Benefit Reserves: Prospective and Retrospective Methods
Compute benefit reserves using prospective and retrospective approaches for Exam LTAM.
Prospective Reserve
The prospective reserve at time t is the expected present value of future benefits minus the expected present value of future premiums, given survival to time t. For a whole life insurance: t_V_x = A_{x+t} minus P_x * a-ddot_{x+t}. This can be rewritten as t_V_x = 1 minus (a-ddot_{x+t} / a-ddot_x), a useful formula that avoids computing A values directly. The reserve starts at zero (by the equivalence principle), increases over time as the insured ages, and approaches the face amount near the end of the mortality table.
Retrospective Reserve
The retrospective reserve at time t equals the accumulated value of past premiums minus the accumulated cost of past insurance, all per surviving policyholder. The formula is t_V_x = (P_x * s-ddot_{x:t} minus k_x * A_{x:t}^1 * (1+i)^t / t_p_x). Under standard assumptions, prospective and retrospective reserves are equal. The retrospective approach is useful when future benefits change or when verifying calculations. Exam LTAM tests both methods and their equivalence under consistent assumptions.