Esscher principle

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