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Tradeoffs for Downside Risk-Averse Decision-Makers and the Self-Protection Decision

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Abstract

In addition to risk aversion, decision-makers tend to be also downside risk averse. Besides the usual size for risk trade-off, this allows several other trade-offs to be considered. The decision to increase the level of self-protection generates five trade-offs each involving an unfavourable downside risk increase and an accompanying beneficial change. Five stochastic orders that correspond to these trade-offs are defined, characterised and used to prove comparative static theorems that provide information concerning the self-protection decision. The five stochastic orders are general in nature and can be applied in any decision model where downside risk aversion is assumed.

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Notes

  1. Liu and Meyer (2013, 2015).

  2. Downside risk aversion is the term used by Menezes et al. (1980). Kimball (1990) uses the term prudence for the risk attitude captured by u‴(x)⩾0.

  3. Decreasing absolute risk aversion (DARA) implies downside risk aversion. Thus, evidence for DARA which goes back to Arrow (1971) and Pratt (1964) supports the downside risk aversion assumption. For some early direct evidence of downside risk aversion, see Menezes et al. (1980). More recently, further evidence of downside risk aversion is provided by Deck and Schlesinger (2010, 2014), Ebert and Wiesen (2011) and Maier and Ruger (2011).

  4. More formally, an increase in size is an FSD improvement, and a decrease in risk is a Rothschild and Stiglitz risk decrease.

  5. Diamond and Stiglitz (1974).

  6. Of the five stochastic orders, only one has been previously discussed. See Denuit et al. (2014).

  7. We find it convenient for writing purposes to sometimes describe or discuss random variables and sometimes their CDFs. Thus, both are used.

  8. Hadar and Russell (1969).

  9. Hanoch and Levy (1969).

  10. Rothschild and Stiglitz (R–S) (1970).

  11. Menezes et al. (MGT) (1980).

  12. A definition of “larger” in the increasing convex order is available in Shaked and Shanthikumar (2007). A related concept, the “stop-loss order” is found in the actuarial science literature (Denuit et al. 2005).

  13. Ross (1981).

  14. Liu and Meyer (2013).

  15. Definition 7 is the Ross more downside risk aversion first defined in Modica and Scarsini (2005). Higher degree extensions of Definition 7 can be found in Jindapon and Neilson (2007), Li (2009), and Denuit and Eeckhoudt (2010a). Third degree Ross more risk aversion in Definition 6 is based on Liu and Meyer (2013).

  16. Ehrlich and Becker (1972).

  17. Dionne and Eeckhoudt (1985).

  18. For example, see Briys and Schlesinger (1990), Lee (1998), Jullien et al. (1999), Chiu (2000), Eeckhoudt and Gollier (2005), Liu et al. (2009) and Meyer and Meyer (2011).

  19. Under some parameter values in the self-protection model, an increase in self-protection could be decomposed into a downside risk increase and another change that also decreases expected utility. These are uninteresting situations because all normal decision-makers who are both risk averse and downside risk averse would prefer less self-protection. In other words, there is no meaningful trade-off to speak of in these situations.

  20. Eeckhoudt and Gollier (2005).

  21. It is reasonable to assume that w−L−e1<w−e2 or equivalently that e2−e1<L, since under no circumstance would a rational individual expend effort on self-protection beyond the size of loss, L.

  22. Which of the two changes occurs first does not alter the implied conditions on F(x) and G(x) or any of the analysis presented here. The notation is always a change from F(x) to G(x) and therefore, it is more natural to consider the change from F(x) to H(x) and then change from H(x) to G(x) even though the order does not matter.

  23. If the convex prudent order were defined instead, the exact same condition obtained in Theorem A1 would hold, but then F(x) is larger than G(x) in the convex prudent order. For discussion of downside risk-averse decision-makers, the terminology chosen seems clearest.

  24. Because there is one definition and two theorems in each of the five subsections, they are labelled in a way that indicates their close association with one another.

  25. In the tradition of Eeckhoudt and Schlesinger (2006), Eeckhoudt (2012) uses this two-step approach to construct two simple binary lotteries that can be ranked by the concave imprudent order (pp. 148–150).

  26. Chiu (2005).

  27. Denuit and Eeckhoudt (2010b) extends Chiu (2005) from 3rd degree to general nth degree.

  28. We focus on the part of Chiu’s (2005) Theorem 1 that is of interest to us in this paper.

  29. Kimball (1990) first uses this condition for comparing the strengths of precautionary saving of different individuals.

  30. Note that throughout the paper, and for these definitions in particular, the outcome variable x is assumed to belong to a bounded interval [a, b]. While this assumption is not restrictive from an empirical point of view, it plays an important theoretical role. As pointed out by Menegatti (2014), a non-satiated individual cannot be both risk averse and imprudent on an unbounded interval [a, ∞). With the assumption of a bounded domain imposed in this paper, Menegatti’s cautions do not apply.

