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.

中文翻译：

---

养老保险模型中的无风险投资及其与简单风险策略的比较：求解积分微分方程的奇异问题

摘要

在无风险投资的情况下，即当每次将保险公司的全部盈余投资于无风险资产（银行账户）时，将研究集体退休金保险（人寿年金）模型。将该策略与先前研究的简单风险进行比较投资策略，根据该策略，不论保险公司的盈余如何，每次盈余的恒定正数都是风险资产（股票），其余部分则投资在银行帐户中。根据传统偿付能力标准（即生存概率）对策略进行比较。原始保险模型是经典Cramér–Lundberg模型的对偶模型：同类型合同的投资组合上的盈余变化是通过与总养老金支出相对应的递减确定性线性函数与具有正数的复合Poisson过程的总和来描述的相应于公司在转让保单持有人财产时获得的收入而跳跃。在指数跳跃大小分布的情况下，无风险投资表明，将生存概率作为非负实半线上定义的初始盈余的函数，是具有非Volterra积分算子的积分微分方程奇异问题的解决方案。得到了所述问题的解决方案，对其特性进行了分析检验，并给出了数值示例。实例用于比较给定模型中有风险和无风险投资对生存概率的影响。