some more straight lines in finance, security market line, capital market line, capital allocation line, security characteristic lines, lines lines lines....
lucfaucheux
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113 slides
Oct 24, 2025
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About This Presentation
some more straight lines in finance, security market line, capital market line, capital allocation line, security characteristic lines, lines lines lines....
Slide Content
Investment Portfolio
Management
FIN 421
Luc Faucheux, PhD
Fall 2025
Management
FIN 421
Luc Faucheux, PhD
Fall 2025
Straight Lines
Part Trois:
More alpha-beta:
CAPM
SML, SCL
CML, CAL
EMH
FRENCH-FAMA
FACTORS
PCA
Part Trois:
More alpha-beta:
CAPM
SML, SCL
CML, CAL
EMH
FRENCH-FAMA
FACTORS
PCA
Straight lines are everywhere
3
Luc Faucheux 2025
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Luc Faucheux 2025
Straight lines are everywhere in Finance 421
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Luc Faucheux 2025
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Luc Faucheux 2025
Straight Lines - I
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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
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…
Straight Lines - II
6
Luc Faucheux 2025
This is not a class in AI, that is another class that I also teach if you are interested
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Luc Faucheux 2025
This is not a class in AI, that is another class that I also teach if you are interested
Straight Lines - III
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Luc Faucheux 2025
CHECKED, we did that in Part UN…
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Luc Faucheux 2025
CHECKED, we did that in Part UN…
Straight Lines - IV
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Luc Faucheux 2025
Still to do
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Luc Faucheux 2025
Still to do
Straight Lines - V
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Luc Faucheux 2025
DONE IN PART DEUX, BUT WILL REVISIT SOME HERE
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Luc Faucheux 2025
DONE IN PART DEUX, BUT WILL REVISIT SOME HERE
Straight Lines - VI
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Luc Faucheux 2025
Still to do
10
Luc Faucheux 2025
Still to do
Straight Lines - VII
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Luc Faucheux 2025
�−�
�=�+�.�
�−�
�+�
�.���+�
??????.���
Still to do
11
Luc Faucheux 2025
�−�
�=�+�.�
�−�
�+�
�.���+�
??????.���
Still to do
Straight Lines - VIII
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Luc Faucheux 2025
•Sampling – Measuring – OLS – Linear regression DONE IN PART UN
•Predicting – Modeling – Security Market Line – CAPM IN THIS DECK
•Evaluating Performance – alpha and beta DONE IN PART DEUX
•Explaining away – Factor Analysis – PCA WILL START IN THIS DECK
•Explaining away – French Fama factor analysis IN THIS DECK
•In this deck we recap the Linear regression, recap a little of the alpha-beta, and start applying it
to ”real-world” in CAPM?
12
Luc Faucheux 2025
•Sampling – Measuring – OLS – Linear regression DONE IN PART UN
•Predicting – Modeling – Security Market Line – CAPM IN THIS DECK
•Evaluating Performance – alpha and beta DONE IN PART DEUX
•Explaining away – Factor Analysis – PCA WILL START IN THIS DECK
•Explaining away – French Fama factor analysis IN THIS DECK
•In this deck we recap the Linear regression, recap a little of the alpha-beta, and start applying it
to ”real-world” in CAPM?
RECAP OF PART UN
Measuring – Sampling – Linear regression
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Luc Faucheux 2025
Measuring – Sampling – Linear regression
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Luc Faucheux 2025
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??????]
14
Luc Faucheux 2025
•Sample Variance Estimator
•
??????
2
=
1
�
.σ
�(�
�−Ƹ??????)
2
•??????
??????
2
=
�−1
�
??????
2
•�
??????
2
=??????
2
Ƹ??????=
??????
4−??????
4
??????
2
−
??????
2
�
•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??????]
14
Luc Faucheux 2025
•Sample Variance Estimator
•
??????
2
=
1
�
.σ
�(�
�−Ƹ??????)
2
•??????
??????
2
=
�−1
�
??????
2
•�
??????
2
=??????
2
Ƹ??????=
??????
4−??????
4
??????
2
−
??????
2
�
RECAP: Measuring – Sampling – Linear regression - II
•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)
•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)
RECAP: Measuring – Sampling – Linear regression - III
•With TWO variables : Sampling a distribution of returns {�
??????,�
??????} with N observations
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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
�
•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
�
RECAP: Measuring – Sampling – Linear regression - IV
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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)
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)
RECAP: Measuring – Sampling – Linear regression - V
18
Luc Faucheux 2025
•Modeling: Y=α+�.�+ℰ
•Sampling: �
�=α+�.�
�+ℰ
�
•Minimizing: �
�=ෝα+መ�.�
�+ℰ
�
•Observing: �
�=α+�.�
�+??????
�
•A quick note here from a discussion in class. I know that a lot of textbooks are a little lazy on the notation. I
try as much as possible to keep:
•Random Variable : CAPITAL LETTER
•Number or “regular” (Newton-Leibniz) continuous variable : lower case
•I think that it helps avoid a lot of confusion, and we can illustrate it from the following example using some
selected students and their height.
18
Luc Faucheux 2025
•Modeling: Y=α+�.�+ℰ
•Sampling: �
�=α+�.�
�+ℰ
�
•Minimizing: �
�=ෝα+መ�.�
�+ℰ
�
•Observing: �
�=α+�.�
�+??????
�
•A quick note here from a discussion in class. I know that a lot of textbooks are a little lazy on the notation. I
try as much as possible to keep:
•Random Variable : CAPITAL LETTER
•Number or “regular” (Newton-Leibniz) continuous variable : lower case
•I think that it helps avoid a lot of confusion, and we can illustrate it from the following example using some
selected students and their height.
RECAP: Measuring – Sampling – Linear regression – V-a
19
Luc Faucheux 2025
•Suppose that I think that the height of people in Miami Dade follows some distribution
•I think for example that this distribution has a mean, which I want to estimate
•The height of people in Miami is a random (stochastic variable) that I will note � (CAPITAL LETTER because it is a random variable)
•If I was to plot the distribution of heights (if I know it for example it is a Normal Distribution Function of mean:
•??????�=??????
•and variance:
• V�=??????
2
,
•I will plot ??????(??????,??????
2
) on the y-axis with � on the x-axis (lower case because it is regular calculus)
•In fact the modern way to define probability is the following:
•From a random variable � you define a function of the continuous variable �, as the probability that the random variable � has a value lower
than �: �(�≤�)
•This illustrates the difference between the CAPITAL LETTER random variable and the lower case continuous (regular, Newton, Leibniz)
variable that we are used to. This distinction is crucial when doing stochastic calculus, less so when dealing with statistics, but when sticking
to this notation you avoid a lot of mistakes, so you might as well do it
•“The right notation is 95% of the work”. K. Godel
19
Luc Faucheux 2025
•Suppose that I think that the height of people in Miami Dade follows some distribution
•I think for example that this distribution has a mean, which I want to estimate
•The height of people in Miami is a random (stochastic variable) that I will note � (CAPITAL LETTER because it is a random variable)
•If I was to plot the distribution of heights (if I know it for example it is a Normal Distribution Function of mean:
•??????�=??????
•and variance:
• V�=??????
2
,
•I will plot ??????(??????,??????
2
) on the y-axis with � on the x-axis (lower case because it is regular calculus)
•In fact the modern way to define probability is the following:
•From a random variable � you define a function of the continuous variable �, as the probability that the random variable � has a value lower
than �: �(�≤�)
•This illustrates the difference between the CAPITAL LETTER random variable and the lower case continuous (regular, Newton, Leibniz)
variable that we are used to. This distinction is crucial when doing stochastic calculus, less so when dealing with statistics, but when sticking
to this notation you avoid a lot of mistakes, so you might as well do it
•“The right notation is 95% of the work”. K. Godel
RECAP: Measuring – Sampling – Linear regression – V-aa
20
Luc Faucheux 2025
•“The right notation is 95% of the work”. K. Godel
•Kurt Godel on the left with an unidentified peasant most likely from Central
Europe on the right..
20
Luc Faucheux 2025
•“The right notation is 95% of the work”. K. Godel
•Kurt Godel on the left with an unidentified peasant most likely from Central
Europe on the right..
RECAP: Measuring – Sampling – Linear regression – V-b
21
Luc Faucheux 2025
•So assume that the height follows a Normal Distribution Function:
•X~??????(??????,??????
2
)
•That means that:
•��≤�=
−∞
??????1
2????????????
2
.exp−
??????−??????
2
2??????
2
.�??????
•
•And the probability density function is:
•��=
??????
????????????
��≤�=
??????
????????????
−∞
??????1
2????????????
2
.exp−
??????−??????
2
2??????
2
.�??????=
1
2????????????
2
.exp−
??????−??????
2
2??????
2
•That is the usual Gaussian on the right-hand side
•The functions on the right-hand side is a regular function of � that you can plot with � on the x-axis and do all kind of regular calculus tht you
are used to, and does not depend on � anymore
21
Luc Faucheux 2025
•So assume that the height follows a Normal Distribution Function:
•X~??????(??????,??????
2
)
•That means that:
•��≤�=
−∞
??????1
2????????????
2
.exp−
??????−??????
2
2??????
2
.�??????
•
•And the probability density function is:
•��=
??????
????????????
��≤�=
??????
????????????
−∞
??????1
2????????????
2
.exp−
??????−??????
2
2??????
2
.�??????=
1
2????????????
2
.exp−
??????−??????
2
2??????
2
•That is the usual Gaussian on the right-hand side
•The functions on the right-hand side is a regular function of � that you can plot with � on the x-axis and do all kind of regular calculus tht you
are used to, and does not depend on � anymore
RECAP: Measuring – Sampling – Linear regression – V-c
22
Luc Faucheux 2025
•��=
1
2????????????
2
.exp−
??????−??????
2
2??????
2
•That is the usual Gaussian on the right-hand side
•The functions on the right-hand side is a regular function of � that you can plot with � on the x-axis and do all kind of
regular calculus tht you are used to, and does not depend on � anymore
•Centered around ??????, you can plot it using the usual z-score :
•�=
??????−??????
