.

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)

In Equation 2.8,

equals expected utility,

equals portfolio expected return,

equals risk aversion, and

equals portfolio variance.

      Utility is a measure of well-being or satisfaction, whereas risk aversion measures how many units of expected return we are willing to sacrifice in order to reduce risk (variance) by one unit. (Chapter 25 includes more detail about utility and risk aversion.) By maximizing this objective function, we maximize expected return minus a quantity representing our aversion to risk times risk (as measured by variance).

Again, assume we have a portfolio consisting of stocks and bonds. Substituting the equations for portfolio expected return and variance (Equations 2.1 and 2.2), we rewrite the objective function as follows:

      (2.11)

This objective function measures the expected utility or satisfaction we derive from a combination of expected return and risk, given our attitude toward risk. Its partial derivative with respect to each asset weight, shown in Equations 2.12 and 2.13, represents the marginal utility of each asset class:

(2.12)

(2.13)

      These marginal utilities measure how much we increase or decrease expected utility, starting from our current asset mix, by increasing our exposure to each asset class. A negative marginal utility indicates that we improve expected utility by reducing exposure to that asset class, whereas a positive marginal utility indicates that we should raise the exposure to that asset class in order to improve expected utility.

      Let us retain our earlier assumptions about the expected returns and standard deviations of stocks and bonds and their correlation. Further, let us assume our portfolio is currently allocated 60% to stocks and 40% to bonds, and that our aversion toward risk equals 2. A risk aversion of 2 means that we are willing to reduce expected return by two units in order to lower variance by one unit.

      If we substitute these values into Equations 2.12 and 2.13, we find that we improve our expected utility by 0.008 units if we increase our exposure to stocks by 1%, and that we improve our expected utility by 0.04 units if we increase our exposure to bonds by 1%. Both marginal utilities are positive. However, we can only allocate 100% of the portfolio. We should therefore increase our exposure to the asset class with the higher marginal utility by 1% and reduce by the same amount our exposure to the asset class with the lower marginal utility. In this way, we ensure that we are always 100% invested.

      Having switched our allocations in line with the relative magnitudes of the marginal utilities, we recompute the marginal utilities given our new allocation of 59% stocks and 41% bonds. Again, bonds have a higher marginal utility than stocks; hence, we shift again from stocks to bonds. If we proceed in this fashion, we find when our portfolio is allocated 1∕3 to stocks and 2∕3 to bonds, the marginal utilities are exactly equal to each other. At this point, we cannot improve expected