The Annuity Puzzle

1. The Annuity Puzzle Richard MacMinn Illinois State University Presentation at L5 The Fifth International Longevity Risk and Capital Market Solutions Conference New York, New York September 25, 2009

2. Literature • Yaari, M. (1965). "Uncertain Lifetime, Life Insurance, and the Theory of the Consumer." The Review of Economic Studies 32: 137-150. • Davidoff, T., J. R. Brown, et al. (2005). "Annuities and Individual Welfare." American Economic Review 95(5): 1573 - 1590. • Warshawsky, M. (1988). "Private Annuity Markets in the United States: 1919-1984." Journal of Risk and Insurance 55(3): 518-528. • Friedman, B. M. and M. J. Warshawsky (1990). "The cost of annuities: implications for saving behavior and bequests." Quarterly Journal of Economics 105(1): 135-154. • Poterba, J. M. (2001). "Annuity Markets and Retirement Security." Fiscal Studies 22: 249-279. • Inkmann, J., P. Lopes, et al. (2007). How deep is the Annuity Market Participation Puzzle?, Working Paper, presented at CESifo Venice Summer Institute. • Purcal, S. and J. Piggott (2008). "Explaining Low Annuity Demand: An Optimal Portfolio Application to Japan." Journal of Risk and Insurance 75(2): 493-516. • Lockwood, L. (2009). Bequest Motives and the Annuity Puzzle. Chicago, University of Chicago. • Sinclair, S. H. and K. A. Smetters (2004). Health Shocks and the Demand for Annuities. Washington, DC, Congressional Budget Office. • Sheshinski, E. (2008). The Economic Theory of Annuities. Princeton, Princeton University Press. • Cannon, E. and I. Tonks (2008). Annuity Markets. Oxford, Oxford University Press. http://www.macminn.org/

3. The Classic Economic Paradigm • Portfolio Model • Dates now and then • The consumer/investor selects a portfolio now of annuities, bonds and life insurance • The portfolio payoff occurs then • The investor survives or not to obtain the portfolio payoff then • The investor exhibits selfish behavior http://www.macminn.org/

4. Portfolio Model The portfolio is The survival probability is The annuity, bond and life insurance prices are The consumption now and then depend on the portfolio choices http://www.macminn.org/

5. Portfolio Model Consumption now and then are Given a utility function u. Expected utility is http://www.macminn.org/

6. First Order Conditions The conditions for an optimal portfolio are: http://www.macminn.org/

7. The Life Insurance Puzzle The classic economic paradigm yields the result that the individual purchases no insurance since http://www.macminn.org/

8. The Annuity Puzzle The annuity puzzle can be demonstrated by noting the expected marginal utility in the direction http://www.macminn.org/

9. A New Economic Paradigm • Suppose the individual is an altruist, at least with respect to one significant other. • Let the individual have preferences defined on the consumption pair ci and the utility v of the significant other • Utility increases in consumption now and then • Utility also increases in the utility of the significant other http://www.macminn.org/

10. Portfolio Theory again Consumption now and then for the individual and significant other http://www.macminn.org/

11. Portfolio Theory again The expected utility in the new paradigm is The first order conditions are http://www.macminn.org/

12. Portfolio Theory again First order conditions continued http://www.macminn.org/

13. The Life Insurance Puzzle • Note that the first two terms in the FOC for insurance say that the expected marginal utility of consumption now is negative since insuring reduces dollars now for the individual and significant other. • The significant other, as beneficiary, receives dollars then in the death event and so the last term in is positive. • If the individual’s utility is increasing in that of the significant other and the significant other’s expected marginal utility of consumption then becomes unbounded as consumption then goes to zero then some life insurance is demanded. • It is the last term in the FOC that is missing in the classic paradigm or equivalently the purely selfish version of the model. http://www.macminn.org/

14. An Annuity Puzzle? Consider the same movement from investing in bonds to investing in annuities that generated the puzzle http://www.macminn.org/

15. Puzzle • Consider loading on the annuity contract • If there is no loading, i.e., then the derivative in the annuitizing direction is zero. If there is loading then the derivative is negative http://www.macminn.org/

16. Extensions of the New Paradigm • Financial Distress • Annuity provider • Insurer • Health Risks http://www.macminn.org/

17. Financial Distress • Consider the risk of insolvency for the annuity provider. • How does this risk affect the demand for annuities? • If p is the probability of insolvency then p (1 – q) is the probability that the individual survives and has an annuity that does not provide the promised payment. • This changes the individual’s expected utility. • The demand for annuities is weakened even without altruism. http://www.macminn.org/

18. Health Risk • Consider an uninsurable health risk that necessitates a medical expenditure now. • The wealth now becomeswhere L represent the expense now that occurs with probability p. Also suppose that the health risk does not affect the mortality rate. • Suppose that the annuity is illiquid. • Such a health risk eliminates or reduces the demand for annuities. http://www.macminn.org/

19. Concluding remarks • The economic paradigm must be changed so that the demand for life insurance can be rationalized. This analysis does that. • This analysis provides the theoretical foundation for the bequest motive. • There is no annuity puzzle in the new paradigm. • Financial distress may weaken the annuity demand. • A health risk may eliminate the annuity demand. http://www.macminn.org/