??????
•Note that the Z-score for the random variable will be noted as :
•�=
�−??????
??????
22
Luc Faucheux 2025
•��=
1
2????????????
2
.exp−
??????−??????
2
2??????
2
•That is the usual Gaussian on the right-hand side
•The functions on the right-hand side is a regular function of � that you can plot with � on the x-axis and do all kind of
regular calculus tht you are used to, and does not depend on � anymore
•Centered around ??????, you can plot it using the usual z-score :
•�=
??????−??????
??????
•Note that the Z-score for the random variable will be noted as :
•�=
�−??????
??????
RECAP: Measuring – Sampling – Linear regression – V-d
23
Luc Faucheux 2025
•��=
1
2????????????
2
.exp−
??????−??????
2
2??????
2
the usual Gaussian centered around ??????, you can plot it using the usual z-score : �=
??????−??????
??????
23
Luc Faucheux 2025
•��=
1
2????????????
2
.exp−
??????−??????
2
2??????
2
the usual Gaussian centered around ??????, you can plot it using the usual z-score : �=
??????−??????
??????
RECAP: Measuring – Sampling – Linear regression – V-e
24
Luc Faucheux 2025
•Back to our illustration with height.
•There are 45 or so students in my class
•I select three of them: Rodrigo, Jaden and Sofia
•I do not know their height until I measure them, so UNTIL I measure them, I have to treat them as random variables which I
will note as :
•Height of Rodrigo: �
���??????���
•Height of Jaden: �
�??????���
•Height of Sofia: �
����??????
•AFTER I measure them, I get a number which I will write with a lower case because it is a number for example �
���??????���
•You can see that if I get tired of writing the full name, I can start indexing them 1,2,3 and we get back to the notation that
we know and have been using �
�
24
Luc Faucheux 2025
•Back to our illustration with height.
•There are 45 or so students in my class
•I select three of them: Rodrigo, Jaden and Sofia
•I do not know their height until I measure them, so UNTIL I measure them, I have to treat them as random variables which I
will note as :
•Height of Rodrigo: �
���??????���
•Height of Jaden: �
�??????���
•Height of Sofia: �
����??????
•AFTER I measure them, I get a number which I will write with a lower case because it is a number for example �
���??????���
•You can see that if I get tired of writing the full name, I can start indexing them 1,2,3 and we get back to the notation that
we know and have been using �
�
RECAP: Measuring – Sampling – Linear regression – V-f
25
Luc Faucheux 2025
•I select three of them: Rodrigo, Jaden and Sofia
•I do not know their height until I measure them, so UNTIL I measure them, I have to treat them as random variables which I
will note as :
•Height of Rodrigo: �
���??????���
•Height of Jaden: �
�??????���
•Height of Sofia: �
����??????
•If I assume that the students are taking from the pool of the Miami Dade population (let’s say ??????�=??????=6 ���� to fix the
ideas)
•What can I say about ??????�
���??????��� BEFORE I measure it ?
•Well since Rodrigo is part of the population, �
���??????��� is a sampling of the random variable �, and so �
���??????��� follows the
same distribution as �, and so :
•??????�
���??????���=??????=6 ����
25
Luc Faucheux 2025
•I select three of them: Rodrigo, Jaden and Sofia
•I do not know their height until I measure them, so UNTIL I measure them, I have to treat them as random variables which I
will note as :
•Height of Rodrigo: �
���??????���
•Height of Jaden: �
�??????���
•Height of Sofia: �
����??????
•If I assume that the students are taking from the pool of the Miami Dade population (let’s say ??????�=??????=6 ���� to fix the
ideas)
•What can I say about ??????�
���??????��� BEFORE I measure it ?
•Well since Rodrigo is part of the population, �
���??????��� is a sampling of the random variable �, and so �
���??????��� follows the
same distribution as �, and so :
•??????�
���??????���=??????=6 ����
RECAP: Measuring – Sampling – Linear regression – V-g
26
Luc Faucheux 2025
•What can I say about ??????�
���??????��� BEFORE I measure it ?
•Well since Rodrigo is part of the population, �
���??????��� is a sampling of the random variable �, and so �
���??????��� follows the
same distribution as �, and so :
•??????�
���??????���=??????=6 ����
•HOWEVER suppose that now I actually measure the height of Rodrigo and find 5.7 feet
•So I will write:
•??????�
���??????���=??????=6 ����
•�
���??????���=5.7 ����
•NOW, I want to get from my measurement an estimate of the population mean ??????
•Since I only have one observation the best I can do is define the sample mean Ƹ?????? as Ƹ??????=�
���??????���
26
Luc Faucheux 2025
•What can I say about ??????�
���??????��� BEFORE I measure it ?
•Well since Rodrigo is part of the population, �
���??????��� is a sampling of the random variable �, and so �
���??????��� follows the
same distribution as �, and so :
•??????�
���??????���=??????=6 ����
•HOWEVER suppose that now I actually measure the height of Rodrigo and find 5.7 feet
•So I will write:
•??????�
���??????���=??????=6 ����
•�
���??????���=5.7 ����
•NOW, I want to get from my measurement an estimate of the population mean ??????
•Since I only have one observation the best I can do is define the sample mean Ƹ?????? as Ƹ??????=�
���??????���
RECAP: Measuring – Sampling – Linear regression – V-h
27
Luc Faucheux 2025
•??????�
���??????���=??????=6 ����
•�
���??????���=5.7 ����
•NOW, I want to get from my measurement an estimate of the population mean ??????
•Since I only have one observation the best I can do is define the sample mean Ƹ?????? as Ƹ??????=�
���??????���
•�
���??????��� is a random variable
•Ƹ?????? is a random variable
•??????�
���??????���=??????
•??????Ƹ??????=??????
•CONVERSELY (just flip the equation), if I do not know the population mean ?????? and try to estimate it with measuring, I know
that ??????�
���??????���=?????? and so by measuring Rodrigo I will get an estimate value for ??????
•It might NOT be the actual true value of ?????? (most likely it will not be), but it will be my current estimate
27
Luc Faucheux 2025
•??????�
���??????���=??????=6 ����
•�
���??????���=5.7 ����
•NOW, I want to get from my measurement an estimate of the population mean ??????
•Since I only have one observation the best I can do is define the sample mean Ƹ?????? as Ƹ??????=�
���??????���
•�
���??????��� is a random variable
•Ƹ?????? is a random variable
•??????�
���??????���=??????
•??????Ƹ??????=??????
•CONVERSELY (just flip the equation), if I do not know the population mean ?????? and try to estimate it with measuring, I know
that ??????�
���??????���=?????? and so by measuring Rodrigo I will get an estimate value for ??????
•It might NOT be the actual true value of ?????? (most likely it will not be), but it will be my current estimate
RECAP: Measuring – Sampling – Linear regression – V-i
28
Luc Faucheux 2025
•You can see how that is easily extendable to more sampling and more observations, but just for the sake of completeness:
•??????�
���??????���=??????=6 ����
•�
���??????���=5.7 ����
•??????�
�??????���=??????=6 ����
•�
�??????���=6.3 ����
•??????�
����??????=??????=6 ����
•�
����??????=5.8 ����
•NOW, I want to get from my measurement an estimate of the population mean ??????
•Since I now I have 3 observations I can define the sample mean Ƹ?????? as Ƹ??????=
1
�
.σ
��
�=
(�
���????????????��+�
????????????���+�
���????????????)
3
28
Luc Faucheux 2025
•You can see how that is easily extendable to more sampling and more observations, but just for the sake of completeness:
•??????�
���??????���=??????=6 ����
•�
���??????���=5.7 ����
•??????�
�??????���=??????=6 ����
•�
�??????���=6.3 ����
•??????�
����??????=??????=6 ����
•�
����??????=5.8 ����
•NOW, I want to get from my measurement an estimate of the population mean ??????
•Since I now I have 3 observations I can define the sample mean Ƹ?????? as Ƹ??????=
1
�
.σ
��
�=
(�
���????????????��+�
????????????���+�
���????????????)
3
RECAP: Measuring – Sampling – Linear regression – V-j
29
Luc Faucheux 2025
•Since I now I have 3 observations I can define the sample mean Ƹ?????? as Ƹ??????=
1
�
.σ
��
�=
(�
���????????????��+�
????????????���+�
���????????????)
3
•My estimate of the population mean is thus:
(??????
���????????????��+??????
????????????���+??????
���????????????)
3
=
(5.7+6.3+5.8)
3
=5.93
•That is NOT 6, but it is my current estimate
•It is actually an UNBIASED estimate because ??????Ƹ??????=??????
•Hope that this little section helped to clarify what is a random variable and what is not
•For example:
•??????�
���??????���=??????=6 ����
•�
���??????���=5.7 ����
•??????�
���??????���=�
���??????���=5.7 ���� because �
���??????��� is just a number ( a scalar)
29
Luc Faucheux 2025
•Since I now I have 3 observations I can define the sample mean Ƹ?????? as Ƹ??????=
1
�
.σ
��
�=
(�
���????????????��+�
????????????���+�
���????????????)
3
•My estimate of the population mean is thus:
(??????
���????????????��+??????
????????????���+??????
���????????????)
3
=
(5.7+6.3+5.8)
3
=5.93
•That is NOT 6, but it is my current estimate
•It is actually an UNBIASED estimate because ??????Ƹ??????=??????
•Hope that this little section helped to clarify what is a random variable and what is not
•For example:
•??????�
���??????���=??????=6 ����
•�
���??????���=5.7 ����
•??????�
���??????���=�
���??????���=5.7 ���� because �
���??????��� is just a number ( a scalar)
RECAP: Measuring – Sampling – Linear regression – V-h
30
Luc Faucheux 2025
•Since I now I have 3 observations I can define the sample mean Ƹ?????? as Ƹ??????=
1
�
.σ
��
�=
(�
���????????????��+�
????????????���+�
���????????????)
3
•My estimate of the population mean is thus:
(??????
���????????????��+??????
????????????���+??????
���????????????)