  31. When H(x) dominates F(x) in SSD, there exists an intermediate CDF J(x) such that J(x) dominates F(x) in the first degree, and J(x) is riskier than H(x). J(x)=F(x) for x<s and J(x)=1 for x⩾1 serves as such an J(x). The value for s is chosen so that the mean of J(x) and the mean of H(x) are equal.

  32. Liu and Meyer (2015).

  33. Dachraoui et al. (2004).

  34. Meyer (1987).

  35. Chiu (2010).

  36. Ebert (2015).

  37. Note that preferences are completely determined by the marginal utility function.

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Acknowledgements

The authors thank Nicolas Treich and an anonymous reviewer for very helpful comments regarding the content and the organisation of the paper.

Author information

Authors and Affiliations

  1. Institut de Sciences Actuarielles and Institut de Statistique, Université Catholique de Louvain, Louvain-la-Neuve, B-1348, Belgium

    Michel M Denuit

  2. IESEG School of Management, LEM, Lille, France

    Louis Eeckhoudt

  3. CORE, Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Louis Eeckhoudt

  4. Private Enterprise Research Center, Texas A&M University, College Station, 77843, TX, U.S.A.

    Liqun Liu

  5. Department of Economics, Michigan State University, East Lansing, 48824, MI, U.S.A.

    Jack Meyer

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  1. Michel M Denuit
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  2. Louis Eeckhoudt
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Appendix A

Appendix A

For many proofs in this appendix the following identity, which is readily derived using integration by parts, serves as a starting point.

Proof of Theorem A1

Proof: “If”—Suppose that

From (*), we have

Condition (A.1) implies that E G u(x)⩾E F u(x) for all u(x) with u″(x)⩽0, and u‴(x)⩽0.

“Only if”—Suppose that E G u(x)⩾E F u(x) for all u(x) with u″(x)⩽0, and u‴(x)⩽0. We need to show that (A.1) holds. First, letting u(x)=x and u(x) =−x respectively implies that G[2](b)−F[2](b)=μ F −μ G =0.

What remains to be shown is G[3](x)−F[3](x)⩾G[3](b)−F[3](b) ∀x∈[a, b].

Proof by contradiction is used. Assume this condition is not satisfied. That is G[3](y)−F[3](y) < G[3](b)−F[3](b)for some y in [a, b]. Then, due to continuity, there exists an interval [α, β]⊂(a, b) such that G[3](y)−F[3](y)<G[3](b)−F[3](b) for all y in [α, β]. Choose a special u(x) such that u″(a)=0, u‴(x) < 0 for x∈(α, β) and u‴(x)=0 otherwise. Then, from (A.2), E F u(x)−E G u(x)>0, which contradicts that E G u(x)⩾E F u(x) for all u(x) with u″(x)⩽0 and u‴(x)⩽0. So (A.1) must hold.  Q.E.D

Proof of Theorem A2

Proof: Only part (a) is demonstrated. Part (b) can be shown in a similar fashion. Assume that G(x) is larger than F(x) in the concave imprudent order, and that E F v(x)⩾E G v(x). Now consider a u(x) who is (3/2)rd degree Ross more risk averse than v(x). Then, there exists a λ>0and φ(x) such that u=λv+φ, where φ″(x)⩾0 and φ‴(x)⩾0 for all x. Therefore,

The second inequality follows because G(x) is larger than F(x) in the concave imprudent order, and φ″(x)⩾0 and φ‴(x)⩾0.  Q.E.D.

Proof of Theorem B1

Proof: “If”—Suppose that

From Theorems C1 and E1, these conditions imply that G(x) is larger than F(x) both in the increasing concave imprudent order and in the increasing convex imprudent order. To show that G(x) is larger than F(x) in the increasing imprudent order, note that for any u(x) with u′(x)⩾0, and u‴(x)⩽0, one of the following three cases must hold.

  1. i)

    u″(x)⩽0 for all x in [a, b]. Then, E G u(x)⩾E F u(x) because G(x) is larger than F(x) in the increasing concave imprudent order;

  2. ii)

    u″(x)⩾0 for all x in [a, b]. Then, E G u(x)⩾E F u(x) because G(x) is larger than F(x) in the increasing convex imprudent order;

  3. iii)

    There exists a<x*<b, such that u″(x)⩾0 for all x in [a, x*] and u″(x)⩽0 for all x in [x*, b]. Define u 1(x) and u 2(x) as followsFootnote 37:

Then

and

For u1(x), u1′(x)⩾0, u1″(x)⩾0 and u1‴(x)⩽0 for all x in [a, b]. So E G u1(x)⩾E F u1(x) because G(x) is larger than F(x) in the increasing convex imprudent order. Similarly, for u2(x), u2′(x)⩾0, u2″(x)⩽0 and u2‴(x)⩽0 for all x in [a, b]. So E G u2(x)⩾E F u2(x) because G(x) is larger than F(x) in the increasing concave imprudent order. Then, noting that u′(x)=u1′(x)+u2′(x)−u′(x*) for all x in [a, b], that both E G u1(x)⩾E F u1(x) and E G u2(x)⩾E F u2(x) leads to E G u(x)⩾E F u(x).

