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This article is cited in 3 scientific papers (total in 3 papers)
Ordinary differential equations
Risk-free investments and their comparison with simple risky strategies in pension insurance model: solving singular problems for integro-differential equations
T. A. Belkinaa, N. B. Konyukhovab, B. V. Slavkoc a Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow, 117418 Russia
b Federal Research Center "Computer Science and Control", Russian Academy of Sciences, Moscow, 119333 Russia
c University of Sydney, Sydney, Australia
Abstract:
A collective pension insurance (life annuity) model is investigated in the case of risk-free investments, i.e., when the whole surplus of an insurance company at each time is invested in risk-free asset (bank account). This strategy is compared with previously studied simple risky investment strategies, according to which, irrespective of the surplus of an insurance company, a constant positive fraction of this surplus at each time consists of risky assets (stocks), while the remaining fraction is invested in a bank account. The strategies are compared in terms of a traditional solvency criterion, namely, the survival probability. The original insurance model is dual to the classical Cram–Lundberg model: the variation in the surplus over the portfolio of same-type contracts is described by the sum of a decreasing deterministic linear function corresponding to total pension payments and a compound Poisson process with positive jumps corresponding to the income gained by the company at the moments of transferring policyholders' property. In the case of an exponential jump size distribution and risk-free investments, it is shown that the survival probability regarded as a function of the initial surplus defined on the nonnegative real half-line is a solution of a singular problem for an integro-differential equation with a non-Volterra integral operator. The solution of the stated problem is obtained, its properties are analytically examined, and numerical examples are given. Examples are used to compare the influence exerted by risky and risk-free investments on the survival probability in the given model.
Key words:
pension insurance, dual risk model, survival probability, investments, risk-free assets, exponential premium size distribution, integro-differential equation, singular problem.
Received: 26.12.2019 Revised: 25.02.2020 Accepted: 09.06.2020
Citation:
T. A. Belkina, N. B. Konyukhova, B. V. Slavko, “Risk-free investments and their comparison with simple risky strategies in pension insurance model: solving singular problems for integro-differential equations”, Zh. Vychisl. Mat. Mat. Fiz., 60:10 (2020), 1676–1696; Comput. Math. Math. Phys., 60:10 (2020), 1621–1641
Linking options:
https://www.mathnet.ru/eng/zvmmf11143 https://www.mathnet.ru/eng/zvmmf/v60/i10/p1676
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