3
=
(5.7+6.3+5.8)
3
=5.93
•That is NOT 6, but it is my current estimate
•It is actually an UNBIASED estimate because ??????Ƹ??????=??????
•A quick note: I could also define an estimator called ��� such as ���=
(2∗�
���????????????��−�
????????????���+�
���????????????)
2
•I claim that this is an estimate of the population mean
•In our case it will be when measured :
(2∗??????
���????????????��−??????????????????���+??????
���????????????)
3
=
(2∗5.7−6.3+5.8)
3
=5.45
•I am actually correct because:
•??????���=??????
(2∗�
���????????????��−�
????????????���+�
���????????????)
2
=
(2∗??????[�
���????????????��]−??????[�
????????????���]+??????[�
���????????????])
2
=
(2∗??????−??????+??????)
2
=
(2∗??????)
2
=??????
•So this estimator is indeed UNBIASED and an estimate of the population mean
•You can show that it has a higher standard error than the sample mean Ƹ??????, and as such is not the “best”
•Remember the sample mean Ƹ?????? is “BLUE”: Best Linear Unbiased Estimator
30
Luc Faucheux 2025
•Since I now I have 3 observations I can define the sample mean Ƹ?????? as Ƹ??????=
1
�
.σ
��
�=
(�
���????????????��+�
????????????���+�
���????????????)
3
•My estimate of the population mean is thus:
(??????
���????????????��+??????
????????????���+??????
���????????????)
3
=
(5.7+6.3+5.8)
3
=5.93
•That is NOT 6, but it is my current estimate
•It is actually an UNBIASED estimate because ??????Ƹ??????=??????
•A quick note: I could also define an estimator called ��� such as ���=
(2∗�
���????????????��−�
????????????���+�
���????????????)
2
•I claim that this is an estimate of the population mean
•In our case it will be when measured :
(2∗??????
���????????????��−??????????????????���+??????
���????????????)
3
=
(2∗5.7−6.3+5.8)
3
=5.45
•I am actually correct because:
•??????���=??????
(2∗�
���????????????��−�
????????????���+�
���????????????)
2
=
(2∗??????[�
���????????????��]−??????[�
????????????���]+??????[�
���????????????])
2
=
(2∗??????−??????+??????)
2
=
(2∗??????)
2
=??????
•So this estimator is indeed UNBIASED and an estimate of the population mean
•You can show that it has a higher standard error than the sample mean Ƹ??????, and as such is not the “best”
•Remember the sample mean Ƹ?????? is “BLUE”: Best Linear Unbiased Estimator
RECAP: Measuring – Sampling – Linear regression – V-i
31
Luc Faucheux 2025
•“The right notation is everything” K. Godel
31
Luc Faucheux 2025
•“The right notation is everything” K. Godel
RECAP: Measuring – Sampling – Linear regression – V-j
32
Luc Faucheux 2025
•“The right notation is everything” K. Godel
•Unfortunately in finance, you cannot find two textbooks with the same consistent notation, or even one
textbook that has a consistent rigorous notation throughout the book
•That does not mean that we should not try
32
Luc Faucheux 2025
•“The right notation is everything” K. Godel
•Unfortunately in finance, you cannot find two textbooks with the same consistent notation, or even one
textbook that has a consistent rigorous notation throughout the book
•That does not mean that we should not try
RECAP: Measuring – Sampling – Linear regression – V-k
33
Luc Faucheux 2025
•“The right notation is everything” K. Godel
•A quick glossary:
•Modeling: Y=α+�.�+ℰ
•Sampling: �
�=α+�.�
�+ℰ
�
•Minimizing: �
�=ෝα+መ�.�
�+ℰ
�
•Observing: �
�=α+�.�
�+??????
�
•�
�: Market returns
•�
??????: risk-free returns (when stochastic or random)
•�
??????: risk-free returns (when constant)
33
Luc Faucheux 2025
•“The right notation is everything” K. Godel
•A quick glossary:
•Modeling: Y=α+�.�+ℰ
•Sampling: �
�=α+�.�
�+ℰ
�
•Minimizing: �
�=ෝα+መ�.�
�+ℰ
�
•Observing: �
�=α+�.�
�+??????
�
•�
�: Market returns
•�
??????: risk-free returns (when stochastic or random)
•�
??????: risk-free returns (when constant)
RECAP: Measuring – Sampling – Linear regression – VI
34
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�.ෞ??????
�
34
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�.ෞ??????
�
RECAP: Measuring – Sampling – Linear regression - VII
35
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
??????
��
)
35
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
??????
��
)
RECAP: Measuring – Sampling – Linear regression - VIII
36
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
36
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
RECAP: Measuring – Sampling – Linear regression - XIX
37
Luc Faucheux 2025
•For large sample, drop the hat, but do not drop the ball…and use the Normal Distribution Function unless
told otherwise….
37
Luc Faucheux 2025
•For large sample, drop the hat, but do not drop the ball…and use the Normal Distribution Function unless
told otherwise….
RECAP: ALPHA AND BETA
38
Luc Faucheux 2025
38
Luc Faucheux 2025
RECAP: ALPHA and BETA - I
39
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
39
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
RECAP ALPHA and BETA - II
40
Luc Faucheux 2025
•Sometimes Wiki just says it better than I would try
(*) Wikipedia
40
Luc Faucheux 2025
•Sometimes Wiki just says it better than I would try
(*) Wikipedia
RECAP ALPHA and BETA - III
41
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
41
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
RECAP ALPHA and BETA - IV
42
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,..)
42
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,..)
ALPHA and BETA in CAPM
43
Luc Faucheux 2025
43
Luc Faucheux 2025
ALPHA and BETA in CAPM - I
44
Luc Faucheux 2025 (*) Wikipedia
44
Luc Faucheux 2025 (*) Wikipedia
ALPHA and BETA in CAPM - II
45
Luc Faucheux 2025
•For a given stock return:
•�−�
�=�+�.�
�−�
�+ℇ
•We can also group stocks by sectors and do a linear regression on the different sectors against the overall market
(*) GARP FRM Part I material 2025
45
Luc Faucheux 2025
•For a given stock return:
•�−�
�=�+�.�
�−�
�+ℇ
•We can also group stocks by sectors and do a linear regression on the different sectors against the overall market
(*) GARP FRM Part I material 2025
ALPHA and BETA in CAPM - III
46
Luc Faucheux 2025 (*) GARP FRM Part I material 2025
As an illustration, CAPM is estimated using 30 years of monthly data between 1989 and 2018. The portfolios measure the value-
weighted return to all firms in a sector. The sectors include banks, technology companies, and industrials. The market portfolio
measures the return on the complete US equity market. The risk-free rate proxy is the interest rate of a one-month US T-bill. The Z-
score for ALPHA is computed with a mean of 0 and the Z-score for BETA is computed with a mean of 1
46
Luc Faucheux 2025 (*) GARP FRM Part I material 2025
As an illustration, CAPM is estimated using 30 years of monthly data between 1989 and 2018. The portfolios measure the value-
weighted return to all firms in a sector. The sectors include banks, technology companies, and industrials. The market portfolio
measures the return on the complete US equity market. The risk-free rate proxy is the interest rate of a one-month US T-bill. The Z-
score for ALPHA is computed with a mean of 0 and the Z-score for BETA is computed with a mean of 1
ALPHA and BETA in CAPM - IV
47
Luc Faucheux 2025
•If stocks (or sectors) have a BETA close to 1, they are usually referred to as cyclical
•If stocks (or sectors) have a BETA higher than 1, they are usually referred to as pro-cyclical
•If stocks (or sectors) have a BETA lower than 1 (or potentially even negative), they are usually referred to as counter-cyclical
•This is based on the observation that there seems to be cycles in the economy (expansion and contraction, sometimes when the
contraction is severe it would get qualified as a recession, a depression, a Great Depression, …)
•A lot of investment strategies are built upon the assumption that those cycles are repeated over time with some degree of
consistency that you can identify
47
Luc Faucheux 2025
•If stocks (or sectors) have a BETA close to 1, they are usually referred to as cyclical
•If stocks (or sectors) have a BETA higher than 1, they are usually referred to as pro-cyclical
•If stocks (or sectors) have a BETA lower than 1 (or potentially even negative), they are usually referred to as counter-cyclical
•This is based on the observation that there seems to be cycles in the economy (expansion and contraction, sometimes when the
contraction is severe it would get qualified as a recession, a depression, a Great Depression, …)
•A lot of investment strategies are built upon the assumption that those cycles are repeated over time with some degree of
consistency that you can identify
ALPHA and BETA in CAPM - V
48
Luc Faucheux 2025
48
Luc Faucheux 2025
ALPHA and BETA in CAPM - VI
49
Luc Faucheux 2025
49
Luc Faucheux 2025
ALPHA and BETA in CAPM - VII
50
Luc Faucheux 2025
50
Luc Faucheux 2025
ALPHA and BETA in CAPM - VIII
51
Luc Faucheux 2025
Boethius, Consolation of Philosophy.
Also Tony Wilson….in one of the best movies ever…
51
Luc Faucheux 2025
Boethius, Consolation of Philosophy.
Also Tony Wilson….in one of the best movies ever…
ALPHA and BETA in CAPM - IX
52
Luc Faucheux 2025
•WORDS THAT YOU WILL FIND CONNECTED TO ALPHA:
•Information
•Intercept
•Diversifiable risk
•Unique risk
•WORDS THAT YOU WILL FIND CONNECTED TO BETA:
•Execution
•Slope
•Non-diversifiable risk
•Systematic risk
•Market risk
52
Luc Faucheux 2025
•WORDS THAT YOU WILL FIND CONNECTED TO ALPHA:
•Information
•Intercept
•Diversifiable risk
•Unique risk
•WORDS THAT YOU WILL FIND CONNECTED TO BETA:
•Execution
•Slope
•Non-diversifiable risk
•Systematic risk
•Market risk
PERFORMANCE INDICATORS
53
Luc Faucheux 2025
53
Luc Faucheux 2025
PERFORMANCE INDICATORS – I - SHARPE
54
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX
•���=
(??????�−??????
