1. More on Asset Allocation
Week 5

2. Minimum Variance and Efficient Frontier
When we only have two risky assets, as in our KO + PEP example, it is easy to construct this graph by simply calculating the portfolio returns for all possible weights.
When we have more than 2 assets, it becomes more difficult to represent all possible portfolios, and instead we will only graph only a subset of portfolios. Here, we will choose only those portfolios that have the minimum volatility for a given return. We will call this graph the minimum variance frontier.
The upper half of the minimum variance frontier is called the efficient frontier. The optimal portfolio is one of the portfolios on the efficient frontier.
The next few slides describes how this frontier can be created.
Once the frontier is created, we can find the portfolio with the highest Sharpe ratio by

3. Creating the minimum variance frontier
How to use a spreadsheet to calculate the frontier when there are more than 2 assets

4. The Minimum Variance Frontier
With two assets, as we saw, we can construct the frontier by brute force - by listing almost all possible portfolios.
When we have more than 2 assets, its gets difficult to consider all possible portfolio combinations. Instead, we will make the process simpler by considering only a subset of portfolios: those portfolios that have the minimum volatility for a given return.
When we plot the return and volatilities of these portfolios, the resultant graph will be known as the minimum variance (or volatility) frontier.
We will use Excel’s “Solver” for these calculations (look under Tools. If it is not there, then add it into the menu through Add-in).

5. The Steps We will implement the procedure in three steps:
1. For each asset (and for the time period that you have chosen), calculate the mean return, volatility and the correlation matrix.
2. Set up the spreadsheet so that the Solver can be used. See the sample spreadsheet. Your objective here is to determine the weights of the portfolio that will allow you to achieve a specified required rate of return with the lowest possible volatility.
3. Repeat 2 for a range of returns, and plot the frontier (return vs. volatility).

6. Step 1: Assembling the Data
A. Fix the time period for the analysis. You want a sufficiently long period so that your estimates of the mean return, volatility and correlation are accurate. But you also don’t want too long a period as very old data may not be valid.
B. Estimate the mean return and volatility for each of your assets. Next, calculate the correlation between each pair of assets. If there are N assets, you will have to calculate N(N-1)/2 correlations.

7. Step 2: Setting up the spreadsheet to use the Solver (1/4)
The objective here is to set up the spreadsheet in a manner that is easy to use with the solver.
The estimates of the return, volatility and the correlation matrix are used to set up a matrix for covariances, which is then used to calculate the portfolio volatility for a given set of weights.
To create the frontier, you will ask the solver to find you the weights that gives you the minimum volatility for a required return.

8. Step 2: Using the Solver (2/4)
1.Target Cell: When you call the solver, it will ask you to specify the objective or the “target cell”. Your objective is to minimize the volatility - so in this case, you will specify the cell that calculates the portfolio variance [$B$36]. As you want to minimize the variance, you click the “Min”.
2. Constraints: You will have to specify the constraints under which the optimization must work. There are two constraints that hold, and a third which will usually also apply.

9. Using the Solver: Constraints on the Optimization (3/4)
1. First, the sum of the weights must add up to 1.
2. Second, you have to specify the required rate of return for which you want the portfolio of least volatility. For each level of return, you will solve for the weights that give you the minimum volatility. To construct the frontier, you will vary this required return over a range. Thus, you will have to change this constraint every time you change the required return.
Third, if you want to impose a short-selling constraint, you can specify that each portfolio weight is positive.

10. Step 2: (4/4)
Finally, you specify the arguments that need to be optimized. In this case, you are searching for the optimal weights, so you will have to specify the range in the spreadsheet where the portfolio weights used [A29 to A34].

11. Step 3
The final step is to simply repeat step 2, until you have a sufficiently large data set so that the minimum variance frontier can be plotted. .

12. The Optimal Allocation
We can now use the graph of the minimum variance frontier to figure out the portfolio with the highest Sharpe Ratio. This portfolio will be the portfolio such that the CAL passing through it is tangent to the minimum variance frontier.
The weights of this portfolio determines the optimal allocation within the assets that make up the “risky portfolio”. All investors should opt for this allocation.
The portfolio will always be on the upper portion of the frontier, above the portfolio with the lowest volatility - this portion is called the efficient frontier.

13. Example
The spreadsheet “MinimumVarianceFrontier” provides an example of the computation of the minimum variance frontier for a portfolio of 6 stocks – KO, PEP, WMT, IBM, XOM, MSFT.
The optimal portfolio turns out to have the following weights: 13% in WMT, 3.50% in IBM, 68% in XOM, and 15.50% in MSFT. This portfolio had an average return of 23%, and a volatility of 15.45%.
The return and volatility of S&P 500 are also plotted on the graph for comparison.

14. Minimum Variance Frontier

15. The Sharpe Ratio for the Portfolios

16. The Optimal Portfolio vs. S&P 500
In our example, the optimal portfolio provides a risk-return tradeoff far superior to investing in the S&P 500.
For example, if we invest 48.8% in the optimal portfolio and the remainder in the Treasury Bill, we would expect to earn a return equal to that of the S&P 500 (of 12.14%), but with a volatility of 7.50%, far lower than the S&P 500 volatility of 28%.
On the other hand, if we leverage up (borrowing at the risk-free rate) to make the volatility of the optimal portfolio equal to the volatility of S&P 500, we would expect to earn a return of 38.25%, much higher than the S&P 500 return of 12.14%.

17. In Summary (1/2)
1. The optimal allocation is determined in two steps. First, we decide the allocation between the risky portfolio, and the riskless asset. Second, we determine the allocation between the assets that comprise the risky portfolio.
2. As every portfolio of the risky assets and the riskless asset has the same Sharpe ratio, there is not one optimal portfolio for all investors. Instead, the allocation will be determined by individual-specific factors like risk aversion and the objectives of the investor, taking into account factors like the investor’s horizon, wealth, etc.
3. When we are considering the allocation between different classes of risky assets, it is possible to create a portfolio that has the highest Sharpe Ratio. The weights of the risky assets in this portfolio will determine the optimal allocation between various risky assets. This portfolio can be determined graphically by drawing the capital allocation line (CAL) such that it is tangent to the minimum variance frontier. This portfolio will always lie on the upper part of the frontier (or on the efficient part of the frontier).

18. In Summary (2/2)
4. The extent to which you can decrease the volatility of the portfolio depends also on the correlation. The lower the average correlation of the stocks in your portfolio, the lower you can decrease the volatility of your portfolio.
5. The homework provides you with an exercise to determine the optimal allocations.