  We study the design of an optimal insurance contract in which the insured
maximizes her expected utility and the insurer limits the variance of his risk
exposure while maintaining the principle of indemnity and charging the premium
according to the expected value principle. We derive the optimal policy
semi-analytically, which is coinsurance above a deductible when the variance
bound is binding. This policy automatically satisfies the incentive-compatible
condition, which is crucial to rule out ex post moral hazard. We also find that
the deductible is absent if and only if the contract pricing is actuarially
fair. Focusing on the actuarially fair case, we carry out comparative statics
on the effects of the insured's initial wealth and the variance bound on
insurance demand. Our results indicate that the expected coverage is always
larger for a wealthier insured, implying that the underlying insurance is a
normal good, which supports certain recent empirical findings. Moreover, as the
variance constraint tightens, the insured who is prudent cedes less losses,
while the insurer is exposed to less tail risk.
