# Estimating the Coefficient of Relative Risk Aversion for Consumption

The coefficient of relative risk aversion for consumption is an important parameter that plays a key role in asset allocation, and helps determine how much to allocate to stocks versus how much to allocate to a risk free asset such as cash.

Despite its importance, even at a glance there appears considerable uncertainty regarding its value. For instance one researcher reports (Choi et al., "Is Relative Risk Aversion Greater Than One?"):

Davies (1981) concludes that his “best guess” estimate of the value of relative risk aversion is between 3 and 4. Binswanger’s (1981) experimental investigation suggests considerably higher values for relative risk aversion. He finds that relative risk aversion ranges from between 1 and 2 to over 1000, depending on the gamble size. ... Schluter and Mount’s estimates range from 0.05 to 3, with an average value of 0.8. Similarly, Hansen and Singleton’s estimates range from 0.17 to 1.36, with five out of six estimates being smaller than unity. Using bond return data, Ferson (1983) reports a range from -1.4 to 5.4.

Part of the confusion can be put down to three factors:

• A significant fraction of the work on relative risk aversion is concerned with the relative risk aversion of wealth (W), rather than consumption (C). This is expanded on by Meyer and Meyer (Meyer and Meyer, "Relative Risk Aversion: What Do We Know?") and in their book (Meyer and Meyer, "Measuring Risk Aversion"). Wealth as used here is what economists refer to as Arrow-Pratt wealth, and only includes assets that can be freely re-allocated. So human capital isn't considered, and home equity probably only just gets considered as part of Arrow-Pratt wealth. Roughly 80% of income comes from labor and 20% from capital. If C = 0.8 + 0.2 W, then it can be shown that the coefficient of relative risk aversion for consumption is (0.8 + 0.2) / 0.2 = 5 times that of the coefficient of relative risk aversion for wealth. This is a very crude example. It fails to consider differences in individual circumstances as to how wealth is distributed.

• Relative risk aversion need not be constant, but may be a function of consumption or wealth.

• Framing and mental bracketing is also important. How the question is asked, and the surrounding assumptions and circumstances play a key role in the results obtained. Asking "Would you rather receive \$100, or a 50-50 chance of \$300 or nothing?" is meaningless without knowing what \$100 means to the recipient.

Estimates of the coefficient of relative risk aversion for consumption seem to come from four sources:

• Estimates based on how people invest. These assume people's investment preferences reveal how they feel about risk. They tend to produce large estimates for the coefficient of relative risk aversion. They also lead us to a circular logic that could lead to the blind leading the blind. We are interested in determining the coefficient of relative risk aversion in order to decide how to invest, and we base our estimate on how people currently invest.

• Estimates based on labor elasticity (Chetty, "A New Method of Estimating Risk Aversion") which find a coefficient of relative risk aversion of consumption of about 1. The model used assumes all income is consumed, with no ability to carry over wealth into future time periods.

• Estimates of the value of a statistical life and expenditures that reduce the probability of death (Kaplow, "The Value of a Statistical Life and the Coefficient of Relative Risk Aversion") which also finds small values for the coefficient of relative risk aversion. This model also assumes all wealth is consumed and not carried over into the future.

• Direct surveys. For instance Barsky et al. (Barsky et al., "Preference Parameters and Behavioral Heterogeneity: An Experimental Approach in The Health and Retirement Study"). In this study 13% of people reported a coefficient of relative risk aversion for consumption in the range 0 to 1, 11% reported 1 to 2, 12% reported 2 to 3.76, and 65% reported a coefficient of relative risk aversion for consumption greater than 3.76. This survey may be subject to a status quo bias in that recipients were not asked to report when to scenarios were equal, but only when one was better.

I have been unable to resolve these conflicting results. Depending on which study you adopt either 1 or 4 could be a reasonable value to use for the coefficient of relative risk aversion for consumption. In favor of the smaller value is that they are based on how people actually behave, not how they report they will behave. In favor of the larger value is it would be more consistent with a coefficient of relative risk aversion for wealth of around 1.

Since it is the inverse of the coefficient of relative risk aversion that counts, a reasonable compromise value might be 2. I currently use 3 as a default mainly to be more compatible with others that perform asset allocation, who are possibly motivated by consistency with how people invest, and therefore often use values in the range 4 to 8.