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:

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?