some more musings about the linear regression, and the ALPHA and BETA

some more musings about the linear regression, and the ALPHA and BETA

1. Investment Portfolio
Management
FIN 421
Luc Faucheux, PhD
Fall 2025

2. Straight Lines
Part Deux:
do you care about
alpha-beta ?

3. Straight lines are everywhere
3
Luc Faucheux 2025

4. Straight Lines - I
4
Luc Faucheux 2025
“God does not build in straight lines” Prometheus.. (the movie not the Greek Titan god)
CHECKED, I watched the movie, awesome movie…

5. Straight Lines - II
5
Luc Faucheux 2025
This is not a class in AI, that is another class that I also teach if you are interested

6. Straight Lines - III
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Luc Faucheux 2025
CHECKED, we did that in Part UN…

7. Straight Lines - IV
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Luc Faucheux 2025
Still to do

8. Straight Lines - V
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Luc Faucheux 2025
This is the focus of this deck

9. Straight Lines - VI
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Luc Faucheux 2025
Still to do

10. Straight Lines - VII
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Luc Faucheux 2025
�−??????
??????=�+�.�
�−??????
??????+??????
??????.��??????+??????
??????.??????��
Still to do

11. Straight Lines - VIII
11
Luc Faucheux 2025
•Sampling – Measuring – OLS – Linear regression DONE IN PART UN
•Predicting – Modeling – Security Market Line – CAPM STILL TO DO
•Evaluating Performance – alpha and beta THIS IS THIS DECK : PART DEUX
•Explaining away – Factor Analysis – PCA STILL TO DO
•Explaining away – French Fama factor analysis STILL TO DO
•In this deck we recap the Linear regression and use it on the popular problem of alpha-beta in
returns, can you separate skills from luck?
•Part Trois, and Quatre will be about the CAPM, the SML, factor analysis, PCA and the French
Fama framework

12. RECAP OF PART UN
Measuring – Sampling – Linear regression
12
Luc Faucheux 2025

13. RECAP: Measuring – Sampling – Linear regression - I
•With one variable : Sampling a distribution of returns {�
??????} with N observations
•A reasonable approximation is that the observations are iid
•Independent: ??????
��=0 except when �=�
•Identically distributed: ??????�
�=?????? and V�
�=??????
2
•The Mean Estimator (Sample Mean) is: Ƹ??????=
1
�
.σ
��
�
•??????Ƹ??????=??????
•??????Ƹ??????=??????
2
Ƹ??????=
??????
2
�
•Ƹ?????? is BLUE (Best Linear Unbiased Estimator)
•Ƹ??????՜
??????
??????(??????,
??????
2
�
)
•Standard Deviation (population) ??????=??????[�]
•Standard Error (sample)
??????
�
=??????[Ƹ??????]
13
Luc Faucheux 2025

14. RECAP Measuring – Sampling – Linear regression - II
•With one variable : Sampling a distribution of returns {�
??????} with N observations
•Standard Deviation (population) ??????=??????[�]
•Standard Error (sample)
??????
�
=??????[Ƹ??????]
•Sample Variance Estimator
•
෢
??????
2
=
1
�
.σ
�(�
�−Ƹ??????)
2
•??????
෢
??????
2
=
�−1
�
??????
2
•??????
෢
??????
2
=??????
2
Ƹ??????=
??????
4−??????
4
??????
2
−
??????
2
�
14
Luc Faucheux 2025

15. RECAP: Measuring – Sampling – Linear regression - III
•With TWO variables : Sampling a distribution of returns {�
??????,�
??????} with N observations
15
Luc Faucheux 2025
�
�
•(�
1,�
1)
•(�
2,�
2)
•(�
3,�
3)
•(�
4,�
4)
•(�
5,�
5)
•(�
6,�
6)
•(�
7,�
7)

16. RECAP: Measuring – Sampling – Linear regression - IV
•With TWO variables : Sampling a distribution of returns {�
??????,�
??????} with N observations
16
Luc Faucheux 2025
�
�
•
•
•
•
•
•
•
•So really a linear regression has not
TWO but THREE variables: �, Y, AND ℰ
•Modeling: Y=α+�.�+ℰ
•Sampling: �
�=α+�.�
�+ℰ
�
•outcome: �
�=α+�.�
�+??????
�
(�
�,�
�)
ℰ
�

