μ ε ε σ ε ε σ - University of Washington

Eric Zivot
Econ 424 Winter 2015
Department of Economics
University of Washington
Lab 7
Statistical Analysis of Efficient Portfolios
Due:Tuesday, 3/10/15 at 8 pm via Canvas
Reading
1.
Class slides statistical analysis of efficient portfolios. Ruppert, Chapter 6 (Resampling)
and Chapter 11 (Portfolio Theory, especially section 7).
Data and Programs
1. Data for this lab are on the class homework page in the file econ424lab6returns.csv (same
data as lab 6).
2. R script files: econ424lab7.R and bootstrapPortfolio.R.
Description of Problem
Using the monthly closing price data on the four Northwest stocks (Boeing, Microsoft,
Nordstrom and Starbucks) over the period February 1995 - January 2000, you will analyze the
statistical properties of Sharpe ratios and optimized portfolios.
On the class web page is the script file econ424lab7.r, which walks you through the R
calculations. As in the previous labs, copy and paste all statistical results and graphs into a MS
Word document while you work, and add any comments and answer all questions in this
document. Please make sure to add comments to your output. Don’t just turn in the numbers and
graphs! See the previous solutions on the class homework page for examples of the types of
comments that I am looking for.
1. Using the return data on Boeing, Nordstrom, Starbucks and Microsoft in the matrix ret.mat,
estimate the parameters  i ,  i2 ,  i ,  ij and  ij of the constant expected return (CER) model
Rit  i   it , t  1, , T
 it ~ iid N (0,  i2 )
cov( it ,  jt )   ij
where Rit denotes the simple return on asset i (i = Boeing, Nordstrom, Starbucks, and Microsoft).
Compute estimated standard errors for the means and volatilities and briefly comment. Arrange
these estimates and standard errors nicely in a table.
2. Show the estimated risk-return tradeoff of these assets (i.e., plot the means on the y-axis and
the standard deviations on the horizontal axis. Briefly comment.
a. Assuming a risk free rate of 0.005 (0.5% per month or about 6% per year) compute
the Sharpe ratios for each asset. Which asset has the highest Sharpe ratio?
b. Using the bootstrap, compute estimated standard errors and 95% confidence intervals
for the Sharpe ratios. How well are the Sharpe ratios estimated?
3. Compute the global minimum variance portfolio allowing short-sales. Briefly comment on
the weights. Compute the expected return and standard deviation and add these points to the
risk return graph computed in question 2.
a. Use the bootstrap (with 999 bootstrap samples) to compute estimated standard errors
and 95% confidence intervals for the estimated expected return and estimated
standard deviation of the minimum variance portfolio. Briefly comment on the
magnitude of the standard errors and the widths of the confidence intervals. Are the
estimates precise?
b. Show the bootstrap estimates of minimum variance expected return and volatility on
the risk-return tradeoff graph from question 2. Briefly comment on the plot.
4. Compute the efficient frontier of portfolios allowing short-sales. Plot the frontier along and
show the minimum variance portfolio as well as the four assets.
a. Use the bootstrap (with 999 bootstrap samples) and estimate the efficient frontier on
each bootstrap sample. Plot these efficient frontiers to illustrate the estimation errors
in the means and volatilities of the frontier portfolios. Is the frontier estimated very
precisely?