In this project you will apply the material from Chapters 6 (Risk and Return) and 7 (Efficient Diversification). 

Due date: Thursday, August 1, 2002 in class. No projects will be accepted after the class.

Intermediate deadline: Friday, July 26 in class. Complete questions 1-3.

You may work individually or in a group of two.

Table 1 contains closing prices for four assets (stocks of four companies) over the 5-year period from June 1997 to June 2002.

Use the Treasury Bill rate as the proxy for the risk-free rate. Assume the T-bill rate is 4% per year (0.33% per month). 

Assume that expected return for each stock is equal to its mean monthly return over the 5-year period.

Clearly indicate your answers by underlining or putting them in a box.

1. (a) Calculate monthly returns for each asset from Table 1.

(b) Find the mean and standard deviation of returns for each asset from Table 1 over the 5-year period.

(c) Comment on the difference between monthly expected returns and monthly realized returns. What measure is used to characterize this difference?

(d) Find the total return over the 5-year period for each stock. Which stock had the highest and lowest returns? Notice that a high average monthly return does not necessarily translate into a high total return over a long period.

(e) On one graph, plot the value of $1 invested in each of the four stocks over the 5-year period.

(f) Are the prices given in Table 1 actual closing prices or prices adjusted for stock splits and dividends? How do you know?

2. (a) Calculate the covariance and correlation matrix for all pairs. Display your answers in a 4X4 table. Verify that covariance between the asset and itself is equal to its variance (there may be some rounding error). Verify that the correlation between all pairs of assets falls in the range [-1, 1]. Verify that the correlation with an asset and itself is 1. Example of correlation matrix is here.

(b) Interpret your results. Do all assets move together?

3. Graph the Capital Allocation Line (CAL) individually for each asset from Table 1 and the risk-free asset, i.e. you should graph four CALs. Extend the line past the 100% investment in each asset, i.e. short the risk-free asset (or borrow, and buy on margin the risky asset).

4. (a) Plot the investment opportunity sets for two pairs of assets: (1) JPM and PLXS and (2) PLXS and HSY. To calculate portfolio variance, use the following formula with nonnegative weights:

VAR(a X + b Y) = a ²VAR(X) + b ²VAR(Y) + 2ab COV(X,Y)

(b) Comment on the difference of the two opportunity sets. Given a choice between the two opportunity sets, which one would you prefer? Explain.

Note: For questions 5 - 9 you will need to use portfolio weights given in  Table 2.

5. After verifying that the portfolio weights in Table 2 sum to 1 for each of the 50 portfolios, calculate the expected return for each of the 50 portfolios.

6. Calculate the variance and standard deviation for each portfolio. Since the assets are not independent, i.e. COV<>0, you must use following formula:

VAR(a X + b Y + g Z + d W) =

a ²VAR(X) + b ²VAR(Y) + g ²VAR(Z) + d ²VAR(W) + 2ab COV(X,Y) 

+ 2ag COV(X,Z) + 2ad COV(X,W) + 2bg COV(Y,Z) + 2bd COV(Y,W) 

+ 2gd COV(Z,W),

where weights a, b, g, and d are arbitrary constants that sum to 1.

7. Plot all the portfolios from Table 2 and identify (approximately) the efficient frontier. Is the efficient frontier preferable to the opportunity sets of two stocks considered above? Explain.

8. Identify the following portfolios for the four assets:

1. Four single-security portfolios.

2. Equally-weighted portfolio.

3. The minimum variance portfolio.

4. The "market" portfolio for the four stocks. Hint: To identify the market portfolio, use the property that the capital market line (CML) is the best attainable CAL.

9. Plot the CML using the T-bill rate and report the reward-to-variability ratio offered by the CML.

In conclusion, comment on what you have learned from this project. Comment on anything else that you found interesting or noteworthy while doing this assignment.

Answer the questions briefly and to the point. Do not hand in your actual spreadsheet, just the summary information, graphs, tables, formulas, etc. Remember to circle or underline your final answers.

Instead of using the weights given in Table 2 for questions 7-9 above, calculate the weights for the frontier portfolios composed of all four stocks by choosing expected returns and obtaining the combination of weights that minimizes portfolio variance (or standard deviation). Hint: You can use Excel's Solver to solve this optimization problem. Perform the calculations for monthly portfolio returns ranging from 0.2% to 3% in increments of 0.2%.

First perform the calculations without restricting the weights (i.e. short positions are allowed). Then repeat the process after restricting the weights to be nonnegative (<=1, >=0). Plot the portfolio frontier for both cases. Identify the efficient frontier. Identify and plot the portfolios listed in #8 above. Plot the CML and report the reward-to-variability ratio offered by the CML.

Comment on the difference of results obtained with and without restrictions. How does this difference relate to the separation property discussed in Chapter 7? (Assume that (1) the universe of investable assets consists of these four stocks and the risk-free asset and (2) one can borrow at the risk-free rate.) Would your answer change if borrowing at the risk-free rate were not possible?

What have you learned from this exercise?

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