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Too risk averse to purchase insurance?

A theoretical glance at the annuity puzzle

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This paper suggests a new explanation for the low level of annuitization, which is valid even if one assumes perfect markets. We show that, as soon there is a positive bequest motive, sufficiently risk averse individuals should not purchase annuities. A model calibration accounting for lifetime risk aversion generates a significantly smaller willingness-to-pay for annuities than the one generated by a standard time-additive model. Moreover, the calibration predicts that riskless savings finance one third of consumption, in line with empirical findings.

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  1. Roughly one half of income stems from public pensions, 17% from company sponsored pension payments and one third is financed from savings.

  2. Some studies, such as Benartzi et al. (2011), argue, however, that the observed levels of annuitization are not as puzzling as usually claimed. They provide evidence to show that most individuals actually choose to annuitize their wealth when possible and point out that most retirement plans do not offer this possibility.

  3. See Brown (2007), as well as the following section for a literature review.

  4. Davidoff et al. (2005) consider non-additive separable preferences featuring habit formation and show that it helps explain the annuity puzzle. However, they do not investigate the role of risk aversion.

  5. This was also noticed by Drèze and Rustichini (2004), who provided an example where insurance demand may decrease with risk aversion (see their Proposition 9.1).

  6. See corollary 1 in Davidoff et al. (2005).

  7. Framing effects describe the fact that individuals’ choices may depend on the formulation of alternatives and especially if they are focused on gains or losses.

  8. Gazzale and Walker (2011) reach a similar conclusion using neutral-context laboratory experiments.

  9. See e.g. Ponzetto (2003), Inkmann et al. (2011), or Horneff et al. (2010).

  10. Precautionary savings can be defined as the optimal savings due to the uncertainty of the second-period income.

  11. As discussed in Kihlstrom and Mirman (1974, Section 2 and in particular 2.1, p. 365 and following) and in Epstein and Zin (1989, Section 4, p. 950), studying the role of risk aversion requires leaving ordinal preferences unchanged. Individuals who would rank deterministic outcomes differently cannot be compared in terms of risk aversion.

  12. Lifetime risk aversion is sometimes also called multivariate risk aversion (see for example Richard 1975), or correlation aversion (see for example Epstein and Tanny (1980), Bommier (2007) or Dorfmeister and Krapp (2007) since it refers to risk aversion with many commodities.

  13. The issues of time inconsistency and history independence do not arise in the two-period framework that is considered in the current section. However they would do so in the N-period extension considered in Section 3.

  14. The formal proof can be found in the electronic supplementary material of the on-line version or in Bommier and LeGrand (2013), the working paper of this article posted on SSRN.

  15. We do not consider endogenous retirement decisions. This aspect is formalized in Chai et al. (2011).

  16. The utility function U(c,s) may also be written as


    where the multiplicative structure is explicit.

  17. All computational codes (in Matlab) are available upon request to authors.

  18. See for example the report of Livingston and Cohn (2010) on American motherhood.

  19. With such a risk aversion with respect to life duration, an agent would be indifferent between living 80 years for sure, or living 78 years or 82.34 years with equal probability.

  20. The relative risk aversion with respect to wealth at the age of 65 is \(-W_{0}\frac {\partial ^{2}EU_{65}}{\partial {W_{0}^{2}}}/\frac {\partial EU_{65}}{\partial W_{0}}\), where E U 65 is the expected lifetime utility at the age of 65.

  21. Keeping unchanged the other parameters of our benchmark calibration (Table 1), we find that people never purchase annuities when λ is larger than 0.0133.

  22. The interpretation of empirical evidence on age specific wealth profiles should, however, be subject to caution. Indeed, saving decumulation can also be obtained under the assumption of risk neutrality with respect to lifetime felicity if annuities are not fairly priced.

  23. We re-estimate the values of u 0 and λ (or β in the additive model) so as to match the different values of VSL and of rate of time discounting.

  24. More precisely, Lockwood (2012b) converted the three other model estimations (in addition to his own one) into a common functional form and we in turn adapt his parameters to our functional form in Eq. 28.


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We are grateful to Edmund Cannon, Alexis Direr, Glenn Harrison, Lee Lockwood, Thomas Post, James Poterba, Ray Rees, Harris Schlesinger (the editor), an anonymous referee and seminar participants at ETH Zurich, University of Paris I, University of Zurich, 2011 Summer Meetings of the Econometric Society and CEAR/MRIC Behavioral Insurance Workshop 2012 (LMU, Münich) for their comments.

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Correspondence to Antoine Bommier.

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Both authors gratefully acknowledge financial support from Swiss-Re.

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1.1 A Proof of Proposition 1

When ϕ is linear, ϕ (U A )=ϕ (U D )=ϕ (0)=1. First order conditions in Eqs. (8)–(10) and the budget constraint can now be expressed as follows:

$$\begin{array}{@{}rcl@{}} u_{2}^{\prime}(c_{2}) & \leq&\frac{u_{1}^{\prime}(c_{1})}{1+R}\qquad(=\text{ if }a>0), \end{array} $$
$$\begin{array}{@{}rcl@{}} v^{\prime}((1+R)s) & \leq&\frac{1}{1-p}\left(\frac{u_{1}^{\prime}(c_{1})}{1+R}-u_{2}^{\prime}(c_{2})\right)+u_{2}^{\prime}(c_{2})\qquad(=\text{ if }s>0), \end{array} $$
$$\begin{array}{@{}rcl@{}} v^{\prime}(\tau) & \leq& u_{2}^{\prime}(c_{2})\qquad(=\text{ if }\tau>0), \end{array} $$
$$\begin{array}{@{}rcl@{}} c_{2}+\tau & =&(1+R)s+a\frac{1+R}{p}. \end{array} $$

Let us show that in any case (1+R)s=τ.

  1. 1.

    s=0. The budget constraint in Eq. (37) implies that a>0: Eq. (34) is therefore an equality. Equation 35 implies then that \(v^{\prime }(0)\leq u_{2}^{\prime }(c_{2})\). Suppose that τ>0: we deduce from Eq. 36 that v (0)≤v (τ), which contradicts that v is concave and non-linear. Thus, (1+R)s=τ=0.

  2. 2.

    s>0. From Eq. 35, which is an equality, together with Eq. 34 and 36, we deduce that \(v^{\prime }((1+R)s)\ge u_{2}^{\prime }(c_{2})\ge v^{\prime }(\tau )\) and τ≥(1+R)s>0. The budget constraint (37) implies that a>0. Equation 34, as Eq. 35 and 36 are therefore equalities: we deduce that v ((1+R)s)=v (τ) and (1+R)s=τ.

We always obtain (1+R)s=τ, and thus also \(a=\frac {pc_{2}}{1+R}.\)

1.2 B Calibrations for alternative bequest specifications

We provide here calibrations for measuring the impact of the alternative bequest specifications. All calibrations generate a value of average bequest of 20% of the non-annuitized wealth W 0 and a rate of time discounting of 5.00%. The value of a statistical life is 500 consumptions at 65.

In all cases, the following parameters are fixed:

Table 4

Calibrations lie in Table 4.

Table 4 Calibrations for alternative bequest motives

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Bommier, A., Grand, F.L. Too risk averse to purchase insurance?. J Risk Uncertain 48, 135–166 (2014).

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