Mathematical and Empirical Finance, 2023–2024, Block 5
Assignment 1: Markowitz and CAPM
Note: You should submit your report (written in English) as a PDF file in combination with a code file, via the Assignment upload tool in Canvas. It should be possible to read your report and understand your answers without having to open the code file; but you may refer to the details of particular calculations in the code file. Some questions may require an answer to an earlier question. If you have been unable to fully answer the earlier question, replace this answer by a (reasonable) hypothetical value, and clearly indicate that you will use this guess in the rest of the assignment. Clearly indicate the names of your team (of one, two or three students) on the first page of report, and in the names of both the Excel file and the PDF file (use the format “MEFAssignment1Name1Name2Name3.pdf”). If (and only if) you have problems submitting your work via Canvas, you may e-mail it to [email protected], with “MEF Assignment 1”in the subject line.
The content of this assignment is intellectual property of the Uni- versity of Amsterdam. Any distribution of this material, such as sharing on studeersnel.nl, requires prior written approval by the University of Amsterdam.
In the empirical analysis of asset pricing models, the “size effect” is of- ten found. This refers to the phenomenon that firms with a small market capitalization (= market value of its shares outstanding) often appear to generate higher stock returns than larger firms. In this assignment we try to quantify this effect for the US stock market, based on data that has been downloaded from Ken French’s data library. The file Size.csv contains monthly data, January 1973 – January 2024, on:
❼ the monthly returns on 10 size decile portfolios (named r1 through r10);
❼ the monthly market return (rM) and the risk-free rate (rf);
❼ the Fama-French factors SMB and HML.
All data are monthly percentage returns/interest rates (not annualized). Details on the construction of the decile portfolios can be found on the web site mentioned above. Each year all firms have been sorted on their market capitalization; the smallest 10% are combined in a (value-weighted) portfolio with monthly return r1, the next 10% gives r2, and so on; so r10 is the return on a value-weighted portfolio of the largest 10% of the firms.
1. [10 points] Construct a table where for each of the 10 size portfolios, as well as the market portfolio, you report (a) the mean return; (b) the standard deviation of the return, and (c) the Sharpe ratio. Using this information, comment on the size effect: do small firms indeed have a higher expected return, and does this also correspond to a higher risk?
2. [25 points] The results from slide 23 of week 1 imply the following formula for the mean-variance frontier:
where a, b and c are defined on slide of the slides of Week 1. Cal- culate the sample mean vector and variance-covariance matrix of the ten portfolio returns (i.e., excluding the market return), and use this to obtain numerical values of a, b and c, and to make a figure that plots the efficient frontier (as usual, with standard deviation on the horizontal axis and mean return on the vertical axis). In the same figure, indicate the location of the ten (σi , ¯(r)i) points corresponding to the size portfolios, as well as (σM , ¯(r)M ). Is the market portfolio on the frontier? What does this imply about the validity of (the assumptions underlying) the CAPM?
3. [25 points] Run 10 separate CAPM regressions of the size portfolio excess returns (ri−rf) on the market excess returns (rM−rf). You do not need to print individual regression output, but collect the results in a table, where for each of the size portfolios, you report the estimated αi and βi, as well as the t-statistic for αi = 0. Also carry out the Gibbons-Ross-Shanken test of the joint hypothesis α1 = ... = α 10 = 0. Using this information, answer the following questions:
(a) Do the portfolios with a higher expected return also have a higher exposure to systematic risk (measured by βi), as predicted by the CAPM?
(b) Is the size effect fully explained by variation in exposure to sys- tematic risk?
(c) Do these results imply a (statistical) rejection of the CAPM?
4. [20 points] Make a graph of the empirical security market line, i.e., a plot of the CAPM-implied expected excess return against beta. In the
same figure, also indicate the (β(ˆ)i , ¯(r)i) points of the 10 size portfolios (where ¯(r)i are the sample averages). How can you see Jensen’s alpha for each of the size portfolios from this figure? Can you see a pattern in the 10 alphas?
5. [20 points] Now analyse the Fama-French 3-factor model for these 10 size portfolios, and discuss to what extent it is an improvement of the CAPM as a pricing model. You should decide for yourself which methods are needed to answer this question, and report on the result- ing empirical outcomes to support your conclusion. (Note: it is not sufficient to base your conclusions only on changes in regression R2.)