�)
??????
•Note: In the case where the risk-free rate is ALSO a random variable, it gets a little more complicated but the convention
is to keep the same terminology for the performance indicator
• ���=
(??????�−�
�)
??????
54
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX
•���=
(??????�−??????
�)
??????
•Note: In the case where the risk-free rate is ALSO a random variable, it gets a little more complicated but the convention
is to keep the same terminology for the performance indicator
• ���=
(??????�−�
�)
??????
PERFORMANCE INDICATORS – II - TREYNOR
55
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•TREYNOR PERFORMANCE INDEX
•���=
(??????�−??????
�)
�
•Note: In the case where the risk-free rate is ALSO a random variable, it gets a little more complicated but the convention
is to keep the same terminology for the performance indicator
• ���=
(??????�−�
�)
�
55
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•TREYNOR PERFORMANCE INDEX
•���=
(??????�−??????
�)
�
•Note: In the case where the risk-free rate is ALSO a random variable, it gets a little more complicated but the convention
is to keep the same terminology for the performance indicator
• ���=
(??????�−�
�)
�
PERFORMANCE INDICATORS – III – JENSEN ALPHA
56
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•JENSEN ALPHA
•����=??????�−�
�−�.??????�
�−�
�
•Note: In the case where the risk-free rate is ALSO a random variable, it gets a little more complicated but the convention
is to keep the same terminology for the performance indicator
•����=??????�−�
�−�.??????�
�−�
�
•Remember that ??????ℇ=0
56
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•JENSEN ALPHA
•����=??????�−�
�−�.??????�
�−�
�
•Note: In the case where the risk-free rate is ALSO a random variable, it gets a little more complicated but the convention
is to keep the same terminology for the performance indicator
•����=??????�−�
�−�.??????�
�−�
�
•Remember that ??????ℇ=0
PERFORMANCE INDICATORS – IIIa – JENSEN ALPHA
57
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•JENSEN ALPHA
•����=??????�−�
�−�.??????�
�−�
�
•In practice the Jensen ALPHA is sometimes computed within the context of performing a linear regression, and so you
might see the definition with the funny hat to indicate the result of a regression:
•����=??????�−�
�−�.??????�
�−�
�
•����=ො�
•Remember that ??????ℇ=0
57
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•JENSEN ALPHA
•����=??????�−�
�−�.??????�
�−�
�
•In practice the Jensen ALPHA is sometimes computed within the context of performing a linear regression, and so you
might see the definition with the funny hat to indicate the result of a regression:
•����=??????�−�
�−�.??????�
�−�
�
•����=ො�
•Remember that ??????ℇ=0
PERFORMANCE INDICATORS – IV – INFORMATION RATIO
58
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•INFORMATION RATIO
•����=
??????�−�
�
V(�−�
�)
•Note: In practice for the IRPI the market is being replaced by a Benchmark or Target Portfolio
•??????�−??????�
� is sometimes called “active returns”
•V(�−�
�) is sometimes called “tracking error”
58
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•INFORMATION RATIO
•����=
??????�−�
�
V(�−�
�)
•Note: In practice for the IRPI the market is being replaced by a Benchmark or Target Portfolio
•??????�−??????�
� is sometimes called “active returns”
•V(�−�
�) is sometimes called “tracking error”
PERFORMANCE INDICATORS – V – SORTINO
59
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SORTINO (tweak on SHARPE)
•���=
(??????�−??????
�)
??????
����=
(??????�−??????
�)
??????
�
or with a target return ����=
(??????�−??????)
??????
�
•??????
� is the standard deviations of returns below the target (could be the risk free returns, but it could be a threshold, a
benchmark or anything you decide, sometimes called MAR, Minimum Acceptable Returns)
•Note: In the case where the risk-free rate is ALSO a random variable, it gets a little more complicated but the convention
is to keep the same terminology for the performance indicator
• ����=
(??????�−�
�)
??????
�
59
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SORTINO (tweak on SHARPE)
•���=
(??????�−??????
�)
??????
����=
(??????�−??????
�)
??????
�
or with a target return ����=
(??????�−??????)
??????
�
•??????
� is the standard deviations of returns below the target (could be the risk free returns, but it could be a threshold, a
benchmark or anything you decide, sometimes called MAR, Minimum Acceptable Returns)
•Note: In the case where the risk-free rate is ALSO a random variable, it gets a little more complicated but the convention
is to keep the same terminology for the performance indicator
• ����=
(??????�−�
�)
??????
�
PERFORMANCE INDICATORS – Va – SORTINO
60
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SORTINO (tweak on SHARPE)
•���=
(??????�−??????
�)
??????
����=
(??????�−??????
�)
??????
�
or with a target return ����=
(??????�−??????)
??????
�
•??????
� is the standard deviations of returns below the target
•In practice, you construct a SORTINO curve by varying the target �
•This actually gives you a lot of information about large events and how robust the SHARPE SPI is to the outliers of your
distribution of returns
60
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SORTINO (tweak on SHARPE)
•���=
(??????�−??????
�)
??????
����=
(??????�−??????
�)
??????
�
or with a target return ����=
(??????�−??????)
??????
�
•??????
� is the standard deviations of returns below the target
•In practice, you construct a SORTINO curve by varying the target �
•This actually gives you a lot of information about large events and how robust the SHARPE SPI is to the outliers of your
distribution of returns
PERFORMANCE INDICATORS – VI
61
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
usually when choosing
•TREYNOR PERFORMANCE INDEX ���=
(??????�−??????
�)
�
usually when ranking
•INFORMATION RATIO ����=
??????�−�
�
V(�−�
�)
usually when evaluating (new HFs)
•SORTINO RATIO ����=
(??????�−??????
�)
??????
�
usually when stress testing the SPI
61
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
usually when choosing
•TREYNOR PERFORMANCE INDEX ���=
(??????�−??????
�)
�
usually when ranking
•INFORMATION RATIO ����=
??????�−�
�
V(�−�
�)
usually when evaluating (new HFs)
•SORTINO RATIO ����=
(??????�−??????
�)
??????
�
usually when stress testing the SPI
PERFORMANCE INDICATORS – VII
62
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•Wait a second I hear you say….you keep telling us that ALPHA is the Holy Grail, and none of those definitions actually
contain ALPHA ?
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
usually when choosing
•TREYNOR PERFORMANCE INDEX ���=
(??????�−??????
�)
�
usually when ranking
•INFORMATION RATIO ����=
??????�−��
V(�−��)
usually when evaluating (new HFs)
•Fear not my young ones…we can re-arrange the above formulas to make the ALPHA appears
62
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•Wait a second I hear you say….you keep telling us that ALPHA is the Holy Grail, and none of those definitions actually
contain ALPHA ?
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
usually when choosing
•TREYNOR PERFORMANCE INDEX ���=
(??????�−??????
�)
�
usually when ranking
•INFORMATION RATIO ����=
??????�−��
V(�−��)
usually when evaluating (new HFs)
•Fear not my young ones…we can re-arrange the above formulas to make the ALPHA appears
PERFORMANCE INDICATORS – VIII
63
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
=
�+�.??????[��]−??????
�
??????
=
�
??????
+
�.??????[��]−??????
�
??????
•From the Linear Regression deck, we can write (under some assumptions, large sample, homoskedasticity, Independent,
Identically Distributed, might not work in real life, might work, yadi yadi yada…)
•�=
ෟ??????��
ෟ??????��
=ෞ??????
��.
ෞ??????�
ෞ??????�
becomes with �՜� and �՜�
� and dropping the hat
•�=
??????��
�
??????�
�
�
�
=??????
���.
??????�
??????�
�
•
�.??????[��]−??????
�
??????
=??????
���.
??????�
??????�
�
.
??????[��]−??????
�
??????�
=??????
���.
??????[��]−??????
�
??????�
�
=??????
���.���(�
�)
63
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
=
�+�.??????[��]−??????
�
??????
=
�
??????
+
�.??????[��]−??????
�
??????
•From the Linear Regression deck, we can write (under some assumptions, large sample, homoskedasticity, Independent,
Identically Distributed, might not work in real life, might work, yadi yadi yada…)
•�=
ෟ??????��
ෟ??????��
=ෞ??????
��.
ෞ??????�
ෞ??????�
becomes with �՜� and �՜�
� and dropping the hat
•�=
??????��
�
??????�
�
�
�
=??????
���.
??????�
??????�
�
•
�.??????[��]−??????
�
??????
=??????
���.
??????�
??????�
�
.
??????[��]−??????
�
??????�
=??????
���.
??????[��]−??????
�
??????�
�
=??????
���.���(�
�)
PERFORMANCE INDICATORS – IX
64
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
=
�+�.??????[��]−??????
�
??????
=
�
??????
+
�.??????[��]−??????
�
??????
•�=
??????��
�
??????�
�
�
�
=??????
���.
??????�
??????�
�
•
�.??????[��]−??????
�
??????
=??????
���.
??????�
??????�
�
.
??????[��]−??????
�
??????�
=??????
���.
??????[��]−??????
�
??????�
�
=??????
���.���(�
�)
•����=
??????�−??????
�
??????
=���(�)=
(??????�−??????
�)
??????�
•���(�
�)=
(??????��−??????
�)
??????�
�
•����=
�
??????�
+??????
���.���(�
�)
64
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
=
�+�.??????[��]−??????
�
??????
=
�
??????
+
�.??????[��]−??????
�
??????
•�=
??????��
�
??????�
�
�
�
=??????
���.
??????�
??????�
�
•
�.??????[��]−??????
�
??????
=??????
���.
??????�
??????�
�
.
??????[��]−??????
�
??????�
=??????
���.
??????[��]−??????
�
??????�
�
=??????
���.���(�
�)
•����=
??????�−??????
�
??????
=���(�)=
(??????�−??????
�)
??????�
•���(�
�)=
(??????��−??????
�)
??????�
�
•����=
�
??????�
+??????
���.���(�
�)
PERFORMANCE INDICATORS – X
65
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•TREYNOR PERFORMANCE INDEX ���=
(??????�−??????
