IB2B20

Financial Econometrics

Group Project, 2022

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Your assignment instructions begin below.

Template Updated 4/11/21

Part 1 (40 marks)
Carefully read the 2015 article titled “A five-factor asset pricing model” by Eugene Fama
and Kenneth French (attached). Provide a summary of the article and explain in your
own words what Fama and French are trying to accomplish in the article. In addition,
discuss whether in your opinion the proposed investment strategies are consistent with
the efficient market hypothesis.
Part 2 (60 marks)
You will need to use Matlab in order to carry out the empirical analyses below. Please
include the Matlab code with your submission. You are given three data files in txt format.
The first data file (FF25.txt) contains monthly data on the 25 Fama-French size and book-
to-market ranked portfolio returns from July 1963 until January 2022. The second data
file (RF.txt) contains monthly data on the risk-free rate (the one-month Treasury Bill
rate) from July 1963 until January 2022. The third data file (FF5.txt) contains monthly
data on the five Fama-French factor returns from July 1963 until January 2022. In
FF5.txt, the first column is the excess market return (in excess of the risk-free rate), the
second column is the size factor, the third column is the value factor, the fourth column
is the profitability factor, and the fifth column is the investment factor. The data is in
percent.
a) Load the data into Matlab. Construct the excess returns on the 25 Fama-French
portfolios and denote the matrix of excess returns by Re. This matrix should have
703 rows and 25 columns. Denote the five Fama-French factors by f, with covariance
matrix Vf . Can we reject the null hypothesis of non-stationarity for R
e and f? [10
marks]
b) Estimate the [α, β] matrix from the multiple linear time series regression of Re on
a constant and the five factors; that is, obtain the OLS estimates of α and β from
the multiple linear regression model
Ret = α + βft + t, t = 1, . . . , T, (1)
where T = 703 is the time series sample size. Assume that the t’s are normally
distributed with zero mean and covariance matrix Σ. (The Σ matrix has 25 rows
and 25 columns.) Note that your [αˆ, βˆ] matrix should have N rows and K + 1
columns, where N = 25 and K = 5. For each of the 25 portfolio excess returns,
1
compute the time series R2. Report the portfolio-specific time series R2s and the
average time series R2 across the 25 portfolios. Comment on the results and explain
whether in your opinion the five factors of Fama and French (2015) explain a large
portion of the time series variation in the 25 Fama-French portfolio excess returns.
[10 marks]
c) We are interested in estimating the zero-beta rate and the factor risk premia. In
order to pin down the zero-beta rate (γ0) and the risk premia (γ1), we follow the
asset pricing literature and run the following cross-sectional regression:
R¯e = 1Nγ0 + βˆγ1 + e, (2)
where R¯e are the time series sample means of the excess returns on the 25 Fama-
French portfolios, 1N is a column vector of ones (with N rows and one column), and
βˆ are the estimated betas from the previous time-series multiple linear regression
model.
Estimate γ = [γ0, γ
′
1]
′ as
γˆ ≡ [γˆ0, γˆ′1]′ = (X ′X)−1X ′R¯e, (3)
where X = [1N , βˆ]. In addition, let A = (X
′X)−1X ′ and denote by Σˆ and Vˆf the
sample counterparts of Σ and Vf , respectively. Then, the asymptotic covariance
matrix of γˆ is given by (see the attached Shanken (1992) article)
V S = (1 + γˆ′1Vˆ
−1
f γˆ1)AΣˆA
′ +
[
0 0′K
0K Vˆf
]
, (4)
where 0K is a zero column vector (with K rows and one column). The t-statistics
can be computed as γˆ divided by the square root of the diagonal elements of VS
divided by T.
Report the γ estimates and their t-statistics. Are the signs and magnitudes of the
γ estimates what you would expect? Are these γ estimates statistically significant
at the conventional significance levels? [10 marks]
d) We would like to compute the cross-sectional R2 and determine whether the five
factors of Fama and French (2015) explain a nontrivial part of the cross-sectional
variation in expected excess returns on the 25 Fama-French portfolios. To perform
this task, we define the sample pricing errors of the five-factor Fama-French model
2
as
eˆ = R¯e − 1N γˆ0 − βˆγˆ1 (5)
and the sample cross-sectional R2 as
R2cs = 1−
eˆ′eˆ
eˆ′0eˆ0
, (6)
where eˆ0 = R¯
e − 1N
(
1′N R¯
e
N
)
.
Compute R2cs and comment on the cross-sectional explanatory power of the five-
factor model. [10 marks]
e) We would like to formally test whether the five-factor model of Fama and French
(2015) is rejected by the data. To perform this task, we consider the finite-sample
test of Gibbons, Ross, and Shanken (GRS, 1989). The article describing the GRS
test is attached. I also include a Matlab function for this test (the file’s name is
grs.m). Recall grs.m in your main Matlab code and run it. Report the test statistic
and its p-value, and discuss whether the five-factor model is rejected by the data
at conventional significance levels. [10 marks]
f) Fama and French (2015) argue that their model is very good at explaining the
time-series and cross-sectional variation in asset expected excess returns although
they notice that their model is formally rejected by the GRS test. Overall, based on
your reading of their article in Part 1 of this assignment and on your own empirical
analysis, would you agree with their claim? Provide some brief comments. [10
marks]
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