Portfolio Optimization Assignment

For this assignment, we are going to do some portfolio optimizations.

Background: As we saw in class, optimizing a portfolio involves several steps:

1. Obtain the return series for the assets.

2. Create a covariance matrix for the portfolio.

3. Using solver or an optimizing program, seek the maximum return at various levels of risk by varying the weights across the portfolio assets.

4. Plot the efficient frontier

5. Find the optimal portfolio based on the highest Sharpe ratio.

The problem: We're going to optimize a portfolio containing broad asset classes. To represent the asset classes, we will use mutual funds.

The Assets are:

VFINX - Vanguard 500 Index Fund

VFITX - Vanguard Intermediate Term Treasury Fund

VIMSX - Vanguard Mid-Cap Fund

NAESV - Vanguard Small-Cap Fund

VGTSX - Vanguard Total International Stock Index Fund

VEIEX - Vanguard Emerging Markets Stock Index Fund

VIPSX - Vanguard Inflation Protected Securities Fund

The data for these assets can be found on the Moodle page.

1. Create a covariance matrix for the portfolio and estimate the risk and return for the following portfolios:

a. Equal weighted

b. Minimum risk

2. Estimate the efficient frontier. You should use a range of standard deviations from 4% to 23%. Increase at regular increments - say every 4%, so estimate the it at 4, 8, 12, 16, 20, 23%. You can use more frequent intervals if you like - it will just take more time.

3. Plot the efficient frontier

4. Plot the locations of each of the assets as single points. (You'll need the return and standard deviation for each asset). Hint: the SD is just the square root of the asset variance, and the asset variances are the diagonal of the covariance matrix.

5. What is the optimal portfolio (max Sharpe ratio) if the risk free rate is:

a. 2%

b. 3%

c. 4%

6. Assume that an investor has purchased the optimal portfolio. What would be the return of his/her portfolio if 130% was invested in the portfolio? What would be the risk? What would be the individual asset weights? Note that you need to include the weight on the risk-free asset. Hint: this question takes us back to the single risky asset problem of moving along the Capital Allocation Line. Use a 4% risk free rate

7. What will be the portfolio return at 12% SD if we allow 10% shorting on any of the assets? (you'll need to run the optimizer one more time for this one and allow weights to go to -10%). Discuss your results.

Attachment:- Assignment Files.rar