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# Alpha

#### sipanivishal

##### Manager-Corporate Banking
Hi David,

I have a question

A portfolio underperformed its benchmark by 2%.what can we say about alpha ?
a. alpha is -2%
b.alpha is definitely negative
c.alpha can be positive or negative

answer given is c . I can guess it to be c but can I have a more intuitive explanation from you

Thanks
Sipani

#### sipanivishal

##### Manager-Corporate Banking
Hi David,

In case of sortino's ratio...if Benchmark return and risk free rate is given what should we take for MAR.....MAR is not specified and why ?

Thanks
SIpani

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Sipani,

Glad you posted this question. This question gets to the essence of Grinold and a hefty piece of the investment risk hedge section.

Let's say benchmark is S&P 500 and equity risk premium = 4%.
Then, per CAPM, excess return [return - riskless] on S&P with a beta of 1.0 = (1.0 beta)(4% ERP) = 4%
Now assume our portfolio's excess return = 2%. The active return is -2% (2% portfolio - 4% benchmark = -2% active return)

But maybe that is because our portfolio's exposure wasn't really "tracking" with the benchmark, perhaps the portfolio's ACTIVE BETA was only 0.4. With zero alpha, our portfolio excess return
= (0.4 active beta)*(4% ERP) = 1.6%. But if the portfolio excess return = 2%, then the residual (alpha) must be +0.4%:
2% portfolio excess return = 0.4 active beta * (4% ERP) + 0.4% alpha

But who knows, we don't have enough information, maybe the portfolio's active beta > 1.0. If active beta = 1.2, then alpha is: 2%-(1.2)(4%) = -2.8% and our alpha is negative

So,
active = portfolio - benchmark,

but Grinold breaks active return into pieces:
active = (beta exposure)(beta factor) + alpha.

IMO, this takes a bit of work to sink in. It's unnatural for some of our backgrounds. But it is worthwhile, Grinold goes further with it (specifically, Grinold breaks up the ACTIVE BETA into piecies), but this is the first step.

Re: Sortino, there is no magic MAR, it's up to the user. MAR replaces riskless rate. It could be cost of equity or return needed to fund pension obligations, e.g.

David

#### ahnnecabiles

##### New Member
Hi David,

In Amenc’s reading, alpha is defined as the difference between the portfolio’s expected return over the portfolio’s calculated return as defined by the CAPM, hence:

Alpha = E(Rp) – Rf – Bp(E(Rm) - Rf)

However, looking at Grinold’s reading (including your spreadsheet example), alpha is not defined as above, instead it says that alpha is the difference between the expected portfolio return vis-à-vis the expected benchmark return (which is defined as the excess of market return over the risk-free rate, meaning, E(Rp) – Rf – (E(Rm) - Rf) --- without multiplying the beta of the portfolio in the market excess return. I am a bit confused. Please help.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Chinquee,

I don't know your Grinold source or my XLS source (i'm not saying you are wrong about the XLS, rather can you tell me which xls so i can check? The only, I think?, Grinold sheet on the member page deconstructs the systematic active return so alpha doesn't enter it. So I'm just not sure i follow your XLS reference?)

Re Grinold, I can't find that reference either? It could be beta = 1.0, implicitly, in your second example. Grinold does start with a Jensen's alpha to demonstrate: Jensen's alpha = E(R) - Rf - (beta)(risk premium) = alpha.

But i would just offer the following view. Grinold is totally consistent with Amenc b/c he (Grinold) starts with that definition of (Jensen's) alpha. That is, in classic CAPM/single factor, alpha is the way you wrote it.

Then Grinold goes further. First, and this is the key step that sets the stage for everything, he substitutes the equity risk premium (ERP; market return - riskless rate) in Amenc's classic CAPM with a benchmark, any benchmark. Amenc gives a very specific definition of alpha. Grinold abstracts by allowing the benchmark to be anything; Grinold's alpha is still residual but it has an infinite number of definitions depending on the beta factor definitions. Again, the Amence (traditional CAPM) reconciles with Grinolds in this way

Alpha = E(Rp) – Rf – Bp(E(Rm) - Rf) becomes
Alpha = E(Rp) – Rf – (beta1)(factor1) [- (beta2)(factor2) - ...]