�)
�
•����=
(??????�−??????
�)
�
•����
�=
(??????��−??????
�)
��
•�=
??????
��
�
??????
�
�
�
�
=??????
�
��.
??????
�
??????
�
�
•�
�=
??????
�
�
�
�
??????
�
�
�
�
=??????
�
��
�
.
??????
�
�
??????
�
�
=1
•����=
�
�
+����
�
65
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•TREYNOR PERFORMANCE INDEX ���=
(??????�−??????
�)
�
•����=
(??????�−??????
�)
�
•����
�=
(??????��−??????
�)
��
•�=
??????
��
�
??????
�
�
�
�
=??????
�
��.
??????
�
??????
�
�
•�
�=
??????
�
�
�
�
??????
�
�
�
�
=??????
�
��
�
.
??????
�
�
??????
�
�
=1
•����=
�
�
+����
�
PERFORMANCE INDICATORS – XI
66
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
=
(??????�−??????
�)
??????
�
•����=
�
??????
�
+??????
�
��.���(�
�)
•TREYNOR PERFORMANCE INDEX ���=
(??????�−??????
�)
�
•����=
�
�
+����
�
•Those clearly shows that ALPHA shows up as an IMPROVEMENT on the Performance Indicators of the specific
stock/strategy/portfolio/manager versus the market or benchmark
66
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•SHARPE PERFORMANCE INDEX ���=
(??????�−??????
�)
??????
=
(??????�−??????
�)
??????
�
•����=
�
??????
�
+??????
�
��.���(�
�)
•TREYNOR PERFORMANCE INDEX ���=
(??????�−??????
�)
�
•����=
�
�
+����
�
•Those clearly shows that ALPHA shows up as an IMPROVEMENT on the Performance Indicators of the specific
stock/strategy/portfolio/manager versus the market or benchmark
Back to the Merkowitz Bullet !
67
Luc Faucheux 2025
67
Luc Faucheux 2025
Merkowitz Bullet - I
68
Luc Faucheux 2025 (*) GARP FRM Part I material 2025
68
Luc Faucheux 2025 (*) GARP FRM Part I material 2025
69
•Π=σ(�
�.�
�)
•�
�: notional (size or amount) of the asset �
�
•�
�: price of asset �
�
•??????
� are the individual positions in the portfolio for each asset �
� expressed as a
percentage of the portfolio value.
•??????
�=�
�.�
�/Π
•Π=σ(�
�.�
�)=Π.σ(??????
�)
•And so:
•σ??????
�=1
Luc Faucheux 2025
Merkowitz Bullet - I
•Π=σ(�
�.�
�)
•�
�: notional (size or amount) of the asset �
�
•�
�: price of asset �
�
•??????
� are the individual positions in the portfolio for each asset �
� expressed as a
percentage of the portfolio value.
•??????
�=�
�.�
�/Π
•Π=σ(�
�.�
�)=Π.σ(??????
�)
•And so:
•σ??????
�=1
Luc Faucheux 2025
Merkowitz Bullet - I
70
•Π=(σ??????
�).Π=σ�
�.�
�
•??????
�=�
�.�
�/Π
•Π is in $
•�
� is in $
•�
� is in # (it is dimensionless number)
•??????
� is in % (dimensionless number)
•(σ??????
�)=1
Luc Faucheux 2025
Merkowitz Bullet - II
•Π=(σ??????
�).Π=σ�
�.�
�
•??????
�=�
�.�
�/Π
•Π is in $
•�
� is in $
•�
� is in # (it is dimensionless number)
•??????
� is in % (dimensionless number)
•(σ??????
�)=1
Luc Faucheux 2025
Merkowitz Bullet - II
71
Luc Faucheux 2025
Merkowitz Bullet - III
•�
�: return of asset �
� in %
•The portfolio return is �(Π):
•�(Π)=σ??????
�.�
�
•Portfolio expected return and volatility:
•�=??????�Π=σ??????
��
�
•??????
2
=��Π= ??????{�Π−??????�Π}
2
=??????�Π
2
−??????�Π
2
=σ
�
σ
�??????
�.??????
�.??????
�.??????
�.??????
��
Luc Faucheux 2025
Merkowitz Bullet - III
•�
�: return of asset �
� in %
•The portfolio return is �(Π):
•�(Π)=σ??????
�.�
�
•Portfolio expected return and volatility:
•�=??????�Π=σ??????
��
�
•??????
2
=��Π= ??????{�Π−??????�Π}
2
=??????�Π
2
−??????�Π
2
=σ
�
σ
�??????
�.??????
�.??????
�.??????
�.??????
��
72
Luc Faucheux 2025
Merkowitz Bullet - IV
??????
2
=��Π=
�
�
??????
�.??????
�.??????
�.??????
�.??????
��
Luc Faucheux 2025
Merkowitz Bullet - IV
??????
2
=��Π=
�
�
??????
�.??????
�.??????
�.??????
�.??????
��
73
Luc Faucheux 2025
Merkowitz Bullet - V
•This was the original approach by Merkowitz (careful on the notations his �
� are our �
�)
Portfolio Selection
Harry Markowitz
The Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91
Luc Faucheux 2025
Merkowitz Bullet - V
•This was the original approach by Merkowitz (careful on the notations his �
� are our �
�)
Portfolio Selection
Harry Markowitz
The Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91
Merkowitz Bullet - VI
74
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•??????�=�
�+??????.
??????��−??????
�
??????�
•??????�−�
�=??????.
??????��−??????
�
??????�
•??????�−�
�=
??????
??????�
.(??????�
�−�
�)
(*) GARP FRM Part I material 2025
74
Luc Faucheux 2025
•For a given stock return: �−�
�=�+�.�
�−�
�+ℇ
•� is a random variable, ??????�=?????? and variance: V�=??????
2
,
•??????�=�
�+??????.
??????��−??????
�
??????�
•??????�−�
�=??????.
??????��−??????
�
??????�
•??????�−�
�=
??????
??????�
.(??????�
�−�
�)
(*) GARP FRM Part I material 2025
Merkowitz Bullet - VII
75
Luc Faucheux 2025
•Every portfolio constructed on the CML (Capital
Market Line, by combining the risk-free rate and
the Market Portfolio has a higher return for the
same volatility than any other portfolio built inside
the bullet (on or inside the efficient frontier)
•That is actually not super obvious nor expected,
would you say ?
•This is actually quite surprising and almost counter-
intuitive
(*) GARP FRM Part I material 2025
75
Luc Faucheux 2025
•Every portfolio constructed on the CML (Capital
Market Line, by combining the risk-free rate and
the Market Portfolio has a higher return for the
same volatility than any other portfolio built inside
the bullet (on or inside the efficient frontier)
•That is actually not super obvious nor expected,
would you say ?
•This is actually quite surprising and almost counter-
intuitive
(*) GARP FRM Part I material 2025
Merkowitz Bullet - VIII
76
Luc Faucheux 2025
•CML (Capital Market Line):
•??????�−�
�=
??????
??????
�
.(??????�
�−�
�)
•Here the � is a portfolio built from some
percentage invested at the risk-free rate and the
Market portfolio build from the MVT
•SML (Security Market Line)
•For a given stock:
•??????�−�
�=�+�.??????�
�−�
�
•This is usually for a single security (hence the name
Security Market Line, as it expresses a given
security as a function of the Market, or a
Benchmark, or index or anything you like)
(*) GARP FRM Part I material 2025
76
Luc Faucheux 2025
•CML (Capital Market Line):
•??????�−�
�=
??????
??????
�
.(??????�
�−�
�)
•Here the � is a portfolio built from some
percentage invested at the risk-free rate and the
Market portfolio build from the MVT
•SML (Security Market Line)
•For a given stock:
•??????�−�
�=�+�.??????�
�−�
�
•This is usually for a single security (hence the name
Security Market Line, as it expresses a given
security as a function of the Market, or a
Benchmark, or index or anything you like)
(*) GARP FRM Part I material 2025
So many straight lines !
SML, SCL, CAL, CML,….
77
Luc Faucheux 2025
SML, SCL, CAL, CML,….
77
Luc Faucheux 2025
So many straight lines! SML, SCL, CAL, CAML.. - I
78
Luc Faucheux 2025
•CML (Capital Market Line):
•??????�−�
�=
??????
??????
�
.(??????�
�−�
�)
•Here the � is a portfolio built from some
percentage invested at the risk-free rate and the
Market portfolio build from the MVT
•SML (Security Market Line)
•For a given stock:
•??????�−�
�=�+�.??????�
�−�
�
•This is usually for a single security (hence the name
Security Market Line, as it expresses a given
security as a function of the Market, or a
Benchmark, or index or anything you like)
(*) GARP FRM Part I material 2025
78
Luc Faucheux 2025
•CML (Capital Market Line):
•??????�−�
�=
??????
??????
