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# Empirical Methods in Finance Homework 7

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Empirical Methods in Finance
Homework 7

Please use Matlab/R to solve these problems. You can just hand in one set of solutions
that has all the names of the contributing students on it in each group.
a report for your boss at Goldman when drafting answers these questions. Try to be clear
and precise.]
Principal Component Analysis
Download the 48 industry portfolio data (monthly) from Kenneth Frenchís web site. Use the
data from 1960 through 2015. Use the value-weighted returns. You may drop the industries
that have missing values and are reported as -99.99. Also, download the 3 Fama-French
factors from his web site. Use the monthly risk-free rate series provided by French in the
same FF factor dataset to compute excess returns on these 48 portfolios.
1. Get the eigenvalues for the sample variance-covariance matrix of the excess returns to
the 48 industries. Plot the fraction of variance explained by each eigenvalue in a bar
plot.
2. Choose the 3 Örst (largest) principal components.
(a) How much of the total variance do these 3 factors explain?
(b) Give the mean sample return to these 3 factor portfolios, their standard deviation,
and correlation.
1
(c) Consider a multi-factor model of returns using these three factors as pricing factors. Plot the predicted return from this model for all the industries versus the
realized average industry returns over the sample. That is, estimate the betas of
each industry with respect to these factors, get the expected excess returns as
E^ [R
e
it] = ^
0
iE^ [Ft
] ; (1)
where Ft are the factor returns and E^ [Ft
] is estimated as the sample average of
each factor. This expected return will be on your x-axis. The sample average of
each industryís excess return will be on the y-axis. Add a 45 degree line to this
plot.
[You can get factor loadings (betas) from the eigenvectors. Or, if you like, you
can run the time-series regression of each industryís return on the 3 factors. The
result is the same.]
(d) Give the implied cross-sectional R2 of the plot in c). That is, calculate:
R
2
cross-section = 1
var
Ract R^pred
V ar
Ract
where Ract is the N 1 vector of average industry excess returns. R^pred is the
N 1 vector of predicted industry excess returns from the 3-factor model.
3. Now, download the 25 FF portfolios sorted on size and book-to-market, same sample
period.
(a) Get the eigenvalues for the sample variance-covariance matrix of the excess returns to these 25 F-F portfolios. Plot the fraction of variance explained by each
eigenvalue in a bar plot.
(b) Given (a), how many factors does do you reckon you need to explain average
returns to the 25 F-F portfolios? Empirical Methods in Finance Homework 7
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