Experience-rating systems, such as bonus-malus systems, offer discounts on future premiums to insureds who maintain a claim-free record, thereby generating incentives for insureds to not report certain losses. This paper formulates an optimal loss-reporting problem in a continuous-time framework for an insured with full insurance. The insured follows a barrier strategy for reporting losses and aims to maximize the expected exponential utility of her terminal wealth over a random horizon t. In the special case when t has a constant hazard rate, we obtained the optimal barrier strategy in closed form, which is a strictly positive constant. In the general case of a time-varying hazard rate for t, we obtain the optimal barrier strategy in semi-closed form, subject to solving a system of ordinary differential equations. Additionally, we uncover several noteworthy qualitative insights into both the optimal barrier strategy and the associated value functions.