Unsystematic Risk/Alpha

[Rufolo]

Hello everyone.


As i’ve recently discovered bionicturtle, very great help for me, i’ve been watching some videos which helped me a lot. Last one was :”Alpha (hedge fund alpha)”. I think that fits very much what I was looking for. Let me explain:
I have a portfolio of 40 different stocks. I was trying to find, for each of them, how much of the total return they got within a month was because of the market=>Systematic risk (benchmark) and how much was because of the stock=>Unsystematic risk (skill/luck). Do you think I can use that alpha as the datai am looking for?

Thanks a lot,

Regards.

 

[chiyui]

Yeah I think you can use alpha becoz it's the return figure due to unsystematic risk.

 

[Rufolo]

Let me please quote this:
"
Fund Evaluation in Practice (1) – Legg Mason (using CAPM)

The Legg Mason Value Prim fund returned 27.3% annually from September 1982 to December
1986 while the market only returned 21.6%. The fund manager might claim the excess returns
were due to her exceptional ability at picking stocks. Armed with the CAPM and regression, we
are able to evaluate the fund manager’s claim of superior performance.



.....

The CAPM considers only one-dimensional market risk, so the realized returns must come from
either the fund’s exposure to market risk or the value added by the manager. The monthly returns
that can be attributed to the manager’s ability are captured in alpha. The results imply the fund
manager was able to add 46 basis points to the fund’s return on a monthly basis or about 5.5%
per year above the return expected from a portfolio with a beta of .93. T......
"
Also it says later that with the 3FM from Fama and French we got and alpha of 0.22 .
My question is...from whom i can trust to do my exercise? To calculate how much of the return of a stock belong to the market and how much to the own stock (alpha)
Thanks a lot,

 

[chiyui]

firstly, what do you mean by "from whom i can trust"?

 

[chiyui]

you mean where does the "46 basis points" and "about 5.5% per year" come out from?

 

[Rufolo]

no, sorry, i mean that, with the CAPM the alpha = 0.46, with 3FM = 0.22

But the intercept of the CAPM-based regression indicates the incremental performance of the asset relative to the CAPM benchmark return. BUT, when applied to portfolios, the alpha measures the return attributable to skill or luck of the portfolio manager.

I am very confused about this. So is it different meassuring a portfolio than a specific stock?

thanks!

 

[chiyui]

no, sorry, i mean that, with the CAPM the alpha = 0.46, with 3FM = 0.22

But the intercept of the CAPM-based regression indicates the incremental performance of the asset relative to the CAPM benchmark return. BUT, when applied to portfolios, the alpha measures the return attributable to skill or luck of the portfolio manager.

I am very confused about this. So is it different meassuring a portfolio than a specific stock?

thanks!

OIC......well let me try to answer your question. I'll split the answer into 2 parts.

First, when you use regression to estimate the CAPM equation, it doesn't matter whether the dependent variable should be a specific stock or a stock portfolio. CAPM does not require the dependent variable must be the expected return of a specific stock. It can be any investment. (And that's why it looks so invincible that it deserves a Nobel Prize! Haha...)

So if you want to estimate the CAPM regression of a portfolio in order to see if there is any alpha exists in the regression result, don't worry and just do it. If you really can see a significant alpha in it and if you have confidence in your data and analysis, then just take action on this profit opportunity.

The point is, the CAPM regression estimate contains 3 parts only:
(1) the intercept term,
(2) the benchmark return attached with a beta coefficient,
(3) the residual term.
By definition (well, actually it's by the arguments of the inventors of CAPM), the (2) is return due to systematic risk (i.e. due to betting), and the (3) is return due to the unpredictable fluctuation (i.e. specific risk) of the stock/portfolio/any investment you wanna analyze.
The spirit of CAPM is that, all these two things have encompassed all the randomness we would encounter when we invest in the stock/portfolio/any investment.