�
.(??????�
�−�
�)
•Here the � is a portfolio built from some
percentage invested at the risk-free rate and the
Market portfolio build from the MVT
•SML (Security Market Line)
•For a given stock:
•??????�−�
�=�+�.??????�
�−�
�
•This is usually for a single security (hence the name
Security Market Line, as it expresses a given
security as a function of the Market, or a
Benchmark, or index or anything you like)
(*) GARP FRM Part I material 2025
So many straight lines! SML, SCL, CAL, CAML.. - II
79
Luc Faucheux 2025
•SCL (Security Characteristic Line) comes from measuring/sampling
•Y-axis is expected excess return, X-axis is excess return of market
•Intercept is risk-free rate, slope is the Market Risk Premium
•It is usually a Linear Regression, built on actual historical returns, hence the funny hat
•For a given stock return: �
�−�
�=ො�+መ�.�
��
−�
�+ℇ
�
•The result of the regression is:??????�−�
�=ො�+መ�.??????�
�−�
�
�−�
�
�
�−�
�
•
•
•
•
•
•
(�
��
−�
�,�
�−�
�)
ℰ
�
Slope መ�
Intercept ො�
79
Luc Faucheux 2025
•SCL (Security Characteristic Line) comes from measuring/sampling
•Y-axis is expected excess return, X-axis is excess return of market
•Intercept is risk-free rate, slope is the Market Risk Premium
•It is usually a Linear Regression, built on actual historical returns, hence the funny hat
•For a given stock return: �
�−�
�=ො�+መ�.�
��
−�
�+ℇ
�
•The result of the regression is:??????�−�
�=ො�+መ�.??????�
�−�
�
�−�
�
�
�−�
�
•
•
•
•
•
•
(�
��
−�
�,�
�−�
�)
ℰ
�
Slope መ�
Intercept ො�
So many straight lines! SML, SCL, CAL, CAML.. - III
80
Luc Faucheux 2025
•SML (Security Market Line) used to predict
•Y-axis is expected return, X-axis is the BETA
•Intercept is ALPHA, slope is BETA
•It is a theoretical relationship (but the SCL helps to identify the BETA of a specific security)
•For a given stock: ??????�−�
�=�+�.??????�
�−�
� with �=0 in CAPM
??????�−�
�
�
Slope
??????�
�−�
�
Intercept �
80
Luc Faucheux 2025
•SML (Security Market Line) used to predict
•Y-axis is expected return, X-axis is the BETA
•Intercept is ALPHA, slope is BETA
•It is a theoretical relationship (but the SCL helps to identify the BETA of a specific security)
•For a given stock: ??????�−�
�=�+�.??????�
�−�
� with �=0 in CAPM
??????�−�
�
�
Slope
??????�
�−�
�
Intercept �
So many straight lines! SML, SCL, CAL, CAML.. - IV
81
Luc Faucheux 2025
•CML (Capital Market Line)
•Y-axis is expected return, X-axis is the Standard Deviation (Risk)
•Intercept is risk-free rate, slope is the Sharpe Ratio of the Market portfolio
•For a given portfolio: ??????�−�
�=
??????
??????
�
.(??????�
�−�
�)
??????�
??????
Slope
??????�
�−�
�
??????
�
SPI(Market)
Intercept �
�
81
Luc Faucheux 2025
•CML (Capital Market Line)
•Y-axis is expected return, X-axis is the Standard Deviation (Risk)
•Intercept is risk-free rate, slope is the Sharpe Ratio of the Market portfolio
•For a given portfolio: ??????�−�
�=
??????
??????
�
.(??????�
�−�
�)
??????�
??????
Slope
??????�
�−�
�
??????
�
SPI(Market)
Intercept �
�
So many straight lines! SML, SCL, CAL, CAML.. - IVa
82
Luc Faucheux 2025
•CML (Capital Market Line)
•Y-axis is expected return, X-axis is the Standard Deviation (Risk)
•Intercept is risk-free rate, slope is the Sharpe Ratio of the Market portfolio
•For a given portfolio: ??????�−�
�=
??????
??????
�
.(??????�
�−�
�)
??????�
??????
Market
Portfolio
�
�
Intercept �
�
82
Luc Faucheux 2025
•CML (Capital Market Line)
•Y-axis is expected return, X-axis is the Standard Deviation (Risk)
•Intercept is risk-free rate, slope is the Sharpe Ratio of the Market portfolio
•For a given portfolio: ??????�−�
�=
??????
??????
�
.(??????�
�−�
�)
??????�
??????
Market
Portfolio
�
�
Intercept �
�
So many straight lines! SML, SCL, CAL, CAML.. - V
83
Luc Faucheux 2025
•CAL (Capital Allocation Line)
•Y-axis is expected return, X-axis is the Standard Deviation (Risk)
•Intercept is risk-free rate, slope is the Sharpe Ratio of the risky portfolio
•For a given portfolio: ??????�−�
�=
??????
??????
??????
.(??????�
??????−�
�)
??????�
??????
Slope
??????�
??????−�
�
??????
??????
SPI(Risky
Portfolio)
Intercept �
�
83
Luc Faucheux 2025
•CAL (Capital Allocation Line)
•Y-axis is expected return, X-axis is the Standard Deviation (Risk)
•Intercept is risk-free rate, slope is the Sharpe Ratio of the risky portfolio
•For a given portfolio: ??????�−�
�=
??????
??????
??????
.(??????�
??????−�
�)
??????�
??????
Slope
??????�
??????−�
�
??????
??????
SPI(Risky
Portfolio)
Intercept �
�
So many straight lines! SML, SCL, CAL, CAML.. - Va
84
Luc Faucheux 2025
•CAL (Capital Allocation Line)
•Y-axis is expected return, X-axis is the Standard Deviation (Risk)
•Intercept is risk-free rate, slope is the Sharpe Ratio of the risky portfolio
•For a given portfolio: ??????�−�
�=
??????
??????
??????
.(??????�
??????−�
�)
??????�
??????
Risky Portfolio
�
??????
Intercept �
�
84
Luc Faucheux 2025
•CAL (Capital Allocation Line)
•Y-axis is expected return, X-axis is the Standard Deviation (Risk)
•Intercept is risk-free rate, slope is the Sharpe Ratio of the risky portfolio
•For a given portfolio: ??????�−�
�=
??????
??????
??????
.(??????�
??????−�
�)
??????�
??????
Risky Portfolio
�
??????
Intercept �
�
So many straight lines! SML, SCL, CAL, CAML.. - VI
85
Luc Faucheux 2025
•CAL and CML are almost the same thing.
•CAL uses a risky portfolio inside the efficient frontier of your choice
•CML uses a specific risky portfolio, the Market portfolio.
85
Luc Faucheux 2025
•CAL and CML are almost the same thing.
•CAL uses a risky portfolio inside the efficient frontier of your choice
•CML uses a specific risky portfolio, the Market portfolio.
More Factors!
86
Luc Faucheux 2025
86
Luc Faucheux 2025
More Factors - I
87
Luc Faucheux 2025
•Sometimes a single variable on the X-axis is not enough to explain the behavior of the variable on the
Y-axis (the residuals are just too big)
•So the whole goal of the Factor Analysis is to ”guess” what else we could add to the variable on the X-
axis in order to “explain away”, meaning getting a nice linear regression between the variable on the
Y-axis and our new variable on the X-axis
•Note that we are not constrained to add linear functions on the X-axis
•It is perfectly reasonable for example to perform a Linear Regression between � and for example
(�+�∗�−�����(�)) if you want to
•So the goal is to construct on the X-axis a function of a lot of different things, so that in the end you
can get a nice straight line between your � and this potentially complicated function
87
Luc Faucheux 2025
•Sometimes a single variable on the X-axis is not enough to explain the behavior of the variable on the
Y-axis (the residuals are just too big)
•So the whole goal of the Factor Analysis is to ”guess” what else we could add to the variable on the X-
axis in order to “explain away”, meaning getting a nice linear regression between the variable on the
Y-axis and our new variable on the X-axis
•Note that we are not constrained to add linear functions on the X-axis
•It is perfectly reasonable for example to perform a Linear Regression between � and for example
(�+�∗�−�����(�)) if you want to
•So the goal is to construct on the X-axis a function of a lot of different things, so that in the end you
can get a nice straight line between your � and this potentially complicated function
More Factors - II
88
Luc Faucheux 2025
•Here is a graph I found about some variant of French-Fama
•Essentially before adding the new factors the SCL (Security Characteristic Line) looked something like
that:
88
Luc Faucheux 2025
•Here is a graph I found about some variant of French-Fama
•Essentially before adding the new factors the SCL (Security Characteristic Line) looked something like
that:
More Factors - III
89
Luc Faucheux 2025
•Here is a graph I found about some variant of French-Fama
•After adding the new factors the new SCL (Security Characteristic Line) looked like that:…much
better…
89
Luc Faucheux 2025
•Here is a graph I found about some variant of French-Fama
•After adding the new factors the new SCL (Security Characteristic Line) looked like that:…much
better…
More Factors - IV
90
Luc Faucheux 2025
•And that is kind of really it…
•Of course the devil is always in the details, depends how you construct the function, how do you
measure improvements and so on and so forth…
•A nice little devil below (the Maxwell demon) messing up the details of the Second Law of
Thermodynamics…those little devils are always in the details….
90
Luc Faucheux 2025
•And that is kind of really it…
•Of course the devil is always in the details, depends how you construct the function, how do you
measure improvements and so on and so forth…
•A nice little devil below (the Maxwell demon) messing up the details of the Second Law of
Thermodynamics…those little devils are always in the details….
