Pension Fund
To complete our discussion of objectives, we now consider an application of the previous framework to present the objectives of a defined benefit (DB) pension fund. The following information is available:
Assuming that the preferences of the DB fund can be expressed as in Equation 1.5, the portfolio manager will select the weights, wi, such that the expected utility is maximized. That is, Equation 1.8 expresses the objective function that is maximized by choosing the values of wi. Of course, the portfolio manager must ensure that the weights will add up to one and some or all of the weights will need to be positive.
1.5.8 Finding Investor Risk Aversion from the Asset Allocation Decision
As mentioned previously, the value of the risk aversion has an intuitive interpretation. The expected excess rate of return on the optimal portfolio (E[RP] − Rf) divided by its variance, σ2P, is equal to the degree of risk aversion, λ:
The value of the parameter of risk aversion, λ, is chosen in close consultation with the plan sponsor. There are qualitative methods that can help the portfolio manager select the appropriate value of the risk aversion. The portfolio manager may select a range of values for the parameters and present asset owners with resulting portfolios so that they can see how their level of risk aversion affects the risk-return characteristics of the portfolio under current market conditions.
EXHIBIT 1.3 Hypothetical Risk Returns for Two Portfolios
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.
Aggressive investor: (15 % − 2 %)/(16%2) = 5.1
Moderate investor: (9 % − 2 %)/(8%2) = 10.9
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.
APPLICATION 1.5.8
Suppose that an investor's optimal portfolio has an expected return of 10 %, which is 8 % higher than the riskless rate. If the variance of the portfolio is 0.04, what is the investor's degree of risk aversion, λ?
Using Equation 1.9, λ can be expressed as:
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:
LIQUIDITY. The asset owner may have certain liquidity needs that must be explicitly recognized. For example, a foundation may be anticipating a large outlay in the next few months and therefore would want to have enough liquid assets to cover those outflows. This will require the portfolio manager to impose a minimum investment requirement for cash and other liquid assets. Even if there are no anticipated liquidity events where cash outlays will be needed, the asset owner may require maintaining a certain level of liquidity by imposing minimum investment requirements for cash and cash-equivalent investments, and maximum investment levels for such illiquid assets as private equity and infrastructure.
TIME HORIZON. The asset owner's investment horizon can affect liquidity needs. In addition, it is often argued that investors with a short time horizon should take less risk in their asset allocation decisions, as there is not enough time to recover from a large drawdown. This impact of time horizon can be taken care of by changing the degree of risk aversion or by imposing a maximum limit on allocations to risky assets. Time horizon may impact asset allocation in other ways as well. For instance, certain asset classes are known to display mean reversion in the long run (e.g., commodities). As a result, an investor with a short time horizon may impose a maximum limit on the allocation to commodities, as there will not be enough time to enjoy the benefits of potential mean reversion.
SECTOR AND COUNTRY LIMITS. For a variety of reasons, an asset owner may wish to impose constraints on allocations