean active return to sample active-return dispersion under one declared scaling convention.",
  "version": "1.0.0",
  "published": "2026-10-16",
  "status": "Educational reference",
  "formula": "IR_ann = mean(portfolio - benchmark) / sample_sd(portfolio - benchmark) * sqrt(K)",
  "inputs": [
    [
      "portfolio",
      "2-250 decimal returns",
      "Equal-frequency portfolio series"
    ],
    [
      "benchmark",
      "Same-length decimal returns",
      "Aligned benchmark series"
    ],
    [
      "periods_per_year",
      "Number, 1 to 366",
      "Explicit scaling factor"
    ]
  ],
  "method": [
    "The default mean active return is 0.00625 and the period tracking error is 0.0075. The period ratio is 0.8333333; multiplying by sqrt(12) gives approximately 2.8867513. A short invented example can produce a large ratio without supplying any evidence about future results.",
    "An additive shift to every active observation changes the mean but not dispersion. That is useful as an arithmetic property test, not a way to manufacture a performance conclusion. The benchmark and full sample must be chosen on economic and research grounds, not selected to maximize this output."
  ],
  "limits": [
    "A zero denominator returns null with an explanation. Annualizing a few observations does not create a year of evidence. The measure excludes estimation uncertainty, serial dependence adjustments, fees not already in the input and selection bias."
  ],
  "checks": [
    "Known arithmetic ratio and zero-dispersion withholding tested.",
    "Negative means remain signed; no clipping to flattering values."
  ],
  "related": [
    "information-ratio-denominator",
    "out-of-sample-and-multiple-testing"
  ],
  "sources": [
    {
      "title": "William F. Sharpe: differential-return ratios and time aggregation",
      "url": "https://web.stanford.edu/~wfsharpe/art/sr/sr.htm"
    },
    {
      "title": "Simple Statistics: reference implementation and API",
      "url": "https://simple-statistics.github.io/"
    }
  ],
  "example": {
    "portfolio": [
      0.02,
      -0.01,
      0.03,
      0.01
    ],
    "benchmark": [
      0.01,
      -0.02,
      0.02,
      0.015
    ],
    "periods_per_year": 12
  },
  "frozen_result": {
    "schema_version": 1,
    "model_id": "information-ratio",
    "model_version": "1.0.0",
    "engine_version": "1.0.0",
    "published_at": "2026-10-16",
    "scope": "Synthetic educational reference calculation. Not a fitted investment strategy, forecast, provider model or client return.",
    "inputs": {
      "portfolio": [
        0.02,
        -0.01,
        0.03,
        0.01
      ],
      "benchmark": [
        0.01,
        -0.02,
        0.02,
        0.015
      ],
      "periods_per_year": 12
    },
    "outputs": {
      "active_returns": [
        0.01,
        0.01,
        0.009999999999999998,
        -0.004999999999999999
      ],
      "mean_active_return": 0.00625,
      "sample_tracking_error": 0.0075,
      "annualized_tracking_error": 0.025980762113533156,
      "annualized_information_ratio": 2.8867513459481287
    },
    "notes": [
      "Arithmetic period differences with sample n-1 dispersion. Square-root annualization assumes stationary, uncorrelated active returns; it is not a compounded wealth difference."
    ]
  }
}