On the Time-Consistent Policies for Financial Hedging and Inventory Management: A Dynamic Mean-Variance Analysis
Abstract: We consider a firm purchasing and processing a storable commodity from a spot market with volatile commodity prices. The firm has access to both a commodity spot market and an associated financial derivatives market. The purchased commodity serves as a raw material which is then processed into an end product with uncertain demand and lost sales. The objective of the firm is to coordinate the replenishment and financial hedging decisions to maximize the mean-variance utility of its terminal wealth over a finite horizon. Acknowledging that the mean-variance measure is time-inconsistent, we restrict our analysis to the class of time-consistent policies. We employ a dynamic programming approach to characterize the structure of optimal time-consistent policies for the joint inventory and financial hedging decisions of the firm. We show that the inventory and financial hedging decisions can be separated as long as there exists forward. The optimal inventory policy can be characterized by a myopic state-dependent base-stock level and the optimal hedging policy can be obtained by minimizing the variance of the hedging portfolio, the value of excess inventory and the profit-to-go as a function of future price. In the presence of a continuum of option strikes, we demonstrate how to construct custom exotic derivatives using forwards and options of all strikes to replicate the profit-to-go function. We show that the optimal time-consistent policy under exotic hedge is also optimal in the admissible policy set, which implies that the time-inconsistency problem can be overcome by exotic hedge in the mean-variance framework. Our results also shed new light into the supply chain risk management: inventory replenishment decisions can be separated from the financial hedging decisions as long as forwards are in place, and the dynamic inventory decision problem reduces to a sequence of myopic optimization problems with financial hedging. Empirical implications of our results are (1) that financial hedges can affect firms' operational decisions, and (2) that inventory and financial hedges can be either substitutes. To examine the robustness of our results, we extend our analysis to varieties of settings including the cases with backorders, price-sensitive demand, variable transaction costs, inventory processed upon receipt; and financial arbitrage.