Your manager was so impressed with your work analyzing the return and standard deviations of the

12 stocks from Chapter 10 that he would like you to continue your analysis. Specifically, he wants you

to update the stock portfolio by:

Rebalancing the portfolio with the optimum weights that will provide the best risk and return

combinations for the new 12-stock portfolio.

Determining the improvement in the return and risk that would result from these optimum

weights compared to the current method of equally weighting the stocks in the portfolio. Use the

Solver function in Excel to perform this analysis (the time-consuming alternative is to find the

optimum weights by trial-and-error).

1. Begin with the equally weighted portfolio analyzed in Chapter 10. Establish the portfolio returns for

the stocks in the portfolio using a formula that depends on the portfolio weights. Initially, these weights will all equal 1/12. You would like to allow the portfolio weights to vary, so you will need to list the weights for each stock in separate cells and establish another cell that sums the weights of the

stocks. The portfolio returns for each month must reference these weights for Excel Solver to be useful.

2. Compute the values for the monthly mean return and standard deviation of the portfolio. Convert

these values to annual numbers (as you did in Chapter 10) for easier interpretation.

3. Compute the efficient frontier when short sales are not allowed. Use the Solver tool in Excel (on the

Data tab in the analysis section).2 To set the Solver parameters:

(a) Set the target cell as the cell of interest, making it the cell that computes the (annual) portfolio

standard deviation. Minimize this value.

(b) Establish the 'By Changing Cells' by holding the Control key and clicking in each of the 12 cells

containing the weights of each stock.

(c) Add constraints by clicking the Add button next to the 'Subject to the Constraints' box. One set

of constraints will be the weight of each stock that is greater than or equal to zero. Calculate the

constraints individually. A second constraint is that the weights will sum to one.

(d) Compute the portfolio with the lowest standard deviation. If the parameters are set correctly,

you should get a solution when you click 'Solve'. If there is an error, you will need to

double-check the parameters, especially the constraints.

4. Next, compute portfolios that have the lowest standard deviation for a target level of the expected

return.

(a) Start by finding the portfolio with an expected return , e.g. 2%, higher than that of the minimum

variance portfolio. To do thiss, add a constraint that the (annual) portfolio return equals this

target level. Click 'Solve' and record the standard deviation and mean return of the solution (and

be sure the mean return equals target,if not, check your constraint).

(b) Repeat Step (a) raising the target return in e.g. 2% increments, recording the result for each step.

Continue to increase the target return and record the result until Solver can no longer find a

solution.

(c) At what level does Solver fail to find a solution? Why?

5. Plot the efficient frontier with the constraint of no short sales. To do thiis, create ann XY Scatter Plot

(similar to what you did in Chapter 10), with portfolio standard deviation on the x-axis and the return

on the y-axis, using the data for the minimum variance portfolio and the portfolios you computed in step 4. How do these portfolios compare to the mean and standard deviation for the equally weighted

portfolio analyzed in Chapter 10?

6. Redo yourr analysis to allow for short sales by removing the constraint that each portfolio weight is

greater than or equal to zero. Use Solver to calculate the (annual) portfolio standard deviation for the

minimum variance portfolio, and when the annual portfolio returns are increased in e.g. 2%

increments. Plot the unconstrained efficient frontier on an XY Scatter Plot. How does allowing short

sales affect the frontier?

7. Redo youur analysis adding a new risk-free security that has a return of 0.5% (0.005) each month.

Include a weight for this security when calculating the monthly portfolio returns. That is, there will

now be 13 weights, one for each of the 12 stocks and one for the risk-free security. Again, these weights

must sum to 1. Allow for short sales, and use Solver to calculate the (annual) portfolio standard

deviation when the annual portfolio returns are increased in e.g. 2% increments. Plot the results on the

same XY Scatter Plot, and in addition keep track of the portfolio weights of the optimal portfolio.

What do you notice about the relative weights of the different stocks in the portfolio as you change the

target return? Can you identify the tangent portfolio?

Notes:

• In Eq. 10.4, we showed how to compute returns with stock price and dividend data. The 'adjusted close' series from Yahoo! Finance is already adjusted for dividends and splits, so we may compute returns based on the percentage change in monthly adjusted prices.
• If the Solver tool is not available, you must load it into Excel as follows:

1. On the File Tab, click Excel Options.

2. Click Add-Ins, and then, in the Manage box, select Excel Add-ins.

3. Click Go.

4. In the Add-Ins available box, select the Solver Add-in check box, and then click OK. If Solver Add-in is not listed in the

Add-Ins available box, click Browse to locate the add-in. If you are prompted that the Solver Add-in is not currently

installed on your computer, click Yes to install it.

5. After you load the Solver Add-in, the Solver command is available in the Analysis group on the Data tab.