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# [P2T8: Ang, Chapter 10: Alpha] : Conceptual Clarity on Formula of Alpha

#### ABSMOGHE

##### New Member
Hi @David Harper CFA FRM ,
I am very sorry that I am disturbing you as of this crucial moment. However reading more about the formula of Alpha has confused me to greater lengths than it should have. Hence I wanted a conceptual clarity on the formula of Alpha.

1.
Now as I understand it we get Alpha as the value by Regressing the following Equation
(rp-rf) =α + β (rb-rf)

Thus solving for the above shouldn't we get
α = (rp - rf) - β (rb-rf) ...........(1)

However in various posts, the general formula seems to be as

α = rp - β (rb)

2.
- Further, what does the β in the Equation (1) interpret aside from the fact that it is the slope coefficient ?
- How would it relate to the actual β (ie. the one under CAPM - Systematic factor of Risk) ?
- Next, would the beta in the regression equation 1 have the same value as it would have under CAPM if our benchmark portfolio was the same as Market Portfolio (ie β(p,m)=1) ?
- Also how would the calculation differ if the benchmarked portfolio was not a market portfolio ?

3.
Also, wrt illustration that you drafted here, (which honestly helped a lot to understand)

Imagine the riskfree-rate is 1.0% and the benchmark return is 4.0% such that the benchmark's excess return = 3.0%. Now imagine a portfolio with a beta of 0.80 returns 4.60%, such that the portfolio's excess return is 3.60%.
• The active return is the difference in excess returns = 3.60% - 3.00% = +0.60%; or just the difference in returns, 4.60% - 4.00% = +0.60%; i.e., active return is the portfolio's return relative to the benchmark
• The residual return = 3.60% - 3.0%*0.80 = 1.20%, or in this single-factor model, (jensen's) alpha = 4.60% - (3.00%*0.80) - 1.0% Rf = + 1.20%; i.e., residual return is the portfolio's excess return relative to (beta*benchmark excess return).
- In the above equation what exactly does the beta of 0.80 refer to (ie. Is it the Beta value under regression equation (ie regressed with benchmark), or is it β value under CAPM ie when regressed with market) ?
- Further assuming that the β in the above question corresponds to that under CAPM (ie. β(p,m)) , if the benchmark is different from market and has a β(b,m) =1.25 or say β(b,m) = 0.7. How would we proceed in the above case ?

I realize that my foundation concepts are very weak, and genuinely apologize for being not able to understand this. I've also skimmed through all the articles in this forum which have been tagged as Alpha, however none of them address my points.
Hence, I would really appreciate if you could help me out on this, as I am unable to move forward on my curriculum without getting a grasp of these concepts.

#### Nicole Seaman

Staff member
Subscriber
@David Harper CFA FRM
Hello @ABSMOGHE

Although sometimes David is able to answer questions the same day, many times that is not the case, especially with the exam coming up. I'm sure you've noticed that the forum is EXTREMELY busy with questions right now, and David is only one person. He spends hours in the forum each day answering very complex questions. Most of them require more than just a quick answer. I know that he is aware of your question here, so we really appreciate your patience.

Thank you,

Nicole

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @ABSMOGHE You've asked many questions, some of them refer to entire chapters in the syllabus. Here is my first pass:
1.
Now as I understand it we get Alpha as the value by Regressing the following Equation
(rp-rf) =α + β (rb-rf)

Thus solving for the above shouldn't we get
α = (rp - rf) - β (rb-rf) ...........(1)

