Hi

@Bucephalus It's a

**smart **question, actually

. In a way, you are correct: according to strict implications of CAPM--in the example above--the portfolio's

*expected* return, E[R(p)] is 3.0% + 0.75*5.0% = 6.75%; it's not 9.0%. I did think about denoting portfolio return (in this section) with R(p) rather than E[R(p)], and

**maybe I should**, but it would differ from Amenc (the longtime author assigned here). The source uses expected return. Bodie in T8 is more accurate in using an average (realized) return.

In the case of Jensen's alpha, the 9.0% should be represented not by

*expected *return but rather by

**average **return. A weakness of the readings (syllabus) in T1 is that performance measures are not introduced (or framed) with a key distinction between

**ex post** measures versus

*ex ante *performance measures (something we've talked about often in this forum over the years).

Ex ante is before or expected, but in my estimation

*realistically* most of our risk measures are

**ex ante **(they are calculated after-the-fact with an actual or observed set of returns or values). Jensen's alpha is generally an

*ex post *measure and as such this alpha is a

**regression intercept**. The proper form is ex post Jensen's alpha, α = average (realized) portfolio excess return - β*[R(m) - Rf)].

On the other hand in terms of an

*ex ante* Jensen's alpha, if we are presuming the CAPM (ie, systematic risk is the only risk factor and everything else is skill or luck), then the portfolio's

*expected* excess return = β*[R(m) - Rf] because

**expected alpha, E[α], is likely to be zero**. In the CAPM universe, E[α) is zero. In this CAPM single-factor simplification, we

*expect* portfolios to fall on the SML because we

*expect* zero alpha (even as we understand the actual portfolio won't fall on the line). But that's all inside the CAPM universe of incredibly restrictive assumptions. See how that's all basically agreeing with you?!

For all practical intents and purposes of the Jensen's alpha, and for that matter the Treynor and Jensen's, then, the E[R(p)] in my exhibit above, represents not really an expected return but an actual or average return, R(p).

Why does Amenc show E[R(p)]? Three possibilities that I can think of. One, it's conceivable he fails to appreciate the ex post versus ex ante distinction, although this seems unlikely to me. Second, by E[R(p)] he actually means average actual because E[X] can represent "average," although this is belied by the fact that he uses "expected return" throughout this chapter.

Third and I prefer this explanation, he does mean expected and at the same time he does mean to imply that expected alpha can be non-zero. The problem with regression alpha is parsing skill from luck (hence the statistical test of the same in P2.T8. Bodie). We are allowed to say the portfolio manager has an expected alpha of 2.250%, aka, the manager has

*persistent skill*. Is this compatible with CAPM? Not really but ....

It doesn't need to matter for purposes of the Jensen's alpha. Jensen's is really an ex post measure but more importantly Jensen's alpha is just the

*portfolio's excess return that is not explained by exposure to CAPM's sole systematic risk factor*. It can utilize CAPM without inheriting its theoretical assumptions. Actually, in this line of thinking, the expected non-zero alpha could include "smart beta:" the manager earns alpha by exposure to some non-market factor(s). Jensen's alpha would give this credit. In this line of thinking, we could have two assumptions: E[R(p)] per CAPM = 6.75%, and at the same time E[R(p)] per other factors = 9.0%.

Sorry if that's too long, there is nuance underneath the surface here. It's a smart question. Hopefully, I didn't too much obscure the simple fact that, if asked for the Jensen's alpha,

**under any scenario we are subtracting β*(E[R(m)] - Rf) or β*[R(m) - Rf)] from a portfolio's return** (of some sort!) because the "Jensen" implies we are evaluating the portfolio return by the single-factor CAPM. In my view, the nuance is "what portfolio return are we subtracting from?" The answers include:

- The average (actual, realized or ex post) portfolio return. To your point, this is what is meant by the confusingly labeled E[R(p)] above
- An
*expected *portfolio return that is non-zero owing to a persistent expectation of skill or non-systematic factor contribution

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