Some Suggestions for the Project
STAT 4261/5261
Professor Hammou Elbarmi
Department of Statistics
The project requires you to synthesize all the material from the course. Please note that it
is not a regular Homework and it should be treated differently. The goal is for you to solidify
your understanding of the financial statistics methods that you have learned in this course.
You will present your findings in a written report. You should explain what you did using
simple words. You do not need to explain the terminology in details. No formulas. The final
report should be clear and readable. The maximum number of pages allowed for the report
is 5 (both sides). All figures and tables that are included should be readable, relevant and
well labeled. Figures can relegated to an Appendix (not part of the 5 pages). Make sure to
only include relevant plots. What follows are some suggestions for your final project. You
need to find your own data and a good source for it is yahoofinance.com. You will need to use
exactly 20 assets and you should look for at least 5 years of monthly closing prices. The risk
free rates can be found at this link http://www.federalreserve.gov/releases/h15/data.htm
under Treasury bills (secondary market) (3months, weekly, monthly or annually ). You are
required to hand in your report no later than Friday, December 18 at 10 am. You are
required to email me your data and your R programs by the same day. Your project should
consist of at least the following items:
1. An executive summary, in which you give a brief summary of the main
results using bullet points
2. Descriptive Statistics
3. Sections that summarize the results of your statistical analysis by topic
(see below)
4. A conclusion
1 Summary
This section should be a brief summary of your main results using bullet points
2 Descriptive Statistics
In this section you report sample statistics (Means, standard deviations, Skewness Coef-
ficients, Kurtosis Coefficients and beta of each asset) and comment of your results. You
should also plot your monthly prices and returns and comment on these plots. You need in
particular to check for any unusually large or small returns and to identify any news events
that may explain them? You should also provide an equity curve for each asset (that is, a
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curve that shows the growth of a $1 in each of the asset over the time period you chose)
and comment of your results. You should do the same for S&P 500 and compare it with the
assets. Create histograms, boxplots and qq-plots for each return series and comment. Run
a test for stationarity. Do the returns look normally distributed ? Are there any outliers in
the data? Fit different distributions to your data, which one fits better? Compute Sharpe’s
slope for each asset. Which asset has the highest slope? Convert the monthly sample means
into annual estimates by multiplying by 12 and convert the monthly sample SDs into annual
estimates by multiplying by the square root of 12. Comment on the values of these annual
numbers. Construct pairwise scatter plots between your assets returns and comment on
any relationships you see. You should also compute the sample covariance matrix of the
returns on your assets and comment on the direction of linear association between the asset
returns.
3 Portfolio Theory:
In this part of the project, you construct some of the portfolios that we covered in class.
Compute the minimum variance portfolio (MVP) and estimate its mean return, its standard
deviation, its value at risk and expected shortfall. Comment on the weights of this portfolio
and annualize the monthly mean and risk by multiplying the mean by 12 and the risk by
the square root of 12. Comment on these values relative to those of each asset. Assume
that you have $100,000 to invest. For the MVP, determine the 5% value-at-risk of the
$100,000 investment over a one month investment horizon. Compare this value to the VaR
values for the individual assets. Repeat this with the added restriction that short-sales are
allowed, and calculate the expected return and risk of this portfolio. Using the estimated
means, variances and covariances computed earlier, compute the efficient portfolio frontier,
with and without short sales allowed, for the risky assets using the Markowitz approach
Obtain the value of Sharpe ratio for each asset as well as for the tangency portfolio. Which
asset has the highest Sharpe ratio? Compute the tangency portfolio when short-sales are
not allowed and compute its expected return, variance and standard deviatio. Obtain the
Sharpe ratios and comment on your results.
4 Asset Allocation:
Suppose you wanted to achieve a target expected return of 6% per year (which corresponds
to an expected return of 0.5% per month) using only the risky assets and no short sales
allowed, what is the efficient portfolio that achieves this target return? How much is invested
in each of the assets in this efficient portfolio? Compute the monthly risk on this efficient
portfolio, as well as the monthly 5% value-at-risk and expected shortfall based on an initial
$100,000 investment. Now suppose you wanted to achieve a target expected return of 6%
per year (which corresponds to an expected return of 0.5% per month) using a combination
of T-Bills and the tangency portfolio (that does not allow for short sales). In this allocation,
how much is invested in each of the assets and how much is invested in the risk free asset?
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Compute the monthly risk on this efficient portfolio, as well as the monthly and 5% value-
at-risk and expected shortfall based on an initial $100,000 investment. Compare this with
the VaR computed from the allocation of risky assets without short sales.
5 Principal Component Analysis:
Compute the sample correlation matrix of the returns on your assets. Which assets are
most highly correlated? Which are least correlated? Based on the estimated correlation
values do you think diversification will reduce risk with these assets? Run the PCA analysis
and comment on your results. Run factor analysis and report the number and the loadings
of each factors. Do they have any meaningful interpretation?
6 Risk Management:
Assume that you have $100,000 to invest. For each asset, estimate the 5% value-at-risk of
the and expected shortfall on $100,000 investment over a one month investment horizon
based on the normal distribution using the estimated means and variances of your assets.
Do the same using the nonparametric method we discussed in class. Which assets have the
highest and lowest VaR at a one month horizon? Which assets have the highest and lowest
expected shortfall at a one month horizon? Do the same for all your portfolios. Use the
bootstrap to compute estimated standard errors and 95% confidence intervals for your 5%
VaR and expected short fall. .
7 Copulas:
Use copulas to to model the joint distribution of the returns. Which copula fits better the
data? What are the implications?
8 Conclusion
In this section you give your conclusion
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