Life Insurance Pricing and Reserving

A life insurance benefit (term, whole life, endowment) or annuity (life annuity-due, immediate, deferred) is described and you must compute its APV at age x.

Decompose the benefit into building blocks: A_x = Σ v^{k+1} k|q_x for whole life, A^{1}_{x:n} = Σ_{k=0}^{n-1} v^{k+1} k|q_x for term, and ä_x = Σ_{k=0}^{∞} v^k k_p_x for life annuity-due. Use the identity A_x + d ä_x = 1 to cross-check. For continuous benefits, replace sums with integrals and v^k with e^{-δk}.

A_x + d ä_x = 1; A_{x:n} = A^{1}_{x:n} + A^{1}_{x:\overline{n}|}.

A benefit and benefit period are given and you must compute a level annual premium under the equivalence principle.

Set APV of premiums = APV of benefits. For a whole life with level annual premium P_x, P_x = A_x / ä_x. For a term, P^{1}_{x:n} = A^{1}_{x:n} / ä_{x:n}. For varying premiums or benefits, write APV equations for each pattern and solve. Always identify whether premium is paid annually, m-thly, or continuously and use the matching annuity factor.

Equivalence principle: APV(premium income) = APV(benefit outgo).

A policy is at duration t and you must compute the benefit reserve _tV.

Prospective: _tV = APV of future benefits at age x + t - APV of future premiums at age x + t. Retrospective: accumulate net cash flows from issue to time t under the assumed mortality and interest. For a level-premium whole life, _tV = A_{x+t} - P_x ä_{x+t}, which simplifies to 1 - ä_{x+t} / ä_x.

Prospective: _tV = APV future benefits - APV future premiums. Retrospective: accumulate past cash flows.

A policy has expenses and gross premium loadings, and you must compute the gross premium reserve at duration t.

Net premium reserve uses the net (benefit-only) premium and ignores expenses. Gross premium reserve uses the actual gross premium and includes future expenses on both sides. Gross reserves are typically lower than net reserves at issue (because the gross premium covers expenses), and the difference is the unamortized expense allowance.

Gross reserve = APV future benefits + APV future expenses - APV future gross premiums.