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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 Fama-French 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]

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 ↵i and βi from the multiple linear regression model

where T = 710 is the time series sample size. Assume that the ε’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 [, ] 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 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 factor risk premia (γ1), we follow the asset pricing literature and run the following cross-sectional regression:

where  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),  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, ] 0 as

where  In addition, let  and denote by  and  the sample counterparts of Σ and Vf , respectively. Then, the asymptotic covariance matrix of  is given by (see the attached Shanken (1992) article)

where 0K is a column vector of zeros (with K rows and one column). The t-statistics can be computed as  divided by the square root of the diagonal elements of V S divided by T.

Report the γ estimates () 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]

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

and the sample cross-sectional R2 as

where .

Compute  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]