Jensen’s Alpha

By On October 9, 2010 · 3 Comments · In Reference

Jensen’s alpha is a statistic that is used to test return-predicting factors for potential use in portfolio construction. For example, suppose that we index data for asset i at time t by (i,t). Let r denote return and f^j denote a return predicting factor. Suppose we have data for securities i=1,2,\ldots,I at times t=1,2,\ldots,T, and suppose there are J return predicting factors. Then for each time t, we can regress returns across firms on all return predicting factors. That is, for fixed t, we estimate the equation:

r_{i,t}= \sum_{j=1}^J \beta^t_j f^j_{i,t} +\varepsilon_i

across firms i. Note that by including a constant factor by setting some f^j=1 for each security and time (i,t), we can account for exposure to the market. We now ask the question: on average, how much does unit exposure to factor j increase or decrease returns independently of the other factors? For example, in a factor model that includes both size and earnings, unit exposure to size may also increase exposure to earnings because large companies tend to have higher earnings; however we wish to quantify how much unit exposure to size increases or decreases returns in a way that is not accounted for by earnings. To answer this question, we estimate the equation:

\beta^t_j = \alpha_0 + \sum_{k\neq j} \alpha_k \beta^t_k + \varepsilon_t

across time t. The quantity \alpha_0 is Jensen’s alpha. There are several commonly cited examples of Jensen’s alpha. CAPM alpha refers to Jensen’s alpha that is obtained when the factor model consists only of some factor j that you wish to test, and a constant. CAPM alpha is the additional return you obtain from unit exposure to j that is not accounted for by CAPM. Three-factor alpha, also called Fama-French alpha, is obtained when you include a constant, size, and value factors. Four-factor alpha additionally includes a momentum factor.

Note: The previous article is one of a series of topic summaries I am writing to introduce various topics that are not explained particularly well by online resources such as Wikipedia. I’m tagging all of these posts as “Wikipedia.” Please feel free to adapt these summaries for any use, with citation.

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3 Responses to Jensen’s Alpha

  1. Anonymous says:
    February 15, 2011 at 3:47 pm

    What information does Jensen’s alpha add that a simple test of significance would not? For example, if factor X always significantly predicts returns in the cross-section, even accounting for factor Y, then when would X have a negative alpha? More importantly, even if X has a negative alpha, why would you omit X from your model given that it accurately predicts returns?

    Reply
  2. vlad says:
    July 11, 2011 at 2:38 pm

    When I do have monthly returns of my portfolio A and an other portfolio B with three-montly returns. I have data for 10 years an now I would like to compare my portfolios with respect to the market.
    What is the formula for scaling the Jensen Alpha to one jear – e.g like at Sharp Ratio – by multipying with root 12???

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