= Survival probability under risk pooling
{title2=$\varphi_{\rm pooled}(0)=\sum_i\frac{c_i}{\sum_kc_k}\varphi_i(0)$}
Independent positively loaded portfolios with zero initial capitals have separate joint survival probability $\prod_i(1-\lambda_i\mu_i/c_i)$. Pool their claims and premium incomes to obtain zero-capital merged survival $1-(\sum_i\lambda_i\mu_i)/(\sum_i c_i)$. This is the premium-weighted average of the individual survival probabilities and is at least their product. The event of aggregate solvency permits transfers of surplus between the original portfolios.
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