Jensen's Alpha

CAPM Regression Model

Jensen's Alpha is derived from the single-factor CAPM regression:

where is the portfolio return, is the risk-free rate, is the market return, is Jensen's Alpha, is the portfolio's systematic risk, and is the error term with .

Alpha Extraction

Alpha can be expressed directly as the difference between the actual excess return and the CAPM-predicted excess return:

This expression makes clear that alpha is the portion of excess return not explained by the portfolio's exposure to the market risk premium. It captures the manager's stock selection skill, market timing ability, or any other source of return not attributable to passive market exposure.

OLS Estimation

In practice, alpha and beta are jointly estimated via Ordinary Least Squares (OLS). Setting and , the parameter vector is:

This yields the best linear unbiased estimates under the Gauss-Markov assumptions (linearity, exogeneity, homoscedasticity, no autocorrelation).

What FolioLab Computes

FolioLab fits the market model once per method, on daily excess returns over the dates the portfolio and the benchmark share. The risk-free rate of the run is converted to a daily rate, , and subtracted from both legs. The slope is the sample covariance over the sample variance, and the reported alpha is the daily intercept scaled by 252:

where and are the daily portfolio and benchmark excess returns. The annualisation is arithmetic, not compounded. The portfolio_alpha field stores the value as a fraction, so 0.02 is an alpha of 2% a year. Fewer than 3 aligned observations, or a benchmark variance at or below , gives an alpha and a beta of 0.0.

Statistical Significance: What FolioLab Tests

The test of skill divides alpha by its own standard error, and compares the result with a Student t-distribution on degrees of freedom. FolioLab runs this test, and the same regression gives the test on the slope:

The residual variance takes degrees of freedom, and the standard errors are the classical OLS forms:

The run reports every part of this regression:

• alpha_pvalue and beta_pvalue: the two-sided p-values of the alpha and the beta.
• alpha_stderr: the standard error of the alpha, multiplied by 252 so that it sits beside the annualised alpha. A t-statistic does not change with the scale, so the p-value is the same on either scale.
• beta_stderr: the standard error of the beta.
• residual_volatility: the standard deviation of the regression residual, annualised by .
• regression_observations: the count of daily observations in the fit.

The regression needs at least 5 observations and a non-degenerate benchmark series. When either condition fails, both p-values fall back to 1.0, both standard errors and the residual volatility to 0.0, and the observation count to 0. A standard error of zero with zero observations means that the regression did not run, not that the estimate is exact.

The standard errors assume independent residuals with a constant variance. Daily returns show volatility clustering, so read the p-values as approximate. A p-value above 0.05 on the alpha means that the regression cannot separate the alpha from zero at the 5% level. The results page draws the fit as a Security Market Line chart and states this when it applies.

Benchmark Basis

The portfolio returns include dividends, because the asset prices are adjusted for them. A price-return index omits the dividend income of its constituents, so the benchmark return it implies is too low. Against a price-return index, alpha is biased upward by approximately the index dividend yield. A Total Return Index reinvests those dividends, so both sides of the comparison include them.

• A retail run on an Indian benchmark measures against the price index unless it chooses the Total Return Index. Every Indian benchmark except BSE SENSEX has a Total Return Index from NSE Indices.
• A fund-manager run that names no benchmark measures against the NIFTY 500 Total Return Index. A fund-manager run that names a benchmark uses the Total Return Index of that benchmark where one exists, unless the run chooses the price index.
• The US benchmarks are total-return ETFs (SPY, QQQ and IWV), so they already include distributions.

Each run records the series it used in its benchmark_basis block. On a price-return basis that block lists portfolio_alpha among the biased metrics, and the results page marks the value. The beta is largely unaffected, because a small, steady drift in the benchmark barely changes its covariance with the portfolio. The benchmarks guide gives a measured example.