Choosing and Using Utility Functions in Forming Portfolios

Abstract

Utility functions offer a means to encode objectives and preferences in investor portfolios. The functions allow one to place a score on outcomes and then identify optimal portfolios by maximizing utility. The central theme of this article is that utility functions should be tailored to the investor. I discuss how an appropriate function might be chosen and demonstrate concepts for power utility and reference-dependent utility. A modeling approach is presented that may be applied without resorting to dynamic optimization. The selection of utility functions is illustrated for four investor types.

Disclosure: The author reports no conflict of interest.

Editor’s Note The original version had a typographical error in Figure 4B, which has been corrected in this version.

Submitted 1 October 2018

Accepted 20 March 2019 by Stephen J. Brown

View correction statement:

Correction

Notes

¹ A certainty equivalent is a guaranteed value that an investor would accept now rather than taking a chance on a higher, but uncertain, value in the future.

² Merton (1971) noted that HARA utility is a rich class that accommodates absolute and relative risk aversion that can be increasing, decreasing, or constant in wealth. It includes the power, quadratic, exponential, and log utility functions. The utility function attributed to Epstein and Zin (1989) and Weil (1989) decouples the “elasticity of intertemporal substitution” from risk aversion, allowing any preference for earlier resolution of uncertainty to be evaluated separately from aversion to volatility. Although this function is widely used in the academic literature, the intuition behind this form of utility can be hard to grasp, and the function needs to be solved recursively, which adds to complexity.

³ Log utility can be seen as a limiting case of power utility as CRRA approaches 1. Rubinstein (1976) argued that log utility has many attractive features from a modeling perspective. For many investors, however, it will place an insufficient penalty on low outcomes.

⁴ The first derivative of Equation 1$$U_{PU,t}=\frac{W_t^{(1-CRRA)}}{1-\text{CRRA}}$$(1) with respect to W is $W^{-CRRA}$.

⁵ Prospect theory entails a broader framework than the value function, including a preliminary stage where prospects are “edited” down to a subset for evaluation and application of decision weights that transform the probabilities attached to outcomes. It proposes an S-shaped function that implies risk aversion for gains but risk seeking for losses.

⁶ These utility functions can be treated as subcases of the general form of reference-dependent utility function presented here.

⁷ The general form of the HARA family of utility functions entails three parameters and provides flexibility to manipulate absolute and relative risk aversion (see Merton 1971, p. 389).

⁸ This assumption equates to compounding returns over multiple periods, with ln(1 + Return_t) being independent and normally distributed.

⁹ This standard deviation would arise from investing over 10 years with a yearly standard deviation of 15.8%; that is, $0.50={\left[\left(0.158\times 0.158\right)\times 10\right]}^{0.5}$.

¹⁰ The curvature parameter on losses of 0.88 has been adopted by such authors as Benartzi and Thaler (1995), Blake et al. (2013), and Levy (2016). Levy (2016) also investigated values of between 0.5 and 1.0 for this parameter. He commented, “In practice the loss aversion is presumably more profound than what is estimated in laboratory experiments” (p. 1421).

¹¹ The ratio is exactly 44.4% in the difference form because the same curvature parameter is applied to gains and losses, while the weighting parameter applies a linear transformation. The ratio is nonlinear under the ratio form of the function because taking a ratio imposes a nonlinear transformation on the outcomes before the utility function is applied.

¹² The literature is often concerned that utility functions be continuous and differentiable. Although these characteristics are required for analytical solutions, they are not necessary in numerical approaches that simply assign and aggregate scores attached to outcomes. Utility might even be specified as a schedule of scores, rather than an explicit function, when numerical methods are being used.

¹³ The approach outlined in this study can provide a way of setting the allocation between riskless and risky assets in the Waring–Siegel (2015) method by including their liability-hedging TIPS (Treasury Inflation-Protected Securities) ladder as one of the candidate assets.

¹⁴ A zero withdrawal becomes undefined (infinite) under power utility. This problem might be handled by specifying a minimum utility value—for example, by substituting the utility value of $1 for any withdrawal less than $1. In reference-dependent utility, consideration might be given to whether the maximum negative utility value achieved at $0 imposes an adequate utility penalty for running out of money. In any event, the aim is to ensure that an appropriate utility value be attached to states where the portfolio value and, hence, withdrawals decline to zero.

¹⁵ Another approach would be to conduct the analysis in benchmark-relative terms.

¹⁶ Available at www.econ.yale.edu/~shiller/data.htm.

¹⁷ The reported long bond yield for that period was used in pricing the bond.

¹⁸ Another approach, similar to that described for bequests, is to apply a discount to terminal portfolio value directly within the utility function.

¹⁹ If the problem is sufficiently well behaved, the analysis might be conducted in Microsoft Excel by using the Solver add-in to find the optimal weights.

²⁰ The positive serial correlation for bonds arises from their greater exposure to inflation, which tends to be persistent.

²¹ See www.aaii.com/assetallocationsurvey for the latest figure.

²² The report is available at www.federalreserve.gov/econres/scf_2013.htm.

²³ The 10-year horizon is a compromise. Extending the length of the data blocks drawn from the Shiller data reduces sample size and would not change the result.

²⁴ The pension payments are set at a value of $0.094 per $1 of liability, so that the discounted value of the payments is $1 at the bond yield assumed to prevail in period t = 0 of 4.92%. This yield represents the average long bond yield in the Shiller dataset over the period of analysis. This assumption establishes a baseline that dovetails with the historical data.

²⁵ If the retiree placed a value on any bequest, the terminal portfolio value would also need to be evaluated. Bell et al. (2017a) provided an example of such an application.

²⁶ The findings in this section are hard to square with investor behavior for a range of reasons, one of which is that this analysis reflects a highly simplified setting. Also, data on the strategies followed by retirees are limited, and decumulation strategies are still in the development stage. Indeed, utility-based analysis might be a tool to help develop better retirement strategies than we currently observe in practice.

²⁷ CEI is estimated as a function of expected utility by rearranging the utility function. In the case of power utility, for instance, $\text{CEI},\text{ age }66–95={\left\{\left[E\left(\text{Utility}\right)/30\right]\times \left(1-\text{CRRA}\right)\right\}}^{\left[1/\left(1-CRRA\right)\right]}$.

²⁸ Note that utility functions such as those of Epstein and Zin (1989) and Weil (1989) require recursive solutions.

²⁹ Admitting dynamic portfolio management may give rise to positions aimed at hedging changes in the investment opportunity set, in the sense of Merton (1973).

³⁰ The analysis in the prior section assumes rebalancing back to target asset weights, which amounts to a simple rule of this type.

³¹ Recall that the withdrawals were fixed for this investor.

³² Initially, a grid of optimal equity weights for Year 3 is established by optimizing utility at the end of Year 3 conditional on the Year 2 funding ratio. The next step is to create a similar grid of equity weights for Year 2 conditional on the realized Year 1 funding ratio, again optimizing utility at the end of Year 3 and assuming that the Year 3 equity weights will follow the Year 3 grid. The final step is to estimate the optimal equity weights at the beginning of Year 1 to optimize utility at the end of Year 3, under the assumption that the equity weights for Year 2 and Year 3 will subsequently be updated in line with both grids.

³³ The Year 1 equity weight differences between the dynamic and static strategies were mainly positive and ranged up to about +8%, although the optimal dynamic equity weights were modestly lower for initial funding ratios in the 0.90–1.00 range.

³⁴ Under the reference-dependent utility function, this change might be roughly interpreted as a risk-adjusted change in wealth.