Stability of pension systems when gains/losses are amortized and rates of return are autoregressive

Methods of funding pension funds which amortize inter-valuation gains or losses over a fixed number of years have been considered by Dufresne (1988, 1989) and Haberman (1990, 1991, 1994). We consider the effects of such a method in the case where the rate of return on investments behaves as a first-order autoregressive process. This means that the actuarial loss process has the structure of a non-linear Time Series, where an autoregressive component is multiplied by the autoregressive rate of return. We obtain a recursive formula for the expected actuarial loss in a given year and for the expectation of the square of this quantity and prove that, for suitable values of the parameters, these expectations converge in time to limits which, though not explicitly obtainable, can nevertheless be found by a simple numerical procedure in any given case.