The dynamics of pension funding
In the context of North American and British actuarial practice, a mathematical model is used to study the evolution over time of the fund levels (F) and contributions (C). First, actuarial cost methods are examined in the traditional static framework. Three points are studied (1) comparison of the various methods, (2) inclusion of new entrants in the valuation basis, and (3) the rate at which F(t) reaches its ultimate level. Next, the model is modified to include varying rates of return and of inflation. Two methods of adjusting the normal cost are considered; (1) the adjustment is equal to the unfunded liability divided by the present value of an annuity for a term of "m" years (Spread method); (2) each intervaluation loss is liquidated by a fixed number of payments over the following years (Amortization of Losses method). The core of the thesis has to do with random rates of return. In discrete time, these rates are supposed independent and identically distributed. Recursive equations are derived for the first and second moments of F(t) and C(t), under methods (1) and (2). In the case of the Spread method, an optimal region is specified for "m" it is shown that for m>m* the variances of both F and C are increasing functions of m. The optimal region is thus 1 < m*. The Spread method assuming rates of return to be a white noise process . A proof is given of the convergence of the discrete processes Fn (representing the fund when "n" valuations are performed every year) to a diffusion process F, as n -+ -. Using the Ito calculus of diffusion processes, the first two moment s of F( t) and C(t) are then shown to satisfy some particular differential equations. The final chapter applies similar ideas to the calculation of the moments of annuities-certain, when rates of return are a white noise process.