However in various posts, the general formula seems to be as

α = rp - β(rb)
In most cases the second version simply incorporates excess returns rather than explicate gross returns. (Note: it's true we can use version #2 with gross returns but for purposes of this discussion, i will treat this as an exception and refer the reader to Elton; to my knowledge Bodie assumes excess returns). This is very common. In your second version, if we define R(p) = the portfolio's excess return = R(P) - Riskless_rate, and R(b) = benchmark's excess return = R(B) - Rf, then these expressions are identical. In general, in my opinion, your first is correct, although in practice it's just easier to define returns as excess returns (i.e., in excess of the riskfree rate), as many authors do, all the way around such that your second is arguably more convenient.
• Ex-post alpha is a regression intercept and its single-factor form, where Rp is a gross return, is given by the sample regression function: α = (Rp - Rf) - β*(Rb - Rf) + e(i)
2.
- Further, what does the β in the Equation (1) interpret aside from the fact that it is the slope coefficient ?
- How would it relate to the actual β (ie. the one under CAPM - Systematic factor of Risk) ?
- Next, would the beta in the regression equation 1 have the same value as it would have under CAPM if our benchmark portfolio was the same as Market Portfolio (ie β(p,m)=1) ?
- Also how would the calculation differ if the benchmarked portfolio was not a market portfolio ?
Ah, well let's proceed from the general to the specific. Here, as is often the case, the resolution to a set of seemingly diverse implementations is mostly, simply that some are special cases of the most general form.
• Beta, β, is (merely) the regression's slope coefficient. We can regress against several factors; for example, the Fama-French has two factors, in addition to the market factor, such that a Fama-French regression model has three betas, each are firstly and merely partial slope coefficients in a regression. Notice this general form is purely mathematical-statistical, and further notice that the general case includes multiple factors and therefore multiple betas (beta is the sensitivity to the factor).
• The important, general idea about beta is that is given by: β(P,x) = σ(P,x)/σ^2(x) = σ(P)*σ(x)*ρ(P,x)/σ^2(x) =σ(P)*σ(x)*ρ(P,x)/σ^2(x) = ρ(P,x)*σ(P)/σ(x); i.e., the beta of a Portfolio is the correlation, scaled by cross-volatility, to something; e.g., the market, an index that proxies the market, a benchmark, or even another portfolio.
• We can then move from the general multivariate (multi-factor) regression form and specialize (i.e., go to the special case of) a single-factor model. Bodie Chapter 9 is about the CAPM versus the single-index model, and any summary is likely to be imprecise, but let's be superficial and assume:
• The CAPM is a very special case of a single-factor model where the single factor is the value-weighted Market Portfolio's excess return. However, it requires several assumptions and further is not falsifiable (testable) such that it is, in a sense, "too special." It is theoretical:
• Instead, we approximate the CAPM by using a broad index (its excess return) as the single factor; hence the single-factor index model, or single index model. The difference here is simply that we go from a theoretical factor (market portfolio in CAPM) to an actionable factor (e.g., S&P 500 index). Mathematically, beta doesn't really change. As a regression, we have the portfolio's excess return on the Y-axis and, mostly, what changes here is what we are regressing against; i.e., what's on the X-axis. In terms of an (ex post) measure, beta remains a regression (slope) coefficient. I am mentioning ex post because the CAPM is really about expected (ex ante) measures but I want to avoid theoretical nuances.
3.
Also, wrt illustration that you drafted here, (which honestly helped a lot to understand)

Imagine the riskfree-rate is 1.0% and the benchmark return is 4.0% such that the benchmark's excess return = 3.0%. Now imagine a portfolio with a beta of 0.80 returns 4.60%, such that the portfolio's excess return is 3.60%.
• The active return is the difference in excess returns = 3.60% - 3.00% = +0.60%; or just the difference in returns, 4.60% - 4.00% = +0.60%; i.e., active return is the portfolio's return relative to the benchmark
• The residual return = 3.60% - 3.0%*0.80 = 1.20%, or in this single-factor model, (jensen's) alpha = 4.60% - (3.00%*0.80) - 1.0% Rf = + 1.20%; i.e., residual return is the portfolio's excess return relative to (beta*benchmark excess return).
- In the above equation what exactly does the beta of 0.80 refer to (ie. Is it the Beta value under regression equation (ie regressed with benchmark), or is it β value under CAPM ie when regressed with market) ?
- Further assuming that the β in the above question corresponds to that under CAPM (ie. β(p,m)) , if the benchmark is different from market and has a β(b,m) =1.25 or say β(b,m) = 0.7. How would we proceed in the above case ?
In that illustration of mine, the beta is the beta of the portfolio with respect to the benchmark; i.e., β(P,B). As above, we can even say "it is the volatility-scaled correlation between the portfolio and the benchmark" if we understand that "volatility-scaled" means "multiplied by cross-volatiltiy." Do you see how this is little more that a statistical measure of association between two series of excess returns? There is no CAPM here, no CAPM theory!! (unless our benchmark is a proxy for the Market portfolio and we are making an approximate statement ...)

Now I realize it is true that "beta" might suggest or connote CAPM's beta to some people, but it's not obvious that is should! For just one example, the APT sensitivities are (can) also called betas, or factor betas. In the youtube video below, where i illustrate the information ratio calculation based on a benchmark (over 3 years), the beta is "with respect to the (hypothetical) benchmark."

I hope that's helpful. I don't want to go too much further, there's too much underneath the surface and, at some point, I'm trying to re-synthesize several assigned chapters. I'll discuss specific follow ups but we need to be mindful of the time it takes to answer super-compound questions. I am impressed with your level of detail, and organization (and we appreciate your politeness!) and it's cool to see somebody grappling with the theory, so i really do hope this is helpful! Thanks,

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#### ABSMOGHE

##### New Member
Hi @ABSMOGHE You've asked many questions, some of the refer to entire chapters in the syllabus. Here is my first pass:

In most cases the second version simply incorporates excess returns rather than explicate gross returns. (Note: it's true we can use version #2 with gross returns but for purposes of this discussion, i will treat this as an exception and refer the reader to Elton; to my knowledge Bodie assumes excess returns). This is very common. In your second version, if we define R(p) = the portfolio's excess return = R(P) - Riskless_rate, and R(b) = benchmark's excess return = R(B) - Rf, then these expressions are identical. In general, in my opinion, your first is correct, although in practice it's just easier to define returns as excess returns (i.e., in excess of the riskfree rate), as many authors do, all the way around such that your second is arguably more convenient.
• Ex-post alpha is a regression intercept and its single-factor form, where Rp is a gross return, is given by the sample regression function: α = (Rp - Rf) - β*(Rb - Rf) + e(i)

Ah, well let's proceed from the general to the specific. Here, as is often the case, the resolution to a set of seemingly diverse implementations is mostly, simply that some are special cases of the most general form.
• Beta, β, is (merely) the regression's slope coefficient. We can regress against several factors; for example, the Fama-French has two factors, in addition to the market factor, such that a Fama-French regression model has three betas, each are firstly and merely partial slope coefficients in a regression. Notice this general form is purely mathematical-statistical, and further notice that the general case includes multiple factors and therefore multiple betas (beta is the sensitivity to the factor).
• The important, general idea about beta is that is given by: β(P,x) = σ(P,x)/σ^2(x) = σ(P)*σ(x)*ρ(P,x)/σ^2(x) =σ(P)*σ(x)*ρ(P,x)/σ^2(x) = ρ(P,x)*σ(P)/σ(x); i.e., the beta of a Portfolio is the correlation, scaled by cross-volatility, with something; e.g., the market, an index that proxies the market, a benchmark, or even another portfolio.
• We can then move from the general multivariate (multi-factor) regression form and specialize (i.e., go to the special case of) a single-factor model. Bodie Chapter 9 is about the CAPM versus the single-index model, and any summary is likely to be imprecise, but let's be superficial and assume:
• The CAPM is a very special case of a single-factor model where the single factor is the value-weighted Market Portfolio's excess return. However, it requires several assumptions and further is not falsifiable (testable) such that it is, in a sense, "too special." It is theoretical:
• Instead, we approximate the CAPM by using a broad index (its excess return) as the single factor; hence the single-factor index model, or single index model. The difference here is simply that we go from a theoretical factor (market portfolio in CAPM) to an actionable factor (e.g., S&P 500 index). Mathematically, beta doesn't really change. As a regression, we have the portfolio's excess return on the Y-axis and, mostly, what changes here is what we are regressing against; i.e., what's on the X-axis. In terms of an (ex post) measure, beta remains a regression (slope) coefficient. I am mentioning ex post because the CAPM is really about expected (ex ante) measures but I want to avoid theoretical nuances.

In that illustration of mine, the beta is the beta of the portfolio with respect to the benchmark; i.e., β(P,B). As above, we can even say "it is the volatility-scaled correlation between the portfolio and the benchmark" if we understand that "volatility-scaled" means "multiplied by cross-volatiltiy." Do you see how this is little more that a statistical measure of association between two series of excess returns? There is no CAPM here, no CAPM theory!! (unless our benchmark is a proxy for the Market portfolio and we are making an approximate statement ...)

Now I realize it is true that "beta" might suggest or connote CAPM's beta to some people, but it's not obvious that is should! For just one example, the APT sensitivities are (can) also called betas, or factor betas. In the youtube video below, where i illustrate the information ratio calculation based on a benchmark (over 3 years), the beta is "with respect to the (hypothetical) benchmark."

I hope that's helpful. I don't want to go too much further, there's too much underneath the surface and, at some point, I'm trying to re-synthesize several assigned chapters. I'll discuss specific follow ups but we need to be mindful of the time it takes to answer super-compound questions. I am impressed with your level of detail, and organization (and we appreciate your politeness!) and it's cool to see somebody grappling with the theory, so i really do hope this is helpful! Thanks,

Hi @David Harper CFA FRM
Thank you so much for explaining this. This really helped me a lot, as I was quite perplexed with these concepts with regards to their notations and interpretations.
I also apologize for being too pushy on this, however as the exam was near the clock, I was very anxious about being able to understand this.
Thank you.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @ABSMOGHE It's totally my pleasure! "Alpha" is a hard concept in the FRM, due in part to its appearances by various authors.

Regarding your statement: "I also apologize for being too pushy on this, however as the exam was near the clock, I was very anxious about being able to understand this"

I would like to explain that your re-request had no effect on me and, in my opinion, it's not a fair urgency to push on us; Nicole helped with the interim reply. Everybody is on the same calendar: GARP's exam dates are known several months in advance. If some people want to wait until the last month/week(s) of the exam, that is their responsibility. As the old saying goes, with all due respect, "your lack of planning is not my emergency." It's really not reasonable to attempt to impose urgency on anybody else, given the FRM calendar is absolutely known to everybody (3rd week in May and November). To me, this is just common sense and good manners.

Instead, I answered your question in the same way that we (David and Nicole) try to triage and answer any customer questions that we get, throughout the year. We're here every day, and have been doing this for a decade+ so we really are not rattled or influenced by your urgency to be honest.

With respect to overall timing and planning, for FRM candidates we recommend that you spend the final two to four weeks practicing (aka, revising) questions. If this is good advice, then it implies you will have finished your basic review (i.e., reading the source assignments in the syllabus, reviewing any summary notes and/or watching videos) before this time. Ideally, this basic review should completed approximately (at least) one month prior to the exam. The final month can then be spent in revision (practice). Anybody who follows this plan has extremely good odds to pass. To repeat, in our view it is not ideal to be grappling with fundamental concepts in the final weeks. Your best plan is start early so that you have the "luxury" of revising practice questions (as the majority or exclusive spend of your time) in the final two to (preferably) four weeks before the exam.

Thank you!

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