17. RECAP: Measuring – Sampling – Linear regression - V
17
Luc Faucheux 2025
•So really a linear regression has not TWO but THREE variables: �, Y, AND ℰ
•Modeling: Y=α+�.�+ℰ
•Sampling: �
�=α+�.�
�+ℰ
�
•Minimizing: �
�=ෝα+መ�.�
�+෡ℰ
�
•Observing: �
�=α+�.�
�+??????
�
•So all THREE of those variables have their own distributions, with of course mean and standard deviation,
among many other things….
•If you think about it, it makes a lot of sense, there were TWO variables �, Y, but by trying to find a linear
relationship between those TWO we are obviously introducing a THIRD one ℰ. In many ways you can say tht
we are replacing Y by ℰ by imposing that Y=α+�.�+ℰ
•ෝα and መ� are also random variables (since they are estimators built from other random variables), and so will
also have their own distribution, mean, standard deviation,…you can think of α,� as free parameters and
the specific set (ෝα,መ�) as a solution for those parameters (in OLS the one minimizing the sum of the squares of
the residuals)

18. RECAP: Measuring – Sampling – Linear regression – VI
18
Luc Faucheux 2025
•Under some reasonable ”Reasonable” assumptions, one can show that (OLS, Ordinary Least Squares):
•Sample slope estimator: መ�=
ෟ??????
��
ෟ??????��
=ෞ??????
��.
ෞ??????
�
ෞ??????�
•With:
•ෞ??????
��=
1
�
.σ
��
�−ෞ??????
�.(�
�−ෞ??????
�)
•ෞ??????
�=
1
�
.σ
��
�
•ෞ??????
�=
1
�
.σ
��
�
•ෞ??????
��=
1
�
.σ
��
�−ෞ??????
�.(�
�−ෞ??????
�)
•Sample intercept estimator as: ො�=ෞ??????
�−መ�.ෞ??????
�

19. RECAP: Measuring – Sampling – Linear regression - VII
19
Luc Faucheux 2025
•Sample slope estimator: መ�=
ෟ??????
��
ෟ??????
��
=ෞ??????
��.
ෞ??????
�
ෞ??????
�
=
σ
??????
�
??????−ෞ??????
�.(�
??????−ෞ??????
�)
σ
??????
�
??????−ෞ??????
�.(�
??????−ෞ??????
�)
•Sample intercept estimator as: ො�=ෞ??????
�−መ�.ෞ??????
�
•??????ො�=� ??????መ�=�
•�.ො�−�՜
??????
??????(0,
??????
2
(??????
��+??????
�
2
)
??????
��
) �.መ�−�՜
??????
??????(0,
??????
2
??????
��
)
•In fact ො� and መ� are jointly asymptotically (bivariate) normally distributed:
•�.
ො�−�
መ�−�
՜
??????
??????(
0
0
,
??????
2
(??????
��+??????
�
2
)
??????
��
−
??????
2
??????
�
??????
��
−
??????
2
??????
�
??????
��
??????
2
??????
��
)

20. RECAP: Measuring – Sampling – Linear regression - VIII
20
Luc Faucheux 2025
•Sample mean estimator: ෞ??????
�=
1
�
.σ
��
�
•Sample variance estimator:
෢
??????
�
2
=ෞ??????
��=
1
�
.σ
�(�
�−ෞ??????
�)
2
•Sample slope estimator: መ�=
ෟ??????
��
ෟ??????
��
=ෞ??????
��.
ෞ??????
�
ෞ??????
�
=
1
??????
.σ
??????
�
??????−ෞ??????
�.(�
??????−ෞ??????
�)
1
??????
.σ
??????
�
??????−ෞ??????
�.(�
??????−ෞ??????
�)
•Sample intercept estimator: ො�=ෞ??????
�−መ�.ෞ??????
�
•Sample “model residuals” estimator: ෡ℰ
�=�
� −ො�−መ�.�
�
•Sample “model residual sum of squares” estimator: σ
�
෡ℰ
�
2

21. RECAP: Measuring – Sampling – Linear regression - XIX
21
Luc Faucheux 2025
•For large sample, drop the hat, but do not drop the ball…and use the Normal Distribution Function unless
told otherwise….

22. ALPHA AND BETA
22
Luc Faucheux 2025

23. ALPHA and BETA - I
23
Luc Faucheux 2025
•Some of the most used, and
misused words in finances….
•Just Google news alpha and just
today I got that….

24. ALPHA and BETA - Ia
24
Luc Faucheux 2025
•Some of the most used, and
misused words in finances….
•Just Google news alpha and just
today I got that….