So, the (1) must be something nonrandom - it is the stock's/portfolio's/any asset's uniquely outperforming profit potential.
If you're talking about stock, then the "uniquely outperforming profit potential" is just what you called the "incremental performance of ... benchmark return".
If you're talking about portfolio, then it is the portfolio manager's skill. Why? Because you (or at least, many people in the world) can't think of any other explanations about the existence of (1) in the portfolio.

So your guess is correct: there's no methodological difference in measuring alpha of a stock or a portfolio or any other assets. All of them use Ri = α + βRm + ei as the regression equation to carry out the estimation. The subscript i can stands for any asset you want to deal with.

--------------------------------------------------------

Second, I'll mention a tricky point - strictly speaking, Ri = α + βRm + ei is not a CAPM regression.
WHY? Just have a look at the CAPM equation and compare:

(a) Regression: Ri = α + βRm + ei
(b) CAPM equation: E(Ri) = rf + β(E(Rm) - rf)

They look very similar, but they are different.

Well, you may say "hey, if we just take mathematical expectation on both sides of (a) and simplify the equation, wouldn't we just get (b)?"
This is the tricky point. If we just do this, (a) will become E(Ri) = α + βE(Rm).
If we must let (a) and (b) be equivalent, this implies α = rf - βrf. (You can verify this by doing some algebra on it.)
And I can tell you that the alpha you observe in the real world rarely equals rf - βrf.
So people tend to believe that (a) is not a CAPM regression. It's just a ordinary regression.

(to be continued)

 

[chiyui]

Strictly speaking, CAPM does not allow any alpha (outperforming profit opportunity) exists in the CAPM equation. This is because of the troublesome assumptions behind CAPM - the market is at equilibrium and investors are rational. If both assumptions are met in the reality (actually sometimes they are met, sometimes aren't), then of course you can't observe any alphas, including portfolio managers' alphas.

That's why some people stubbornly believe that portfolio managers don't have alpha at all. They are so funky about rejecting CAPM that, they prefer to believe no alpha exists at all rather than to believe that the real world does not meet the CAPM assumptions. That's why they prefer saying "we can't use CAPM to estimate alpha" rather than "CAPM is wrong". It's a game of logic.

So if you ask me whether you should use CAPM to estimate the alpha of the portfolio (the manager's skill), I will say no if I were those people. And I will say you should use Ri = α + βRm + ei rather than use CAPM and be reminded that these two things are different. And if I were stubborn enough, I would even say "just do it, and your alpha-searching journey will be futile".

---------------------------------------------------

But of course, this is not a very convincing answer to the problem (and that's why CAPM is a hot and controversial research topic). So some people now will say "don't use CAPM to estimate alpha, use 3FM". 3FM is nothing special. It only adds two more independent variables in the CAPM equation. And it also allows not only individial stocks but also stock portfolios to be the dependent variable.

Since those people feel more comfortable saying "use 3FM to estimate alpha" rather than "use CAPM to estimate alpha", you can use 3FM as your regression model to estimate the alpha (if any).

Hope my complicated (I also think so =_=) explanation helps your understanding.

 

[chiyui]

You can treat "incremental performance of the asset relative to the benchmark return" and "the return attributable to skill or luck of the portfolio manager" are both containing the same meaning - they are the abnormal return due to something not able to be explained by the benchmark/market or by the specific risk of the asset (either an individual stock or a portfolio manager).

 

[Rufolo]

Well first of all let me thanks your very nice explanation, and then for taking off your time¡
I would try both, capm and 3fm, and see what i get, having in mind that 3fm is more accurate than capm but also more messi to compute.
In a simple answer, wouls you use intercept formula in excel to calculate what i want?also, would you use daily returns since last year? Lets have a quick example with Google. I would use weekly returns for the last year againt sp500. You agree?
I am ot yet get used to not to have a unique result with these quant formulas, is more the way you read them. I mean, because you obtain different results using Capm or 3FM to obtain alpha. Just comming to my mind, what does a negative alpha mean?

