×

Optimal retirement income tontines. (English) Zbl 1348.91176

Summary: Tontines were once a popular type of mortality-linked investment pool. They promised enormous rewards to the last survivors at the expense of those died early. While this design appealed to the gambling instinct, it is a suboptimal way to generate retirement income. Indeed, actuarially-fair life annuities making constant payments-where the insurance company is exposed to longevity risk-induce greater lifetime utility. However, tontines do not have to be structured the historical way, i.e. with a constant cash flow shared amongst a shrinking group of survivors. Moreover, insurance companies do not sell actuarially-fair life annuities, in part due to aggregate longevity risk. We derive the tontine structure that maximizes lifetime utility. Technically speaking we solve the Euler-Lagrange equation and examine its sensitivity to (i) the size of the tontine pool \(n\), and (ii) individual longevity risk aversion \(\gamma\). We examine how the optimal tontine varies with \(\gamma\) and \(n\), and prove some qualitative theorems about the optimal payout. Interestingly, Lorenzo de Tonti’s original structure is optimal in the limit as longevity risk aversion \(\gamma \to \infty\). We define the natural tontine as the function for which the payout declines in exact proportion to the survival probabilities, which we show is near-optimal for all \(\gamma\) and \(n\). We conclude by comparing the utility of optimal tontines to the utility of loaded life annuities under reasonable demographic and economic conditions and find that the life annuity’s advantage over the optimal tontine is minimal. In sum, this paper’s contribution is to (i) rekindle a discussion about a retirement income product that has been long neglected, and (ii) leverage economic theory as well as tools from mathematical finance to design the next generation of tontine annuities.

MSC:

