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QUANTIFYING THE TRADE-OFF BETWEEN INCOME STABILITY AND THE NUMBER OF MEMBERS IN A POOLED ANNUITY FUND

Published online by Cambridge University Press:  22 October 2020

Thomas Bernhardt Open the ORCID record for Thomas Bernhardt [Opens in a new window]  and
Catherine Donnelly
Thomas Bernhardt
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan, MI48109-1043, USA, E-Mail: bernt@umich.edu
Catherine Donnelly*
Affiliation:
Risk Insight Lab, Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh, EH14 4AS, UK, E-Mail: C.Donnelly@hw.ac.uk
*
E-Mail: C.Donnelly@hw.ac.uk
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Abstract

The number of people who receive a stable income for life from a closed pooled annuity fund is studied. Income stability is defined as keeping the income within a specified tolerance of the initial income in a fixed proportion of future scenarios. The focus is on quantifying the effect of the number of members, which drives the level of idiosyncratic longevity risk in the fund, on the income stability. To do this, investment returns are held constant, and systematic longevity risk is omitted. An analytical expression that closely approximates the number of fund members who receive a stable income is derived and is seen to be independent of the mortality model. An application of the result is to calculate the length of time for which the pooled annuity fund can provide the desired level of income stability.

Keywords

Longevity creditdecumulationtontineunsystematic riskpoolingG11G22
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 51 , Issue 1 , January 2021 , pp. 101 - 130
Copyright
© 2020 by Astin Bulletin. All rights reserved

