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

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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. Friday, May 01, 20152http://www.macminn.org/

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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 Friday, May 01, 2015http://www.macminn.org/3

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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 Friday, May 01, 2015http://www.macminn.org/4

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Portfolio Model Consumption now and then are Given a utility function u. Expected utility is Friday, May 01, 2015http://www.macminn.org/5

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First Order Conditions The conditions for an optimal portfolio are: Friday, May 01, 2015http://www.macminn.org/6

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The Life Insurance Puzzle The classic economic paradigm yields the result that the individual purchases no insurance since Friday, May 01, 2015http://www.macminn.org/7

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The Annuity Puzzle The annuity puzzle can be demonstrated by noting the expected marginal utility in the directiondirection Friday, May 01, 2015http://www.macminn.org/8

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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 c i 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 Friday, May 01, 2015http://www.macminn.org/9

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Portfolio Theory again Consumption now and then for the individual and significant other Friday, May 01, 2015http://www.macminn.org/10

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Portfolio Theory again The expected utility in the new paradigm is The first order conditions are Friday, May 01, 2015http://www.macminn.org/11

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Portfolio Theory again First order conditions continued Friday, May 01, 2015http://www.macminn.org/12

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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. Friday, May 01, 2015http://www.macminn.org/13

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An Annuity Puzzle? Consider the same movement from investing in bonds to investing in annuities that generated the puzzlepuzzle Friday, May 01, 2015http://www.macminn.org/14

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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 Friday, May 01, 2015http://www.macminn.org/15

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Extensions of the New Paradigm Financial Distress ◦ Annuity provider ◦ Insurer Health Risks Friday, May 01, 2015http://www.macminn.org/16

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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. Friday, May 01, 2015http://www.macminn.org/17

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Health Risk Consider an uninsurable health risk that necessitates a medical expenditure now. The wealth now becomes where 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. Friday, May 01, 2015http://www.macminn.org/18

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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.puzzle Financial distress may weaken the annuity demand. A health risk may eliminate the annuity demand. Friday, May 01, 2015http://www.macminn.org/19

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