Variance Contracts

TL;DR

This paper designs an optimal insurance contract balancing the insured's utility maximization with the insurer's variance risk limit, deriving a coinsurance policy with implications for moral hazard and insurance demand.

Contribution

It provides a semi-analytical solution for variance-constrained insurance contracts, highlighting conditions for deductible absence and analyzing effects of wealth and risk bounds.

Findings

Optimal policy is coinsurance above a deductible when variance constraint is binding.

Expected coverage increases with insured's wealth, indicating insurance as a normal good.

Tighter variance constraints lead to less loss ceding by prudent insureds and reduced tail risk for insurers.

Abstract

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.