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References

Bengen, W. (2001) Conserving client portfolios during retirement, Part IV. Journal of Financial Planning, 5, 110–119.Google Scholar
Billingsley, P. (1999) Convergence of Probability Measures, 3rd edn. Wiley Series in Probability and Statistics, New York, USA: Wiley.CrossRefGoogle Scholar
Birnbaum, Z. and Lientz, B. (1969) Exact distributions for some Rényi-type statistics. Applicationes Mathematicae, 10, 179–192.CrossRefGoogle Scholar
Bruhn, K. and Steffensen, M. (2013) Optimal smooth consumption and annuity design. Journal of Banking and Finance, 37, 2693–2701.CrossRefGoogle Scholar
Chen, A., Hieber, P. and Klein, J.K. (2019) Tonuity: A novel individual-oriented retirement plan. ASTIN Bulletin, 49(1), 5–30.CrossRefGoogle Scholar
Chen, A. and Rach, M. (2019) Options on tontines: An innovative way of combining tontines and annuities. Insurance: Mathematics and Economics, 89, 182–192.Google Scholar
Constantinides, G. (1990) Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy, 98, 519–543.CrossRefGoogle Scholar
Continuous Mortality Investigation (2008) S1PFL. Technical report, Institute and Faculty of Actuaries, UK. Data downloaded at https:/www.actuaries.org.uk/learn-and-develop/continuous-mortality-investigation/cmi-mortality-and-morbidity-tables/s1-series-tables on 28 Sept 2019.Google Scholar
Csörgö, M. (1965) Exact probability distribution functions of some Rényi type statistics. Proceedings of the American Mathematical Society, 16(6), 1158–1167.CrossRefGoogle Scholar
Curatola, G. (2017) Optimal portfolio choice with loss aversion over consumption. The Quarterly Review of Economics and Finance, 66, 345–358.Google Scholar
Devroye, L. (1986) Non-Uniform Random Variate Generation. New York: Springer-Verlag.CrossRefGoogle Scholar
Donnelly, C. (2015) Actuarial fairness and solidarity in pooled annuity funds. ASTIN Bulletin, 45(1), 49–74.CrossRefGoogle Scholar
Donnelly, C., Guillén, M. and Nielsen, J. (2014) Bringing cost transparency to the life annuity market. Insurance: Mathematics and Economics, 56, 14–27.Google Scholar
Donnelly, C. and Young, J. (2017) Product options for enhanced retirement income. British Actuarial Journal, 22(3), 636–656.CrossRefGoogle Scholar
Donsker, M. (1952) Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems. The Annals of Mathematical Statistics, 23(2), 277–281.CrossRefGoogle Scholar
Forman, J. and Sabin, M. (2015) Tontine pensions. University of Pennsylvania Law Review, 163(3), 755–831.Google Scholar
Guyton, J. (2004) Decision rules and portfolio management for retirees: Is the ‘safe’ initial withdrawal rate too safe? Journal of Financial Planning, 17(10), 54–62.Google Scholar
Guyton, J. and Klinger, W. (2006) Decision rules and maximum initial withdrawal rates. Journal of Financial Planning, 19(3), 50–58.Google Scholar
He, L. and Liang, Z. (2013) Optimal dynamic asset allocation strategy for ELA scheme of DC pension plan during the distribution phase. Insurance: Mathematics and Economics, 52(2), 404–410.Google Scholar
Human Mortality Database (2018) United Kingdom, Life tables (period 1x1), Total (both sexes). Technical report, University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Data downloaded at https://www.mortality.org on the 28 Sep 2019.Google Scholar
Karatzas, I. and Shreve, S. (1988) Brownian Motion and Stochastic Calculus. New York, USA: Springer-Verlag.CrossRefGoogle Scholar
Kerman, J. (2011) A closed-form approximation for the median of the Beta distribution. Available at arXiv. https://arxiv.org/pdf/1111.0433.Google Scholar
Milevsky, M. and Salisbury, T. (2015) Optimal retirement income tontines. Insurance: Mathematics and Economics, 64, 91–105.Google Scholar
Milevsky, M.A. and Salisbury, T.S. (2016) Equitable retirement income tontines: Mixing cohorts without discriminating. ASTIN Bulletin, 46(3), 571–604.CrossRefGoogle Scholar
Munk, C. (2008) Portfolio and consumption choice with stochastic investment opportunities and habit formation in preference. Journal of Economic Dynamics and Control, 32(11), 3560–3589.CrossRefGoogle Scholar
Pfau, W. and Kitces, M. (2014) Reducing retirement risk with a rising equity glide path. Journal of Financial Planning, 27, 38–45.Google Scholar
Piggott, P., Valdez, E. and Detzel, B. (2005) The simple analytics of a pooled annuity fund. The Journal of Risk and Insurance, 72(3), 497–520.CrossRefGoogle Scholar
Qiao, C. and Sherris, M. (2013) Managing systematic mortality risk with group self-pooling and annuitization schemes. Journal of Risk and Insurance, 80(4), 949–974.CrossRefGoogle Scholar
Sabin, M. (2010) Fair tontine annuity. Available at SSRN. http://dx.doi.org/10.2139/ssrn.1579932.CrossRefGoogle Scholar
Shorack, G. and Wellner, J. (2009) Empirical Processes with Applications to Statistics. Philadelphia, USA: SIAM.CrossRefGoogle Scholar
Stamos, M.Z. (2008) Optimal consumption and portfolio choice for pooled annuity funds. Insurance: Mathematics and Economics, 43, 56–68.Google Scholar
Valdez, E.A., Piggott, J. and Wang, L. (2006) Demand and adverse selection in a pooled annuity fund. Insurance: Mathematics and Economics, 39(2), 251–266.Google Scholar
Van Bilsen, S., Laeven, R. and Nijman, T. Consumption and portfolio choice under loss aversion and endogenous updating of the reference level. Management Science, 66(9), 3927–3955.CrossRefGoogle Scholar
Vrbik, J. (2018) Small-sample corrections to Kolmogorov-Smirnov test statistic. Pioneer Journal of Theoretical and Applied Statistics, 15(1–2), 15–23.Google Scholar
Work and Pensions Committee (2018) Pension freedoms. Technical report, House of Commons (United Kingdom). Ninth Report of Session 2017-19, HC917.Google Scholar