Can an Optimal Insurance Contract Violate the Principle of Indemnity?

The principle of indemnity is well-accepted in the field of insurance and is also commonly assumed in the literature of optimal contract design. However, should this principle be imposed on all insurance contracts without exceptions? Does there exist an insurance contract that is optimal for both the insurer and the insured even as the principle of indemnity is violated? The answer is yes. This paper demonstrates that in the insurance market there do exist contracts in which optimality is accompanied by a violation of the indemnity principle if the insurer and insured have deviant beliefs in loss distribution. We further use retrospective rating as an example to explain why an insurance contract, violating the principle of indemnity, commonly exists in commercial insurance, especially where moral hazard and adverse selection is less severe.