answers to the assignment deck, MVT, VaR, Sampling correlation, causality and all that good stuff

answers to the assignment deck, MVT, VaR, Sampling correlation, causality and all that good stuff

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

2. VaR
(shorter deck with
answers to the
assignment)

3. Sampling/Measuring
vs
Guessing/Predicting/Modeling
3
Luc Faucheux 2025

4. 4
Luc Faucheux 2025
Measuring vs Guessing - I
•Guessing / Predicting / Modeling the Mean and Variance of a
portfolio
•Π=σ(??????
�.�
�)
•??????
�: notional (size or amount) of the asset �
�
•�
�: price of asset �
�
•�
�: return of asset �
�
•The portfolio return is: �(Π)=σ??????
�.�
� with ??????
�=??????
�.�
�/Π
•Portfolio expected return and volatility:
•??????=??????�Π=σ??????
�??????
�
•??????
2
=??????�Π= ??????{�Π−??????�Π}
2
= ??????�Π
2
−??????�Π
2
•??????
2
=σ
�
σ
�??????
�.??????
�.??????
�.??????
�.??????
��
•Sampling a distribution of returns {??????
??????} with N
observations
•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??????]

5. 5
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Measuring vs Guessing – II – Guessing the Portfolio Variance
??????
2
=??????�Π=෍
�
෍
�
??????
�.??????
�.??????
�.??????
�.??????
��
Weight ??????
� from portfolio construction
Weight ??????
� from portfolio construction
Correlation ??????
�� between the returns from
asset �
� and �
� , can be historical, or
stressed from a baseline level
Volatility??????
� of the returns from asset
�
�, can be historical, or stressed from a
baseline levelVolatility ??????
� of the returns from asset
�
�, can be historical, or stressed from a
baseline level

6. 6
Luc Faucheux 2025
Measuring vs Guessing - III
•Guessing / Predicting / Modeling the Mean and Variance of a
portfolio
•??????
2
=??????�Π=σ
�
σ
�??????
�.??????
�.??????
�.??????
�.??????
��
•Weight ??????
� and ??????
� from portfolio construction: ??????
�=??????
�.�
�/Π
•Volatility ??????
� of the returns from asset �
�, can be historical, or
stressed from a baseline level, or estimated
•??????
�
2
=??????�
�= ??????{�
�−??????�
�}
2
= ??????�
�
2
−??????�
�
2
•Correlation ??????
�� between the returns from asset �
� and �
� , can be
historical, or stressed from a baseline level
•??????�
�=??????
�
•??????�
�=??????
�
2
•෩�
�=�
�−??????�
�
•??????෩�
�.෩�
�=??????
�.??????
�.??????
��
•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
•Sample Variance Estimator
•
෢
??????
2
=
1
??????
.σ
�(�
�−Ƹ??????)
2
•??????
෢
??????
2
=
??????−1
??????
??????
2
•??????
෢
??????
2
=??????
2
Ƹ??????=
??????
4−??????
4
??????
2
−
??????
2
??????

7. 7
Luc Faucheux 2025
Measuring vs Guessing – IV – Sampling and measuring
•HOW DOES IT WORK?
•Get a sample with N observations {�
�}
•Compute the sample mean ො??????=
1
??????
.σ
��
�
•??????ො??????=??????
•The sample mean ො?????? is your estimate of the “true” population mean ??????
•That is an estimate, that might not be the actual value
•Compute the sample standard deviation
෢
??????
2
=
1
??????
.σ
�(�
�−ො??????)
2

8. 8
Luc Faucheux 2025
Measuring vs Guessing – IV – Sampling and measuring b
•HOW DOES IT WORK?
•Compute the sample standard deviation
෢
??????
2
=
1
??????
.σ
�(�
�−ො??????)
2

•??????
෢
??????
2
=
??????−1
??????
??????
2
•??????
2
=??????
෢
??????
2
.
??????
??????−1
•So the value
෢
??????
2
.
??????
??????−1
is your estimate of the “true” population variance ??????
2
•That is an estimate, that might not be the actual value
•But hey, that is the best that you can do…

