Equilibrium reporting strategy: Two rate classes and full insurance

We propose a multiperiod insurance model under a bonus–malus system with two rate classes and consider an insured who has purchased full insurance for her losses. To explore the potential advantage of underreporting her insurable losses, the insured follows a barrier strategy and only reports lossses above the barrier to the insurer. We obtain a unique equilibrium declaration strategy in closed form for a risk-neutral insured who maximizes her expected wealth, and in semiclosed form for a risk-averse insured who maximizes her expected exponential utility of wealth, both over an exogenous random horizon. We find that the equilibrium barriers for the two classes are equal and strictly greater than zero, offering a theoretical explanation for the underreporting of insurable losses, a form of ex post moral hazard. Finally, we consider the case of three rate classes and show, through numerical examples, that the equilibrium barriers are not equal.