Stochastic Actuarial Modelling of a Defined-Benefit Social Security Pension Scheme: An Analytical Approach

Among the systems in place in different countries for the protection of the population against the long-term contingencies of old-age (or retirement), disability and death (or survivorship), defined-benefit social security pension schemes, i.e. social insurance pension schemes, by far predominate, despite the recent trend towards defined-contribution arrangements in social security reforms. Actuarial valuations of these schemes, unlike other branches of insurance, continue to be carried out almost exclusively on traditional, deterministic lines. Stochastic applications in this area, which have been restricted mainly to occasional special studies, have relied on the simulation technique. This paper develops an analytical model for the stochastic actuarial valuation of a social insurance pension scheme. Formulae are developed for the expected values, variances and covariances of and among the benefit expenditure and salary bill projections and their discounted values, allowing for stochastic variation in three key input factors, i.e., mortality, new entrant intake, and interest (net of salary escalation). Each deterministic output of the valuation is thus supplemented with a confidence interval, that is, a range with an attached probability. The treatment covers the premiums under the different possible financial systems for these schemes, which differ from the funding methods of private pensions, as well as the testing of the level of the Fund ratio when the future contributions schedule is pre-determined. Although it is based on a relatively simplified approach and refers only to retirement pensions, with full adjustment in line with salary escalation, the paper brings out the stochastic features of pension scheme projections and illustrates a comprehensive stochastic valuation. It is hoped that the paper will stimulate interest in further research, both of a theoretical and a practical nature, and lead to progressively increasing recourse to stochastic methods in social insurance pension scheme valuations.