BUFN 640 – Problem Set 2
Due on Sunday, October 10 at 11:59 pm
This assignment will show you how to estimate and test the Fama-French 3-factor model
and solve the Markowitz portfolio choice problem in Python. Please use the template Python
notebook (Problem Set 2 – template.ipynb) in the shared Google Drive folder or on Canvas
to load the data. Implement the rest of the file yourself.
Important: Please submit your homework using Canvas. Your submission needs to in-
clude a single file – a copy of your Python notebook (.ipynb Jupyter notebook file). To
produce the latter, please click File → Download .ipynb in Google Colab, then save and
upload the file on Canvas. Multiple-choice questions are to be answered in the notebook file,
too. Each student has to submit his/her individual assignment.
Part I: Multiple-choice questions (25 points)
1. (5 points) If our regression equation is y = Xβ+u, where we have T observations and
K regressors, what will be the dimension of β using the standard matrix notation
(a) T ×K
(b) T × 1
(c) K × 1
(d) K ×K.
2. (5 points) Suppose that the value of R2 for an estimated regression model is exactly
one. Which of the following are true?
i. All of the data points must lie exactly on the line
ii. All of the residuals must be zero
iii. All of the variability of y around its mean has been explained by the model
iv. The fitted line will be horizontal with respect to all of the explanatory variables.
(a) (ii) and (iv) only
(b) (i) and (iii) only
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(c) (i), (ii), and (iii) only
(d) (i), (ii), (iii), and (iv).
3. (5 points) Which of these is a mathematical expression of the residual sum of squares?
i. u?′u?
ii. [u?1, u?2, ..., u?T ]
iii. u?1 + u?2 + ...+ u?T
(a) (i) only
(b) (i) and (ii) only
(c) (i) and (iii) only
(d) (i), (ii), and (iii).
4. (5 points) Why is R2 a commonly used and perhaps better measure of how well a
regression model fits the data than the residual sum of squares (RSS)?
(a) The RSS is often too large
(b) The RSS does not depend on the scale of the dependent variable whereas the R2
does
(c) The RSS depends on the scale of the dependent variable whereas the R2 does not
(d) The RSS depends on the scale of the independent variable whereas the R2 does
not.
5. (5 points) In the following regression estimated on 64 observations:
yt = β1 + β2X2,t + β3X3,t + β4X4,t + ut,
Which of the following null hypotheses could we test using an F -test?
i. β2 = 0
ii. β2 = 1 and β3 + β4 = 1
iii. β3β4 = 1
iv. β2 ? β3 ? β4 = 1.
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(a) (i) and (ii) only
(b) (ii) and (iv) only
(c) (i) and (iii) only
(d) (i), (ii), (iii), and (iv)
(e) (i), (ii), and (iv) only.
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Part II: The Fama and French Model (40 points)
CAPM In the previous assignment you used a univariate linear regression to test the
Capital Asset Pricing Model (CAPM). CAPM relates the expected return on portfolio p –
denoted by E[Rp] – and the expected return on the market – denoted by E[Rm] – as follows:
E [Rp]?Rf = βp (E[Rm]?Rf) ,
where Rf is the risk-free rate and βp =
Cov[Rp,Rm]
V ar[Rm]
. An empirical test of the CAPM is based
on the following regression:
Rp,t ?Rf,t = αp + βp (Rm,t ?Rf,t) + up,t, t = 1, . . . , T
implying that, if CAPM holds, the intercept αp should be equal to 0.
Fama-French 3-factor (FF3) Another very popular asset pricing model in the empirical
finance literature is the Fama-French 3-factor (FF3) that was published in 1993. Eugene
Fama and Kenneth French found that value stocks tend to outperform growth stocks (i.e.,
value), and that small-cap stocks outperform large-cap stocks (i.e., size). Thus, the FF3
mode adds in size and value as risk factors to the model as shown below
E [Rp]?Rf = αp + βp,M (Rm,t ?Rf,t) +βp,SRSMB,t + βp,hRHML,t︸ ︷︷ ︸
new factors
+up,t.
Setup
Use 25 Fama and French portfolios from Ken French’s website at the monthly frequency,
which contain returns on portfolios sorted on size (5 portfolios) and book-to-market (5 port-
folios). The total is, therefore, 25 portfolios. Start your sample in 1926 (to present). The
portfolios are named according to their first and second sorting variable. In particular:
? SMALL LoBM, ME1 BM1, ..., ME1 HiBM are Small portfolios (split into 5 book-to-
market portfolios)
? ...
? Big LoBM, ME5 BM2, ..., ME5 HiBM are Large portfolios (split into 5 book-to-market
portfolios)
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You may use the example from class to load data into Python using the DataReader
interface (don’t forget to change to monthly frequency). Specifically, the following piece of
code downloads the data from the website, unpacks it, and load into Python as a Pandas
table:
import pandas datareader.data as web
r = web.DataReader(’25 Portfolios 5x5’, ’famafrench’, start=1900)[0]/100
I have also prepared a Colab template notebook which guides you through these steps.
