Risk Comparisons of Premium Rules: Optimality and a Life Insurance Study
Sören Asmussen and Jakob R. Möller
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2001
ISSN 14039338

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,
GompertzMakeham law, large deviations,

life annuities, loading premium, shotnoise, third order
basis, whole life insurance