91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Alter, G., How to bet on lives: A guide to life contingent contracts in early modern Europe, Res. Econ. Hist., 10, 1-53, (1986)
[2] Barberis, N. C., Thirty years of prospect theory in economics: A review and assessment, J. Econ. Perspect., 27, 1, 173-196, (2013)
[3] Bellhouse, D. R., De moivre: setting the stage for classical probability and its applications, (2011), CRC Press New York · Zbl 1235.01020
[4] Bernard, C.; He, X.; Yan, J.-A.; Zhou, X. Y., Optimal insurance design under rank-dependent utility, Math. Finance, 25, 1, 154-186, (2015) · Zbl 1314.91134
[5] Cannon, E.; Tonks, I., Annuity markets, (2008), Oxford University Press UK
[6] Chancellor, E., Life long and prosper, The Spectator, (2001), 24 March
[7] Chung, J.; Tett, G., Death and the salesmen: As people live longer, pension funds struggle to keep up, which is where a new, highly profitable market will come in — one that bets on matters of life and death, The Financial Times, 26, (2007)
[8] Ciecka, J. E., The first mathematically correct life annuity valuation formula, J. Legal Econ., 15, 1, 59-63, (2008)
[9] Ciecka, J. E., Edmond halley’s life table and its uses, J. Legal Econ., 15, 1, 65-74, (2008)
[10] Cooper, R., An historical analysis of the tontine principle with emphasis on tontine and semi-tontine life insurance policies, (1972), S.S. Huebner Foundation for Insurance Education University of Pennsylvania
[11] Dai, M.; Kwok, Y. K.; Zong, J., Guaranteed minimum withdrawal benefit in variable annuities, Math. Finance, 18, 3, 595-611, (2008) · Zbl 1214.91052
[12] Dickson, P. G.M., The financial revolution in england A study in the development of public credit, 1688-1756, (1967), Macmillan
[13] Donnelly, C., Actuarial fairness and solidarity in pooled annuity funds, ASTIN Bull., 45, 49-74, (2015) · Zbl 1390.91177
[14] Donnelly, C.; Guillen, M.; Nielsen, J. P., Exchanging mortality for a cost, Insurance Math. Econom., 52, 1, 65-76, (2013) · Zbl 1291.91103
[15] Donnelly, C.; Guillén, M.; Nielsen, J. P., Bringing cost transparency to the life annuity market, Insurance Math. Econom., 56, 1, 14-27, (2014) · Zbl 1304.91101
[16] Elsgolc, L. D., Calculus of variations, (2007), Dover Publications Mineola, New York
[17] Finlaison, J., Report of John finlaison, actuary of the national debt, on the evidence and elementary facts on which the tables of life annuities are founded, (1829), House of Commons London, UK
[18] Gelfand, I. M.; Fomin, S. V., Calculus of variations, (2000), Dover Publications Mineola, New York, translated by R.A. Silverman · Zbl 0964.49001
[19] Goldsticker, R., A mutual fund to yield annuity-like benefits, Financ. Anal. J., 63, 1, 63-67, (2007)
[20] Hald, A., (History of Probability and Statistics and Their Applications Before 1750, Wiley Series in Probability and Statistics, (2003), John Wiley & Sons New Jersey)
[21] Homer, S.; Sylla, R., A history of interest rates, (2005), John Wiley and Sons New Jersey
[22] Jennings, R. M.; Swanson, D. F.; Trout, A. P., Alexander hamilton’s tontine proposal, William Mary Q., 45, 1, 107-115, (1988)
[23] Jennings, R. M.; Trout, A. P., The tontine: from the reign of Louis XIV to the French revolutionary era, 91, (1982), S.S. Huebner Foundation for Insurance Education University of Pennsylvania
[24] Kopf, E. W., The early history of the annuity, Proc. Casualty Actuar. Soc., 13, 28, 225-266, (1927)
[25] Lewin, C. G., Pensions and insurance before 1800: A social history, (2003), Tuckwell Press East Lothian, Scotland
[26] McKeever, K., A short history of tontines, Fordham J. Corp. Financ. Law, 15, 2, 491-521, (2009)
[27] Milevsky, M. A., Portfolio choice and longevity risk in the late 17th century: a re-examination of the first English tontine, Financ. Hist. Rev., 21, 3, 225-258, (2014)
[28] Milevsky, M. A., King william’s tontine: why the retirement annuity of the future should resemble its past, (2015), Cambridge University Press New York City
[29] Milevsky, M.A., Salisbury, T.S., 2015. On the choice between tontines and annuities under stochastic and asymmetric mortality, Manuscript (in preparation).
[30] Moody’s Investor Services, 2013. European Insurers: Solvency II — volatility of regulatory ratios could have broad implications for European insurers, retrieved May 2013 from www.moodys.com.
[31] O’Donnell, T., History of life insurance in its formative years; compiled from approved sources, (1936), American Conservation Co. Chicago
[32] Pechter, K., Possible tontine revival raises worries, Annuity Market News, (2007), 1 May, SourceMedia
[33] Piggott, J.; Valdez, E. A.; Detzel, B., The simple analytics of a pooled annuity fund, J. Risk Insur., 72, 3, 497-520, (2005)
[34] Pitacco, E.; Denuit, M.; Haberman, S.; Olivieri, A., Modeling longevity dynamics for pensions and annuity business, (2009), Oxford University Press UK · Zbl 1163.91005
[35] Poitras, G., The early history of financial economics: 1478-1776, (2000), Edward Elgar Cheltenham UK
[36] Poterba, J. M., Annuities in early modern Europe, (Goetzmann, W. N.; Rouwenhorst, K. G., The Origins of Value: The Financial Innovations that Created Modern Capital Markets, (2005), Oxford University Press New York)
[37] Promislow, S. D., Fundamentals of actuarial mathematics, (2011), John Wiley & Sons United Kingdom · Zbl 1205.62159
[38] Qiao, C.; Sherris, M., Managing systematic mortality risk with group self-pooling and annuitization schemes, J. Risk Insur., 80, 4, 949-974, (2013)
[39] Ransom, R. L.; Sutch, R., Tontine insurance and the Armstrong investigation: A case of stifled innovation, 1868-1905, J. Econ. Hist., 47, 2, 379-390, (1987)
[40] Richter, A.; Weber, F., Mortality-indexed annuities managing longevity risk via product design, N. Am. Actuar. J., 15, 2, 212-236, (2011)
[41] Rotemberg, J.J., 2009. Can a Continuously-liquidating Tontine (or Mutual Inheritance Fund) Succeed where Immediate Annuities have Floundered?, Harvard Business School: Working Paper.
[42] Rothschild, C., Adverse selection in annuity markets: evidence from the british life annuity act of 1808, J. Publ. Econ., 93, 5-6, 776-784, (2009)
[43] Sabin, M.J., 2010. Fair tontine annuity, SSRN abstract #1579932.
[44] Stamos, M. Z., Optimal consumption and portfolio choice for pooled annuity funds, Insurance Math. Econom., 43, 1, 56-68, (2008) · Zbl 1140.91411
[45] Steele, J. M., (Probability Theory and Combinatorial Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 69, (1997), SIAM Philadelphia, Pennsylvania)
[46] Tonti, L., (Haberman, S.; Sibbett, T. A., Edict of the King for the Creation of the Society of the Royal Tontine, History of Actuarial Science, vol. V, (1995), William Pickering London), Translated by V. Gasseau-Dryer
[47] Valdez, E. A.; Piggott, J.; Wang, L., Demand and adverse selection in a pooled annuity fund, Insurance Math. Econom., 39, 2, 251-266, (2006) · Zbl 1098.91078
[48] Weir, D. R., Tontines, public finance, and revolution in France and england, J. Econ. Hist., 49, 1, 95-124, (1989)
[49] Yaari, M. E., Uncertain lifetime, life insurance and the theory of the consumer, Rev. Econ. Stud., 32, 2, 137-150, (1965)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.