Gary Parker

Stochastic pension fund modelling

Abstract This paper considers the stochastic behaviour of the funding level of a defined benefit pension plan through time and its relationship with the plan contribution rate. First, we investigate the effect of the valuation basis and of the amortization period on the variability of funding levels and contribution rates and this introduces the concept of the efficient frontier as a means of choosing an optimal funding strategy. Second, we consider models with dependent rates of return and provide a sufficient condition for the funding level to be ergodic. Upon considering the AR(1) model we derive a recursive method for calculating the conditional distribution of the funding level and provide further insight into the main factors which influence the behaviour of the funding level.

Two Stochastic Approaches for Discounting Actuarial Functions

Two approaches used to model interest randomness are presented. They are the modeling of the force of interest accumulation function and the modeling of the force of interest. The expected value, standard deviation and coefficient of skewness of the present value of annuities-immediate are presented as illustrations. The implicit behavior of the force of interest under the two approaches is investigated by looking at a particular conditional expectation of the force of interest accumulation function.

Limiting Distribution of the Present Value of a Portfolio

An approximation of the distribution of the present value of the benefits of a portfolio of temporary insurance contracts is suggested for the case where the size of the portfolio tends to infinity. The model used Is the one presented in PARKER (1922b) and involves random interest rates and future hfenmes Some justifications of the approximation are given. Illustrations for hmttmg portfolios of temporary insurance contracts are presented for an assumed Ornstem-Uhlenbeck process for the force of interest

Moments of the present value of a portfolio of policies

Abstract A model for the present value of insurance benefits where the interest rates and future lifetimes are random is presented. Recursive calculation methods involved in finding the first three moments of the present value of benefits for a portfolio of identical policies are suggested. Illustrations of these moments when the force of interest is modeled by an Ornstein-Uhlenbeck process are presented.

A portfolio of endowment policies and its limiting distribution

Two methods for approximating the limiting distribution of the present value of the benefits of a portfolio of identical endowment insurance contracts are suggested. The model used assumes that both future lifetimes and interest rates are random. The first method is similar to the one presented in Parker (1994b). The second method is based on the relationship between temporary and endowment insurance contracts.

Stochastic analysis of a portfolio of endowment insurance policies

Abstract Parker (1994) presents a model for the present value of insurance benefits where the interest rates and future lifetimes are random. This paper presents a generalization of this model which can be used for portfolios of identical endowment insurance contracts. Illustrations of these moments when the force of interest is modeled by an Ornstein-Uhlenbeck process are presented. When of interest, some comparisons with the corresponding moments for a portfolio of temporary insurance contracts are made.

A second order stochastic differential equation for the force of interest

Abstract In this paper, we model the force of interest by a linear second order stochastic differential equation. We use this model in the discounting process and apply it to immediate annuities certain for which we illustrate the first three moments. We obtain explicit results for the expected value and autocovariance function of the force of interest and of the force of interest accumulation function. The three cases for the roots of the characteristic equation, namely, real and distinct, real and equal, and complex conjugate roots are treated.

Stochastic interest rates with actuarial applications

This paper presents recursive double integral equations to obtain the distribution of the discounted value or accumulated value of deterministic cash flows. The double integrals have to be evaluated numerically at each iteration. Those distributions are useful when studying the investment risk of portfolios of insurance contracts. The methods suggested take advantage of the Markovian property of the Gaussian process used to model the future rates of return. We start with the first cash flow and successively add the other cash flows while keeping track of the latest information about the rate of return in order to update the distribution at each step. Various means and covariances of bivariate normal distributions which are required if one wants to apply the results in practice are given. In the paper, the Ornstein–Uhlenbeck process is chosen to model the rate of return but the results could be extended to a second order differential equation. Copyright