Questions
Using Python:
1. (10 points) Estimate the intercept αp and the slope βp for all 25 portfolios using OLS
for both the CAPM and the FF3. Are the annualized αp’s economically large? (You
can annualize monthly αp simply by multiplying it by 12). How different are the
magnitudes of alpha in the CAPM vs FF3 model? How different are the R2 for the
two models? Hint: you can use my regression function from the FamaFrench notebook.
2. (10 points) For the Small HiBM portfolio test the null that βSMB = βHML using the
F -test. What’s your conclusion, can you reject the hypothesis that the two coefficients
are equal? Hint: we have tested a similar restriction in the FamaFrench notebook.
3. (5 points) Test the null hypothesis H0 : αp = 0 for each of the 25 portfolios, using
a two-sided alternative hypothesis and the FF3 model. Specify the test statistic for
a level of significance equal to 5%. Compute standard errors and the p-values for all
αp. Hint: this should be easy and requires no manual calculations. You may rely
on the output of a Python’s regression function or my regression function from the
FamaFrench notebook.
4. (5 points) Are the estimated betas in the FF3 model what you expected them to be
for all the portfolios? That is, what are the average values of βp,M , βp,s, βp,h? Are some
betas higher or lower for small/big/value/growth stocks?
5. (5 points) Which portfolios seem to be the best performing in terms of expected re-
turns? Which ones are the worst performing? Do you see any pattern between size
and/or book-to-market and performance?
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6. (5 points) If you were an asset manager, what kind of portfolios would you invest in?
If you could short some of the portfolios, would you do that?
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Part III: An Optimal Portfolio of Industries (35 points)
Modern Portfolio Theory Modern Portfolio Theory (MPT) is an investment theory
developed by Harry Markowitz and published under the title ”Portfolio Selection” in the
Journal of Finance in 1952.
Fundamentally, this is a theory of “risk-return trade-off.” Higher risk is associated with
greater probability of higher return and lower risk with a greater probability of smaller
return. MPT assumes that investors are risk-averse, meaning that given two portfolios that
offer the same expected return, investors will prefer the less risky one. Thus, an investor will
take on increased risk only if compensated by higher expected returns.
Another factor comes in to play in MPT is “diversification.” Modern portfolio theory
says that it is not enough to look at the expected risk and return of one particular stock. By
investing in more than one stock, an investor can reap the benefits of diversification – chief
among them, a reduction in the riskiness of the portfolio.
What you need to understand is “risk of a portfolio is not equal to average/weighted-
average of individual stocks in the portfolio.” In terms of return, yes, it is the aver-
age/weighted average of individual stock’s returns, but that’s not the case for risk. The
risk is about how volatile the asset is, if you have more than one stock in your portfolio,
then you have to take count of how these stocks movement correlates with each other. The
beauty of diversification is that you can even get lower risk than a stock with the lowest risk
in your portfolio, by optimising the allocation.
Setup
Start by downloading and importing monthly industry data from Ken French’s website.
This is the data you will use in your group project as well. To load the data, you can use
the DataReader interface. Specifically, the following piece of code downloads the data from
the website, unpacks it, and load into Python as a Pandas table:
import pandas datareader.data as web
r = web.DataReader(’12 Industry Portfolios’, ’famafrench’, start=1900)[0]/100
Next load the risk-free rate, merge it with returns, and compute excess returns on every
portfolio (in excess of the risk-free rate). The template Colab file shows how to perform
these steps.
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Questions
Using Python:
1. (5 points) Now that have obtain excess returns on every industry, compute and show
their means. Which industries have performed well?
2. (15 points) Compute the minimum variance portfolio using these 12 industries. Recall,
the minimum variance portfolio is the portfolio that delivers the lowest variance of any
possible investment portfolio. Use the formula I showed in class to find this portfolio’s
weights. Which industries have the highest/lowest weights? Perhaps you found that
some industries that had high mean returns in the previous question command a neg-
ative weight in the minimum-variance portfolio (that is, we would want to short sell
these industries). Explain why this could be the case.
3. (15 points) Compute the weights of the portfolio with the highest possible Sharpe
ratio using the formula I showed in class. Compare the weights to the weights of the
minimum variance portfolio. Recall, the highest Sharpe ratio portfolio is the portfolio
that delivers the highest possible expected excess return per unit of risk (as measured by
the standard deviation of portfolio returns). Which industries have the highest/lowest
weights? Perhaps you found that some industries that had high mean returns in the
previous question command a negative weight in the maximum Sharpe Ratio portfolio
(that is, we would want to short sell these industries). Explain why this could be the
case. Hint: you may use the Efficient Frontier and Modern Portfolio Theory notebook
I discussed in class. Specifically, the section “The Markowitz unconstrained solution
(using matrix algebra)” performs this computation.
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