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Theory and Decision

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05.12.2019

On temperance and risk spreading

verfasst von: Christophe Courbage, Béatrice Rey

Erschienen in: Theory and Decision | Ausgabe 4/2020

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Abstract

This paper shows that temperance is the highest order risk preference condition for which spreading N independent and unfair risks provides the highest level of welfare than any other possible allocations of risks. These results are also interpreted through the concept of N-superadditivity of the utility premium. This paper provides a novel application of temperance, not in terms of two risks as it is common, but in terms of N risks.

Vorheriger Artikel On the instability of majority decision-making: testing the implications of the ‘chaos theorems’ in a laboratory experiment

Nächster Artikel Supergrading: how diverse standards can improve collective performance in ranking tasks

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Ebert et al. (2017) and Courbage et al. (2018), respectively, reinterpret Eeckhoudt and Schlesinger (2006)’s results by showing that temperance ensures mutual aggravation, respectively, mutual mitigation, of risk changes.

Obviously, the lottery \((0,X_{1}+X_{2},X_{3})\) is equivalent in terms of welfare to the lottery \((X_{1}+X_{2},0,X_{3})\) that is equivalent to the lottery \((X_{1}+X_{2},X_{3},0)\) since lotteries are three state equiprobable lotteries.

Risks spreading refers, throughout the paper, to spreading the N risks each over N states.

Note that there exists cases where each risk \(X_{i}\) is such that \( E(X_{i})\le 0\) and where it is impossible to compare random variables \(X_{i} \) with zero via a statistical rule. See Denuit et al. (1998) and Denuit et al. (2001) for more details.

Note that under simple and usual regularity conditions (u defined over \( \mathbb {R}^{+}\), non-satiation and bounded marginal utility tending to plus infinity), item (b) of Proposition 1 holds when the DM is only temperate without requiring risk aversion and prudence (see Menegatti 2014, Propositions 2 and 3). Indeed, under these simple conditions, Menegatti (2014) shows that prudence implies risk aversion and that temperance implies prudence.

In item (b) of Proposition 1, we assume that each \(X_{i}\) is dominated by 0 via second-order stochastic dominance as it is usual in economics since Rothschild and Stiglitz (1970). Note that second-order stochastic dominance implies stochastic dominance of higher orders. Consequently, our results hold in cases where \(X_{i}\preceq _{{\text {SD}}-s}0\)\(\forall s\ge 3\). If, instead of the Rothschild and Stiglitz assumption, we assume that \(X_{i}\preceq _{{\text {SD}}-s}0\) with \(s\ge 3\), it is easy to show that item (b) of Proposition 1 writes then as follows: for all mixed risk averse from 1 to 2s decision-makers (\((-1)^{n+1}u^{(n)}>0\)\(\forall n=1,\ldots ,2s\)) when \( X_{i}\preceq _{{\text {SD}}-s}0\).

In the case where the L background risks are not introduced in each state, they can be then considered as additional risks. The DM faces then \(N+L\) risks. If we assume \(\tilde{\epsilon }_{l}\preceq _{{\text {SD}}_{k}}0\) with \(k=1,2,\) i.e. if we assume that these risks share the same properties as \(X_{i}\) for all i, this case brings us back to the previous remark (i.e. the case with a number of risks greater than the number of states).

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Courbage, C., Loubergé, H., & Rey, B. (2018). On the properties of high-order non-monetary measures for risks. Geneva Risk and Insurance Review, 43, 77–94.CrossRef

Denuit, M., Lefevre, C., & Scarsini, M. (2001). On s-convexity and risk aversion. Theory and Decision, 50(3), 239–248.CrossRef

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Ebert, S. (2013). Even (mixed) risk lovers are prudent: Comment. American Economic Review, 103(4), 1536–1537.CrossRef

Ebert, S., Nocetti, D. C., & Schlesinger, H. (2017). Greater mutual aggravation. Management Science, 64(6), 2473–2972.

Eeckhoudt, L., & Schlesinger, H. (2006). Putting risk in its proper place. American Economic Review, 96, 280–289.CrossRef

Eeckhoudt, L., Schlesinger, H., & Tsetlin, E. (2009). Apportioning risks via stochastic dominance. Journal of Economic Theory, 113, 1–31.

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Kimball, M. S. (1992). Precautionary motives for holding assets. In P. Newman, M. Milgate, & J. Falwell (Eds.), The New Palgrave Dictionary of Money and Finance. London: Macmilan.

Menegatti, M. (2014). New results on the relationship among risk aversion, prudence and temperance. European Journal of Operational Research, 232, 613–617.CrossRef

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Titel: On temperance and risk spreading
verfasst von: Christophe Courbage
Béatrice Rey
Publikationsdatum: 05.12.2019
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 4/2020
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-019-09737-0

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