Actuarial Mathematics (Life Contingencies)
Actuarial Mathematics (Life Contingencies)
This domain tests survival models, life-table functions, insurance and annuity present values, premiums, and reserves. The exam expects consistent timing conventions and use of the equivalence principle.
Core rules and concepts
- Survival: S(x)=P(T>x); force of mortality μ(x)=−S′(x)/S(x); t p x=S(x+t)/S(x).
- Curtate future lifetime K is the integer part of T; complete expectation e̊_x and curtate e_x differ by roughly 0.5.
- Whole life insurance at end of year: A_x=Σv^(k+1) k p x q_{x+k}.
- Whole life annuity-due: ä_x=(1−A_x)/d, where d=i/(1+i).
- Net premium by the equivalence principle: P=A/ä for a whole life policy paid annually.
- Reserve: prospective V=APV(future benefits)−APV(future premiums); retrospective and prospective reserves agree under the same basis.
Worked example
For a 1,000 whole life policy on (x) with A_x=0.30 and ä_x=12, the annual net premium is 1,000×0.30/12=25.00. If the policy is still in force after one year and the then-reserve is 18.00, the insurer holds 18.00 per policy as the present value of future net obligations.
Traps that cost marks
- Using q_x where 1−p_x or a survival factor is required.
- Discounting a death benefit one year too few or too many.
- Mixing annuity-immediate and annuity-due formulas.
- Ignoring select mortality in the first years after underwriting.
- Computing a reserve without specifying the premium basis; reserves are basis-dependent.