Life Contingencies
Scope
Life contingencies tests survival models, life tables, commutation functions, life insurance and annuity present values, net premiums, and reserves. Korean questions frequently ask you to convert between insurance and annuity values under a stated mortality and interest basis, or to compute a reserve prospectively.
Core rules
- p_t = exp(-∫μ), and t q_x = 1 - p_x.
- Commutation: D_x = v^x l_x, N_x = ΣD, C_x = v^{x+1} d_x, M_x = ΣC.
- Whole life: A_x = 1 - d ä_x, with d = i/(1+i).
- Net annual premium for whole life: P_x = A_x / ä_x when premiums are payable annually in advance.
- Prospective reserve = PV(future benefits) - PV(future net premiums); retrospective reserve = accumulated past premiums - accumulated past costs.
- Fractional-age assumptions matter: UDD, constant force, and Woolhouse approximations give different answers.
Worked example
At i=5%, a whole life annuity-due on (x) has ä_x = 15. Find A_x and the net annual premium per unit of sum insured.
d = 0.05/1.05 = 0.04761905. A_x = 1 - d ä_x = 1 - 0.04761905 × 15 = 1 - 0.71428575 = 0.28571425. The premium is A_x/ä_x = 0.28571425/15 = 0.0190476 per unit, i.e. about KRW 19.05 per KRW 1,000 sum insured. Reasoning: the annuity-due factor prices a payment stream starting immediately; the identity converts it into the equivalent single-premium death benefit; dividing by the annuity factor spreads that single premium over the premium-paying period.
Traps
Do not use a_x instead of ä_x when premiums are payable in advance. Confusing d with i shifts the answer. Selection effects mean the mortality basis at issue differs from the ultimate table; exam wording such as “select” or “ultimate” must be followed. For reserves, check whether the question asks for net or gross premium reserve and whether expenses are included.