Too risk averse to purchase insurance?

A theoretical glance at the annuity puzzle


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

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

  2. 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. 3.

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

  4. 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. 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. 6.

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

  7. 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. 8.

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

  9. 9.

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

  10. 10.

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

  11. 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. 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. 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. 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. 15.

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

  16. 16.

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


    where the multiplicative structure is explicit.

  17. 17.

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

  18. 18.

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

  19. 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. 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. 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. 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. 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. 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.


  1. Agnew, J. R., Anderson, L. R., Gerlach, J. R., Szykman, L. R. (2008). Who chooses annuities? An experimental investigation of the role of gender, framing, and defaults. American Economic Review, 98(2), 418–422.

    Article  Google Scholar 

  2. Ameriks, J., Caplin, A., Laufer, S., Van Nieuwerburgh, S. (2011). The joy of giving or assisted living? Using strategic surveys to separate bequest and precautionary motives. Journal of Finance, 66(2), 519–561.

    Article  Google Scholar 

  3. Andersen, S., Harrison, G., Lau, M., Rutstroem, E. (2011). Multiattribute utility theory, intertemporal utility and correlation aversion. CEAR Working Paper 2011-04, Georgia State University.

  4. Benartzi, S., Previtero, A., Thaler, R. H. (2011). Annuitization puzzles. Journal of Economic Perspectives, 25(4), 143–164.

    Article  Google Scholar 

  5. Bleichrodt, H., & Eeckhoudt, L. (2005). Saving under rank-dependent utility. Economic Theory, 25(2), 505–511.

    Article  Google Scholar 

  6. Bommier, A (2006). Uncertain lifetime and intertemporal choice: risk aversion as a rationale for time discounting. International Economic Review, 47(4), 1223–1246.

    Article  Google Scholar 

  7. Bommier, A. (2007). Risk aversion, intertemporal elasticity of substitution and correlation aversion. Economics Bulletin, 4(29), 1–8.

    Google Scholar 

  8. Bommier, A. (2013). Life cycle preferences revisited. Journal of European Economic Association, 11(6), 1290-1319.

  9. Bommier, A., & LeGrand, F. (2013). Too risk averse to purchase insurance? A theoretical glance at the annuity puzzle. CER-ETH Working Paper 12/157, ETH Zurich.

  10. Bommier, A., Chassagnon, A., LeGrand, F. (2012). Comparative risk aversion: a formal approach with applications to saving behaviors. Journal of Economic Theory, 147(4), 1614–1641.

    Article  Google Scholar 

  11. Brown, J.R. (2007). Rational and behavioral perspectives on the role of annuities in retirement planning. NBER Working Paper 13537, National Bureau of Economic Research.

  12. Brown, J. R., Kling, J. R., Mullainathan, S., Wrobel, M. V. (2008). Why don’t people insure late-life consumption? A framing explanation of the under-annuitization puzzle. American Economic Review, 98(2), 304–309.

    Article  Google Scholar 

  13. Chai, J., Horneff, W., Maurer, R., Mitchell, O. S. (2011). Optimal portfolio choice over the life cycle with flexible work, endogenous retirement, and lifetime payouts. Review of Finance, 15(4), 875–907.

    Article  Google Scholar 

  14. Cook, P.J., & Graham, D. A. (1977). The demand for insurance and protection: the case of irreplaceable commodities. Quarterly Journal of Economics, 91(1), 143–156.

    Article  Google Scholar 

  15. Davidoff, T., Brown, J .R., Diamond, P. A. (2005). Annuities and individual welfare. American Economic Review, 95(5), 1573–1590.

    Article  Google Scholar 

  16. De Nardi, M. (2004). Wealth inequality and intergenerational links. Review of Economic Studies, 71(3), 743–768.

    Article  Google Scholar 

  17. De Nardi, M., French, E., Jones, J. B. (2010). Why do the elderly save? The role of medical expenses. Journal of Political Economy, 118(1), 39–75.

    Article  Google Scholar 

  18. Dorfleitner, G., & Krapp, M. (2007). On multiattributive risk aversion: some clarifying results. Review of Managerial Science, 1(1), 47–63.

    Article  Google Scholar 

  19. Drèze, J., & Rustichini, A. (2004). State-dependent utility and decision theory, 2: extensions. Dordrecht In S. Barbera PH, & C. Seild (Eds.), Handbook of utility theory, (pp. 839–892): Kluwer Academic Publishers.

  20. Drèze, J. H., & Modigliani, F. (1972). Consumption decisions under uncertainty. Journal of Economic Theory, 5(3), 308–335.

    Article  Google Scholar 

  21. Eden, B (2008). Substitution, risk aversion and asset prices: an expected utility approach. Working Paper 0803, Department of Economics, Vanderbilt University.

  22. Epstein, L. G., & Tanny, S. M. (1980). Increasing generalized correlation: a definition and some economic consequences. Canadian Journal of Economics, 13(1), 16–34.

    Article  Google Scholar 

  23. Epstein, L. G., & Zin, S. E. (1989). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica, 57(4), 937–969.

    Article  Google Scholar 

  24. Fernández-Villaverde, J., & Krueger, D. (2007). Consumption over the life cycle: facts from consumer expenditure survey data. The Review of Economics and Statistics, 89(3), 552–565.

    Article  Google Scholar 

  25. Finkelstein, A., & Poterba, J. (2002). Selection effects in the United Kingdom individual annuities market. Economic Journal, 112(476), 28–50.

    Article  Google Scholar 

  26. Finkelstein, A., & Poterba, J. (2004). Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market. Journal of Political Economy, 112(1), 183–208.

