Vadiveloo et al (1982) introduced a criterion for choosing, among the various net risk premium payment plans, the so-called optimum one. They considered a life insurance situation where benefits were payable at the end of the year of death. Of course, premiums cease after death! This optimum criterion was the minimization of the discounted net profit variance. Their model can be mathematically described as follows. Consider an insured life aged exactly x having a death benefit with present value, if death occurs at time t, denoted by B(t). If the present value of the total premiums paid up to time t is R(t) and the random variable T denotes the time of the death of x, then the profit at death has present value Z(T) where Z(T)=R(T)−B(T).
The optimum net premium payment plan, R*(t), is the one that minimizes Var[Z(T)] subject to E[Z(T)] = 0.