Risk comparisons of premium rules: optimality and a life insurance study

Consider a risk Y1(x) depending on an observable 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 ) + η p 1 ( x ) . The corresponding adjustment coefficient γ is the solution of E   exp { γ [ Y 1 ( A ) - p ( A ) ] } = 1 , and we characterize the rule for the loading premium p1(·) which maximizes γ subject to the constraint E p 1 ( A ) = 1 . ife insurance study, the optimal p 1 * ( · ) 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. Bonus is not included either.