Survival probability under risk pooling (${\phi }_{\mathrm{pooled}}(0)={\sum }_i\frac{c_i}{{\sum }_k{c}_k}{\phi }_i(0)$)

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.