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Final Project:
Winter 2016
Note: Please check this page for updates
and corrections every few days. Additions will be dated.
Last updated:
January 11, 2016
Selecting Data
For this project, everyone will be
using the same data.

S&P 500 index: vfinx

European stock index: veurx

Emerging markets fund: veiex

Longterm bond fund: vbltx

Shortterm bond fund: vbisx

Pacific stock index: vpacx
Information on these funds is available
on the Yahoo! finance site. After typing in the sticker symbol and
retrieving the quote data, choose Profile to get a summary of the
fund. Please review each fund before doing any of the analysis below.
Downloading Data
For the project you
will analyze 5 years of monthly
closing price data from the
end of December 2010 through
the end of December 2015.
The following R script file guides
you through the creation of the necessary R objects for the analysis of
the data in R
Organization of
Results
As in the homework assignments,
summarize your R work in a Word file. You will find it helpful to organize
your Excel results in a spreadsheet by task. That is, put all of the data in one
worksheet tab, put all the graphs in another, put the portfolio analysis
in another tab, etc. This will make it easier for you to print out
results. It is also helpful to use
names for your data and for certain results. This makes working with
formulas much easier and it also helps to eliminate errors in formulas
etc.
You will find it helpful to add text boxes in
your spreadsheet to organize comments etc.
Remember to save your work often as
Excel has a tendency to crash with large complicated spreadsheets. Also,
keep a backup copy of your project.
Formal Writeup
I want you to give a formal writeup,
separate from the Excel spreadsheet analysis
and R statistical analysis. Treat this
writeup as a termpaper project. Typically, the
writeup is between 10 and 20 pages (double spaced with graphs and
tables). Your write up should consist of:

An executive
summary, which gives a brief summary of the main results using bullet
points

Sections that summarize the
results of your statistical analysis by topic (see below).
You may find it helpful to include parts of your spreadsheet and computer
output as part of your writeup. Alternatively, you
can refer to your spreadsheets for the quantitative results, graphs
etc.
You only need to turn in the formal
writeup. Turning in printouts of your Excel spreadsheets
and R output is optional.
I have boxes of
projects from previous classes. Feel free to come by my office to look at
them.
Exerpts from an
example class project:
424projectExample.pdf
Analysis
Return calculations and Sample
Statistics
Compute time plots of monthly
prices and continuously compounded returns
and comment. Are there any unusually large or small returns? Can you
identify any news events that may explain these unusual values? Give a
plot showing the growth of $1 in each of the funds over the five year
period (recall, this is called an "equity curve"). Which fund gives the highest future value?
Are you surprised?
Create four panel
diagnostic plots containing histograms,
boxplots, qqplots, and SACFs for each return
series and comment. Do
the returns look normally distributed? Are there any outliers in the
data? Is there any evidence of linear time dependence? Also, create a boxplot showing the distributions of all of the
assets in one graph.
Compute
univariate descriptive statistics (mean,
variance, standard deviation, skewness, kurtosis, quantiles) for each return series and comment.
Which funds have the highest and lowest average return? Which funds
have the highest and lowest standard deviation? Which funds look most
and least normally distributed?
Using a monthly
risk free rate equal to
0.0004167 per month (which corresponds to a continuously
compounded annual rate of 0.5%),
compute Sharpe's slope/ratio for each asset. Use
the boostrap to calculate estimated standard errors for the Sharpe
ratios. Arrange
these values nicely in a table. Which asset has the highest
slope? Are the Sharpe slopes estimated precisely?
Compute estimated standard errors
and form 95% confidence intervals for
the the estimates of the mean and standard deviation. Arrange these
values nicely in a table. Are these means
and standard deviations estimated very precisely? Which estimates are
more precise: the estimated means or standard deviations?
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. Using these
values, compute annualized Sharpe ratios. Are the asset rankings the
same as with the monthly Sharpe ratios? Assuming
you get the average annual return every year for 5 years, how much
would $1 grow to after 5 years? (Remember, the annual return you
compute is a cc annual return).
Compute and plot all pairwise
scatterplots between your 6 assets. Briefly comment
on any relationships you see.
Compute the sample
correlation matrix
of the returns on your six assets and plot this correlation matrix
using the R corrplot package function corrplot(). 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?

Added
August 4, 2015. Use the JB statistic and a
significance level of 5% to test the null hypothesis that each monthly
asset return is normally distributed. What do you conclude?
ValueatRisk Calculations

Assume that you have $100,000 to invest
starting at
June 30, 2015. For each asset, determine the 1% and 5%
valueatrisk of the $100,000 investment over a one month investment
horizon based on the normal distribution using the estimated means and variances of your assets.

Use the bootstrap to compute estimated
standard errors and 95% confidence intervals for your 1% and 5% VaR estimates.
Create a table showing the 1% and 5% VaR estimates along with the bootstrap
standard errors and 95% confidence intervals.

Using these results, comment on the precision of your VaR estimates.
Which assets have the highest and lowest VaR at each horizon?

Using the monthly mean and standard deviation estimates, compute the
annualized mean (12 time monthly mean) and standard deviation (square root
of 12 time monthly std dev) and determine the 1% and 5% valueatrisk
of the $100,000 investment over a one year
investment horizon. Arrange these results nicely in a
table.

Repeat the VaR
analysis (but skip the bootstrapping and the annualized VaR calculation), but this time use the empirical 1% and 5% quantiles of the return
distributions (which do not assume a normal distribution  this method is
often called historical simulation). How different are the results from
those based on the normal distribution?
Rolling Analysis of the
CER Model Parameters

For each asset, compute 24 month
rolling estimates of the mean and
standard deviation of the continuously compounded returns using the
R function
rollapply() (see the script file
for lab 5 for examples). For each asset, graph these rolling estimates
together with the returns (so that you have just one graph for each
asset). Briefly comment on the stability of the mean and SD parameters
of
the constant expected return model.

