Arrow's theorem and optimal reinsurance (stop loss)

In 1960, the Norwegian economist Karl Borch looks for the best form of reinsurance for a cedant that wants to reduce the variance of its results. He finds that the stop loss, which pays everything above a threshold, is optimal. In 1963, Kenneth Arrow, in an article on the economics of medical care, proves the same result for direct insurance, by an expected utility argument: the risk-averse buyer facing a risk-neutral insurer, whose premium is the expected indemnity plus a proportional loading, should choose a deductible and full cover above it. Intuitively, the proportional premium makes cover of small losses expensive, where the smoothing gain is small, and cover of large losses, where it is high, deserves the loading. The result gives theoretical justification for the excess of loss treaty and the stop loss. It assumes little: a neutral insurer, a proportional premium, a single risk, no moral hazard. In practice, deviations are many. Minimum deductibles create thresholds, the quota share remains the commonest reinsurance contract because it also brings capital and know-how, the loading is not proportional for heavy tails, and the reinsurer's neutrality is debatable. Work by Raviv in 1979 and by Gollier and Schlesinger in the 1990s extended or qualified the framework, with administrative costs, moral hazard or risky counterparties.