More Factors - V
91
Luc Faucheux 2025
•If you include macro-economic factors, your model becomes a “Macro-economic factors model”
•For example, Chen, Roll, and Ross introduced the first macroeconomic factor model in the 1980s, by adding some
factors on the X-axis such as inflation, industrial production or the steepness of the yield curve
•In that case you just add more BETAS, you can solve and minimize recursively or all at once, recursively is usually
better controlled
•�−�
�=�+�.�
�−�
�+ℇ
•That does not work great so you define: ℇ=ℇ
1+�
�??????.�� (for example if you are introducing the Industrial
Production IP)
•�−�
�=�+�.�
�−�
�+�
�??????.��+ℇ
1
•And you perform the regression (OLS if you want) on the new residuals ℇ
1
•Congrats ! You have a Macroeconomic Two Factors model ! Of course that works if you can reduce the size of the
residuals (you explain better the stock returns in this case as a function of “other stuff”)
91
Luc Faucheux 2025
•If you include macro-economic factors, your model becomes a “Macro-economic factors model”
•For example, Chen, Roll, and Ross introduced the first macroeconomic factor model in the 1980s, by adding some
factors on the X-axis such as inflation, industrial production or the steepness of the yield curve
•In that case you just add more BETAS, you can solve and minimize recursively or all at once, recursively is usually
better controlled
•�−�
�=�+�.�
�−�
�+ℇ
•That does not work great so you define: ℇ=ℇ
1+�
�??????.�� (for example if you are introducing the Industrial
Production IP)
•�−�
�=�+�.�
�−�
�+�
�??????.��+ℇ
1
•And you perform the regression (OLS if you want) on the new residuals ℇ
1
•Congrats ! You have a Macroeconomic Two Factors model ! Of course that works if you can reduce the size of the
residuals (you explain better the stock returns in this case as a function of “other stuff”)
More Factors - VI
92
Luc Faucheux 2025
•If you include fundamental factors, your model becomes a “Fundamental factors model”
•For example, French and Fama introduced the first fundamental factor model in the 1990s, by adding some factors
on the X-axis such as what is usually known as SMB and HML
•In that case you just add more BETAS, you can solve and minimize recursively or all at once, recursively is usually
better controlled
•�−�
�=�+�.�
�−�
�+ℇ
•Becomes:
•�−�
�=�+�.�
�−�
�+�
��??????.���+�
���.���+ ℇ
2
92
Luc Faucheux 2025
•If you include fundamental factors, your model becomes a “Fundamental factors model”
•For example, French and Fama introduced the first fundamental factor model in the 1990s, by adding some factors
on the X-axis such as what is usually known as SMB and HML
•In that case you just add more BETAS, you can solve and minimize recursively or all at once, recursively is usually
better controlled
•�−�
�=�+�.�
�−�
�+ℇ
•Becomes:
•�−�
�=�+�.�
�−�
�+�
��??????.���+�
���.���+ ℇ
2
More Factors - VII
93
Luc Faucheux 2025
•FRENCH – FAMA FUNDAMENTAL THREE FACTORS MODEL:
•�−�
�=�+�.�
�−�
�+�
��??????.���+�
���.���+ ℇ
2
1.Small Minus Big (SMB) = (the difference between excess returns from small stocks and those from large stocks)
2.High Minus Low (HML) = (the difference between the excess returns on stocks with high book-to-market values and
those of stocks that have low book-to-market values)
•In 2015, French and Fama extended the model by adding two new factors (making it the French-Fama
Fundamental Five Factors Model)
3. Robust Minus Weak (RMW) = (the difference between the excess returns of companies with high (robust) and low
(weak) operating profitability)
4. Conservative Minus Aggressive (CMA) = (the difference between the excess returns of companies that invest
conservatively and those that invest aggressively)
93
Luc Faucheux 2025
•FRENCH – FAMA FUNDAMENTAL THREE FACTORS MODEL:
•�−�
�=�+�.�
�−�
�+�
��??????.���+�
���.���+ ℇ
2
1.Small Minus Big (SMB) = (the difference between excess returns from small stocks and those from large stocks)
2.High Minus Low (HML) = (the difference between the excess returns on stocks with high book-to-market values and
those of stocks that have low book-to-market values)
•In 2015, French and Fama extended the model by adding two new factors (making it the French-Fama
Fundamental Five Factors Model)
3. Robust Minus Weak (RMW) = (the difference between the excess returns of companies with high (robust) and low
(weak) operating profitability)
4. Conservative Minus Aggressive (CMA) = (the difference between the excess returns of companies that invest
conservatively and those that invest aggressively)
More Factors - VIII
94
Luc Faucheux 2025
•FRENCH – FAMA FUNDAMENTAL THREE FACTORS MODEL:
•�−�
�=�+�.�
�−�
�+�
��??????.���+�
���.���+ ℇ
2
1.Small Minus Big (SMB) = (the difference between excess returns from small stocks and those from large stocks)
2.High Minus Low (HML) = (the difference between excess the returns on stocks with high book-to-market values and
those of stocks that have low book-to-market values)
For example here are the results of the linear regression for two companies (Coke and JPM) using monthly returns
from January 2011 through April 2019
(*) GARP FRM Part I material 2025
COCA-COLA JP MORGAN
Alpha 0.08 0.16
Beta (Market) 0.53 1.45
Beta (SMB) −0.74 -0.14
Beta (HML) -0.10 1.29
94
Luc Faucheux 2025
•FRENCH – FAMA FUNDAMENTAL THREE FACTORS MODEL:
•�−�
�=�+�.�
�−�
�+�
��??????.���+�
���.���+ ℇ
2
1.Small Minus Big (SMB) = (the difference between excess returns from small stocks and those from large stocks)
2.High Minus Low (HML) = (the difference between excess the returns on stocks with high book-to-market values and
those of stocks that have low book-to-market values)
For example here are the results of the linear regression for two companies (Coke and JPM) using monthly returns
from January 2011 through April 2019
(*) GARP FRM Part I material 2025
COCA-COLA JP MORGAN
Alpha 0.08 0.16
Beta (Market) 0.53 1.45
Beta (SMB) −0.74 -0.14
Beta (HML) -0.10 1.29
More Factors - IX
95
Luc Faucheux 2025
•FRENCH – FAMA FUNDAMENTAL THREE FACTORS MODEL:
•�−�
�=�+�.�
�−�
�+�
��??????.���+�
���.���+ ℇ
2
•As an illustration, CAPM is extended to French-Fama (3) by using 30 years of monthly data between 1989 and 2018
(*) GARP FRM Part I material 2025
FRENCH - FAMA 3 FACTORS
ALPHA BETA (MARKET) BETA (SMB) BETA (HML)
Banking -0.17 1.23 -0.16 0.84
Beer and Liquor 0.52 0.63 -0.39 -0.03
Chemicals -0.05 1.10 -0.03 0.46
Computers 0.06 1.30 0.22 -0.67
Consumer Goods 0.22 0.68 -0.17 0.12
Electrical Equipment 0.10 1.26 -0.03 0.12
Retail 0.22 0.93 -0.04 0.01
Shipping Containers 0.00 1.01 -0.01 0.28
Transportation -0.01 1.00 0.05 0.44
Wholesale -0.10 0.88 0.26 0.28
95
Luc Faucheux 2025
•FRENCH – FAMA FUNDAMENTAL THREE FACTORS MODEL:
•�−�
�=�+�.�
�−�
�+�
��??????.���+�
���.���+ ℇ
2
•As an illustration, CAPM is extended to French-Fama (3) by using 30 years of monthly data between 1989 and 2018
(*) GARP FRM Part I material 2025
FRENCH - FAMA 3 FACTORS
ALPHA BETA (MARKET) BETA (SMB) BETA (HML)
Banking -0.17 1.23 -0.16 0.84
Beer and Liquor 0.52 0.63 -0.39 -0.03
Chemicals -0.05 1.10 -0.03 0.46
Computers 0.06 1.30 0.22 -0.67
Consumer Goods 0.22 0.68 -0.17 0.12
Electrical Equipment 0.10 1.26 -0.03 0.12
Retail 0.22 0.93 -0.04 0.01
Shipping Containers 0.00 1.01 -0.01 0.28
Transportation -0.01 1.00 0.05 0.44
Wholesale -0.10 0.88 0.26 0.28
More Factors - X
96
Luc Faucheux 2025
For comparison the original CAPM table
(*) GARP FRM Part I material 2025
96
Luc Faucheux 2025
For comparison the original CAPM table
(*) GARP FRM Part I material 2025
More Factors - XI
97
Luc Faucheux 2025
And side by side (note that I dropped the standard errors for sake of clarity, but they are crucial in estimating how much you
can “trust” your estimate of the parameters)
(*) GARP FRM Part I material 2025
FRENCH - FAMA 3 FACTORS CAPM
ALPHA BETA (MARKET) BETA (SMB) BETA (HML) ALPHA BETA (MARKET)
Banking -0.17 1.23 -0.16 0.84 0.03 1.10
Beer and Liquor 0.52 0.63 -0.39 -0.03 0.52 0.57
Chemicals -0.05 1.10 -0.03 0.46 0.05 1.04
Computers 0.06 1.30 0.22 -0.67 -0.10 1.41
Consumer Goods 0.22 0.68 -0.17 0.12 0.25 0.63
Electrical Equipment 0.10 1.26 -0.03 0.12 0.13 1.24
Retail 0.22 0.93 -0.04 0.01 0.22 0.92
Shipping Containers 0.00 1.01 -0.01 0.28 0.06 0.98
Transportation -0.01 1.00 0.05 0.44 0.09 0.95
Wholesale -0.10 0.88 0.26 0.28 -0.04 0.89
97
Luc Faucheux 2025
And side by side (note that I dropped the standard errors for sake of clarity, but they are crucial in estimating how much you
can “trust” your estimate of the parameters)
(*) GARP FRM Part I material 2025
FRENCH - FAMA 3 FACTORS CAPM
ALPHA BETA (MARKET) BETA (SMB) BETA (HML) ALPHA BETA (MARKET)
Banking -0.17 1.23 -0.16 0.84 0.03 1.10
Beer and Liquor 0.52 0.63 -0.39 -0.03 0.52 0.57
Chemicals -0.05 1.10 -0.03 0.46 0.05 1.04
Computers 0.06 1.30 0.22 -0.67 -0.10 1.41
Consumer Goods 0.22 0.68 -0.17 0.12 0.25 0.63
Electrical Equipment 0.10 1.26 -0.03 0.12 0.13 1.24
Retail 0.22 0.93 -0.04 0.01 0.22 0.92
Shipping Containers 0.00 1.01 -0.01 0.28 0.06 0.98
Transportation -0.01 1.00 0.05 0.44 0.09 0.95
Wholesale -0.10 0.88 0.26 0.28 -0.04 0.89
More Factors - XII
98
Luc Faucheux 2025
•The tough part is now to prove that you are doing a “better” job (meaning overall smaller residuals) in the new more
complicated model than in the simpler one. That involves some statistical measures like the �
2
, which I will not get in
details (I can if you want, it is just that we are already near the end of October and the class ends in a month or so and
we still need to cover a lot), but rest assured that there are plenty of ways to measure if a new model with more factors
perform better (explain away the residuals) or worse than another model….lifetime guaranteed employment for people
who know statistics….
