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Moodle 
Final Project:
Summer
2013
Note: Please check this page for updates
and corrections every few days. Additions will be dated.
Last updated:
August 15, 2013
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 May 2008 through
the end of May 2013.
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.
Turn in the formal
writeup, as well as a printout of your Excel spreadsheets
and R output, in a bound folder.
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 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 (This is called an "equity curve"). Which fund gives the highest future value?
Are you surprised?
Create four panel
diagnostic plots containing histograms, smoothed density plots,
boxplots and qqplots for each return
series and comment. Do
the returns look normally distributed? Are there any outliers in the
data?
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 for each asset. Which asset has the highest
slope?
Compute estimated standard errors
and form 95% confidence intervals for
the the estimates of the mean and standard deviation. 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. Assuming
you get the average annual return every year for 5 years, how much
would $1 grow to after 5 years?
Compute and plot all pairwise
scatterplots between your 6 assets. Briefly comment
on any relationships you see.
Compute the sample covariance matrix
of the returns on your six assets and comment on
the direction of linear association between the asset returns.
Compute the sample
correlation matrix
of the returns on your six assets. Which assets are most highly
correlated? Which are least correlated?
Compute estimated standard errors and 95% confidence
intervals for your estimates. How precise are these correlation
estimates? Finally, based on the estimated correlation values do you
think diversification will reduce risk with these assets?
ValueatRisk Calculations

Assume that you have $100,000 to invest
starting at May 31, 2013. 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.
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.

Use the bootstrap to compute estimated
standard errors and 95% confidence intervals for your 5% VaR estimates.
Using these results, comment on the precision of your VaR estimates.

Repeat the VaR
analysis (but skip the bootstrapping), 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? Note: it may not seem obvious how to
annualize the empirical quantile.
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.
Portfolio Theory
Use all 6 assets and the descriptive
statistics computed above 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. Briefly comment on
these values relative to those for each asset.
Assume that you have $100,000 to invest
starting at May 31, 2013. 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.

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. 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 May 31, 2013. 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 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? If so, interpret the negative weights.

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

Give the value
of Sharpe's slope for each asset as well as for the tangency portfolio. Which
asset has the highest Sharpe's slope?

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. Briefly
comment.
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.

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 6% per year (which
corresponds to an expected return of 0.5% per month) using a combination of TBills and the tangency portfolio
(that does not allow for short sales). In this allocation,
how much is invested in each of the six Vanguard funds and how
much is invested in TBills?

Compute the
monthly SD on this efficient portfolio, as well as the monthly 1%
and 5% valueatrisk based on an initial $100,000 investment.
Compare this with the VaR computed from the allocation of risky
assets without short sales.
Single Index Model
and CAPM (August 15, 2013  last addition)
Using the R
function lm,
estimate the
Single Index model, using the S&P 500 index (vfinx) as the
market index, for each of the 5 remaining funds (see the example
script singleIndex.r or lab8.r). For each regression,
give the estimated single index model equation showing the estimated intercept and slope and indicate the
estimated standard errors of the estimates in parentheses below the estimates. Also present the
estimated Rsquare and the estimated standard deviation of the residuals
(called
the regression standard error in the R output). Interpret the estimated slope coefficients (betas),
comment on the precision of the estimates and comment on the values of
Rsquare and 1  (Rsquare). (Hint: see the solutions for lab
9 on the
homework page)

For each single
index model regression, test the hypotheses H_{0}:
beta = 1 vs. H_{1}: beta (not equal to) 1 using a 5% test.
What does it mean if a asset has a beta equal to 1?

Which of your stocks are high beta stocks and which are low
beta assets? Are there any surprises?

Make a
scatterplot of the average monthly returns
for each asset (on the yaxis) against the
estimated betas (on the xaxis). Does there appear to be a linear
relationship average return and beta as
predicted by the CAPM?

Using the estimated betas for your
6 assets (recall, the
beta of the S&P 500 is one by definition),
the weights in the
global minimum variance portfolio and the weights in the tangency
portfolio (allowing for short sales), compute the beta of the global
minimum variance portfolio and the beta of the tangency portfolio (recall, the
beta of a portfolio is a weighted average of the betas of the assets
in the portfolio). Comment.
