Читать книгу Asset Allocation - William Kinlaw, Mark P. Kritzman - Страница 58

To begin, we define a portfolio's expected return and risk.

As noted earlier, the expected return of a portfolio is simply the weighted average of the assets' expected returns. Equation 2.1 shows expected return for a portfolio consisting of only stocks and bonds:

(2.1)

In Equation 2.1, equals the portfolio's expected return, equals the expected return of stocks, equals the expected return of bonds, equals the percentage of the portfolio allocated to stocks, and equals the percentage allocated to bonds.

As noted earlier, portfolio risk is a little trickier. It is defined as volatility, and it is measured by the standard deviation or variance (the standard deviation squared) around the portfolio's expected return. To compute a portfolio's variance, we must consider not only the variance of the asset class returns, but also the extent to which they covary. The variance of a portfolio of stocks and bonds is computed as follows:

(2.2)

Here equals portfolio variance, equals the standard deviation of stocks, equals the standard deviation of bonds, and equals the correlation between stocks and bonds.

Our objective is to minimize portfolio risk subject to two constraints. Our first constraint is that the weighted average of the stock and bond returns must equal the expected return for the portfolio. We are also faced with a second constraint: we must allocate our entire portfolio to some combination of stocks and bonds. Therefore, the fraction we allocate to stocks plus the fraction we allocate to bonds must equal 1.

We combine our objective and constraints to form the following objective function:

(2.3)

The first term of Equation 2.3 up to the third plus sign equals portfolio variance, the quantity to be minimized. The next two terms that are multiplied by represent the two constraints. The first constraint ensures that the weighted average of the stock and bond returns equals the portfolio's expected return. The Greek letter lambda () is called a Lagrange multiplier. It is a variable introduced to facilitate optimization when we face constraints, and it does not easily lend itself to economic interpretation. The second constraint guarantees that the portfolio is fully invested. Again, lambda serves to facilitate a solution.

Our objective function has four unknown values: (i) the percentage of the portfolio to be allocated to stocks, (ii) the percentage to be allocated to bonds, (iii) the Lagrange multiplier for the first constraint, and (iv) the Lagrange multiplier for the second constraint. To minimize portfolio risk given our constraints, we must take the partial derivative of the objective function with respect to each asset weight and with respect to each Lagrange multiplier and set it equal to zero, as shown below:

(2.4)

(2.5)

(2.6)

(2.7)

Given assumptions for expected return, standard deviation, and correlation (which we specify later), we wish to find the values of and associated with different values of , the portfolio's expected return. The values for and are merely mathematical by-products of the solution.

Next, we express Equations 2.4, 2.5, 2.6, and 2.7 in matrix notation, as follows:

(2.8)

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%