The expected utility insurance premium principle with fourth-order statistics: Does it make a difference?

The expected utility principle is often used to compute the insurance premium through a second-order approximation of the expected value of the utility of losses. We investigate the impact of using a more accurate approximation based on the fourth-order statistics of the expected loss and derive the premium under this expectedly more accurate approximation. The comparison between the two approximation levels shows that the second-order-based premium is always lower (i.e., an underestimate of the correct one) for the commonest loss distributions encountered in insurance. The comparison is also carried out for real cases, considering the loss parameters values estimated in the literature. The increased risk of the insurer is assessed through the Value-at-Risk