Abstract
Risk aversion has long played a key role in examining decision making under uncertainty. But we now know that prudence, temperance, and other higher-order risk attitudes also play vital roles in examining such decisions. In this chapter, we examine the theory of these higher-order risk attitudes and show how they entail a preference for combining “good” outcomes with “bad” outcomes. We also show their relevance for non-hedging types of risk-management strategies, such as precautionary saving. Although higher-order attitudes are not identical to preferences over moments of a statistical distribution, we show how they are consistent with such preferences. We also discuss how higher-order risk attitudes might be applied in insurance models.
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Notes
- 1.
The lottery A 2 is easily seen to be a simple mean-preserving spread of the lottery B 2.
- 2.
- 3.
The rationale for statistical independence here should be apparent. For example, if \(\tilde{\varepsilon }_{1}\) and \(\tilde{\varepsilon }_{2}\) were identically distributed and perfectly negatively correlated, every risk averter would prefer to have the two risks in the same state, since they would then “cancel” each other.
- 4.
It is easy to see in Fig. 2.2 that the means and variances for A 3 and B 3 are identical, but B 3 has a higher skewness (is more right skewed). For two distributions with the same first two moments, it can be shown that it is impossible for every prudent individual to prefer the distribution with a lower skewness. If the two zero-mean risks in Fig. 2.3 are symmetric, then the first three moments of A 4 and B 4 are identical, but with A 4 having a higher kurtosis (fatter tails). For two distributions with the same first three moments, it can be shown that it is impossible for every temperate individual to prefer the distribution with a higher kurtosis.
- 5.
These higher orders are already known to be important in various contexts. For example, standard risk aversion as defined by Kimball (1993), as well as risk vulnerability as defined by Gollier and Pratt (1996), each require temperance. Lajeri-Chaherli (2004) looks at a rationale to use 5th-order risk attitudes. Both of these higher-order risk preferences are also given intuitive economic interpretations by Courbage and Rey (2010).
- 6.
Although utility-based models can also be derived without differentiability, most of the literature assume that these derivatives exist.
- 7.
For the mathematically astute, we admit that this is a slight exaggeration. Strict risk aversion also allows for u ′ ′ = 0 at some wealth levels, as long as these wealth levels are isolated from each other. See Pratt (1964) for more details.
- 8.
An article by Hanson and Menezes (1971) made this same observation more than 40 years ago!
- 9.
In the original article by Friedman and Savage (1948), the risks that were considered had positive expected payoffs and could thus have a positive utility premium, even for a risk averter. In this chapter, we only consider zero-mean risks.
- 10.
The utility function is only unique up to a so-called affine transformation. See Pratt (1964).
- 11.
Replacing the second condition in the definition with F (i)(b) ≤ G (i)(b) yields a definition of Nth-order stochastic dominance. The results in this section easily extend to stochastic dominance, as shown by Eeckhoudt et al. (2009).
- 12.
Kimball (1993) refers to the two risks in this case as “mutually aggravating.” Pratt and Zeckhauser (1987) came very close to making this same observation. Their basic difference was considering independent risks \(\varepsilon _{i}\) that were disliked by a particular individual, rather than zero-mean risks, which are disliked by every risk averter. Menezes and Wang (2005) offer an example that is also quite similar and refer to this case as “aversion to outer risk.”
- 13.
These authors also provide a proof of this result, which we do not reproduce here.
- 14.
This example is adapted from Fei and Schlesinger (2008).
- 15.
The second-order sufficient condition for a maximum follows trivially if we assume risk aversion.
- 16.
For another interesting application, see Gollier (2010), who lets h denote the quality of the planet’s environment.
- 17.
This analysis is based on a generalization and extension of the results in Tsetlin and Winkler (2009), who confine themselves to expected-utility models.
- 18.
For a generalization of the multiplicative case to any arbitrary order n, see Wang and Li (2010).
- 19.
Note that for commonly used CRRA utility functions, relative prudence always equals the measure of relative risk aversion plus one, so that relative risk aversion exceeding one is equivalent to relative prudence exceeding two.
- 20.
Caballé and Pomansky (1996) further extended these measures to arbitrarily high orders.
- 21.
A short summary of these existing measures is provided by Eeckhoudt (2012).
- 22.
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Eeckhoudt, L., Schlesinger, H. (2013). Higher-Order Risk Attitudes. In: Dionne, G. (eds) Handbook of Insurance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0155-1_2
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