Risk Comparisons of Premium Rules: Optimality and a Life Insurance Study

Sören Asmussen and Jakob R. Möller

Abstract:: Consider a risk $Y_1(x)$ depending on a covariate $x$ which is the outcome of a random variable $A$; with a known distribution, and consider a premium $p(x)$ of the form $p(x)=E Y_1(x)+\eta p_1(x)$.; The corresponding adjustment coefficient $\gamma$ is the solution of $E{\rm exp}\{\gamma[Y_1(A)-p(A)]\}$ $=1$, and we; characterize the rule for the loading premium $p_1(\cdot)$ which maximizes $\gamma$ subject to the constraint $E p_1(A)=1$.
: In a life insurance study, the optimal $p^*_1(\cdot)$ is compared to other premium principles like the expected value--,
the variance-- and the standard deviation principles as well as the practically important rules based on safe mortality rates: (i.e., using the first order basis rather than the third order one). The life insurance model incorporates premium reserves,
discounting, and interest return on the premium reserve but not on the free reserve.

Key words:: Adjustment coefficient, convex ordering, delayed claims, first order basis, Gompertz-Makeham law, large deviations,
life annuities, loading premium, shot-noise, third order basis, whole life insurance