Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 40 / 2 / a / Solution

Under quota share reinsurance, $S_I=\alpha S$ and $S_R=(1-\alpha )S$. Write $v=\mathrm{Var}(S)=\lambda \mathbb{E}[X^2]$. ThenThe variance-minimizing quota share split is thereforeFor positive claims and $\lambda >0$ this minimizer is unique. If the aggregate has zero variance, every retained fraction gives the same zero objective.