Actuarial Present Value of Life Insurance Benefits

APV Concept

The actuarial present value (APV) is the expected present value of a future contingent payment. For a life insurance benefit b_t payable at time t upon death of (x), the APV is E[b_{T_x} * v^{T_x}] where v = 1/(1+i) is the discount factor and T_x is the future lifetime random variable. In the continuous case, the APV equals the integral from 0 to infinity of b_t * v^t * t_p_x * mu_{x+t} dt. In the discrete case, it equals the sum over k of b_{k+1} * v^{k+1} * k_p_x * q_{x+k}.

Key Insurance APV Symbols

Standard actuarial notation uses A with decorations to denote APVs. A_x is the APV of a whole life insurance paying 1 at death. The bar over A indicates continuous payment (at the moment of death). A_{x:n} is an n-year term insurance. A with a subscript x:n with a special symbol denotes endowment insurance. The second moment E[v^{2T}] is denoted with a superscript 2 on A, computed at force of interest 2*delta. The variance of the present value random variable equals the second moment minus the square of the first moment.