On Numerical Problems in Computing Life Annuities Based on the Makeham–Beard Law
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Fredy Castellares
ABSTRACT
Analytic expressions for the single and joint life annuities based on the Makeham–Beard mortality law have been derived recently in the literature, which depend on special mathematical functions such as hypergeometric functions. We verify that the arguments of the hypergeometric functions in the analytic expressions for the single and joint life annuities may assume values very close to unity (boundary of the convergence radius), and so numerical problems may arise when using them in practice. We provide, therefore, alternative analytic expressions for the single and joint life annuities where the arguments of the hypergeometric functions in the new analytic expressions do not assume values close to one.
Funding statement: Artur Lemonte acknowledges the financial support of the Brazilian agency CNPq (grant 303554/2022-3).
Funding statement: Fredy Castellares gratefully acknowledges the financial support from FAPEMIG (Belo Horizonte/MG, Brazil).
Acknowledgement
The authors would like to thank an anonymous reviewer for the insightful comments and suggestions.
REFERENCES
[1] Appell, P.: Sur les sries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérives partielles, Comptes rendus hebdomadaires des séances de l’Académie des sciences 90 (1880), 296–298.Search in Google Scholar
[2] Beard, R. E.: Note on some mathematical mortality models. In: The Lifespan of Animals (G.E.W. Wolstenholme & M. O’Connor, eds.), Little, Brown, Boston, 1959, pp. 302–311.10.1002/9780470715253.app1Search in Google Scholar
[3] Bowie, D.: Analytic expressions for annuities based on Makeham–Beard mortality laws, Ann. Actuar. Sci. 15 (2021), 1–13.10.1017/S1748499520000032Search in Google Scholar
[4] Gompertz, B.: On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Philosophical Transactions of the Royal Society of London 115 (1825), 513–585.10.1098/rstl.1825.0026Search in Google Scholar
[5] Lebedev, N.: Special Functions and Their Applications, Prentice-Hall, INC, 1965.10.1063/1.3047047Search in Google Scholar
[6] Lehmann, E. L.—Casella, G.: Theory of Point Estimation, 2nd ed. Springer-Verlag, New York, USA, 1998.Search in Google Scholar
[7] Makeham, W. M.: On the law of mortality and the construction of annuity tables, The Assurance Magazine and J. Inst. Actuar. 8 (1860), 301–310.10.1017/S204616580000126XSearch in Google Scholar
[8] Perks, W.: On some experiments in the graduation of mortality statistics, J. Inst. Actuar. 63 (1932), 12–57.10.1017/S0020268100046680Search in Google Scholar
[9] Picard, E.: Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques, Annales Scientifiques de l’École Normale Superieure 10 (1881), 305–322.10.24033/asens.203Search in Google Scholar
[10] Ponnusamy, S.—Vuorinen, M.: Asymptotic expansions and inequalities for hypergeometric function, Mathematika 44 (1997), 278–301.10.1112/S0025579300012602Search in Google Scholar
[11] R CORE TEAM: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2021.Search in Google Scholar
[12] Rainville, E. D.: Special Functions, The Macmillan Company, New York, USA, 1960.Search in Google Scholar
[13] Richards, S.: Applying survival models to pensioner mortality data, Br. Actuar. J. 14 (2008), 257–326.10.1017/S1357321700001720Search in Google Scholar
[14] Richards, S.: The cascade model of mortality, Longevitas Information Matrix, https://www.longevitas.co.uk/site/informationmatrix/thecascademodelofmortality.html.Search in Google Scholar
[15] Schlosser, M. J.: Multiple hypergeometric series – Appell Series and beyond, https://arxiv.org/abs/1305.1966.Search in Google Scholar
© 2023 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Remembering Professor Štefan Znám, 9.2.1936–17.7.1993
- Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures
- Enlargements of Quantales
- Multiplicative Dependent Pairs in the Sequence of Padovan Numbers
- Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions
- On the Rational Parametric Solution of Diagonal Quartic Varieties
- Theory of Certain Non-Univalent Analytic Functions
- Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)
- A Conjecture on H3(1) for Certain Starlike Functions
- Coefficient Problems of Quasi-Convex Mappings of Type B on the Unit Ball in Complex Banach Spaces
- Complete Monotonicity and Inequalities Involving the k-Gamma and k-Polygamma Functions
- Study of Oscillation Criteria of Odd-Order Differential Equations with Mixed Neutral Terms
- On a System of Difference Equations Defined by the Product of Separable Homogeneous Functions
- On the Existence of Bi-Lipschitz Equivalent Metrics in Semimetric Spaces
- The Lehmann Type II Teissier Distribution
- Asymptotic Predictive Inference of Negative Lower Tail Index Distributions
- On Numerical Problems in Computing Life Annuities Based on the Makeham–Beard Law
- The Rational Zero-Divisor Cup-Length of Oriented Partial Flag Manifolds
