S&P 500 index: vfinx

Class Project
Due Mar 14 by 8pm Points 4 Submitting a file upload
Available Feb 20 at 8am – Mar 15 at 8pm 23 days
Submit Assignment
Selecting Data
For this project, everyone will be using the same data.
1. S&P 500 index: vfinx
2. European stock index: veurx
3. Emerging markets fund: veiex
4. Long-term bond fund: vbltx
5. Short-term bond fund: vbisx
6. Pacific stock index: vpacx
Information on these funds is available on the Yahoo! finance site (http://finance.yahoo.com/) . 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 January
2014 through the end of January 2019.
The following R script file guides you through the creation of the necessary R objects for the analysis of the
data in R
• 424projectWinter2019.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 back-up copy of your project.
Formal Write-up
I want you to give a formal write-up, separate from the Excel spreadsheet analysis and R statistical analysis.
Treat this write-up as a term-paper project. Typically, the write-up is between 10 and 20 pages (double
spaced with graphs and tables). Your write up should consist of:
1. An executive summary, which gives a brief summary of the main results using bullet points
2. 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 write-up.
Alternatively, you can refer to your spreadsheets for the quantitative results, graphs etc.
You only need to turn in the formal write-up. Turning in print-outs of your Excel spreadsheets and R
output is optional.
Excerpts from an example class project: 424projectExample.pdf
(http://faculty.washington.edu/ezivot/econ424/424projectexample.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, qq-plots, 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 pair-wise 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 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?
Value-at-Risk Calculations
• Assume that you have $100,000 to invest starting at June 30, 2016. For each asset, determine the 1%
and 5% value-at-risk 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% value-at-risk 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?
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?
◦ Graph the weights of the 6 assets in this portfolio using a bar chart.
• 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 January 31, 2019. For the global minimum variance
portfolio, determine the 1% and 5% value-at-risk 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 short-sales 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 short-sales.
• Assume that you have $100,000 to invest for a year starting at January 31. For the global minimum
variance portfolio with short-sales not allowed, determine the 1% and 5% value-at-risk 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).
◦ Create a plot (based on monthly frequency) with portfolio expected return on the vertical axis and
portfolio standard deviation on the horizontal axis showing the efficient portfolios. Indicate the location
of the global minimum variance portfolio (with short sales allowed) as well as the locations of your six
assets.
• 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 T-bills and the tangency portfolio on a plot
with the Markowitz bullet. That is, compute the efficient portfolios consisting of T-bills 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 short-sales are not
allowed.
◦ Compute the expected return, variance and standard deviation of the tangency portfolio.
◦ Give the value of Sharpe’s slope for the no-short 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 short-sales 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% value-at-risk 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% value-at-risk based
on an initial $100,000 investment.