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Interpreting Alpha in the Carhart Four-Factor Model

Article Quant Q&A · Author: Kaiser_Soze23
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Summary

The document clarifies the intercept in a time-series regression of portfolio excess returns on the Carhart four-factor returns. The intercept, or estimated alpha, is the portion of the portfolio’s average excess return that remains unexplained by its covariance with the selected factors. It is therefore not the portfolio’s total return above the risk-free rate; factor exposures may explain much of that return.

The answer places this interpretation in the broader logic of factor models: investors may require compensation for bearing risks represented by factors, so returns associated with those exposures are not counted as alpha. A negative estimate means the portfolio’s mean return is below what the fitted factor model implies. The question mentions daily data and annualization, but the response does not resolve that frequency-specific issue or explain how to convert daily alpha to an annual figure. It also cautions that the economic interpretation of factor risk compensation is nuanced.

Key ideas

  • A factor model intercept measures average excess return unexplained by the included factors.
  • Portfolio returns linked to factor exposures are distinct from alpha.
  • A negative alpha indicates a mean return below the level implied by the fitted factor model.
  • The response does not specify how to annualize an intercept estimated from daily data.

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Factor investingStatisticsEquities
Full text
# Carhart 4-Factor Model intercept interpretation


# Carhart 4-Factor Model intercept interpretation












I've been following studies such as Kempf & Osthoff (2007) and Statman & Glushkov (2009) in building a methodology measuring ESG portfolio performance centred around the Carhart 4-Factor Model. I've been using factors from the Kenneth French database and regressing in Microsoft Excel to get my coefficients and p-values.

My understanding of the intercept term in the Model is that it is representative of the alpha, or excess return of the portfolio. However, when running these regressions on returns and factor data over a 5 year period, my alpha is statistically significant, but is miniscule, at -.40%, when the true return in excess of R(f) is much greater. As such, my primary question is how to interpret this alpha, and what is the reasoning behind its significant difference from the true return in excess of R(f)?

Find both above referenced studies below:

https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1468-036X.2007.00402.x?saml_referrer

https://www.tandfonline.com/doi/pdf/10.2469/faj.v65.n4.5?needAccess=true

Edit: As I am running the regression on daily returns and factor datapoints, is the intercept indicative of daily alpha, in which case I should annualise returns for the yearly periods under investigation?

## Answer by Matthew Gunn (score 3)

https://quant.stackexchange.com/a/65673

The idea behind any of these factor models (whether it be the CAPM, Fama-French 3 Factor Model, Carhart 4 Factor Model etc...) is that expected returns are linear in covariance with variables of hedging concern to investors. The economic idea is that there are macroeconomic risks investors do not wish to hold, and to entice investors to hold these risks, investors are compensated in the form of higher expected returns. Covariance with variables of hedging concern is risk that obtains a positive market price. The more covariance, the more risk, and the higher the expected return. (How this theory meshes with reality is of course highly nuanced and debatable.)

Anyway, in a time-series regression of excess returns on other returns series (i.e. tradable factors), the intercept term (i.e. the alpha estimate), is the difference between the average return of the portfolio and what can be explained by covariance with some set of factors. A negative alpha implies that the mean return is lower than what would be expected given factor model.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.