Many papers in the literature have adopted the expected utility paradigm to analyze insurance decisions. Insurance companies manage policies by growing, by adding independent risks. Even if adding risks generally ultimately decreases the probability of insolvency, the impact on the insurer's expected utility is less clear. Indeed, it is not true that the risk aversion toward the additional loss generated by a new policy included in an insurance portfolio always decreases with the number of contracts already underwritten. The present paper derives conditions under which zero-utility premium principles are subadditive for independent risks. It is shown that subadditivity is the exception rather than the rule: the zero-utility premium principle generates a superadditive risk premium for most common utility functions. For instance, all completely monotonic utility functions generate superadditive zero-utility premiums. The main message of the present paper is thus that the zero-utility premium for a marginal policy is generally not sufficient to guarantee the formation of insurance portfolios without additional capital.

For a large class of subsets $\varOmega\subset\mathbb{R}^{N}$ (including unbounded domains), we discuss the Fredholm and properness properties of second-order quasilinear elliptic operators viewed as mappings from $W^{2,p}(\varOmega;\mathbb{R}^{m})$ to $L^{p}(\varOmega;\mathbb{R}^{m})$ with $N\ltp\lt\infty$ and $m\geq1$. These operators arise in the study of elliptic systems of $m$ equations on $\varOmega$. A study in the case of a single equation ($m=1$) on $\mathbb{R}^{N}$ was carried out by Rabier and Stuart.

AMS 2000 Mathematics subject classification: Primary 35J45; 35J60. Secondary 47A53; 47F05