In both, alpha is the residual. The expected (ex ante) or realized (ex post) return piece that cannot be explained by some linear combination of [beta exposure * factor]. And please note this is very similar to Andrew Lo's hedge fund replication method.

David

#### ahnnecabiles

##### New Member
Hi David,

Thanks for the prompt reply and thanks so much for all the efforts. Even if this is just a distant learning, I really really felt your utmost sincerity in helping us pass this difficult exam. I am simultaneously attending a formal FRM review (company sponsored), but to be honest, I am learning more here, because you have really put so much efforts in explaining at least down to the layman's level, the rationale for most of the principles that we have to understand, especially the NUMEROUS formulas that we have to memorize. Thanks so much, really really appreciate your efforts.

I just presumed from Grinold's readings and the spread sheet example that "excess benchmark return" is the market excess return, since in the spread sheet example, we computed "average portfolio return" as the "average active beta" multiplied by "average benchmark returns". Hence, to my understanding, "benchmark returns" here refer to "excess market returns" since we still have to multiply the portfolio's active beta to the benchmark returns to compute for the portfolio's excess returns as observed by the CAPM formula. I was just deciphering what alpha really is (since it says it is the difference between the excess Rp over the excess benchmark returns).

I find Amenc's and Grinold's readings very difficult, I am a bit loss and can't really appreciate the concepts. I cannot really play around with those ratios. Is it because it's far from the usual discipline? Thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Chinquee,

You are pretty close with: "Since we still have to multiply the portfolio’s active beta to the benchmark returns to compute for the portfolio’s excess returns as observed by the CAPM formula" Yes, that is pretty much what Grinold is getting at. The spreadsheet illustrates the ACTIVE SYSTEMIC RETURN which equals, as you suggest, active beta * benchmark (excess) return. The XLS breaks this into three pieces but, as there is no alpha in the active systemic return, there is no alpha in the pieces

This is the Grinold (page 500):
Active portfolio return = Active Portfolio Beta * (excess) benchmark + [fator1*exposure + factor2*exposure +...] + ALPHA

The XLS breaks out only the first part, what he calls Total Active Systemic Return [i.e., Active Portfolio Beta * (excess) benchmark], into three further pieces. This active systemic return contains no alpha just like (beta * ERP) does not include the alpha in CAPM. But it does includes some SKILL. Ergo, ALPHA is skill but skill is more than alpha.

We can generically view the above as:
return = exposure*factor + exposure*factor + exposure*factor + .... + ALPHA

such that the first term, the single factor, is similar to CAPM:
CAPM: Excess portfolio return [i.e., return - riskless] = beta*ERP + alpha, except now substitute "benchmark" for ERP:

portfolio return = beta*benchmark + alpha

so we are back to the essential definition of alpha, it's what cannot be (or is not) explained by the exposure (i.e., beta) to the factors, where the benchmark is a factor. But unlike the CAPM, Grinold does not stop at only the single market factor, he adds more "in between" factors:

portfolio return = beta*benchmark + (common beta*common factor + common beta * commmon factor + ... ) + alpha

THANK YOU very much for your kind appreciation, it is my pleasure to help. The Grinold reading (Ch 17) is sort of unfair, I needed to read the whole book to understand this chapter, I really don't think it stands alone at all...

David

#### Nikjam

##### New Member
Hi David,
1 Alpha is a measure of the performance of the portfolio above or below the benchmark (expected return) right?
2 How do we relate alpha with active risk?
3 What exactly is neutralization?Please explain Benchmark, Cash and Risk Factor Neutral Alpha.
4 I didnt understand the Scaling and trimming process too.
5 How important wrt to weightage in exam is this topic?
I got a very rough idea on the above topic from the readings.
I know I am asking too many questions at one time but the problem is I am unable to follow this part and iits taking a lot of time.

Regards,

#### Nikjam

##### New Member
Hi David,
I read the above posts of year 2008 and have got even more confused.
What is the benchmark return? Isnt it the return expected or targeted by the manager? Is this benchmark return , the return(benchmark) that the portfolio gives in excess of the expected return? Please explain.

Thanx.