MGT 595--Moskowitz
Applied Quantitative Finance
Problem Set 1
Instructions: Problem sets should be done in groups of 1-3 people. Each group hands in one
typed copy of their answers. Be brief and to the point, but be sure to explain your logic. Do
not print data, entire spreadsheets, or programs – instead, copy the relevant statistics to a
table. All tables and charts should have legends and explanations. Answers (excluding tables
and figures) should not exceed a maximum of 5 pages. Exceeding these limits will draw a
penalty.
This problem set has two parts. These are the basics of the course and consist of simple
data analysis and statistics questions. Much of this should be review (even if you have not
used it in a while). You should at least be familiar with all of the concepts in this problem
set, even if you don’t have all of the answers.
Note: the problems below require very little (if any) background in Finance. This is what
I will teach you. The problems are designed to test your statistics and math skills since it
is imperative that you have the mathematical and, more importantly, statistical rigor to
handle the course. These exercises will also give you some practice in using Matlab.
Part1:
The first is a simple data exercise, which examines how the variances of portfolios
comprised of randomly selected stocks are reduced as a result of diversification. There are
also some simple sample statistics and regressions on the data you are asked to perform
and interpret.
In order to proceed, you need Microsoft Excel and the file "Problem_Set1_2017.xls",
which can be downloaded from the course website.
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I do not care if you use Excel. In fact, many other software packages are superior to Excel
for this class (namely Matlab, Minitab, or any programming language such as Fortran or C).
Warning: for most assignments Excel will take much longer than other software. Feel free to
use any programming language or software you wish for the rest of the course. I and the TA
will support Matlab. Therefore, you are encouraged to use Matlab. The data will always be
provided in Excel for convenience, but can be easily read into almost any software package.
I strongly encourage you to do this assignment in Matlab to get up to speed quickly in that
software language as it will come in handy for the rest of the course.
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MGT 595--Moskowitz
Applied Quantitative Finance
a) Using the file "Problem_Set1_2017.xls ", form equal-weight portfolios using the first
5, first 10, first 25, and all 50 stocks. Calculate the sample mean and standard
deviation of returns for each of the four equal-weight portfolios. Plot estimated
standard deviations as a function of the number of stocks in the equal-weight
portfolio. Comment on the shape of the function. Are the results consistent with what
you would expect theoretically? Eye-balling the graph, does it look like adding more
and more stocks will diversify away all of the standard deviation? Why or why not?
b)
For all four equal-weight portfolios, decompose the estimated portfolio variance into
its two components (the contributions of variances and covariances). Hint: you do
not have to estimate the pairwise covariances in order to compute the decomposition.
Plot the percentage of the portfolio’s variance due to the variances of individual
security returns as a function of the number of stocks in the portfolio. Comment on
the shape of the function. Are the results consistent with what you would expect
theoretically?
c)
Suppose instead of equal-weighted portfolios, we computed value-weighted (e.g.,
weighted by market capitalization) portfolios. Would you expect the value-weighted
portfolios to exhibit more or less variance relative to equal-weighted portfolios?
What might it depend on? (Do not make any calculations here, just answer what it
will depend on.)
d)
Compute the test statistics for whether the mean return of each of the four equalweight portfolios you calculated in part b) is different from zero. What statistical
distribution do these test statistics follow? Do you reject or fail to reject the null
hypothesis that each of the mean returns on the portfolios is different from zero?
e)
Choose the first stock and test whether its returns follow a normal distribution (hint:
compute the sample studentized range, skewness, and kurtosis and compare them to
what you would expect under a normal distribution. A simple chi-square test can be
performed using the sample skewness and kurtosis measures to see if the null of
normally distributed returns is rejected). Repeat this test for the equal-weighted
portfolio of all 50 stocks as well as the market portfolio index of all NYSE, AMEX,
and Nasdaq stocks (which is also included in the spreadsheet).
f)
Regress the returns of the first ten stocks in the spreadsheet (CTL through WFT) on
the value weighted market index (a cap-weighted index of all NYSE, AMEX, and
Nasdaq stocks) return, and report intercepts () and slope coefficients (), in a table.
1. Interpret the slope coefficients of this model.
2. Interpret the intercepts of the model, in terms of the effect of the market return on
the securities’ returns.
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MGT 595--Moskowitz
Applied Quantitative Finance
3. What does the R2 of the regressions tell you? What does it mean when the R2 is
low in this case?
Part 2:
This part provides a brief introduction to some of the tools that you will need for this
class. Specifically, you are asked to estimate a market model regression
~
~
( Rit  i   i Rmt  ~it ) for a sample of stocks.
Questions:
a) For CSCO and the market, calculate volatility (return standard deviation) using two
different methods. First method: for each date, calculate volatility using all the past data
up to that date. Second method: use only one year of past data. For both CSCO and the
market, plot the two volatility estimates on the same chart. Why does the estimate using
only the most recent year of data move around more than the estimate using all data?
Give two reasons why this might be the case.
b) Run a market model regression with CSCO in which you calculate OLS beta using the
following two methods. First method: For each date, calculate OLS beta using all the
past data up to that date. Second method: Use only one year of past data. Plot the two
OLS beta estimates on the same chart along with a 95% confidence interval for each.
Comparing the one-year estimates versus those using the full sample, does the chart
indicate that betas move through time? Or, is there another explanation consistent with
this picture?
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