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日期：2024-02-25 08:25

1 Group Project Financial Econometrics 2024

Students are encouraged to refer to other related literauture if thought useful/relevant and may

find the following two referenes, describing the testing of the Capital Asset Pricing (CAPM),

useful:

Introductory Econometrics for Finance, 4th Edition, C.Brooks, Cambridge University

Press. (Note: 2nd and 3rd editions also applicable). Chapter 14, section 14.2, pages

586-592. This has a more introductory approach.

The Econometrics of Financial Markets, J. Campbell, A.W. Lo and A. Craig MacKinlay,

Princeton University Press. Chapter 5, section 5.1 and 5.8. in particular. Note this

chapter is very detailed and technical.

1.1 Part 1 (40 marks)

Carefully read the attached article titled“A five-factor asset pricing model,” by Eugene Fama

and Kenneth French (2015, Jounral of Fianncial Economics). 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.

Groups have to decide for themselves how many words to allocate to this part, a useful guide

is 600 words (40% of the 1500 word limit). Therefore it cannot be a description of everything,

nor can it realistically cover all the results/tables, but requires students to identify the key

parts, salient points and conclusions.

1.2 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 August 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 August 2022. The third data file (FF5.txt) contains monthly data on the five FamaFrench factor returns from July 1963 until August 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 (in excess of the risk-free rate)

on the 25 Fama-French portfolios and denote the matrix of excess returns by Re

. This

matrix should have 710 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 Re

and f?

[10 marks]

1

b) Estimate the [α, β] matrix from separate multiple linear OLS time series regressions for

each portfolio (i = 1, ..., N), Re on a constant and the five factors; that is, obtain the

OLS estimates of αi and βi

from the multiple linear regression model

R

e

it = αi + βift + ?it, t = 1, . . . , T, (1)

where T = 710 is the time series sample size. Assume that the ?’s are normally distributed

with zero mean and covariance matrix Σ. (The Σ matrix has 25 rows and 25 columns

and is calculated the OLS residuals from each regression ) Note that your [?α, β?] matrix

should have N rows and K + 1 columns, where N = 25 and K = 5. For each of the 25

portfolio excess returns, compute the time series R2

. Report the portfolio-specific time

series R2

s 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 factor risk premia (γ1), we follow the asset

pricing literature and run the following cross-sectional regression:

Rˉe = 1N γ0 + βγ?

1 + e, (2)

where Rˉe 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), β? are the

estimated betas from the previous time-series multiple linear regression model, and e are

the unobservable model’s pricing errors (a column vector with N rows and one column).

Estimate γ = [γ0, γ′

1

]

′ as

γ? ≡ [?γ0, γ?

′

1

]

′ = (X?′X?)

?1X?′Rˉe

, (3)

where X? = [1N , β?]. In addition, let A? = (X?′X?)

?1X?′ and denote by Σ and ? V?

f the sample

counterparts of Σ and Vf , respectively. Then, the asymptotic covariance matrix of ?γ is

given by (see the attached Shanken (1992) article)

V S = (1 + ?γ

′

1V? ?1

f

γ?1)A?Σ?A?′ +



0 0′

K

0K V?

f



, (4)

where 0K is a column vector of zeros (with K rows and one column). The t-statistics can

be computed as ?γ divided by the square root of the diagonal elements of V S divided by

T.

Report the γ estimates (?γ) and their t-statistics. Are the signs and magnitudes of the γ

estimates what you would expect? Compare the γ estimates for the five factors with the

corresponding time series sample means of the factors. Do the time series sample means

differ from their corresponding γ estimates? And if your answer to the latter question

is yes, why is this the case? Finally, determine whether the γ estimates are statistically

significant at the conventional significance levels.

[20 marks]

2

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 as

e? = Rˉe ? 1N γ?0 ? β?γ?1 (5)

and the sample cross-sectional R2 as

R

2

cs = 1 ?

e?

′

e?

e?

′

0

e?0

, (6)

where ?e0 = Rˉe ? 1N



1

′

N Rˉe

N



.

Compute R2

cs and comment on the cross-sectional explanatory power of the five-factor

model.

[10 marks]

e) Fama and French (2015) argue that their model is very good at explaining the time-series

and cross-sectional variation in equity 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]

3