= Solution
For a single claim $x$, <quota share reinsurance> with retained fraction $\alpha\in[0,1]$ makes the direct insurer pay $\alpha x$ and the reinsurer pay $(1-\alpha)x$. Under <excess of loss reinsurance> with retention $M\geq0$, the direct insurer pays $\min(x,M)$ and the reinsurer pays the <positive part> $(x-M)_+$. Thus the concise payout pairs are
$$
\boxed{(x_I,x_R)=(\alpha x,(1-\alpha)x)quad\hbox{or}\quad(\min(x,M),(x-M)_+).}
$$
The cap in <excess of loss reinsurance> applies separately to every claim; it is not a cap on the entire annual aggregate.
For the following <variance> calculations take $\lambda>0$ and $\mathbb E[X^2]<\infty$, so the displayed <variances> are finite. For any per-claim payout $Y$, the <law of total variance> in a <compound Poisson distribution> gives
$$
\operatorname{Var}\left(\sum_{j=1}^NY_j\right)
=\mathbb E[N]\operatorname{Var}(Y)+\operatorname{Var}(N)(\mathbb E[Y])^2
=\lambda\mathbb E[Y^2].
$$
The final term is the raw <second moment>, not the single-claim <variance>. Both parties' totals are <retained compound Poisson aggregates>, with different payout functions of the same claims; they are generally dependent.
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