Portfolio 2: Minimum Variance Portfolios with Targeted Return

Whereas portfolio 1 did not stipulate a return (it earns a return but this is the return to the minimum variance portfolio), Portfolio 2 does. This stipulation takes the form of another constraint (see the example in Chapter 5), and therefore the Lagrangian has form:

From the first order conditions, we write the system as:

Suppose we stipulate the portfolio return as 0.036. This is not entirely arbitrary; from the spreadsheet, the two mean returns are 0.03 and 0.04. The portfolio return has to be a weighted average of these. Set it too high (above 0.04) and the optimizer will have to short the lower return asset in an attempt to achieve this higher return. You can demonstrate this using the spreadsheet. Assuming independence in the returns so that , we have the following setup:

The calculations are in the spreadsheet. It makes intuitive sense that the portfolio would be tilted toward the higher returning asset since the portfolio return is greater than what the lower returning asset can deliver.