Premium and Policy Value Calculations
This domain tests the equivalence principle, net and gross premiums, and the prospective and retrospective calculation of policy values.
Premium principles
- The net premium ignores expenses and uses the equivalence principle: EPV premiums = EPV benefits. For a whole-life assurance with unit sum assured, P_x = A_x / a-due_x.
- A gross premium also covers initial, renewal and claim expenses, with each expense discounted at the time it is incurred. The gross premium reserve allows for future expenses and future gross premiums.
- The prospective policy value at duration t is EPV future benefits plus future expenses minus EPV future premiums. The retrospective value accumulates past premiums minus past benefits and expenses. At the same basis they are equal.
- The recursive relationship links successive reserves: (V_t + G - e_t)(1+i) = q_{x+t}(S - V_{t+1}) + V_{t+1}, for a unit death benefit S payable at the end of the year.
Worked example
For a whole-life assurance of 100,000 with A_x = 0.30 and a-due_x = 15, the annual net premium is 100,000 x 0.30/15 = 2,000. At duration 0 the prospective reserve is 100,000 A_x - 2,000 a-due_x = 30,000 - 30,000 = 0, as required by the equivalence principle. If mortality is heavier than assumed, the insurer makes a mortality loss because the reserve released is less than the death benefit paid.
Traps that cost marks
Do not put expenses at the wrong time: initial expenses usually occur at the start, renewal expenses at each premium date, and claim expenses at death. Do not use an annuity-immediate when premiums are annual in advance. A negative reserve can be valid early in a contract. Gross premiums are not found by adding expenses to the net premium without discounting.