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Derive a stock's alpha and beta by regressing its excess returns on those of the market

CFA® Level I · Quantitative Methods · Applications of Simple Linear Regression in Finance · QM.19.a

The idea

Regressing a stock's excess return on the market's excess return, Ri - Rf = a + b x (Rm - Rf) + e, estimates the inputs of the capital asset pricing model. The slope is beta, cov(stock, market) / var(market), which can also be written correlation x sd(stock) / sd(market); it measures systematic risk. The intercept is alpha, the average excess return not explained by market exposure, and because the fitted line passes through the means, a = mean stock excess return - b x mean market excess return. A positive, significant alpha suggests risk-adjusted outperformance. Jensen's alpha applies the same idea to realised returns: actual return minus the CAPM return using the estimated beta.

A practice question

From the question bank, with the reason each wrong option is wrong.

The covariance between a stock's excess returns and the market's excess returns is 0.0024. The variance of the market's excess returns is 0.0020 and of the stock's is 0.0036. The estimated beta is:

  1. 0.67Why not: Divided the covariance by the stock's variance instead of by the market's.
  2. 0.83Why not: Inverted the ratio, dividing the market variance by the covariance.
  3. 1.20 — correct

The regression slope is cov(stock, market) / var(market) = 0.0024 / 0.0020 = 1.20.

A harder one

A stock's mean monthly excess return is 1.1%, the market's is 0.8% and the estimated beta is 1.25. The estimated alpha is closest to:

  1. 0.30%Why not: Subtracted the market excess return directly, treating beta as 1.
  2. 0.10% — correct
  3. 0.46%Why not: Divided the market excess return by beta instead of multiplying by it.

The least-squares line passes through the means, so alpha = 1.1% - 1.25 x 0.8% = 0.10%.

The hardest tier

Over four months, the market's excess returns were 2%, -1%, 3% and 0%, and a stock's were 3%, -2%, 4% and 1%. The least-squares estimates of beta and alpha (per month) are closest to:

  1. beta 1.40, alpha 0.10% — correct
  2. beta 0.67, alpha 0.83%Why not: Divided the cross-products by the stock's squared deviations (14 / 21) instead of the market's, then used that slope for alpha.
  3. beta 1.40, alpha 0.50%Why not: Took alpha as the stock's mean less the market's mean, ignoring beta.

Means are 1% (market) and 1.5% (stock). Sum of cross-products = 14; sum of squared market deviations = 10. Beta = 14 / 10 = 1.40; alpha = 1.5 - 1.40 x 1 = 0.10%.

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