Proportional and excess-of-loss reinsurance under investment gains

On the assumption that investment fund follows the logarithm-normal distribution, the paper derives the forms of proportional and excess-of-loss reinsurance contracts which make the convex combination of the insurer’s rate of return v₁ and the reinsurer’s rate of return v₂ exceeds R at the probability of f. In the whole paper, the premium takes the expectation principle.