= Total variance stationary condition for excess of loss
{title2=$M\overline F(M)=\mathbb E[(X-M)_+]$}
For positive claim sizes with finite <second moment> in a <compound Poisson distribution> aggregate of parameter $\lambda$, the two <excess of loss reinsurance> payouts are $\min(X,M)$ and $(X-M)_+$. The derivative of their total aggregate <variance> is $2\lambda(M\overline F(M)-\mathbb E[(X-M)_+])$. Thus the displayed condition characterizes stationarity. When the tail probability is positive it says that the retention equals the <mean residual life>. Global minimality requires an additional sign or comparison argument.
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