= Variance-minimizing retention for shape-three Pareto claims
{title2=$M^*=3d/4,\quad\min V=15\lambda d^2/8$}
For a <Pareto distribution> of shape three and minimum $d>0$ with count rate $\lambda>0$, the sum of insurer and reinsurer aggregate <variances> is $V(M)=\lambda(3d^2-3dM+2M^2)$ for $0\le M\le d$, and $\lambda(3d^2-d^3/M)$ for $M\ge d$. The first branch has its unique minimum at $3d/4$; the second is increasing. The branches agree in value and slope at $d$, giving the displayed global minimum. This criterion minimizes the sum of party <variances>; the total portfolio <variance> includes their <covariance> and remains $3\lambda d^2$.
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