Esscher principle

The Esscher principle is an insurance premium principle. It is given by ${\displaystyle \pi [X,h]=E[Xe^{hX}]/E[e^{hX}]}$, where ${\displaystyle h}$ is a strictly positive parameter. This premium is the net premium for a risk ${\displaystyle Y=Xe^{hX}/m_{X}(h)}$, where ${\displaystyle m_{X}(h)}$ denotes the moment generating function.

The Esscher principle is a risk measure used in actuarial sciences that derives from the Esscher transform. This risk measure does not respect the positive homogeneity property of coherent risk measure for ${\displaystyle h>0}$.