(*) GARP FRM Part I material 2025
FRENCH - FAMA 3 FACTORS CAPM
ALPHA BETA (MARKET) BETA (SMB) BETA (HML) ALPHA BETA (MARKET)
Banking -0.17 1.23 -0.16 0.84 0.03 1.10
Beer and Liquor 0.52 0.63 -0.39 -0.03 0.52 0.57
Chemicals -0.05 1.10 -0.03 0.46 0.05 1.04
Computers 0.06 1.30 0.22 -0.67 -0.10 1.41
Consumer Goods 0.22 0.68 -0.17 0.12 0.25 0.63
Electrical Equipment 0.10 1.26 -0.03 0.12 0.13 1.24
Retail 0.22 0.93 -0.04 0.01 0.22 0.92
Shipping Containers 0.00 1.01 -0.01 0.28 0.06 0.98
Transportation -0.01 1.00 0.05 0.44 0.09 0.95
Wholesale -0.10 0.88 0.26 0.28 -0.04 0.89
98
Luc Faucheux 2025
•The tough part is now to prove that you are doing a “better” job (meaning overall smaller residuals) in the new more
complicated model than in the simpler one. That involves some statistical measures like the �
2
, which I will not get in
details (I can if you want, it is just that we are already near the end of October and the class ends in a month or so and
we still need to cover a lot), but rest assured that there are plenty of ways to measure if a new model with more factors
perform better (explain away the residuals) or worse than another model….lifetime guaranteed employment for people
who know statistics….
(*) GARP FRM Part I material 2025
FRENCH - FAMA 3 FACTORS CAPM
ALPHA BETA (MARKET) BETA (SMB) BETA (HML) ALPHA BETA (MARKET)
Banking -0.17 1.23 -0.16 0.84 0.03 1.10
Beer and Liquor 0.52 0.63 -0.39 -0.03 0.52 0.57
Chemicals -0.05 1.10 -0.03 0.46 0.05 1.04
Computers 0.06 1.30 0.22 -0.67 -0.10 1.41
Consumer Goods 0.22 0.68 -0.17 0.12 0.25 0.63
Electrical Equipment 0.10 1.26 -0.03 0.12 0.13 1.24
Retail 0.22 0.93 -0.04 0.01 0.22 0.92
Shipping Containers 0.00 1.01 -0.01 0.28 0.06 0.98
Transportation -0.01 1.00 0.05 0.44 0.09 0.95
Wholesale -0.10 0.88 0.26 0.28 -0.04 0.89
More Factors - XIII
99
Luc Faucheux 2025
•The tough part is now to prove that you are doing a “better” job (meaning overall smaller residuals) in the new more
complicated model than in the simpler one. That involves some statistical measures like the �
2
, which I will not get in
details (I can if you want, it is just that we are already near the end of October and the class ends in a month or so and
we still need to cover a lot), but rest assured that there are plenty of ways to measure if a new model with more factors
perform better (explain away the residuals) or worse than another model….lifetime guaranteed employment for people
who know statistics….
•As a visual reminder, it is not that obvious from the naked eye that we are doing such a good job…
(*) GARP FRM Part I material 2025
99
Luc Faucheux 2025
•The tough part is now to prove that you are doing a “better” job (meaning overall smaller residuals) in the new more
complicated model than in the simpler one. That involves some statistical measures like the �
2
, which I will not get in
details (I can if you want, it is just that we are already near the end of October and the class ends in a month or so and
we still need to cover a lot), but rest assured that there are plenty of ways to measure if a new model with more factors
perform better (explain away the residuals) or worse than another model….lifetime guaranteed employment for people
who know statistics….
•As a visual reminder, it is not that obvious from the naked eye that we are doing such a good job…
(*) GARP FRM Part I material 2025
More Factors - XIV
100
Luc Faucheux 2025
•If you include statistical factors, your model becomes a “Statistical factors model”
•This is essentially (running the risk of maybe oversimplifying a wee little bit tad) a PCA (Principal Component
Analysis).
•It is guessing what kind of statistics (combination of the sampling observations) can be the best factors to explain
away the residuals.
•You usually rank those factors in the order of decreasing contribution to the ??????
�
100
Luc Faucheux 2025
•If you include statistical factors, your model becomes a “Statistical factors model”
•This is essentially (running the risk of maybe oversimplifying a wee little bit tad) a PCA (Principal Component
Analysis).
•It is guessing what kind of statistics (combination of the sampling observations) can be the best factors to explain
away the residuals.
•You usually rank those factors in the order of decreasing contribution to the ??????
�
How good of a job are you doing? ??????
�
101
Luc Faucheux 2025
�
101
Luc Faucheux 2025
How good of a job are you doing? ??????
�
- I
102
Luc Faucheux 2025 (*) Wikipedia
�
- I
102
Luc Faucheux 2025 (*) Wikipedia
How good of a job are you doing? ??????
�
- II
103
Luc Faucheux 2025
•�−�
�=�+�.�
�−�
�+ℇ
•Define the Total Sum of Squares as: ���=σ(�
�−??????�)
2
=�.�[�]
•Define the Residual Sum of Squares as: ���=σℇ
�
2
•In OLS, you are minimizing the ���
•The measure �
2
is defined as: �
2
=1−
���
���
•Ideally, �
2
is as close to 1 as possible
•Sometimes �
2
is called the “coefficient of determination”
�
- II
103
Luc Faucheux 2025
•�−�
�=�+�.�
�−�
�+ℇ
•Define the Total Sum of Squares as: ���=σ(�
�−??????�)
2
=�.�[�]
•Define the Residual Sum of Squares as: ���=σℇ
�
2
•In OLS, you are minimizing the ���
•The measure �
2
is defined as: �
2
=1−
���
���
•Ideally, �
2
is as close to 1 as possible
•Sometimes �
2
is called the “coefficient of determination”
How good of a job are you doing? ??????
�
- III
104
Luc Faucheux 2025
•Let’s compare the CAPM with the FRENCH-FAMA 3-factors… still a long way to go to ??????
�
=�
(*) GARP FRM Part I material 2025
FRENCH - FAMA 3 FACTORS
CAPM IMPROVEMENT
ALPHA BETA (MARKET) BETA (SMB) BETA (HML) R^2 ALPHA BETA (MARKET) R^2 dR^2
Banking -0.17 1.23 -0.16 0.84 0.76 0.03 1.10 0.58 0.18
Beer and Liquor 0.52 0.63 -0.39 -0.03 0.31 0.52 0.57 0.25 0.06
Chemicals -0.05 1.10 -0.03 0.46 0.68 0.05 1.04 0.62 0.06
Computers 0.06 1.30 0.22 -0.67 0.66 -0.10 1.41 0.58 0.08
Consumer Goods 0.22 0.68 -0.17 0.12 0.45 0.25 0.63 0.42 0.03
Electrical Equipment 0.10 1.26 -0.03 0.12 0.72 0.13 1.24 0.71 0.00
Retail 0.22 0.93 -0.04 0.01 0.61 0.22 0.92 0.61 0.00
Shipping Containers 0.00 1.01 -0.01 0.28 0.50 0.06 0.98 0.48 0.02
Transportation -0.01 1.00 0.05 0.44 0.65 0.09 0.95 0.59 0.06
Wholesale -0.10 0.88 0.26 0.28 0.72 -0.04 0.89 0.67 0.05
�
- III
104
Luc Faucheux 2025
•Let’s compare the CAPM with the FRENCH-FAMA 3-factors… still a long way to go to ??????
�
=�
(*) GARP FRM Part I material 2025
FRENCH - FAMA 3 FACTORS
CAPM IMPROVEMENT
ALPHA BETA (MARKET) BETA (SMB) BETA (HML) R^2 ALPHA BETA (MARKET) R^2 dR^2
Banking -0.17 1.23 -0.16 0.84 0.76 0.03 1.10 0.58 0.18
Beer and Liquor 0.52 0.63 -0.39 -0.03 0.31 0.52 0.57 0.25 0.06
Chemicals -0.05 1.10 -0.03 0.46 0.68 0.05 1.04 0.62 0.06
Computers 0.06 1.30 0.22 -0.67 0.66 -0.10 1.41 0.58 0.08
Consumer Goods 0.22 0.68 -0.17 0.12 0.45 0.25 0.63 0.42 0.03
Electrical Equipment 0.10 1.26 -0.03 0.12 0.72 0.13 1.24 0.71 0.00
Retail 0.22 0.93 -0.04 0.01 0.61 0.22 0.92 0.61 0.00
Shipping Containers 0.00 1.01 -0.01 0.28 0.50 0.06 0.98 0.48 0.02
Transportation -0.01 1.00 0.05 0.44 0.65 0.09 0.95 0.59 0.06
Wholesale -0.10 0.88 0.26 0.28 0.72 -0.04 0.89 0.67 0.05
Straight lines are everywhere
105
Luc Faucheux 2025
105
Luc Faucheux 2025
Straight Lines - I
106
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…
106
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…
Straight Lines - II
107
Luc Faucheux 2025
This is not a class in AI, that is another class that I also teach if you are interested
107
Luc Faucheux 2025
This is not a class in AI, that is another class that I also teach if you are interested
Straight Lines - III
108
Luc Faucheux 2025
CHECKED, we did that in Part UN…
108
Luc Faucheux 2025
CHECKED, we did that in Part UN…
Straight Lines - IV
109
Luc Faucheux 2025
WE JUST DID IT
109
Luc Faucheux 2025
WE JUST DID IT
Straight Lines - V
110
Luc Faucheux 2025
DONE IN PART DEUX, BUT WILL REVISIT SOME HERE
110
Luc Faucheux 2025
DONE IN PART DEUX, BUT WILL REVISIT SOME HERE
Straight Lines - VI
111
Luc Faucheux 2025
We sort of did it without explaining it in the “Statistical Factors Model”, if you guys
want I can put it in another deck as we are close to 100 slides….
111
Luc Faucheux 2025
We sort of did it without explaining it in the “Statistical Factors Model”, if you guys
want I can put it in another deck as we are close to 100 slides….
Straight Lines - VII
112
Luc Faucheux 2025
�−�
�=�+�.�
�−�
�+�
�.���+�
??????.���
WE JUST DID IT
112
Luc Faucheux 2025
�−�
�=�+�.�
�−�
�+�
�.���+�
??????.���
WE JUST DID IT
So at least for now…
113 Luc Faucheux 2025
113 Luc Faucheux 2025
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