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When to use: The user wants to understand where returns come from:is it alpha or factor exposure? Any query about “factor analysis”, “decompose returns”, “where does alpha come from”, “factor exposure”, “Fama-French”, or “what’s driving the returns” should use this tool. What it returns: Alpha (daily and annualized) with t-statistic, factor loadings (market, size, value, momentum) with t-statistics, R-squared, residual volatility, and interpretation hints.

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Parameters

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equity_curve
list[object]
required
List of {date, value} objects representing portfolio value over time. Typically the full_equity_curve field from backtest_strategy. Requires 30+ data points.

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Example Input

{
  "equity_curve": [
    { "date": "2023-01-03", "value": 100000 },
    { "date": "2023-01-04", "value": 100500 },
    { "date": "2023-01-05", "value": 100234 }
  ]
}

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Example Response

{
  "alpha_daily": 0.000182,
  "alpha_annualized": 0.0469,
  "alpha_tstat": 1.87,
  "factors": {
    "Mkt-RF": { "loading": 0.8234, "tstat": 12.41 },
    "SMB":    { "loading": 0.2156, "tstat": 3.22 },
    "HML":    { "loading": 0.4891, "tstat": 5.67 },
    "Mom":    { "loading": -0.0812, "tstat": -1.03 }
  },
  "r_squared": 0.7823,
  "residual_vol": 0.1241,
  "interpretation": {
    "alpha": "Alpha is NOT statistically significant (t-stat=1.87, need |t| >= 2)",
    "factors": "Dominant factor: HML (loading=0.4891)",
    "r_squared": "High R-squared (0.7823):most return variance explained by known factors"
  }
}

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Interpreting the Output

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Alpha

For a single pre-specified strategy, |t| >= 2 (~95% confidence) is a reasonable significance threshold. If the strategy was discovered by testing many parameter combinations, a higher bar applies — Harvey, Liu & Zhu (2016) suggest t > 3.0 after multiple testing correction.
  • Significant alpha (|t| >= 2):The strategy generates returns beyond systematic factor exposure. The alpha cannot be easily replicated with factor ETFs.
  • Insignificant alpha (|t| < 2):Returns may be explained by factor exposure alone. Extending the backtest period may resolve this.

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Factor Loadings

FactorWhat it means
Mkt-RFMarket beta. Loading > 1 amplifies both gains and losses.
SMBSize factor. Positive = small-cap tilt. Negative = large-cap tilt.
HMLValue factor. Positive = value tilt (cheap stocks). Negative = growth tilt.
MomMomentum factor. Positive = momentum tilt (recent winners).
The magnitude of each loading indicates exposure strength. A loading of 0.5 on HML means 0.5 units of value factor exposure per unit of portfolio return.

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R-squared

  • High R-squared (> 0.7):Most return variance is explained by the four factors. The strategy is largely replicable with factor ETFs.
  • Low R-squared (< 0.4):Significant idiosyncratic return. The strategy is doing something the factors don’t capture.

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What to do next

High R-squared, low alpha

The strategy is a factor bet. Consider whether you can get equivalent exposure cheaper with factor ETFs (e.g. VLUE for HML, MTUM for momentum).

Significant alpha

The strategy has genuine edge. Consider increasing allocation, then validate with out-of-sample data or paper trading.

Compare strategies

Run factor analysis on multiple strategies to find the one with the most differentiated return source.

Extend the backtest

If alpha t-stat is close to 2 but not quite there, run a longer backtest to increase statistical power.