Last thing bothering you... Any video, document,... To read/watch about ANOVA results? I would like to understand all its factors.

Thanks a lot for your explanations and time!

 

[chiyui]

Well if you use the single factor model (i.e. what you called using CAPM), you will get an alpha larger than if you use the multi-factor model (i.e. 3FM).
This is because the larger part of the alpha is due to the two extra factors in 3FM. When you did not include these two extra factors in the CAPM, of course the alpha will be inflated.

I don't know what is the "intercept formula" in excel so I won't use it unless I know what it is.
For me, I will use the VBA Analysis Toolpak installed in excel. The tool pack consists of many calculation procedures, including regression.
Because you gotta estimate the single factor or multi-factor model in terms of regression so as to read up the alpha parameter in the result, you just use the regression procedure in the tool pack and read up the alpha estimate as shown in the spreadsheet, and mission completed.

For data, to be frank I also don't really remember how should I deal with the time frame of the data. For me, I don't mind using daily, weekly, monthly or yearly data. In my opinion it is only an empirical issue. No one should be blamed if he/she use daily data instead of monthly data, for example.

But what you said about S&P 500 should be questioned a little bit. If your dependent variable is a weekly return of a stock of S&P 500's constituent, then you can use S&P 500. Otherwise, you should use the appropriate stock index that your dependent variable stock belongs to. For example, you should use Nasdaq 100 if you're regressing Facebook's stock return.

Of course you'll get different result when you use,
1. different equation (single factor or multi-factor?),
2. different time frame (daily or monthly?),
3. different sample period (using 1980~1990 or 2000~2010?).
This is typical.

That's why I don't like CAPM or 3FM. All these models just give too different results even though I change my data setting only a little bit. I don't believe them much.
But who cares? They're the industry standard. So I can't help but just getting endure to them.
(Actually there're many more reliable models than CAPM or 3FM in the literature. But none of them are recognized as textbook standard so no one learns them compulsorily.)

If you get a negative alpha, that means the stock/portfolio/asset provides an underperformance than the benchmark. Say if the alpha is -2%, and the benchmark is 10%. Then you'd better invest in the benchmark portfolio (e.g. S&P 500 ETF), and you'll get an expected return of 10% in the long run. If you invest in the -2% alpha asset, you'll get an expected return of only 8% in the long run.
You can also do this if you can: sell short the -2% alpha asset and use the proceeds to invest in S%P 500 ETF. Then in the long run you'll lose 8% in the asset (you're short in it) and you'll gain 10% in S&P 500 ETF. The net gain is then 2%. But you did not put any money of your own!

Just a spoiler, there're many literature claiming that they find a negative alpha in most of the active mutual funds in the earth. These guys say that those funds charge a so high management fee that makes their alpha negative. So they claim that we shouldn't invest in these funds and instead we should invest in passive ETFs......

I'm not expert in CAPM/EMH/Fama-French, so maybe I have something incorrect and to the admins, please correct my comment if necessary.

 

[chiyui]

I don't have any idea of video/document about ANOVA, so I can only suggest you to read a book which was used to be my favorite: John E. Freund's Mathematical Statistics
This is a good book. Clear mathematical procedure, rigorous argument, and most importantly - very incisive, no rubbish sentences.
Chapter 15 devotes wholly to ANOVA. I used to learn from it when I was taking the Hong Kong Statistical Society exams........

 

[Rufolo]

Hello again,


So as I have collected these past days I’d say that:

1.- Alpha is the result of substracting the portfolio’s return from benchmark’s return.
2.- Also we can use the simple factor model (through CAPM formula) to obtain the alpha for a portfolio, which it would be the difference between the return we estimated for the portfolio and the one really realized. This can show us how much of the return is from manager’s skills.
We can make it more accurate through the 3FM which includes also 2 more factor than the previous formula. This’ll made the alpha goes down because we explain it through 2 more factors.

What I am a little bit confuse is the number 1.- and number 2.- . They shouldn’t be named the same right? They don’t represent same data.
As we can measure it with single factor model, 3F, 4FM, monthly, weekly,… I came to the conclusion that best accurate and easy way is single factor (although I am not agree at all with any of these methods)

Last thing I’d like to know…as Fama and French did “discover” the 3FM with the HML and SBM factors…how did they discover them? Through how many stocks, indices,…? It is not published right?

Thanks a lot once more!

 

[chiyui]

If you subtract the portfolio return from the benchmark return, the difference is called tracking error, not alpha.
Alpha must be calculated by reading the intercept of your regression model.
So of course they're different.

Tracking error is the distance of portfolio return from the benchmark.
So as you can guess, it measures how strong the portfolio performance is anchored to the benchmark performance.
It does not measure how well the manager has done his/her job.
FYI, you can also calculate the standard deviation of the tracking error. It is called tracking error volatility.
And it can be shown that the tracking error volatility is just equal to the standard error of the residual in your regression model (also called non-systematic risk, specific risk, idiosyncratic risk).

But well, tracking error volatility does have some relationship with alpha.
The ratio of alpha to the tracking error volatility is called the information ratio.
It is a common measurement of fund manager's performance.

The information ratio measures the alpha per unit of tracking error volatility (non-systematic risk). But what does it mean?
Well, suppose you have a stock fund specializing in small-cap companies. Because the fund covers small-caps only, it is not well-diversified and as a result it contains a significant portion of non-systematic risk.
How should you measure this fund's manager's performance? Because the fund is not well-diversified, you will expect the fund manager to do a good job only when he/she provides a significantly high alpha. This is a compensation of intentionally not to well-diversify the fund portfolio.
So you should use information ratio in this case. If the information ratio is large, this means the manager does a good job because he/she can generate a high alpha per unit of non-systematic risk. Putting in another way, he/she utilizes the non-systematic risk very well.

-------------------------------------------

Of coz Fama and French has published the 3FM in the Journal of Finance.
See this -> Fama, Eugene F.; French, Kenneth R. (1992). "The Cross-Section of Expected Stock Returns". Journal of Finance 47 (2): 427–465.

But if you asked me how they discover them.........to be frank, I think they only came across with it all of a sudden XD
In layman's term, HML and SBM is just "value stock" and "growth stock". This has been many mutual funds' investment theme for a long time already.
So I think Fama and French only make a research on this already-existing concept among the investment community. And I think they just happened to be successfully capturing this concept by using math equation (i.e. the 3 factor model). Nothing special.

 

[Rufolo]

Ok i am finishing asking! Last question.
I am not agree about tracking error. I calculate it measuring the std deviation of the diferrence of the returns between portfolio and benchmark... You agree?
For your previous explanation,if i have a portfolio full of smallcaps... Do i need to use information ratio instead raw alpha from the one factor formula?

And in order to calculate a more accurated alpha should i use 3fm because it covers more range of explanation with 3 variables instead just one?

Thanks!,

 

[chiyui]

According to Phillipe Jorion's 2011 FRM Handbook Plus Test Bank, he defines the tracking error as the difference between portfolio return and benchmark return. Then he defines the tracking error volatility as the std deviation of the difference between portfolio return and benchmark return.

Well it depends on what are you going to do with the small-cap fund.
If you're just going to compare the fund manager's skill with other similar funds in absolute term, then you just use alpha.
If you're going to compare the fund manager's skill with other similar funds in risk-adjusted term, then you use information ratio.
This question is similar to asking whether one should use expected return or Sharpe ratio as the performance criteria.

You can use 3FM but I can tell you that some people don't believe that 3FM's alpha is more accurate, especially when your fund sample does not specialized in value strategy or growth strategy.
Their common argument is that value and growth strategy becomes less significant drivers of fund performance in the recent years becoz there're so many fund managers pursuing this strategy. This crowd will make the strategy less reliable due to the curse of the Efficient Market Hypothesis......
So if I were you, I'd use 3FM if I believed it indeed, while I would bear in mind those criticisms at the same time.