    Article  Google Scholar 

  27. Gazzale, R. S., & Walker, L. (2011). I’ll cross that bridge if I get to it: focusing on the near future. Working Paper, University of Toronto.

  28. Horneff, W. J., Maurer, R. H., Mitchell, O. S., Stamos, M. Z. (2010). Variable payout annuities and dynamic portfolio choice in retirement. Journal of Pension Economics and Finance, 9(2), 163–183.

    Article  Google Scholar 

  29. Hu, W. Y., & Scott, J. S. (2007). Behavioral obstacles in the annuity market. Financial Analysts Journal, 63(6), 71–82.

    Article  Google Scholar 

  30. Hurd, M. D., & Smith, J. P. (2002). Expected bequests and their distribution. NBER Working Paper 9142, National Bureau of Economic Research.

  31. Inkmann, J., Lopes, P., Michaelides, A. (2011). How deep is the annuity market participation puzzle. Review of Financial Studies, 24(1), 279–319.

    Article  Google Scholar 

  32. James, E., & Song, X. (2001). Annuities markets around the world: money’s worth and risk intermediation. CeRP Working Paper 16/01, Center for Research on Pensions and Welfare Policies.

  33. Jappelli, T. (1999). The age-wealth profile and the life-cycle hypothesis: a cohort analysis with time series of cross-sections of italian households. Review of Income and Wealth, 45(1), 57–75.

    Article  Google Scholar 

  34. Johansson, P. O. (2002). On the definition and age-dependency of the value of a statistical life. Journal of Risk and Uncertainty, 25(3), 251–263.

    Article  Google Scholar 

  35. Johnson, R. W., Burman, L. E., Kobes, D. I. (2004). Annuitized wealth at older ages. Evidence from the health and retirement study. Final report to the Employee Benefits Security Administration of the U.S. Department of Labor, The Urban Institute.

  36. Kihlstrom, R. E., & Mirman, L. J. (1974). Risk aversion with many commodities. Journal of Economic Theory, 8(3), 361–388.

    Article  Google Scholar 

  37. Kimball, M. S., & Weil, P. (2009). Precautionary saving and consumption smoothing across time and possibilities. Journal of Money Credit and Banking, 41(2–3), 245–284.

    Article  Google Scholar 

  38. Kopczuk, W., & Lupton, J. P. (2007). To leave or not to leave: the distribution of bequest motives. Review of Economic Studies, 74(1), 207–235.

    Article  Google Scholar 

  39. Livingston, G., & Cohn, D. (2010). The new demography of American motherhood. A Social and Demographic Trends Report, Pew Research Center.

  40. Lockwood, L. M. (2012a). Bequest motives and the annuity puzzle. Review of Economic Dynamics, 15(2), 226–243.

    Article  Google Scholar 

  41. Lockwood, L. M. (2012b). Bequest motives and the choice to self-insure late-life risks. Working Paper, Northwestern University.

  42. Milevsky, M. A., & Young, V. (2007). Annuitization and asset allocation. Journal of Economic Dynamics and Control, 31(9), 3138–3177.

    Article  Google Scholar 

  43. Mitchell, O. S., Poterba, J. M., Warshawsky, M. J., Brown, J. R. (1999). New evidence on the money’s worth of individual annuities. American Economic Review, 89(5), 1299–1318.

    Article  Google Scholar 

  44. Pang, G., & Warshawsky, M. J. (2010). Optimizing the equity-bond-annuity portfolio in retirement: the impact of uncertain health expenses. Insurance: Mathematics and Economics, 46(1), 198–209.

    Article  Google Scholar 

  45. Pashchenko, S. (2013). Accounting for non-annuitization. Journal of Public Economics, 98, 53–67.

    Article  Google Scholar 

  46. van der Ploeg, F. (1993). A closed-form solution for a model of precautionary saving. Review of Economic Studies, 60(2), 385–395.

    Article  Google Scholar 

  47. Ponzetto, G. (2003). Risk aversion and the utility of annuities. CeRP Working Paper 31/03, Center for Research on Pensions and Welfare Policies.

  48. Richard, S. F. (1975). Multivariate risk aversion, utility independence and separable utility functions. Management Science, 22(1), 12–21.

    Article  Google Scholar 

  49. Sinclair, S. M., & Smetters, K. A. (2004). Health shocks and the demand for annuities. Technical Paper 2004-9, Congressional Budget Office.

  50. Van den Heuvel, S. J. (2008). Temporal risk aversion and asset prices. Working Paper 2008-37, Federal Reserve Board.

  51. Viscusi, W. K., & Aldy, J. E. (2003). The value of a statistical life: a critical review of market estimates throughout the world. Journal of Risk and Uncertainty, 27(1), 5–76.

    Article  Google Scholar 

  52. Yaari, M. E. (1965). Uncertain lifetime, life insurance, and the theory of the consumer. Review of Economic Studies, 32(2), 137–150.

    Article  Google Scholar 

  53. Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica, 55(1), 95–105.

    Article  Google Scholar 

  54. Yogo, M. (2009). Portfolio choice in retirement: health risk and the demand for annuities, housing and risky assets. NBER Working Paper 15307, National Bureau of Economic Research.

Download references


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.

Author information



Corresponding author

Correspondence to Antoine Bommier.

Additional information

Both authors gratefully acknowledge financial support from Swiss-Re.

Electronic supplementary material

Below is the link to the electronic supplementary material.

(PDF 309 KB)



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 \(a=\frac {pc_{2}}{1+R}.\)

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


  • Annuity puzzle
  • Insurance demand
  • Bequest
  • Intergenerational transfers
  • Risk aversion
  • Multiplicative preferences

JEL Classifications

  • D11
  • D81
  • D91