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# Benchmark neutral alpha question

#### JesusZ

##### New Member
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Sorry If this is answered somewhere else and I missed it.

I have a simple question, why is the benchmark neutral alpha computed as:

alpha neutral = active alpha - active beta * benchmark alpha ?

Any explanation / help will be highly appreciated

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Hi @JesusZ I'm not sure if that's from T9 Grinold (in which case this is maybe helpful https://www.bionicturtle.com/forum/...ha-in-portfolio-construction.23783/post-88532) but you have it in T5. Is it a T5 reference? I will say there is a pattern precedent. If we step down from alpha to CAPM, there is a sense in which we "neutralize" a portfolio's excess return with respect to beta in order to identify its (Jensen's) alpha.

For example, say that while the the riskfree rate = 3.0%, the market's excess return = 5.0%, our portfolio generates an excess return of 8.0% (i.e., 11.0% gross return) with a beta of 1.40. Jensen's alpha = 8.0% - 1.40*5.0% = +1.0%. To rephrase this operation, residual return (aka, alpha) = 8.0% active return - 1.40 beta * 5.0% benchmark return. But we could call the Jensen's alpha a beta-neutralized return because it is an excess return that is neutralized with respect to beta. So, while I don't have the exact context, the pattern in familiar. The difference between mine and yours is, in typical Grinold fashion, yours is breaking down the alpha (aka, residual return) itself into a factor contribution and, perhaps, a pure (skill?) alpha. I don't know why you have in T5, it smacks of Grinold.

So, like I'm just musing here okay? Say the benchmark's alpha = 2.0%, and the active beta = 1.20, while our porfolio's active alpha = 3.0%, then this says the alpha neutral = 3.0% - 2.0%*1.2% = 0.60%. The naïve numerical interpretation is easy: the 3.0% active alpha deconstructs into 2.40% contribution by the benchmark plus 0.60% as "pure" alpha.

That said, I would never write those words myself, for several reasons. For one, "active alpha" is not a term that I myself define precisely; I can think of three attempts to define and one contradiction (in the IR, we typically use either active return or alpha; aka, residual return). So the harder part is interpretation. I hope that's helpful

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#### JesusZ

##### New Member
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Hi @David Harper CFA FRM thank you for the prompt reply!

Sorry I wasn't sure how to post this in the correct place.

My question is a reference to Grinold as you correctly pointed out. Specifically it's about "Chapter 14 of Active Portfolio Management: A Quantitative Approach for Providing Superior Returns and Controlling Risk, Second Edition by Richard C. Grinold and Ronald N .Khan".

In the GARP study materials this is referenced in Table 4.2, section on Benchmark and Cash-Neutral Alphas where the authors point out that "In this example, the benchmark alpha is only 1.6 basis points, so subtracting beta_n *alpha_B from each modified alpha does not change the alpha very much."

He uses the formula I was asking about to build Table 4.2, hence my question.

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @JesusZ Thanks, I must have forgotten that section: so indeed Grinold is just doing basically what I said. If the benchmark happens itself to generate alpha = 1.6 bps, probably he regressed each stock's alpha against the benchmark in order to determine each stocks beta with respect to the benchmark's alpha, β(n, B), then subtracts from each individual stock the quantity β(n, B) * α(B) = β(n, B) * 1.6 bps. So if a stock has an alpha of +3.0 bps but also a high beta with respect to the benchmark, β(n, B), then this 3.0 is "getting to much credit" for alpha that really the benchmark generated. If the β(n, B) = 1.0, then 3 - 1.0*1.6 = +1.4 bps is an alpha that is benchmark neutral; i.e., is not getting credit for the benchmark or "benchmark timing" as the text says. Hope that's helpful,