On the accumulated aggregate surplus of a life portfolio

The present paper considers a simple stochastic model for a multi-period (non-profit) life portfolio with two sources of uncertainty, the returns on investments and the claims. Within this model the bonus of a portfolio is determined numerically using the concept of stable reserve associated to a financial gain. The accumulated aggregate surplus of the portfolio after a finite number of insurance periods is expressed as the difference between the accumulated value on investments and the accumulated actuarial reserves and aggregate claims. It is shown that replacing the dependence between the aggregate claims of successive periods by an assumption of independence increases the riskiness in stop-loss order of the accumulated aggregate surplus. This results in higher stable reserves under the independence assumption. It is shown that the stable reserve can be determined numerically by solving an implicit expected value equation, which involves Black–Scholes formula for the stop-loss premiums on the accumulated return on investments and De Pril’s two-stage recursive formulas for the distribution of the accumulated aggregate claims. The obtained results are illustrated numerically at a portfolio of endowment insurance policies.