Risk-free investments and their comparison with simple risky strategies in pension insurance model: solving singular problems for integro-differential equations

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.