Production Planning with Risk Hedging Under a Conditional Value at Risk Objective

Abstract

A central problem in planning production capacity is how to effectively manage demand risk. We develop a model that integrates capacity planning and risk hedging decisions under a popular risk measure, conditional value at risk (CVaR). The CVaR objective generalizes the usual risk-neutral objective (such as the expected payoff) and allows for explicit modeling of the degree of aversion to downside risk (associated with low demand). The starting point of our model is to incorporate the impact on demand from a financial asset (including for instance, a tradable market index as a proxy for the general economy). This way, in addition to the capacity decision at the beginning of the planning horizon, there is also a dynamic hedging strategy throughout the horizon, and the latter plays the role of both mitigating demand risk and supplementing the payoff. The hedging strategy is restricted to partial information and constrained with a cap on loss (pathwise). To find the optimal hedging strategy, we construct and solve a dual problem to derive the optimal terminal wealth from hedging; the real-time hedging strategy is then mapped out via the martingale representation theorem. With the hedging strategy optimized, we show that optimizing the production quantity is a concave maximization problem. With both production and hedging (jointly) optimized, we provide a complete characterization of the efficient frontier and quantify the improvement over the production-only model. Furthermore, via sensitivity and asymptotic analyses, we spell out the impacts of the loss cap and the risk aversion level, along with other qualitative insights.