Net Premium Calculation for Life Insurance Products
Derive net premiums using the equivalence principle for life insurance products on Exam LTAM.
Equivalence Principle
The net premium is the amount that makes the expected present value of premiums equal to the expected present value of benefits at policy inception. For a fully discrete whole life insurance with annual premiums, P_x = A_x / a-ddot_x. This can also be written as P_x = (1/a-ddot_x) minus d, using the relationship A_x = 1 minus d * a-ddot_x. The loss-at-issue random variable is L_0 = v^{K+1} minus P * a-ddot_{K+1}, and setting E[L_0] = 0 yields the equivalence principle premium.
Premiums for Other Products
For n-year term insurance: P_{x:n}^1 = A_{x:n}^1 / a-ddot_{x:n}. For endowment insurance: P_{x:n} = A_{x:n} / a-ddot_{x:n}. For h-payment whole life (premiums limited to h years): h_P_x = A_x / a-ddot_{x:h}. The variance of the loss variable is Var(L_0) = (1 + P/d)^2 * (2_A_x minus A_x^2) for whole life. Exam LTAM tests premium calculation for various benefit and payment patterns, including increasing, decreasing, and modified premium structures.