Recursion Formulas for Insurance and Annuity Values

Insurance Recursions

Whole life insurance satisfies A_x = v*q_x + v*p_x*A_{x+1}. This decomposes the value into the cost of one year of coverage plus the value of deferred coverage. For term insurance: A_{x:n}^1 = v*q_x + v*p_x*A_{x+1:n-1}^1 with boundary A_{x:0}^1 = 0. For endowment insurance: A_{x:n} = v*q_x + v*p_x*A_{x+1:n-1} with boundary A_{x+n:0} = 1 (the pure endowment payoff). These recursions enable backward computation from the end of the mortality table.

Annuity and Reserve Recursions

The annuity-due recursion is a-ddot_x = 1 + v*p_x*a-ddot_{x+1}, reflecting the current payment plus the discounted, survival-weighted continuation. For reserves, the recursion is: (t_V + P) * (1+i) = q_{x+t} * b_{t+1} + p_{x+t} * (t+1)_V. This states that the reserve plus premium, accumulated with interest, funds the expected death benefit and the next year's reserve. Exam LTAM uses recursions extensively for building tables of values and for understanding how reserves develop over time. Know these recursions and practice applying them.