9. Assignment
9
Luc Faucheux 2025

10. 10
Luc Faucheux 2025
Assignment – portfolio construction - I
•Stock �
1 exhibits a return �
1
•The expected value, mean, average, location of �
1 is: ??????�
1=??????
1
•The standard deviation of �
1 is ??????
1=??????[�
1]
•Stock �
2 exhibits a return �
2
•The expected value, mean, average, location of �
2 is: ??????�
2=??????
2
•The standard deviation of �
2 is ??????
2=??????[�
2]

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Assignment – portfolio construction - II
•The correlation between the stock returns is:
•??????
12=
??????[෪??????
1.෪??????
2]
??????
1.??????
2
=
??????[??????
1−??????[??????
1].(??????
2−??????[??????
2])]
??????
1.??????
2
=
??????[??????
1−??????
1.(??????
2−??????
2)]
??????
1.??????
2
•??????෪�
1.෪�
2=??????
12.??????
1.??????
2=??????
12
•෪�
1=�
1−??????[�
1]
•෪�
2=�
2−??????[�
2]

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Assignment – portfolio construction - III
•The portfolio that you are constructing has weight ?????? of stock �
1 and weight (1−??????) of
stock �
2
•The expected value of the returns of the portfolio is:
•??????=σ??????
�??????
�=??????
1??????
1+??????
2??????
2=??????.??????
1+1−??????.??????
2
•The expected standard deviation of the portfolio return
•??????
2
= (σ
�
σ
�??????
�.??????
�.??????
�.??????
�.??????
��)
•??????
2
=??????
2
.??????
1
2
+(1−??????)
2
.??????
2
2
+2.??????.1−??????.??????
1.??????
2.??????
12

13. 13
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Assignment – portfolio construction - IVw1 w2 r var stdev
0% 100% 3.00% 2% 15%
5.00% 95% 3.35% 2% 14%
10.00% 90% 3.70% 2% 13%
15.00% 85% 4.05% 1% 12%
20.00% 80% 4.40% 1% 12%
25.00% 75% 4.75% 1% 11%
Return_1 10%30.00% 70% 5.10% 1% 12%
Return_2 3%35.00% 65% 5.45% 1% 12%
40.00% 60% 5.80% 2% 13%
Volatility_1 30%45.00% 55% 6.15% 2% 14%
Volatility_2 15%50.00% 50% 6.50% 2% 15%
55.00% 45% 6.85% 3% 16%
Correlation -30.00%60.00% 40% 7.20% 3% 17%
65.00% 35% 7.55% 3% 19%
70.00% 30% 7.90% 4% 20%
75.00% 25% 8.25% 5% 22%
80.00% 20% 8.60% 5% 23%
85.00% 15% 8.95% 6% 25%
90.00% 10% 9.30% 7% 27%
95.00% 5% 9.65% 8% 28%
100.00% 0%10.00% 9% 30%
w2=1-w1
r=w1*r1+w2.r2
var=w1^2*v1^2+w2^2.v2^2+2.rho.v1.v2
stdev=sqrt(var)

14. 14
Luc Faucheux 2025
Assignment – portfolio construction - V

15. 15
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Assignment – portfolio construction - VIw1 w2 r var stdev
0% 100% 3.00% 2% 15%
5.00% 95% 3.35% 2% 14%
10.00% 90% 3.70% 2% 13%
15.00% 85% 4.05% 1% 12%
20.00% 80% 4.40% 1% 12%
25.00% 75% 4.75% 1% 11%
Return_1 10%30.00% 70% 5.10% 1% 12%
Return_2 3%35.00% 65% 5.45% 1% 12%
40.00% 60% 5.80% 2% 13%
Volatility_1 30%45.00% 55% 6.15% 2% 14%
Volatility_2 15%50.00% 50% 6.50% 2% 15%
55.00% 45% 6.85% 3% 16%
Correlation -30.00%60.00% 40% 7.20% 3% 17%
65.00% 35% 7.55% 3% 19%
70.00% 30% 7.90% 4% 20%
75.00% 25% 8.25% 5% 22%
80.00% 20% 8.60% 5% 23%
85.00% 15% 8.95% 6% 25%
90.00% 10% 9.30% 7% 27%
95.00% 5% 9.65% 8% 28%
100.00% 0%10.00% 9% 30%
w2=1-w1
r=w1*r1+w2.r2
var=w1^2*v1^2+w2^2.v2^2+2.rho.v1.v2
stdev=sqrt(var)
The portfolio has (yearly) returns with :
Mean : 5.1%
Standard deviation : 12%

16. 16
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Assignment – portfolio construction - VII
The portfolio has (yearly) returns with :
Mean : 5.1%
Standard deviation : 12%
•What is the probability that the portfolio will suffer in any given year a negative return of
magnitude greater than 10%?
•What is the probability that the portfolio will suffer in any given year a negative return of
magnitude greater than 5%?
•What is the probability that the portfolio will suffer in any given year a negative return?

