{
  "slug": "beta",
  "title": "Single-benchmark return beta",
  "topic": "estimation",
  "summary": "Calculate a descriptive raw-return sensitivity to one aligned benchmark series.",
  "version": "1.0.0",
  "published": "2026-10-16",
  "status": "Educational reference",
  "formula": "beta = sample_cov(benchmark, asset) / sample_var(benchmark)",
  "inputs": [
    [
      "asset",
      "3-250 decimal returns",
      "Chronological raw asset returns"
    ],
    [
      "benchmark",
      "Same length, nonconstant decimal returns",
      "Aligned reference returns"
    ]
  ],
  "method": [
    "The default asset series is each benchmark return plus 0.005. Its slope is exactly one and its period intercept is 0.005. The intercept is a property of this invented relationship, not a demonstrated strategy premium.",
    "A beta label is incomplete without the benchmark, sample dates, frequency and return definition. This fixture contains no real calendar or instrument identity. A real study must preserve them outside the array and ensure that each pair describes the same interval."
  ],
  "limits": [
    "No risk-free series is deducted. Do not describe this raw intercept as CAPM alpha or interpret in-sample R-squared as investment quality. Nonlinear exposure, stale prices and changing factor sensitivities require a broader specification."
  ],
  "checks": [
    "Slope reconciled with covariance/variance and a known affine series.",
    "Zero benchmark variance and unequal lengths rejected."
  ],
  "related": [
    "beta-window-and-benchmark",
    "regression-fit-versus-forecast"
  ],
  "sources": [
    {
      "title": "SciPy: least-squares regression",
      "url": "https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.linregress.html"
    },
    {
      "title": "Simple Statistics: reference implementation and API",
      "url": "https://simple-statistics.github.io/"
    }
  ],
  "example": {
    "asset": [
      0.015,
      -0.005,
      0.025,
      0.005
    ],
    "benchmark": [
      0.01,
      -0.01,
      0.02,
      0
    ]
  },
  "frozen_result": {
    "schema_version": 1,
    "model_id": "beta",
    "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": {
      "asset": [
        0.015,
        -0.005,
        0.025,
        0.005
      ],
      "benchmark": [
        0.01,
        -0.01,
        0.02,
        0
      ]
    },
    "outputs": {
      "slope": 1,
      "intercept": 0.005,
      "r_squared": 1,
      "residual_sum_squares": 3.76158192263132e-36,
      "observations": 4
    },
    "notes": [
      "Descriptive same-period raw-return beta and intercept. No risk-free adjustment; intercept is not a causal alpha estimate."
    ]
  }
}