Stability of pension systems when gains/losses are amortized and rates of return are autoregressive. (English) Zbl 0914.62089
Summary: Methods of funding pension funds which amortize inter-valuation gains or losses over a fixed number of years have been considered by D. Dufresne [see ibid. 8, No. 1, 71-76 (1989; Zbl 0666.62105)] and S. Haberman [see ibid. 14, No. 3, 219-240 (1994; Zbl 0808.62097)]. 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 nonlinear 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.
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.
MSC:
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
Keywords:
funding with autoregressive rates of return; autoregressive multiplicative noise; nonlinear time seriesReferences:
| [1] | Dufresne, D., Moments of pension contributions and fund levels when rates of return are random, Journal of the Institute of Actuaries, 115, 535-544 (1988) |
| [2] | Dufresne, D., Stability of pension systems when rates of return are random, Insurance: Mathematics and Economics, 8, 71-76 (1989) · Zbl 0666.62105 |
| [3] | Haberman, S., Stochastic approach to pension funding methods, (Proceedings of First AFIR Colloquium, Paris, Volume 4 (1990)), 93-112 |
| [4] | Haberman, S., Pension funding methods and autoregressive interest rate models, (Proceedings of Second AFIR Colloquium, Brighton, Volume 2 (1991)), 181-204 |
| [5] | Haberman, S., Autoregressive rates of return and the variability of pension contributions and fund levels for a defined benefit pension scheme, Insurance: Mathematics and Economics, 14, 219-240 (1994) · Zbl 0808.62097 |
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