Summarising, for cases (i), (ii) and (iii), for any u(x) with u′(x)⩾0, and u‴(x)⩽0, we have E G u(x)⩾E F u(x). Therefore, G(x) is larger than F(x) in the increasing imprudent order.

“Only if”—Suppose that G(x) is larger than F(x) in the increasing imprudent order. Then G(x) must be larger than F(x) both in the increasing concave imprudent order and in the increasing convex imprudent order. Then the conditions in Theorems C1 and E1 hold and combined these are (A.3).  Q.E.D.

Proof of Theorem B2

Proof: Again only part (a) is demonstrated because the proof of part (b) is similar. Assume that G(x) is larger than F(x) in the increasing imprudent order and that E F v(x)⩾E G v(x). Now consider a u(x) who is (3/1)rd degree Ross more risk averse than v(x). Then, there exists a λ>0and φ(x) such that u=λv +φ, where φ′(x)⩽0 and φ‴(x)⩾0 for all x. Therefore,

The second inequality follows because G(x) is larger than F(x) in the increasing imprudent order, and φ′(x)⩽0 and φ‴(x)⩾0.  Q.E.D.

Proof of Theorem C1

Proof: “If”—Suppose that

From (*), we have (A.2), which is copied below for convenience.

Condition (A.4) with (A.2) implies that E G u(x)⩾E F u(x) for all u(x) with u′(x)⩾0, u″(x)⩽0, and u‴(x)⩽0. That is, G(x) is larger than F(x) in the increasing concave imprudent order.

“Only if”—Suppose that E G u(x)⩾E F u(x) for all u(x) with u′(x)⩾ 0, u″(x)⩽0 and u‴(x)⩽0. We need to show that (A.4) holds. First, letting u(x)=x implies that G[2](b)−F[2](b)=μ F −μ G ⩽0. So what remains to be shown is G[3](x)−F[3](x)⩾G[3](b)−F[3](b), ∀x∈[a, b].

Proof by contradiction is used. Assume that G[3](y)−F[3](y)<G[3](b)−F[3](b) for some y in [a, b]. Then, due to continuity, there exists an interval [α, β]⊂(a, b) such that G[3](y)−F[3](y)<G[3](b)−F[3](b) for all y in [α, β]. Choose a special u(x) such that u′(b)=0 u″(a)=0, u‴(x)<0 for x∈(α, β) and u‴(x)=0 otherwise. Then, from (A.2), we have E G u(x)<E F u(x), contradicting the assumption that E G u(x)⩾E F u(x) for all u(x) with u′(x)⩾ 0, u″(x)⩽0 and u‴(x)⩽0. Therefore (A.4) holds.  Q.E.D.

Proof of Theorem C2

Proof: Only part (a) is demonstrated. Assume that G(x) is larger than F(x) in the increasing concave imprudent order, and that E F v(x)⩾E G v(x). Now consider a u(x) who is both (3/1)rd degree and (3/2)rd degree Ross more risk averse than v(x). By definition, there exist λ1>0 and λ2>0 such that (u‴(x))/(v‴(x))⩾λ1⩾(u′(y))/(v′(y)) and (u‴(x))/(v‴(x))⩾λ2⩾(u″(y))/(v″(y)) for all x and y.

Let λ=max{λ1, λ2}>0 and define φ(x) by u=λv+φ. It is the case that φ′=u′−λv′⩽0, φ″=u″−λv″⩾0 and φ‴=u‴−λv‴⩾0 for all x in [a, b]. Therefore,

The second inequality holds because G(x) is larger than F(x) in the increasing concave imprudent order, and φ′(x)⩽0 φ″(x)⩾0 and φ‴(x)⩾0.  Q.E.D.

Proof of Theorem D1

Proof: “If”—Suppose that

From (*), we have

Using condition (A.5), it is readily seen from (A.6) that E G u(x)⩾E F u(x) for all u(x) with u′(x)⩽0, u″(x)⩽0, and u‴(x)⩽0.

