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# Jensen's Alpha Formula - Please Confirm

#### chatty06

##### New Member
Hi All,

Can someone confirm the correct Jensen's Alpha formula? I've seen two versions used throughout the material and I don't understand why. Maybe I'm just overlooking something. One version adds the Risk Free Rate to the Price of Risk while the other one subtracts it. Examples below

Version 1 = Expected Return (portfolio) - (Risk Free Rate + Beta (portfolio) [Expected Return (Market) - Risk Free Rate]
7:12 minutes into the Instructional Video: Amenc, Chatper 4.

Version 2 = Expected Return (portfolio) - (Risk Free Rate - Beta (portfolio) [Expected Return (Market) - Risk Free Rate]
PG 3 of R9.P1.T1.Amenc_v7_592b10f43df06.pdf

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
The version 1 seems correct. The formula is Jensen's Alpha= Expected Return (portfolio) - (Risk Free Rate + Beta (portfolio) [Expected Return (Market) - Risk Free Rate])
WE compare the portfolio return Expected Return (portfolio) and the CAPM implied portfolio return Risk Free Rate + Beta (portfolio) [Expected Return (Market) - Risk Free Rate].
Version 2 seems to have flaw of the - sign.
thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @chatty06 It's your Version 1 because alpha is the difference between the portfolio return and the return predicted by CAPM, Rf+β*ERP. So it assumes the E(P) = α + [Rf+β*ERP] or α = E(P) - [Rf+β*ERP] = E(P) - Rf - β*ERP. I actually do not see your Version 2 in the notes (sorry, I could be wrong, see below): I would like to mention that this week we will publish a much improved version of this Amenc study note , yay! e.g., see below the explication of alpha in the new note: The new note is also cool (I think) because I improved the dynamic spreadsheets that support these calculations, here is the image for the upcoming note (the text labels are dynamic to the assumptions). Thanks! Last edited:

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@chatty06 can you reply about the 2nd version, so that if we do have a mistake, we can fix it? Thanks,

#### chatty06

##### New Member
Thanks for the clarification.

It looks like my algebra is rusty. I was having difficulty getting the right answer using version 1.

α = E(RP) – [RF+BP(E(RM)-RF)] = E(RP) – RF-BP(E(RM)-RF)

So when using the inputs from the "Summary of Risk-Adjusted Performance Measures" I get the below, which look to agree. I just needed the clarification that these were both the same and to work through the math in the correct sequence.

Using the first version
α = E(RP) – [RF+BP(E(RM)-RF)]
α = .09 – [.03+.75(.08-.03)]
α = .09 – [.03+.75(.05)]
α = .09 – [.03+.0375]
α = .09 – [.0675]
α = .0225

Using the second version
α = E(RP) – RF-BP(E(RM)-RF)
α = .09 - .03-.75(.08-.03)
α = .09 - .03-.75(.05)
α = .09 - .03-.0375
α = .0225

Thanks All!

#### Bucephalus

##### Member
Subscriber
Hi @David Harper CFA FRM

Sorry if this question sounds a little dumb as I am not able to get around this in my head. The expected return in the above example is assumed to be 9% while the expected return calculated per CAPM comes to 6.75%. While calculating Jensen's alpha, shouldn't we consider 6.75 as the expected return of the portfolio instead of 9%. Why do we take the assumed value instead of the risk adjusted return?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
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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#### Bucephalus

##### Member
Subscriber
Thank @David Harper CFA FRM for the detailed response. Is it fair to conclude that Jensen's alpha is always an ex post value and not an ex ante value? Will the ex ante value always be zero as we invariably calculate it using the CAPM model? In other words as CAPM fails to recognize the unsystematic risk, it fails to measure the E(α)? And, hence it is impossible to measure the future excess returns a manager will be able to deliver however good or bad his/her past performances have been?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Bucephalus Please note the theory here is deep, which is my way of saying that somebody else, or a finance professor, might disagree with me
• Re: Is it fair to conclude that Jensen's alpha is always an ex post value and not an ex ante value?
I would prefer to say that Jensen's alpha is an ex post measure in general, because it is a regression intercept (citing, eg, Carl Bacon https://www.amazon.com/dp/1118369742/?tag=bt077d-20). Exceptions imply a deviation from (violations of) the CAPM and its decidedly unrealistic assumptions
• Will the ex ante value always be zero as we invariably calculate it using the CAPM model? In other words as CAPM fails to recognize the unsystematic risk, it fails to measure the E(α)? And, hence it is impossible to measure the future excess returns a manager will be able to deliver however good or bad his/her past performances have been?
Yes, I think so. If we hold the CAPM assumptions, CAPM tells us that the expected (ex ante) excess return is a function only of beta exposure to the market risk factor (aka, single factor) such that CAPM says E[α] = 0. If we expect a security/portfolio to fall on the SML, that is equivalent to expecting (ex ante) zero alpha.
• In other words as CAPM fails to recognize the unsystematic risk, it fails to measure the E(α)? And, hence it is impossible to measure the future excess returns a manager will be able to deliver however good or bad his/her past performances have been?
While I empathize with this seemingly logical step, I think I disagree here. The CAPM universe is restrictive and unrealistic, saying that only one factor (aside from luck) explains excess returns. Jensen's alpha, as I perceive it, does not inherit all of the CAPM assumptions. Rather, it is just using the CAPM to evaluate residual return (residual return = regression alpha). As mentioned, my exhibit above probably should say "average (realized over the period) portfolio return = 9.0%" rather than E[R(p)]. The finding that Jensen's alpha of 2.250% is saying something like "because CAPM assumes non-systematic risk factors are diversified to zero and eliminated--leaving only the systematic risk factor--CAPM says the portfolio should only earns excess return due to exposure to the market premium ("as if" CAPM were true), and, in this case the portfolio should have earned 6.75% (I just realized that I actually do have an E[return] per CAPM = 6.75% line in my XLS ) however it earned 9.0%, which is therefore generally unexplained by CAPM's single-factor view and we will call this vertical distance from the SML alpha (aka, residual), which is here + 2.25%." In this way, CAPM is just exploited to help us measure (evaluate). I don't think it has any further claim on the interpretation of the +2.25%. It can be some combination of: luck (random sampling), skill, nonsystematic risk (that did not actually get diversified away), or very likely: non-market factor exposure (hidden beta), like a momentum or value factor. I hope that's helpful!

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#### jealaign

##### New Member
Hi everyone,

I read that Jensen's alpha measures the abnormal return of an asset over the theoretical return given by the CAPM.

So, does this alpha explains some of the idiosyncratic risk, or does it only explains systemic risk (which means that some systemic risk is not explained by the beta ?)

And also, it seems to be an ex-post measure so why is it defined with regard to E(Rp) - Rf and not Rp - Rf in the study notes of the reading of Amenc ?

Thank you,
Jean

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#### Flashback

##### Active Member
Jensen's alpha is a measure of manager's additional realized return over expected return on market portfolio measured by CAPM.
Therefore, it explains return over market systematic risk, not a specific risk. Such risks are supposed to be eliminated by portfolio diversification.
Correct. It is an ex-post measure based on a (positive) difference on ex-ante expected return basis. This explains E(Rp) variable in formula.