Insurance purchasers obtain varied discounts for insurance. This paper examines what drives these differences, specifically whether the loss probability and the wealth of the insured affect the size of the premium discount in automobile insurance. To describe a bargain between a client and an insurer over premiums and coverage, we first develop a sequential insurance bargaining game where the client has an outside option to bargain with another insurer. We find that the equilibrium involves full coverage and, based on the results of comparative statics, we propose hypotheses regarding the effects of the loss probability and the wealth of the insured on the size of the premium discount. We then use a unique data set of 85,806 observations of Taiwanese automobile liability insurance for property damage to empirically test the predictions. After controlling for underwriting and macroeconomic variables, we find that both (1) the insured with a lower claim probability (as a proxy for the insured with a lower loss probability) and (2) the insured with a higher salvage value car (as a proxy for the wealthier insured) receive a greater premium discount. These results support our theoretical results.