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BUFN 640 – Problem Set

创建日期：2024-04-17 16:43:51

所属分类：计算机computer science

标签： BUFN 640

Python

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BUFN 640 – Problem Set 2

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
1
(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.
2
(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.
3
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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