# Global Asset Allocation and Stock Selection

Campbell R. Harvey

## Assignment 2: Global Optimal Portfolio Control: Strategic Asset Management

### Partial work needed by beginning of Lecture 4

All students should be familiar with the concept of portfolio control from the introductory finance course. The idea of this first assignment is to apply these concepts in developing your own portfolio selection software. I want you to develop a program that you could use for your own investments or to show to perspective clients or employers. I recommend that you develop this software in EXCEL where it is easy to use the quadratic optimizer in SOLVER.

Form groups of four for this project and the other projects. Group members should make it clear to other group members at the offset how many assignments you plan to turn in. Each group needs a name, i.e. Positive Skewness Asset Management.

When the project is finished, copy it to the class directory on the student drive Harvey_BA453 (see below for file naming convention). I will run the program. It must be very user friendly. Assume the user has no knowledge of Excel.

## Question 1: Markowitz Mean-Variance Optimization

The basic problem is to maximize expected return, w′E, where w is an (n x 1) (rows x colums) vector of portfolio weights and E is an (n x 1) vector of expected returns on n assets, subject to at least two constraints. [Remember w′ takes a (n x 1) vector and transposes it into a (1 x n) vector.] The first constraint sets the level of variance w′ V w, where V is an (n x n) variance-covariance matrix (variances on the diagonal). The second constraint forces the weights to sum to unity, 1′w, where 1 is an (n x 1) vector of ones. Your program should be general enough to allow for asset specific constraints. For example, there should be limits to short-selling and long positions that the user can input before the program is run.

The program requires three sets of data: the means E, the correlation matrix, and the standard deviations (σ). The your program must convert the correlation matrix into the variance-covariance matrix. Remember from Finance 350, the covariance between i and j (Cij)is:

Cij=Corrij × σi × σj.

In terms of the correlation matrix, multiply row1 by σ1, row2 by σ2, and so on. Then multiply column1 by σ1, column2 by σ2, and so on. The diagonal elements are the variances (i.e. σi2, for each asset i).

You will find it useful to use the vector and matrix commands in EXCEL to do w′V w operation (i.e. MMULT). You can add as many options as desired such as a bar graph in the right-hand corner that graphs the asset weights as SOLVER is working.

To start the exercise, download returns on at least five but not more than eight asset classes (at least 3 of the assets must be international). One of the assets must be the U.S. equity return I have put up on the CD and the student drive a number of files. In particular, look for files with the preference Ibb (for Ibbotson Associates) or MSCI (for Morgan Stanley Capital International -- to be updated). Feel free to draw data from DATASTREAM too. Note you might want to design your program to take more assets. This will make it easier to do future assignments. Use total return (not price return) country indices (such as MSCI or IFC for equities and J.P. Morgan or Salomon for bonds, the U.S. equity return is the S&P 500) or regional indices. Decide on your currency of choice; i.e. if you choose the U.S. dollar, returns from all countries must be converted to U.S. dollars. Using monthly data, calculate the unconditional means (average), standard deviations and the correlations of the returns.

Calculate the standard deviation (converted to your currency of choice) of the U.S. equity return. Call this σUS-equity. Also download (if it is not in your set of assets) a U.S. government bond portfolio total return series (medium or long term) (in Ibb file) and calculate its standard deviation. Call this σUS-bonds.

### Steps

1. Optimize the weights for a portfolio standard deviation of σUS-equity (exactly the U.S. equity standard deviation).

2. Compare the expected portfolio return to the strategy of just holding the U.S. equity portfolio and comment on this.

3. Reoptimize with no short-sales allowed. Comment how this might affect the efficient frontier (that is, assuming the constraint is binding).

4. Reoptimize (not allowing short-sales) with a portfolio standard deviation of σUS-bond (exactly the long-term government bond standard deviation). How does the expected return compare with just holding the U.S. bond? [If this standard deviation is not feasible, just use the minimum variance portfolio.]

5. Reoptimize allowing limited short sales (20%) and limited long positions (<50%) with a portfolio standard deviation of σUS-equity but this time change the expected returns to reflect your subjective assessment the expected asset performance over the next year. Interpret the results. Does the allocation make sense compared to the first allocation?
##### Important
Excel-Solver may not work correctly, if the quadratic optimization problem contains both equality and inequality constraints (such as box-constraints or even refusals of short-sellings by x>=0). This is too much for the classic Lagrange-method. To avoid this problem, it is best to make use of a feasable sequential quadratic programming-solver. Frontline Systems provides a Quadratic engine, that will handle these types of problems. You can test this, by downloading a free 15-day trial version of their Premium Solver Plus. Just visit the private forum at website.

## Question 2: Resampled Frontier

I would like you to attempt to resample your frontier. Given your means and variance-covariance matrix, use a multivariate normal distribution to "resample" your returns. This should deliver a new randomly generated set of returns with the same number of observations. Let's assume no short selling. You might want to investigate AnalyCorp's simulation engine. Free Trial of XLSim.
• Please look at the tutorial for simulating investment returns. In particular, I would install the trial version of XLSim. I loaded it indirectly by Start, Programs, XLSIM2e, XLSim 2.0. Alternatively, you could use it as an add-in. Follow the tutorial (PDF). You need means and a variance-covariance matrix. Suppose you have 4 assets. Hence, the means will have four elements. The variance-covariance matrix will have 16. Highlight four new cells. Use the function wizard, under statistics find, gen_MVNormal. Follow the steps. However, I anchor the cells for the means and variance-covariance, i.e. \$A\$1:\$D\$1 for the means. Same for variance-covariance. When done, don't hit OK, hit control shift enter. You can then drag this row of four cells down, say one hundred rows. You have simulated the portfolio returns. You can double check the means and correlations. They should be close but not exactly equal to what you started with. Everytime you hit F9 (or load new data anywhere), a new simulation runs. Here is an example file. Note there are other ways to do this via Crystal Ball, for example.
1. I only require a simplifed version of the resampling. Given your five assets, means, variance-covariance matrix, choose a point on the frontier that you are comfortable with. For example, you could choose the expected return that is the average of the maximum and minimum expected frontier return. Present the weights.
2. Resample the data (i.e. generate a new set of data). Rerun the optimizer. Solve for the weights that match the expected return (or close to it) that you chose. Alternatively, given my example above, solve for the maximum expected return and the minimum. Take the expected return that is the average. Save the weights.
3. Repeat the resampling. Depending on your computer platform this could get painful. Hence, you only need to do this 25 times.
4. Average the weights. Each asset should have 25 weights to average. [Make sure the averaged weights sum to one, if not, rescale them.]
5. Present a graph of the original data and the original efficient frontier. Mark on the graph the original portfolio you chose. Mark on the graph the resampled average portfolio (i.e. the resampled weights imply a portfolio using the original data that is different from your initial portfolio). Why is the resampled portfolio inside the original efficient frontier?
6. Write down (but do not do the calculations) how you would tackle this problem if instead of just choosing one portfolio, you chose 10 portfolios in step one. Please write this down step by step so that a computer programer could understand how to create the full resampling.
7. Bonus: The Michaud resampling method averages the weights in the final step. Can you suggest an alternative to the simple average.

## Question 3: Strategic Views for the U.S.

Question 1 is the easy part of the assignment. Although Assignment 2 is due at the beginning of lecture 5, I expect each group to make a contribution to lecture 4 (one slide showing the best model so far in forecasting the five-year S&P 500 return.) I will likely put your slide on the overhead. Be sure to have your group's name on the slide.

I have a number of Excel files on CD. The key file is us_5yr_historical_data-updated_dec2005.XLS. There are more than fifty columns of data in this monthly file with some data going back to the 1850s. You should expect to do some work updating and adding new data (for example, see the far right fields which are a partially updated version of Robert Shiller's famous dataset). Develop a regression forecasting model using variables such as dividend yield, Aaa-Tbill, Baa-Aaa, various interest rates and spreads. Be sure to lag the forecasting variables. That is, the dividend yield in December 1998 should be matched with the five year return from January 1999 to December 2003. You might find it useful to look at the Federal Reserve Bank of St. Louis' website which contains good historical interest rate data.

If there are data updates, I will post on the Class site.

You can use overlapping data (by year) or non-overlapping data (only one observation for every five years). Provide a an out of sample forecast for 2006-2010. Comment on your forecast. Would you adjust your forecast? What would your adjusted (using judgement forecast be). What is the motivation for your adjustment? What is wrong with just using the dividend yield for expected returns? How would you adjust the forecasting model (different specification of variables or new variables)?

Reestimate your model through December 2000. What were the forecasts for 2001-2005?

Apply a simple valuation model to the S&P 500, the dividend growth model. I suggest the following, collect forecasted earnings for the next three to ten years. Use a version of the dividend growth model: [E(r)-g]=d/p. I have included a S&P 500 operating earnings file in the data directory on the CD. You can update with the following link. I have also put in a template that you can play with called Dividend_discount_model.xls (this sheet is rough and is only meant as an example of what could be done). Comment on the expected return that you get relative to the forecasted return based on the regression model. Is the market over or undervalued according to the model? Why?

1. What do you think the value of the S&P500 should be today?
2. What is you 80% confidence range, i.e. give me the 10% and 90% tails in your valuation.
3. On a scale of zero (least) to seven (most), rate how competent you feel about your ability to value the market.

### File send format:

Please do not email files to my account.

Word document: A2_ (underscore) + first letters of group name .DOC (i.e. A2_PSAM.DOC for Positive Skewness Asset Management). Copy to Assignment_2 subdirectory of the BA453 directory on the student drive.

Excel document: A2_ (underscore) + first letters of group name .XLS (i.e. A2_PSAM.XLS for Positive Skewness Asset Management. Copy to Assignment_2 subdirectory of the BA453 directory on the student drive.

kewness Asset Management. Copy to Assignment_2 subdirectory of the BA453 directory on G.