Asset Allocation without Unobservable Parameters

Some asset allocation advice for long-term investors is based on maximization of expected utility. Most commonly used investor utilities require measurement of a risk-aversion parameter appropriate to the particular investor. But accurate assessment of this parameter is problematic at best. Maximization of expected utility is thus not only conceptually difficult for clients to understand but also difficult to implement. Other asset allocation advice is based on minimizing the probability of falling short of a particular investor's long-term return target or of an investable benchmark. This approach is easier to explain and implement, but it has been criticized by advocates of expected utility. These seemingly disparate criteria can be reconciled by measuring portfolio returns relative to the target (or benchmark) and then eliminating the usual assumption that the utility's risk-aversion parameter is not also determined by maximization of expected utility. Financial advisors should not be persuaded by advocates of the usual expected-utility approach. Asset allocation advice for long-term investors is based on a variety of criteria. Some advice is based on maximization of expected utility. The most commonly used utility functions are (1) quadratic or exponential, which yield the ubiquitous mean-variance utility underlying modern portfolio theory, and (2) the constant relative risk-aversion (CRRA) power utility. Both utilities require measurement of a risk-aversion parameter appropriate to a particular investor. But no validated procedures exist for reliably assessing an individual's risk-aversion parameter, and some investigators have suggested that all such procedures are doomed to failure because the risk aversion of an individual can depend on the scale of risks encountered.Other asset allocation advice for long-term investors is based on a different criterion: minimizing the probability of falling short of a particular investor's targeted long-term return or an investable benchmark. This approach is grounded in the findings of behavioral finance, may be easier to explain to investors than maximization of expected utility, and obviates the need to assess a risk-aversion parameter.The article presents a description of the two criteria and illustrates specifically how to implement each in the three-step asset-allocation process: (1) choosing a criterion function to maximize, (2) using historical time-series (or some other) data on asset class returns to estimate optimal asset allocations consistent with the chosen criterion function, and (3) using specific investor information to select the asset allocation appropriate for the particular investor. Then, I argue that the CRRA-utility and shortfall-probability analyses can be reconciled. Surprisingly, the seemingly disparate conventional CRRA-utility-maximization and shortfall-probability-minimization methods can be reconciled by completely maximizing the expected CRRA utility of the ratio of the portfolio's return to the investor's target return. This maximization requires unconventionally maximizing the expected utility by selection of both the portfolio's asset allocation weights and the utility's risk-aversion parameter (as opposed to conventional maximization over the weights alone with the use of some fixed value of the risk-aversion parameter). This unconventional formulation of minimizing long-run target shortfall probability retains the framework of expected-utility maximization while eliminating the conventional but problematic requirement that the advisor fix a value of the risk-aversion parameter that is most appropriate for the investor. Instead, in an interactive feedback process, the advisor and the investor mutually determine the most appropriate target rate of return.I use a simple two-asset allocation problem to illustrate this approach. The results are quite sensible and lead to a reexamination of the arguments put forth by advocates of the conventional use of expected utility and of the arguments against the minimization of shortfall probability.Criticisms of the use of shortfall probability are either overstated or not applicable to target-shortfall minimization (target-outperformance maximization) as described in this article. Theorists who believe that this criterion is inferior to risk aversion parameter-dependent expected utility need to reevaluate that position in light of the implementation and the risk-scaling problems highlighted in this article.