17. 17
Luc Faucheux 2025
Assignment – portfolio construction – VII - ANSWER
The portfolio has (yearly) returns with :
Mean : 5.1%
Standard deviation : 12%

18. 18
Luc Faucheux 2025
Assignment – portfolio construction - VIII
The portfolio has yearly returns with :
Mean : 5.1%
Standard deviation : 12%
•There is a 10% probability that the portfolio will suffer a negative return of magnitude greater than �
90%.
•There is a 5% probability that the portfolio will suffer a negative return of magnitude greater than �
95%.
•There is a 1% probability that the portfolio will suffer a negative return of magnitude greater than �
99%.
•Compute �
90%, �
95% and �
99% and their associated Z-scores �
90%, �
95% and �
99%
•If the portfolio has a value estimated in $ of 13.5 millions, there is a 1% probability that the portfolio will
suffer a negative drawdown of magnitude greater than ??????
99%($). Compute ??????
99%.
•??????
99% is expressed in ($) and is referred to as the 99% Value-at-Risk

19. 19
Luc Faucheux 2025
Assignment – portfolio construction – VIII - ANSWER
The portfolio has yearly returns with :
Mean : 5.1%
Standard deviation : 12%

20. 20
Luc Faucheux 2025
Assignment – portfolio construction - IX
•If the portfolio has a value estimated in $ of 13.5 millions, there is a 1% probability that the portfolio will
suffer a negative drawdown of magnitude greater than ??????
99%($). Compute ??????
99%.
•??????
99% is expressed in ($) and is referred to as the 99% Value-at-Risk
•You change the weight from 30% to 40%. Compute the new ??????
99%

21. 21
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Assignment – portfolio construction – IX - ANSWERS

22. 22
Luc Faucheux 2025
Assignment – Sampling - I
•You are an analyst working at an allocator and you are interviewing an emerging manager who is telling
you the following.
•”My strategy is amazing, I achieve 10% returns on average per year with a volatility of 2%, my Sharpe
ratio is 5, and you should invest a lot of money in my fund, and I want a couple of Veyrons for me and my
staff.”
•“Oh by the way I have no track record, but if I did backtest my strategy over the past 8 years I get the
following returns: {10%, 12%, 16%, -2%, 5%, 8%, 10%, 3%}

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Assignment – Sampling - II
•“Oh by the way I have no track record, but if I did backtest my strategy over the past 8 years I get the
following returns: {10%, 12%, 16%, -2%, 5%, 8%, 10%, 3%}
•What is your estimate of the average returns of the fund?
•If we define the margin of error as one standard deviation away from the mean on either side, what is
your margin of error on your estimate?
•What is your estimate of the standard deviation of the returns of the fund?
•What would be the probability that the fund on any given year will return a negative return greater in
magnitude than 10%?
•What would be the probability that the fund on any given year will return a negative return greater in
magnitude than 5%?
•What would be the probability that the fund on any given year will return a negative return?

24. 24
Luc Faucheux 2025
Assignment – Sampling – II - ANSWER

25. Some useful formulae (from last class)
25
Luc Faucheux 2025

26. 26
Luc Faucheux 2025
Expectations - I
•Discrete case: ??????�=σ
�??????
�.�
�
•??????
� is the probability to observe the output �
� for the stochastic variable �
•σ
�??????
�=1
•Continuous case: ??????�=׬
−∞
+∞
????????????.??????.????????????
•???????????? is the probability density function to observe the output ?????? for the
stochastic variable �
•׬
−∞
+∞
????????????.????????????=1

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Expectations - II
•The expectation operator is a linear operator
•??????�+�=??????�+??????�
•????????????.�=??????.??????� where ?????? is a scalar (number)
•????????????=?????? where ?????? is a scalar (number)
•CAREFUL
•????????????
2
=??????
2
=????????????
2
where ?????? is a scalar (number)
•??????�
2
≠??????�
2
where � is a stochastic (random) variable

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Luc Faucheux 2025
Variance - I
•??????
2
=??????�= ??????{�−??????�}
2
•??????
2
=??????�= ??????{�
2
+??????�
2
−2.??????�.�}
•??????
2
=??????�= ??????[�
2
]+??????[??????�
2
]−??????[2.??????�.�]
•Again, because ??????� and ??????�
2
are numbers and NOT stochastic variables,
they can be taken out of the expectation operator
•??????
2
=??????�= ??????�
2
+??????�
2
−2.??????�.??????[�]
•??????
2
=??????�= ??????�
2
+??????�
2
−2.??????�
2
•??????
2
=??????�= ??????�
2
−??????�
2

29. 29
Luc Faucheux 2025
Variance - II
•??????
2
=??????�= ??????{�−??????�}
2
•??????
2
=??????�= ??????�
2
−??????�
2
•A great illustration that: ??????�
2
≠??????�
2
otherwise ??????
2
=??????� would always
be equal to zero, which we know is not true

30. 30
Luc Faucheux 2025
Correlation - I
•The correlation between the stock returns �
1 and �
2 is:
•??????
12=
??????[෪??????1.෪??????2]
??????1.??????2
=
??????[??????1−??????[??????1].(??????2−??????[??????2])]
??????1.??????2
=
??????[??????1−??????1.(??????2−??????2)]
??????1.??????2
•??????෪�
1.෪�
2=??????
12.??????
1.??????
2=??????
12
•෪�
1=�
1−??????[�
1]
•෪�
2=�
2−??????[�
2]
•??????�
1=??????
1
•??????�
2=??????
2

31. 31
Luc Faucheux 2025
Correlation - II
•The correlation between the stock returns �
1 and �
2 is:
•??????
1.??????
2.??????
12=??????�
1−??????�
1.�
2−??????�
2
•??????
1.??????
2.??????
12=??????�
1.�
2−??????�
1.�
2−??????�
2.�
1+??????�
1.??????�
2
•??????
1.??????
2.??????
12=??????�
1.�
2]−??????[??????�
1.�
2]−??????[??????�
2.�
1]+??????[??????�
1.??????�
2
•Because ??????�
1 is a number (and not a stochastic variable), we can take it
out of the expectation operator
•??????
1.??????
2.??????
12=??????�
1.�
2]−??????�
1.??????�
2−??????�
2.??????[�
1]+??????[??????�
1.??????�
2

32. 32
Luc Faucheux 2025
Correlation - III
•??????
1.??????
2.??????
12=??????�
1.�
2]−??????�
1.??????�
2−??????�
2.??????[�
1]+??????[??????�
1.??????�
2
•Because ??????�
1. ??????�
2 is also number (and not a stochastic variable), we can
take it out of the expectation operator
•????????????�
1.??????�
2=??????�
1.??????�
2.??????1=??????�
1.??????�
2
•??????
1.??????
2.??????
12=??????[�
1.�
2]−??????�
1.??????�
2−??????�
2.??????[�
1]+??????�
1.??????�
2
•??????
1.??????
2.??????
12=??????[�
1.�
2]−??????�
1.??????�
2

33. 33
Luc Faucheux 2025
Correlation - IV
•??????
12=
??????[෪??????
1.෪??????
2]
??????
1.??????
2
=
??????[??????
1−??????[??????
1].(??????
2−??????[??????
2])]
??????
1.??????
2
=
??????[??????
1−??????
1.(??????
2−??????
2)]
??????
1.??????
2
•??????
12=
??????[෪??????
1.෪??????
2]
??????
1.??????
2
=
??????[??????
1.??????
2]−????????????
1.????????????
2
??????
1.??????
2
=
????????????
1.??????
2−??????
1.??????
2
??????
1.??????
2
•Obviously ??????
12=??????
21

34. 34
Luc Faucheux 2025
Correlation - V
•Correlation versus Causality
•“Causality is the Holy Grail of signal creation, correlation is a pitfall of Risk Management”
•“"You see, there is only one constant, one universal, it is the only real truth: causality. Action, reaction.
Cause and effect””

35. 35
Luc Faucheux 2025
Correlation - VI
•Correlation versus Causality

36. 36
Luc Faucheux 2025
Correlation - VII
•Correlation versus Causality (thanks to Tom Brackbill for pointing this out)

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