With 6 assets there are 6*5/2 = 15
pairwise correlations. To keep things simple,
compute and plot 24 month rolling estimates of the sample correlation
between the S&P 500 index (vfinx) and the longterm bond index
(vbltx). Is this correlation stable over time? When is the correlation
the highest and when is it the lowest?
Portfolio Theory
Use all 6 assets and the
CER model estimates computed above (from the continuously compounded
returns) for the following computations.
Compute the global minimum
variance portfolio and calculate the expected return and SD of this
portfolio. Are there any negative weights in the global minimum
variance portfolio?
Annualize the the monthly mean
and SD by multiplying
the mean by 12 and the SD by the square root of 12. Compute the annual
Sharpe ratio from these values. Briefly comment on
these values relative to those for each asset.
Assume that you have $100,000 to
invest starting at
June 30, 2015. For the global minimum variance portfolio,
determine the 1% and 5% valueatrisk of the $100,000 investment over
a one month investment horizon. Remember that returns are
continuously compounded, so you have to convert the 1% and 5% quantiles
to simple returns (see the example in the lecture notes on
Introduction to Portfolio Theory). Compare this
value to the VaR values for the individual assets.
Compute the global minimum
variance portfolio with the added restriction that shortsales are not
allowed, and calculate the expected return and
SD of this portfolio. This is the relevant portfolio for
you because you cannot short mutual funds in your 401K account.

Graph the weights of the
6
assets in this portfolio.

Annualize the the monthly estimates by
multiplying the ER by 12 and the SD by the square root of 12. Compute
the annual Sharpe ratio from these values. Compare
this portfolio with the global minimum variance portfolio that allows
shortsales.
Assume that you have $100,000 to
invest for a year starting at
June 30. For the global minimum variance portfolio
with shortsales not allowed, determine the 1% and 5% valueatrisk of
the $100,000 investment over
a one month investment horizon. Compare your
results with those for the global minimum variance that allows short
sales.
Using the estimated means, variances
and covariances computed earlier, compute and plot the efficient portfolio frontier,
allowing for short sales, for the 6 risky assets using the Markowitz algorithm. That is, compute the Markowitz bullet.
Recall, to do this you only need to find two efficient portfolios and
then every other efficient portfolio is a convex combination of the
two efficient portfolios. Use the global minimum variance portfolio as
one efficient portfolio. For the second efficient portfolio, compute
the efficient minimum variance portfolio with a target return equal to the
maximum of the average returns for the six assets (see
example from lecture notes).
Compute the tangency portfolio using
a monthly risk free rate equal to
0.0004167 per month (which corresponds to an annual rate of
0.5%).
Recall, we need the risk free rate to be smaller than the average return
on the global minimum variance portfolio in order to get a nice graph.

Graph the weights of the
6
assets in this portfolio. In the tangency portfolio, are any of
the weights on the 6 funds negative?

Compute the
expected return, variance and standard deviation of the tangency portfolio.

Compare the
Sharpe ratio of the tangency portfolio with those of the
individual assets.

Show the tangency
portfolio as well as combinations of Tbills and the tangency
portfolio on a plot with the Markowitz bullet. That
is, compute the efficient portfolios consisting of Tbills and
risky assets.

Annualize the the monthly ER and SD of
the tangency portfolio by
multiplying the ER by 12 and the SD by the square root of 12. Compute
the annual Sharpe ratio from these values. Briefly
comment.

Compute and plot
the efficient portfolio frontier this time not allowing for short sales, for the 6 risky assets using the Markowitz algorithm.
Recall, to do this you need to create a grid of target return values,
between the mean of the no short sales global minimum variance portfolio
and the mean of the asset with the highest average return, and solve
the Markowitz algorithm with the no short sales restriction.

Compare the no short sale frontier
with the frontier allowing short sales (try to plot them on the
same graph)

Consider a portfolio with a target
volatility of 0.02 or 2% per month. What is the approximate cost in
expected return of investing in a no short sale efficient
portfolio versus a short sale efficient portfolio?
Using a monthly
risk free rate equal to
0.0004167 per month and the
estimated means, variances and covariances compute the tangency
portfolio imposing the additional restriction that shortsales are not
allowed.

Compute the
expected return, variance and standard deviation of the tangency portfolio.

Give the value
of Sharpe's slope for the noshort sales tangency portfolio.

Annualize the the monthly ER and SD of
the tangency portfolio by
multiplying the ER by 12 and the SD by the square root of 12. Compute
the annual Sharpe ratio from these values. Briefly
comment.

Compare this tangency
portfolio with the tangency portfolio where shortsales are allowed.
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 (6 Vanguard portfolios) and
no short sales. Recall, you cannot
short a mutual fund. What
is the efficient portfolio that achieves this target return? How
much is invested in each of the Vanguard funds in this efficient
portfolio?

Compute the
monthly SD on this efficient portfolio, as well as the monthly 1%
and 5% valueatrisk based on an initial $100,000 investment.

Now suppose you
wanted to achieve a target expected return of 12% per year (which
corresponds to an expected return of 1% per month) using only
the risky assets (6 Vanguard portfolios) and
no short sales. What
is the efficient portfolio that achieves this target return? How
much is invested in each of the Vanguard funds in this efficient
portfolio? How does this portfolio differ from the efficient
portfolio with a 6% annual target?

Compute the
monthly SD on this efficient portfolio, as well as the monthly 1%
and 5% valueatrisk based on an initial $100,000 investment.
