Esscher principle

The Esscher principle is an insurance premium principle. It is given by $\pi [X,h]=E[X{e}^{hX}]/E[e^{hX}]$, where $h$ is a strictly positive parameter. This premium is the net premium for a risk $Y=X{e}^{hX}/m_X(h)$, where $m_X(h)$ 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 $h>0$.