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Eric Zivot
Economics 483
Introduction to Computational Finance and Financial Econometrics

Final Project

Fall 2004

Note: Please check this page for updates and corrections every few days. Additions will be dated.

Last updated: December 3, 2004

Selecting Data

For this project, everyone will be using the same data. These are prices and continuously compounded returns on 10 Vanguard mutual funds

  1. S&P 500 index: vfinx

  2. European stock index: veurx

  3. Emerging markets fund: veiex

  4. Extended market index: vexmx

  5. Growth equity fund: vgeqx

  6. Value equity fund: vivax

  7. Long-term bond fund: vbltx

  8. Short-term bond fund: vbisx

  9. Pacific stock index: vpacx

  10. Small capitalization stock index: naesx

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. 

Downloading Data

 

For the project I have downloaded 5 years of monthly data from the end of September 1999 through the end of September 2004. These are available in the following Excel spreadsheets

The following S-PLUS script file guides you through the creation of the necessary S-PLUS objects for the analysis of the data in S-PLUS

Organization of Results

As in the homework assignments, summarize your S-PLUS 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 S-PLUS 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.

I want you to turn in the formal write-up, as well as a print-out of your Excel spreadsheets and S-PLUS output, in a bound folder.

Analysis

Return calculations and Sample Statistics (added October 27, 2004)

  • Compute time plots of monthly returns and comment. Are there any unusually large or small returns? Can you identify any news events that may explain these unusual values?

  • Create four panel diagnostic plots containing histograms, smoothed density plots, boxplots and qq-plots 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. How are these estimates related to the parameters of the constant expected return (CER) model?

  • Using a monthly risk free rate equal to 0.083% (0.00083) per month (which corresponds to a continuously compounded annual rate of 1%), 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 pair-wise scatterplots between your 10 assets. Briefly comment on any relationships you see. 

  • Compute the sample covariance matrix of the returns on your ten assets and comment on the direction of linear association between the asset returns. 

  • Compute the sample correlation matrix of the returns on your ten 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?

Value-at-Risk Calculations (added October 27, 2004)

  • Assume that you have $100,000 to invest starting at October 1, 2004.  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. Which assets have the highest and lowest VaR at each 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 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 (added November 22, 2004)

Use all 10 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? 

    • Graph the weights of the 10 assets in this 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 November 1, 2003. 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)

  • 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.  

    • Graph the weights of the 10 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 short-sales. 

  • Assume that you have $100,000 to invest for a year starting at November 1, 2003. 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. 

  • Using the estimated means, variances and covariances computed earlier, compute the efficient portfolio frontier, allowing for short sales, for the 10 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 ten assets (just like in lab 7). 

    • 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 ten assets.

  • Compute the tangency portfolio using a monthly risk free rate equal to 0.083% (0.00083) per month (which corresponds to an annual rate of 1%)

    • Graph the weights of the 10 assets in this portfolio. In the tangency portfolio, are any of the weights on the 10 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. 

    • 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. Briefly comment.

  • Using a monthly risk free rate equal to 0.083% (0.00083) 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. 

    • Compare this tangency portfolio with the tangency portfolio where short-sales are allowed.

Asset Allocation (Added December 3, 2004)

  • Suppose you wanted to achieve a target expected return of 6% per year (which corresponds to an expected return of 0.5% per month).  What is the combination of T-Bills and the tangency portfolio (efficient portfolio) that achieves this target expected return?  In this allocation, how much is invested in each of the ten Vanguard funds?

  • Compute the monthly SD on the efficient portfolio, as well as the monthly 1% and 5% value-at-risk based on an initial $100,000 investment.

Rolling Analysis of the CER Model Parameters (Added December 3, 2004)

  • For each asset, compute 24 month rolling estimates of the mean and standard deviation of the continuously compounded returns using the S-PLUS function aggregateSeries (see the script file rolling.ssc or lab8.ssc 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.

  • Given what you see regarding the stability of the mean and the SD parameters, how confident are you about the portfolio analysis performed using the full five years of monthly data?

Single Index Model and CAPM (Added December 3, 2004)

  • Using the S-PLUS function lm, estimate the Single Index model, using the extended market index (vexmx) as the market index, for each of the 9 remaining funds (see the example script singleIndex.ssc or lab8.ssc). 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 R-square and the estimated standard deviation of the residuals (called the regression standard error in the S-PLUS output). Interpret the estimated slope coefficients (betas), comment on the precision of the estimates and comment on the values of R-square and 1 - (R-square). (Hint: see the solutions for lab 8 on the homework page)

    • For each single index model regression, test the hypotheses H0: beta = 1 vs. H1: beta (not equal to) 1 using a 5% test. What does it mean if a stock has a beta equal to 1? What does it mean if a stock has a beta equal to zero?

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

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

    • Using the estimated betas for your 10 stocks, 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.