Risk-Free Investments and Their Comparison with Simple Risky Strategies in Pension Insurance Model: Solving Singular Pro
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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,**, and B. V. Slavkoc,*** a
b
Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow, 117418 Russia Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, Moscow, 119333 Russia c University of Sydney, Sydney, Australia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] Received December 26, 2019; revised February 25, 2020; accepted June 9, 2020
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 Cramér–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. Keywords: pension insurance, dual risk model, survival probability, investments, risk-free assets, exponential premium size distribution, integro-differential equation, singular problem DOI: 10.1134/S096554252010005X
INTRODUCTION We continue the study begun in [1] concerning the influence exerted by investments on the survival probability of an insurance company in the dual risk model. This model is dual to the classical Cramér– Lundberg (CL) collective risk model in insurance and can be treated as a collective pension insurance model. The dual risk model, which is also known as the life annuity insurance model (see [2]), is obtained from the CL one by reversing the signs of the components of the stochastic process describing the dynamics of the surplus
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