When to Sell the Car? Actual Replacement Decisions of a Rental Car Company versus Predictions of an Optimal Stopping Model

We study a large, successful rental car company that buys new vehicles and rents them to a series of customers who choose long or short term rental contracts. The company earns extraordinarily high rates of return on its rental cars, with average internal rates of return between purchase and sale of approximately 50%. We develop a stochastic semi-Markov duration model of the company’s operations, and via stochastic simulations, show that the model closely mimics the outcomes experienced by this company under its current operating strategy, which is to charge “flat” rental rates (i.e. rates that do not decline with vehicle age or odometer) and to replace old cars with new vehicles after approximately three years of operation. In particular, our model replicates the high internal rates of return earned by this company on its rental vehicles. Using this model, we can evaluate the profitability of a wide range of alternative replacement and rental pricing strategies. We characterize the profit maximizing replacement strategy by formulating the problem of periodic replacement of vehicles in the company’s fleet as a regenerative optimal stopping problem, and solve the problem numerically. Depending on the vehicle type, we find that the company’s expected discounted profits would be between 6% to over 140% higher under an alternative operating strategy where vehicles are kept longer and rental rates of older vehicles are discounted to induce customers to rent them.