 

[David Harper CFA FRM]

Great conversation, thank you chiyui for some your awesome insights. I just wanted to share what we have learned about the information ration merely with respect to the FRM

First, I personally consider tracking error to be settled (in the FRM): TE is active risk, not residual risk; i.e., the standard deviation of the series of the difference between the portfolio and the benchmark/index. Within the FRM, imo, this has never really changed (e.g., Grinold). In this way, I consider accurate to be @Rufolo's definition of tracking error: "I calculate it measuring the std deviation of the difference of the returns between portfolio and benchmark"
... although Jorion and Bodie do employ TE = port return - bench return (i.e., as merely the difference), so that StdDeviation(port return - bench return) becomes Jorion's TEV and Bodie's Benchmark Risk, this is a semantic inconvenience that never seems to get in the way of the fact that Tracking Error tends to be treated like a volatility not just a difference.

Information ratio is the definition that we've asked to be standardized. Details on the conflicting usage are found in my note to GARP here at http://forum.bionicturtle.com/threads/information-ratio-definition.5554/ , but where GARP has settled on, I think, a fine solution: IR = active return/TE or alternatively (the more technically pure) residual return/residual risk (i.e., notice the ratio consistency) but, unless otherwise specified, is not the ratio inconsistent active/residual. To support this conclusion, here is answer to 2012 GARP Practice Exam Question 3:

Question 3. "Explanation: The information ratio may be calculated by either a comparison of the residual return to residual risk, or the excess return to tracking error." -- GARP's 2012 Part 1 Practice Exam

 

[chiyui]

Actually I'm just 依書直說 (making interpretation only according to existing books/literatures and making as less subjective comments involved as possible) only. So I'm only stating those authors' insights. Thanks for your comments.

Now I get what you're saying. I've only read Bodie/Kane/Marcus's Investment and Jorion's FRM Handbook. So I got my first impression on distinguishing between the difference itself and the volatility of the difference. But I agree with you - no one use the difference itself straightly. In all practical aspects people use its volatility (stdev) but not itself. So it really makes the communication confusing as you said, a semantic inconvenience.

That's why I like to use symbols instead of words - IR = α/σ(e).
There may be confusion on what the term "residual return/excess return" means if one is not familiar with the communication method. But there's a unique meaning of α in this context. The same reasoning applies on σ(e).
By looking at the symbolic definition of IR, there will be no need to argue for the calculation method.

Thank you for correcting my limited cognition on this subject.

 

[David Harper CFA FRM]

I could not agree with you more: α and σ(e) are unique and, in the context of a regression, there is really only one way to compute them: alpha is the intercept. And, arguably, the only true IR is the one you've just defined.

Further, by viewing these as regression symbols, it is easier to appreciate the difference between an ex ante perspective, which is the expectation that an asset/portfolio will fall "on the line" (e.g., CAPM manifest as the SML tells us the expected alpha is zero) compared to a ex post perspective, which is the recognition that we are regressing on a realized sample (and also depends on an single or multi-factor model; highly simplified, we should not be surprised if the residual contains un-regressed factors, so called "alpha disguised as beta") such that the intercept is a sample alpha which can be some combination of luck (sampling variation) and skill (genuine non-correlated contribution to the return).

The regression context has a way of highlight the fragility of alpha:

• depends on the regression assumptions; omitted variable bias?
• highly susceptible to sampling variation (infected by luck). Statistical evidence of skill in the alpha requires a depressingly long time series

Thanks,

 

[Rufolo]

Well, thanks both for these great arguments. Finally, i would like you both to give me your opinion.
As my portfolio is based on low value and high growth, i definitely think i should use the 3FM. Please correct me if my thoughts are wrong.

Thanks for these great explanation, moreover thanks to chiyui for being the first in answering my question and kept going on on this post.

Thanks,