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Example: Consider the information for two well-diversified portfolios shown in Exhibit 1.3. The riskless rate is 2 % per year.

      Assuming that these are optimal portfolios for two asset owners, what are their degrees of risk aversion?

We know from Equation 1.9 that the expected excess return on each portfolio divided by its variance will be equal to the degree of the risk aversion of the investor who finds that portfolio optimal.

      As expected, the aggressive portfolio represents the optimal portfolio for a more risk-tolerant investor, while the moderate portfolio represents the optimal portfolio for a more risk-averse investor.

      1.5.9 Managing Assets with Risk Aversion and Growing Liabilities

      As mentioned earlier in the chapter, most asset owners are concerned with funding future obligations using the income generated by the assets. In the previous example, the DB plan has liabilities that will need to be met using the fund's assets. Suppose the current value of these liabilities is L euros. Further, suppose the rate of growth in liabilities is given by G, which could be random. In this case, the objective function of Equation 1.8 can be restated as:

(1.10)

      In this case, the DB plan wishes to maximize the expected rate of return on the fund's assets, subject to its aversion toward deviations between the return on the fund and the growth in the fund's liabilities. In other words, the risk of the portfolio is measured relative to the growth in liabilities. Later in this chapter, we will demonstrate how this problem can be solved.

      One final comment about evaluating investment choices: Although the framework outlined here is a flexible and relatively sound way of modeling preferences for risk and return, the presentation considered only one-period investments and decisions. Economists have developed methods for extending the framework to more than one period, where the investor has to withdraw some income from the portfolio. These problems are extremely complex and beyond the scope of this book. However, in many cases, the solutions that are based on the single-period approach provide a reasonable approximation of the solutions obtained under approaches that are more complex.

      1.6 Investment Policy Constraints

      The previous section introduced the expected utility approach as a simple and yet flexible approach to modeling risk-return objectives of asset owners. This section discusses the typical set of constraints that must be taken into account when trying to select the investment strategy that maximizes the expected utility of the asset owner.

      1.6.1 Investment Policy Internal Constraints

      Internal constraints refer to those constraints that are imposed by the asset owner as a result of its specific needs and circumstances. Some of these internal constraints can be incorporated into the objective function previously discussed. For example, we noted how the constraint that allocations with positive skewness are preferred could be incorporated into the model. However, there are other types of constraints that may be expressed separately. Some examples of these internal constraints are: