Stochastic analysis of life insurance surplus

The behaviour of insurance surplus over time for a portfolio of homogeneous life policies in an environment of stochastic mortality and rates of return is examined. We distinguish between stochastic and accounting surpluses and derive their first two moments. A recursive formula is proposed for calculating the distribution function of the accounting surplus. We then examine the probability that the surplus becomes negative in any given insurance year. Numerical examples illustrate the results for portfolios of temporary and endowment life policies assuming an AR(1) process for the rates of return.