25. ALPHA and BETA - Ib
25
Luc Faucheux 2025
•Some of the most used, and
misused words in finances….
•Just Google news alpha and just
today I got that….

26. ALPHA and BETA - II
26
Luc Faucheux 2025
•A lot of people are “Seeking ALPHA”

27. ALPHA and BETA - III
27
Luc Faucheux 2025
•A lot of people are “Seeking ALPHA”

28. ALPHA and BETA - IV
28
Luc Faucheux 2025
•If I was to be very simplistic
•ALPHA is the INTERCEPT
•BETA is the SLOPE
•Of plotting something as a
function of something else.
•If you plot the return of a stock
as function of an index or group
of stocks, you are trying to
identify some characteristics of
that stock

29. ALPHA and BETA - V
29
Luc Faucheux 2025
•You could also plot a return as a function of a group of stocks (for example beer and liquors) see if that
specific stock behaves as the cluster does
•You could plot the return of a specific hedge fund against a cluster of similar hedge funds in terms of their
description (macro, trend following, CTA, relative value, index arbitrage,..) to see if that hedge fund is lying to
you
•You could also make up that line based on what you think the stock, or your trading strategy is
•For example CAPM (Capital Asset Pricing Model) is really nothing more than a straight line

30. ALPHA and BETA - VI
30
Luc Faucheux 2025
•Sometimes Wiki just says it better than I would try

31. ALPHA and BETA - VII
31
Luc Faucheux 2025
•In the Efficient Market Hypothesis (EMH) the ALPHA has to be zero
•You can think of ALPHA as being linked to INFORMATION
•If a manager has better INFORMATION than the market (i.e. the other market participants), the manager
should be able to realize positive ALPHA (INFORMATION could be legal or not by the way)
•If a manager is better at analyzing INFORMATION (faster, more accurate) than the market (i.e. the other
market participants), the manager should be able to realize positive ALPHA
•This is why the article with Ken Griffin is so relevant, hedge funds are seeking ALPHA, and AI should in theory
help them achieve better performance extracting, treating, analyzing INFORMATION (or maybe not
according to Ken)

32. ALPHA and BETA - VIII
32
Luc Faucheux 2025
•You can think of BETA as being linked to EXECUTION
•A manager of a 2x leverage ETF for example should have a BETA of 2 with respect to the underlying index or
asset of the ETF
•BETA will deteriorate with EXECUTION costs (mistakes, timing errors, outright costs of running the ETF or the
fund,..)

33. Do you trust ALPHA and BETA ?
33
Luc Faucheux 2025

34. Do you trust ALPHA and BETA - I
34
Luc Faucheux 2025
•We saw in Part UN that the ALPHA and BETA are estimators and as such as random variables with their own
mean and standard deviation.
•So how well can you trust ALPHA and BETA ?
•Let’s take the example that we had in Part UN

35. Do you trust ALPHA and BETA - II
35
Luc Faucheux 2025
•We saw in Part UN that the ALPHA and BETA are estimators and as such as random variables with their own
mean and standard deviation.
•So how well can you trust ALPHA and BETA ?
•Let’s take the example that we had in Part UN

36. Do you trust ALPHA and BETA - III
36
Luc Faucheux 2025
•You can replace X and Y by anything you want along the lines of what we talked before, but for the purpose
of the exercise let’s say that it is a hedge fund claiming to have a beta of 2, and an ALPHA of 10bps
•Let’s assume everything that we need to assume in order to be able to make some predictions (the returns
are iid, constant in time, …)

37. Do you trust ALPHA and BETA - IV
37
Luc Faucheux 2025
•We computed in Part UN a number of quantities, in particular the following:
•So it looks like the manager was right on the BETA, but maybe a wee little bit tad optimistic on the ALPHA
•But how right ? Or how wrong ?
•There again we use some of the results we went over in Part UN:
•�.
ො�−�
መ�−�
՜
??????
??????(
0
0
,
??????
2
(??????
��+??????
�
2
)
??????
��
−
??????
2
??????
�
??????
��
−
??????
2
??????
�
??????
��
??????
2
??????
��
)
ALPHA BETA
0.0164052.001427

38. Do you trust ALPHA and BETA - V
38
Luc Faucheux 2025
�.
ො�−�
መ�−�
՜
??????
??????(
0
0
,
??????
2
(??????
��+??????
�
2
)
??????
��
−
??????
2
??????
�
??????
��
−
??????
2
??????
�
??????
��
??????
2
??????
��
)
•Remember, drop the funny hat, and use the estimates for the population
•Just for example, the numbers in our Excel example turned out to be:

39. Do you trust ALPHA and BETA - VI
39
Luc Faucheux 2025
•What this shows is that for you to claim that
the manager is wrong and that there is only a
5% probability that the true ALPHA is greater
or equal to 10bps, you would need a lot more
points to sample, in fact almost 6 times more
points than you were given.
•Similarly, there is only a 75% probability that
the actual BETA is greater than 1.85
•In order to get this probability above 95% you
would need also roughly 6 times more
samples or roughly 35 samples

40. Do you trust ALPHA and BETA - VII
40
Luc Faucheux 2025
•So in our case we got 6 observations
•There is (again under some fairly reasonable but having to be justified assumptions) a 25% probability
that the BETA will be under 1.85
•For this probability to go under 5%, we would need 6 times more observations (roughly because as you
get more observations, the actual values of those observations will impact your estimates, say for
example the next 10 observations are lining up on the horizontal axis, this will obviously push your
running estimate of BETA down, and might turn the “6 times more” into a “20 times more” very easily
•CAREFUL, what I just wrote above is not exactly correct, I told you that statistics are really hard because
it is one of the branches of mathematics that rely the most on words, and it is very hard to use words
without being impacted by some of the meaning that those words have…
•And sometimes the meaning of words is actually not in the words…

41. Do you trust ALPHA and BETA - VIII
41
Luc Faucheux 2025
•In Statistics, sometimes the meaning of the word is not in the word…
•Or as Wittgeinstein would say, the meaning of the world does not reside in the world….

42. Do you trust ALPHA and BETA - IX
42
Luc Faucheux 2025
•CAREFUL, what I just wrote above is not exactly correct, I told you that statistics are really hard because
it is one of the branches of mathematics that rely the most on words, and it is very hard to use words
without being impacted by some of the meaning that those words have…
•So really I should not be saying: “There is a 25% probability that the BETA will be under 1.85” even
though we kind of all get what I am trying to say
•But really I should be saying something more along the lines of:
•Suppose that I can repeat the sampling of 6 observations as many times as I want
•Suppose that I can build a histogram of all those different estimates of BETA
•Suppose now that the true BETA is indeed equal to 2 (also called Null Hypothesis)
•In the limit of a very large number of repeated samples, there should be (again making a bunch of
assumptions that might or might not hold in the real world) a 25% probability that the estimate of BETA
for a given sample is lower than 1.85
•I think that I am not wrong in saying that “the left-one-sided p-value of BETA=1.85 is 25%”
•But then again with statistics you have to be very careful about what you say…

43. Do you trust ALPHA and BETA - X
43
Luc Faucheux 2025
•But in any case you get the gist of it…
•When evaluating a trading strategy, a fund manager, a hedge fund, even say a stock, you need enough
samples
•If there is not enough samples, there is no track record and you cannot prove or disprove the claims of
the fund manager
•The number of samples that you really need to get to a 95% ”confidence level” (again term being used
very loosely here, and I should be a lot more careful), tends to be a lot more than what you can get
•On the example below, that would be roughly 40 or so samples, if you are assuming yearly returns this
is obviously impractical
•Even more to the point, you would need at least a three-year track record to be even given 5 minutes of
the time of an allocator
•But the allocator cannot say anything that would be statistically sound unless they have at least 10
years (in our example to get to a 5% level we would need around 40years)
•AND crucially, over that period of time, most likely things will change in such a way that your statistical
analysis is now flawed (say for example you cannot rely on IID anymore)

44. Do you trust ALPHA and BETA - XI
44
Luc Faucheux 2025
•So you may ask ”Why even drill a fund manager about their ALPHA and BETA if you know that you will
not be able to use this information in a statistically sound or significant manner?”
•Very good question my little ones. We cover that in class in the Due Diligence section
•It is more like a detective questioning a suspect and asking questions that the detective already knows
the answer to, or irrelevant questions
•It is more about gauging the reaction of the suspect (the fund manager) and their ability to produce
reports, numbers, almost on the fly, and observe and measure their processes
•So ALPHA and BETA are still going to be around for while, with all their tweaks and variations such as
“Smart BETA” for example, that are really just rebucketting some returns in a different category

45. Nest deck: factor analysis, French-Fama,
PCA and all that good stuff….
45
Luc Faucheux 2025

46. So at least for now…
46 Luc Faucheux 2025