William Kinlaw

Asset Allocation


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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)

Target Portfolio Return 9% 10% 11% 12%
Stock Allocation 25% 50% 75% 100%
Bond Allocation 75% 50% 25% 0%

      (2.10)

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).

      (2.11)

      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