Abstract
This paper exploits the representation of the conditional mean risk sharing allocations in terms of size-biased transforms to derive effective approximations within insurance pools of limited size. Precisely, the probability density functions involved in this representation are expanded with respect to the Gamma density and its associated Laguerre orthonormal polynomials, or with respect to the Normal density and its associated Hermite polynomials when the size of the pool gets larger. Depending on the thickness of the tails of the loss distributions, the latter may be replaced with their Esscher transform (or exponential tilting) of negative order. The numerical method then consists in truncating the series expansions to a limited number of terms. This results in an approximation in terms of the first moments of the individual loss distributions. Compound Panjer-Katz sums are considered as an application. The proposed method is compared with the well-established Panjer recursive algorithm. It appears to provide the analyst with reliable approximations that can be used to tune system parameters, before performing exact calculations.
Similar content being viewed by others
References
Asmussen S, Goffard PO, Laub P (2018) Orthonormal polynomial expansions and lognormal sum densities. In: Risk and Stochastics: Ragnar Norberg at 70. Mathematical Finance Economics. World Scientific
Denuit M (2019) Size-biased transform and conditional mean risk sharing, with application to P2P insurance and tontines. ASTIN Bull 49:591–617
Denuit M (2020a) Size-biased risk measures of compound sums. N Am Actuar J 24:512–532
Denuit M (2020b) Investing in your own and peers’ risks: The simple analytics of P2P insurance. Eur Actuar J 10:335–359
Denuit M, Dhaene J (2012) Convex order and comonotonic conditional mean risk sharing. Insur Math Econ 51:265–270
Denuit M, Dhaene J, Goovaerts MJ, Kaas R (2005) Actuarial theory for dependent risks: measures orders and models. Wiley, New York
Denuit M, Robert CY (2020a) Large-loss behavior of conditional mean risk sharing. ASTIN Bull 50:1093–1122
Denuit M, Robert CY (2020b) Conditional tail expectation decomposition and conditional mean risk sharing for dependent and conditionally independent risks. Available at https://dial.uclouvain.be
Denuit M, Robert CY (2021a) From risk sharing to pure premium for a large number of heterogeneous losses. Insur Math Econ 96:116–126
Denuit M, Robert CY (2021b) Collaborative insurance with stop-loss protection and team partitioning. N Am Actuar J, in press
Denuit M, Robert CY (2021c) Risk sharing under the dominant peer-to-peer property and casualty insurance business models. Risk Manag Insur Rev, in press
Denuit M, Robert CY (2021d) Stop-loss protection for a large P2P insurance pool. Insur Math Econ 100:210–233
Gil A, Segura J, Temme NM (2017) Efficient computation of Laguerre polynomials. Comput Phys Commun 210:124–131
Gil A, Segura J, Temme NM (2020) Asymptotic computation of classical orthogonal polynomials. arXiv:2004.05038
Goffard PO, Laub P (2020) Orthogonal polynomial expansions to evaluate stop-loss premiums. J Comput Appl Math 370:112648
Goffard PO, Loisel S, Pommeret D (2016) A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model. J Comput Appl Math 296:499–511
Jin T, Provost S, Ren J (2014) Moment-based density approximations for aggregate losses. Scand Actuar J 2014:216–245
Kaas R, Goovaerts MJ, Dhaene J, Denuit M (2008) Modern actuarial risk theory using R. Springer, New York
Kendall M, Stuart A, Ord JK (1987) Kendall’s advanced theory of statistics, vol 1. Oxford University Press, Inc, Griffin London
Morris CN (1982) Natural exponential families with quadratic variance functions. Ann Stat 10:65–80
Munkhammar J, Mattsson L, Ryden J (2017) Polynomial probability distribution estimation using the method of moments. PLOS ONE 12:1–14
Nadarajah S, Chu J, Jiang X (2016) On moment based density approximations for aggregate losses. J Comput Appl Math 298:152–166
Provost S (2005) Moment-based density approximants. Math J 9:727–756
Sundt B (2003) Some recursions for moments of compound distributions. Insur Math Econ 33:487–496
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Denuit, M., Robert, C.Y. Polynomial Series Expansions and Moment Approximations for Conditional Mean Risk Sharing of Insurance Losses. Methodol Comput Appl Probab 24, 693–711 (2022). https://doi.org/10.1007/s11009-021-09881-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-021-09881-7
Keywords
- Conditional expectation
- Size-biased transform
- Esscher transform
- Exponential tilting
- Laguerre polynomials
- Hermite polynomials