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.

[The quality of the write-up matters for your grade. Please imagine that youíre writing

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?