Life Contingencies and Actuarial Mathematics
This domain examines life-table mechanics and the valuation of life insurance and annuities: commutation-style relationships, fractional-age assumptions, reserves, and premium structures, using the mortality basis approved for Iranian insurers' reserve calculations.
Core rules and concepts
- From l_x: d_x = l_x − l_{x+1}, p_x = l_{x+1}/l_x, q_x = d_x/l_x, and the curtate expectation e_x = Σ_{k≥1} kp_x.
- The whole-life identity A_x = 1 − d·ä_x with d = i/(1+i) connects insurance and annuity values; it holds for whole-life, not for term or endowment products.
- Fractional ages: under UDD, tq_x = t·q_x and the expected payment at death is approximated at the mid-point of the year of death; under constant force the results differ.
- Reserves are prospective (EPV future benefits minus EPV future net premiums) or retrospective; both agree for net premium reserves. Recursive reserves roll forward with interest and mortality.
- Gross premiums add expense loadings and a profit margin to net premiums; reserves are normally computed on a net or modified basis.
Worked example
With q_x = 0.004 and q_{x+1} = 0.0045 at i = 5%, the EPV of 1 payable at the end of the year of death within 2 years is A¹ = 0.004/1.05 + (0.996)(0.0045)/1.05² = 0.003810 + 0.004065 = 0.007875. For a 1,000 sum assured the single premium is about 7.87.
Traps
- Discounting from the year end when the benefit is payable at the moment of death.
- Applying A_x = 1 − d·ä_x to term or endowment products.
- Mixing UDD and constant-force assumptions within one calculation.
- Confusing net premium reserves with gross premium (Zillmer-type) reserves.
- Sign errors in recursive reserves and ignoring negative early reserves.