with respect to each asset weight and with respect to each Lagrange multiplier and set it equal to zero, as shown below:
Given assumptions for expected return, standard deviation, and correlation (which we specify later), we wish to find the values of
Next, we express Equations 2.4, 2.5, 2.6, and 2.7 in matrix notation, as follows:
We next substitute estimates of expected return, standard deviation, and correlation for domestic equities and Treasury bonds shown earlier in Tables 2.1 and 2.2.
With these assumptions, we rewrite the coefficient matrix as follows:
Its inverse equals:
Because the constant vector includes a variable for the portfolio's expected return, we obtain a vector of formulas rather than values when we multiply the inverse matrix by the vector of constants, as follows:
(2.9)
We are interested only in the first two formulas. The first formula yields the percentage to be invested in stocks in order to minimize risk when we substitute a value for the portfolio's expected return. The second formula yields the percentage to be invested in bonds. Table 2.3 shows the allocations to stocks and bonds that minimize risk for portfolio expected returns ranging from 9% to 12%.
TABLE 2.3 Optimal Allocation to Stocks and Bonds
| Target Portfolio Return | 9% | 10% | 11% | 12% |
|---|---|---|---|---|
| Stock Allocation | 25% | 50% | 75% | 100% |
| Bond Allocation | 75% | 50% | 25% | 0% |
THE SHARPE ALGORITHM
In 1987, William Sharpe published an algorithm for portfolio optimization that has the dual virtues of accommodating many real-world complexities while appealing to our intuition.8 We begin by defining an objective function that we wish to maximize:
(2.10)
