Study On The Theory Of Life Annuities Under Random Rates Of Interest

Usually the traditional actuarial theory is based on a fixed interest rate with a purpose to simplify calculations. However, since the life annuity is a long-term economic action, the factors of government policy and economic cycles may cause interest rate to be uncertain during the period of insurance. So the study on the theory of life annuities under random rates of interest has gradually become one of the heated and major problems of actuarial science.Currently, the stochastic interest rate model is divided into two, continuous and discrete. Some statistics properties of the present value of life annuities under both models are studied in this thesis. The main works obtained here can be summarized as follows:(1) Life annuities under continuous interest rate model are discussed. Firstly, this thesis discusses temporary life annuities-immediate policies under the assumption that the force of interest is modeled by Wiener process or Ornstein-Uhlenbeck process, proves that the limit of average cost of the policies tends in distribution to the present value of temporary life annuities-immediate where payments are one when the number of the policies approaches infinity, and meanwhile obtains the approximate formula of distribution function of the present value. Secondly, under the force of interest accumulation function modeled by Wiener process, through applying the fundamental results on the integral of Geometric Brownian motion, all orders moment of present value of continuous life annuities are calculated and the concise expressions of all orders moment of present value are given under certain mortality distribution.(2) Life annuities under discrete interest rate model are discussed. In order to make the interest rate model more realistic, the existing AR(p) model of force of interest and MA(q) model of force of interest are improved using time series theory. Conditional steady ARMA(p,q) model and generalized conditional ARMA(p,q) model are provided for the force of interest of every year δ_i(i = 1,2,…) and the actuarial present value of discrete lifeannuities are derived under both models of force of interest. Finally, a case analysis is presented based on the above models and actuarial present value.