Policy Combination Of Actuarial Problems

Mortality risk ,expense risk and interest rate risk is three main risks for life insurance company ;mortality risk and expense risk are easy to reduce ,but interest rate risk is difficult to control .Reserve is the most important liability for life insurance company ,eighty-five percent of the liability is prepared for the reserve of life insurance ,health insurance and annuity .Interest rate is critical to the calculation of reserve .The calculation of reserve for a portfolio of joint life annuities is given in this article .The joint life annuity is designed with a last survivor payout rule .This rule specifies a periodic payment that the annuitants will receive provided both of them are still alive .In addition ,it specifies a fraction of this payout ,that will be paid to the survivor after the death of one member of the couple .The effect of stochastic interest rate and stochastic mortality on the reserve of the portfolio is also considered . The stochastic interest rate model exists in many empirical situations,the rate of interest will remain fixed at a certain level (random instaneous rate i)for a random period of time ti until a new random rate i+1 should be considered,that will remain for ti+1 waiting time until the next change in the rate of interest.The formulas for the moments of the present value pf future payment streams for a policy or a portfolio of policies are quite extensive ,especially for discrete time case ,so in the last section ,a matrix model for the first two moments of present value of future payment streams for a portfolio of policies is given ,where the interest rate and future lifetime are random.