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.
References
Arrow K (1965) Aspects of the theory of risk bearing. Yrjo Jahnssen Foundation, Helsinki
Bernoulli D (1738) Specimen Theoriae Novae de Mensura Sortis, Commentarii Academiae Scientiarum Imperialis Petropolitanae V:175–192. (Translated to English by L. Sommer, 1954, as “Exposition of a New Theory on the Measurement of Risk.” Econometrica 22, 23–36)
Bewley T (1977) The permanent income hypothesis: a theoretical formulation. J Econ Theory 16:252–292
Caballé J, Pomansky A (1996) Mixed risk aversion. J Econ Theory 71:485–513
Chiu WH (2005) Skewness preference, risk aversion, and the precedence relations on stochastic changes. Manag Sci 51:1816–828
Courbage C, Rey B (2010) On non-monetary measures in the face of risks and the signs of the derivatives. Bull Econ Res 62:295–304
Deck C, Schlesinger H (2010) Exploring higher-order risk effects. Rev Econ Stud 77:1403–1420
Ebert S, Wiesen D (2011) Testing for prudence and skewness seeking. Manag Sci 57:1334–1349
Ebert S, Wiesen D (2012) Joint measurement of risk aversion, prudence and temperance: a case for prospect theory. Working Paper, University of Bonn
Eeckhoudt L (2012) Beyond risk aversion: why, how and what’s next? Geneva Risk and Insurance Review 37:141–155
Eeckhoudt L, Etner J, Schroyen F (2009) The values of relative risk aversion and prudence: a context-free interpretation. Math Soc Sci 58:1–7
Eeckhoudt L, Gollier C, Schlesinger H (1995) The risk averse (and prudent) newsboy. Manag Sci 41:786–974
Eeckhoudt L, Gollier C, Schlesinger H (1996) Changes in background risk and risk-taking behavior. Econometrica 64:683–690
Eeckhoudt L, Gollier C, Schneider T (1995) Risk-aversion, prudence and temperance: a unified approach. Econ Lett 48:331–336
Eeckhoudt L, Rey B, Schlesinger H (2007) A good sign for multivariate risk taking. Manag Sci 53:117–124
Eeckhoudt L, Schlesinger H (2006) Putting risk in its proper place. Am Econ Rev 96:280–289
Eeckhoudt L, Schlesinger H (2008) Changes in risk and the demand for saving. J Monet Econ 55:1329–336
Eeckhoudt L, Schlesinger H (2009) On the utility premium of Friedman and Savage. Econ Lett 105:46–48
Eeckhoudt L, Schlesinger H, Tsetlin I (2009) Apportioning of risks via stochastic dominance. J Econ Theory 144:994–1003
Ekern S (1980) Increasing Nth degree risk. Econ Lett 6:329–333
Epstein L, Tanny S (1980) Increasing generalized correlation: a definition and some economic consequences. Can J Econ 12:16–34
Fei W, Schlesinger H (2008) Precautionary insurance demand with state-dependent background risk. J Risk Insuran 75:1–16
Friedman M, Savage L (1948) The utility analysis of choices involving risk. J Polit Econ 56:279–304
Gollier C (2001) The economics of risk and time. MIT, Cambridge
Gollier C (2010) Ecological discounting. J Econ Theory 145:812–829
Gollier C, Pratt J (1996) Risk vulnerability and the tempering effect of background risk. Econometrica 64:1109–1124
Hanson DL, Menezes CF (1971) On a neglected aspect of the theory of risk aversion. West Econ J 9:211–217
Keynes JM (1930) A treatise on money. AMS, New York
Kimball MS (1990) Precautionary savings in the small and in the large. Econometrica 58:53–73
Kimball MS (1992) Precautionary motives for holding assets. In: Newman P, Milgate M, Falwell J (eds) The new palgrave dictionary of money and finance. MacMillan, London
Kimball MS (1993) Standard risk aversion. Econometrica 61:589–611
Lajeri-Charerli F (2004) Proper prudence, standard prudence and precautionary vulnerability. Econ Lett 82:29–34
Leland HE (1968) Saving and uncertainty: the precautionary demand for saving. Quart J Econ 82:465–473
Maier J, Rüger M (2011) Higher order risk preferences: an experimental investigation. Working Paper, University of Hamburg
Menezes CF, Geiss C, Tressler J (1980) Increasing downside risk. Am Econ Rev 70:921–932
Menezes CF, Wang XH (2005) Increasing outer risk. J Math Econ 41:875–866
Noussair C, Trautmann S, van de Kuilen G (2013) Higher order risk attitudes, demographics and financial decisions. Review of Economics Studies, forthcoming
Pratt J (1964) Risk aversion in the small and in the large. Econometrica 32:122–136
Pratt J, Zeckhauser R (1987) Proper risk aversion. Econometrica 55:143–154
Rey B, Rochet J-C (2004) Health and wealth: how do they affect individual preferences? Geneva Papers Risk Insurance Theory 29:43–54
Richard SF (1975) Multivariate risk aversion, utility independence and separable utility functions. Manag Sci 22:12–21
Rothschild M, Stiglitz JE (1970) Increasing risk: I. A definition. J Econ Theory 2:225–243
Sandmo A (1970) The effect of uncertainty on saving decisions. Rev Econ Stud 37:353–360
Tarazona-Gomez M (2004) Are individuals prudent? an experimental approach using lotteries. Working Paper, University of Toulouse (http://www2.toulouse.inra.fr/lerna/english/cahiers2005/05.13.177.pdf)
Tsetlin I, Winkler R (2009) Multiattribute utility satisfying a preference for combining good with bad. Manag Sci 55:1942–1952
Wang J, Li J (2010) Multiplicative risk apportionment. Math Soc Sci 60:79–81
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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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