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Disclaimer: I come from an academic finance perspective and hence I will definitely have my inherent biases in this question.

How does one think about "alpha" in portfolio management? In particular, in some practitioner's literature, there's this discussion of an "alpha factor" in the linear factor models. Taking an instance out of numerous examples out there, see http://www.iijournals.com/doi/abs/10.3905/jpm.2008.709976 https://www.msci.com/documents/10199/c6e5e3f7-cd44-4322-aeb5-331e20e2afb7

As I understand it, the practitioner has in mind a linear factor model of the form, $$R_k = \alpha_k + \beta_{k1} R_{k1} + \ldots + \beta_{kN} R_{kN} + \epsilon_k $$, where $R_{kn}$ are the $n = 1, \ldots, N$ factor excess returns and $R_k$ is the excess return of asset $k$, and $\epsilon_k$ is the usual idiosyncratic risk. But what is this discussion of $\alpha_k$ being "spanned" by risk factors (i.e., it can be rewritten as a linear form of $R_{kn}$) or that it represents "return forecasts"?

To my (perhaps limited) understanding, no top academic finance journal would ever interpret $\alpha_k$ as an "alpha factor" that is spanned by risk factors (i.e. this would imply that such an "alpha" can be collapsed into the linear span of ${ R_{k1}, ..., R_{kN} }$ and hence is risky; or that it can be viewed as a forecast of future returns, since the expectation would simply be $E R_k$, which when $R_{kn}$ have nonzero expectations, is not equal to $\alpha_k$. In our usual terminology, the presence of a nonzero $\alpha_k$ represents mispricings in the market, skills of a portfolio manager, or an error in the model that drives returns. Indeed, going way back to tests of flaws in the CAPM, this was exactly what was done.

In all: is "alpha" as understood by academics and practitioners equivalent?

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