Your manager was so impressed with your work analyzing the return and standard deviations of the 12 stocks in case 8 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).
- 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.
- 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.
- Compute the efficient frontier when short sales are not allowed. Use the Solver tool in Excel (on the Data tab in the analysis section).* To set the Solver parameters:
- Set the objective to be the cell that computes the (annual) portfolio standard deviation. Minimize this value.
- Set the “By Changing Variable Cells” to the cells containing the portfolio weights. (Hold the Control key and click in each of the 12 cells containing the weights of each stock.)
- Add constraints by clicking the Add button next to the “Subject to the Constraints” box. The first constraint is that the cell containing the sum of all the portfolio weights must equal one. The next set of constraints is that each portfolio weight is non-negative. You can enter these constraints individually, or check the box “Make Unconstrained Variables Non-Negative.”
- 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 doublecheck the parameters, especially the constraints.
- Next, compute portfolios that have the lowest standard deviation for a target level of the expected return.
- Start by finding the portfolio with an expected return of 2% higher than the annual return for the minimum variance portfolio you computed in Step 3, rounded to the nearest whole percentage. To do this, 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).
- Repeat Step (a) raising the target return in 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. Next, repeat Step (a) by lowering the target return in 2% increments from the return of the minimum variance portfolio, again recording each result.
- At what level does Solver fail to find a solution? Why?
- Plot the efficient frontier with the constraint of no short sales. To do this, create an 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?
- Redo your 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 annual returns in 5% increments from 0% to 40%. Plot the unconstrained efficient frontier on an XY Scatter Plot. How does allowing short sales affect the frontier?
- Redo your analysis adding a new risk-free security that has a 3% annual return, or 0.25% (0.0025) 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 set to 3%, 10%, 20%, 40%. 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?
*If the Solver tool is not available, you must load it into Excel as follows:
- On the File Tab, click Excel Options.
- Click Add-Ins, and then, in the Manage box, select Excel Add-ins.
- Click Go.
- In the Add-Ins available box, select the Solver Add-in check box, and then click OK.
Tip: 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.
- After you load the Solver Add-in, the Solver command is available in the Analysis group on the Data tab.