Читать книгу Asset Allocation - William Kinlaw, Mark P. Kritzman - Страница 59

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 utility any further by changing the allocation between stocks and bonds. We have maximized our objective function.

By varying the values we assign to we identify mixes of stocks and bonds for many levels of risk aversion, thus enabling us to construct the entire efficient frontier of stocks and bonds. Figure 2.1 shows the efficient frontier based on the expected returns, standard deviations, and correlations shown in Tables 2.1 and 2.2. All the portfolios along the efficient frontier offer a higher level of expected return for the same level of risk than the portfolios residing below the efficient frontier.

FIGURE 2.1 Efficient frontier.

Now let us consider the efficient frontier that is composed of the six asset classes specified in Tables 2.1 and 2.2. In particular, let us now focus on three efficient portfolios that lie along this efficient frontier: one for a conservative investor, one for an investor with moderate risk aversion, and one for an aggressive investor. Table 2.4 shows the three efficient portfolios.

TABLE 2.4 Conservative, Moderate, and Aggressive Efficient Portfolios

Asset Classes	Conservative (%)	Moderate (%)	Aggressive (%)
US Equities	15.9	25.5	36.8
Foreign Developed Market Equities	14.5	23.2	34.5
Emerging Market Equities	5.6	9.1	15.8
Treasury Bonds	15.5	14.3	0.0
US Corporate Bonds	11.4	22.0	8.9
Commodities	4.0	5.9	4.0
Cash Equivalents	33.2	0.0	0.0
Expected Return	6.0	7.5	9.0
Standard Deviation	6.8	10.8	15.2

The composition of these portfolios should not be surprising. The conservative portfolio has nearly a 1∕3 allocation to cash equivalents.

The moderate portfolio is well diversified and not unlike many institutional portfolios. The aggressive portfolio, by contrast, has more than an 86% allocation to US and foreign equities. It should be comforting to note that we did not impose any constraints in the optimization process to arrive at these portfolios. The process of employing an equilibrium perspective for estimating expected returns yields nicely behaved results.