“Only if”—Suppose that E G u(x)⩾E F u(x) for all u(x) with u′(x)⩽0, u″(x)⩽0, and u‴(x)⩽0. We need to show that (A.5) holds. First, letting u(x)=−x, we have G[2](b)−F[2](b)=μ F −μ G ⩾0. What remains to be shown is

We use proof by contradiction. Assume that G[3](y)−F[3](y)+[G[2](b)−F[2](b)](b−y)<[G[3](b)−F[3](b)]for some y in [a, b]. Then, due to continuity, there exists an interval [α, β]⊂(a, b) such that G[3](y)−F[3](y)+[G[2](b)−F[2](b)](b−y) < [G[3](b)−F[3](b)] for all y in [α, β]. Choose a special u(x) such that u′(a)=0, u″(a)=0, u‴(x)<0 for x∈(α, β) and u‴(x)=0 otherwise. Then, from (A.6), E F u(x)−E G u(x)>0, which contradicts that E G u(x)⩾E F u(x) for all u(x) with u′(x)⩽0, u″(x)⩽0, and u‴(x)⩽0. Therefore (A.5) holds.  Q.E.D.

Proof of Theorem D2

Proof: Only part (a) is demonstrated because the proof of part (b) is similar. Assume that G(x) is larger than F(x) in the decreasing concave imprudent order, and that E F v(x)⩾E G v(x). Now consider a u(x) who is both (2/1)nd degree Ross less risk averse and (3/2)rd degree Ross more risk averse than v(x). By definition, there exist λ1>0 and λ2>0 such that (u″(x))/(v″(x))⩽λ1⩽(u′(y))/(v′(y)) and (u‴(x))/(v‴(x))⩾λ2⩾(u″(y))/(v″(y)) for all x and y.

Let λ=min{λ1, λ2}>0 and define φ(x) by u=λv+φ. This implies that φ′=u′−λv′⩾0, φ″=u″−λv″⩾0 and φ‴=u‴−λv‴⩾0 for all x in [a, b]. Therefore,

The second inequality follows because G(x) is larger than F(x) in the decreasing concave imprudent order, and φ′(x)⩾0φ″(x)⩾0 and φ‴(x)⩾0.  Q.E.D.

Proof of Theorem E1

Proof: “If”—Suppose that

From (*), we have

Condition (A.7) along with (A.8) implies that E G u(x)⩾E F u(x) for all u(x) with u′(x)⩾0, u″(x)⩾0, and u‴(x)⩽0.

“Only if”—Suppose that E G u(x)⩾E F u(x) for all u(x) with u′(x)⩾0, u″(x)⩾0, and u‴(x)⩽0. We need to show that (A.7) holds. Letting u(x)=x yields G[2](b)−F[2](b)=μ F −μ G ⩽0.

What remains to be shown is G[3](x)−F[3](x)⩾[G[2](b)−F[2](b)](x−a), ∀x∈[a, b].

Proof by contradiction is used. Assume that G[3](y)−F[3](y) < [G[2](b)−F[2](b)](y−a) for some y in [a, b]. Then, due to continuity, there exists an interval [α, β]⊂(a, b) such that G[3](y)−F[3](y) < [G[2](b)−F[2](b)](y−a) for all y in [α, β]. Choose a special u(x) such that u′(a)=0, u″(b)=0, u‴(x) < 0 for x∈(α, β) and u‴(x)=0 otherwise. Then, from (A.8), E F u(x)−E G u(x)>0, which contradicts that E G u(x)⩾E F u(x) for all u(x) with u′(x)⩾0, u″(x)⩾0, and u‴(x)⩽0. Hence (A.7) holds.   Q.E.D.

Proof of Theorem E2

Proof: Again only part (a) is demonstrated. Assume that G(x) is larger than F(x) in the increasing convex imprudent order, and that E F v(x)⩾E G v(x). Now consider a u(x) who is both (2/1)nd degree and (3/1)rd degree Ross more risk averse than v(x). By definition, there exist λ1>0 and λ2>0 such that (u″(x))/(v″(x))⩾λ1⩾(u′(y))/(v′(y)) and (u‴(x))/(v‴(x))⩾λ2⩾(u′(y))/(v′(y)) for all x and y.

Let λ=min{λ1, λ2}>0 and define φ(x) by u=λv+φ. This implies that φ′=u′−λv′⩽0, φ″=u″−λv″⩽0 and φ‴=u‴−λv‴⩾0 for all x in [a, b]. Therefore,

The second inequality holds because G(x) is larger than F(x) in the increasing convex imprudent order, and φ′(x)⩽0 φ″(x)⩽0 and φ‴(x)⩾0.  Q.E.D.

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Denuit, M., Eeckhoudt, L., Liu, L. et al. Tradeoffs for Downside Risk-Averse Decision-Makers and the Self-Protection Decision. Geneva Risk Insur Rev 41, 19–47 (2016). https://doi.org/10.1057/grir.2015.3

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  • DOI: https://doi.org/10.1057/grir.2015.3

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