Esscher principle

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