Abstract
This paper examines the behavior of the competitive firm under correlated price and background risk when a futures market exists for hedging purposes. We show that imposing the background risk, be it additive or multiplicative, on the firm has no effect on the separation theorem. The full-hedging theorem, however, holds if the background risk is independent of the price risk. In the general case of the correlated price and background risk, we adopt the concept of expectation dependence to describe the bivariate dependence structure. When the background risk is additive, the firm finds it optimal to opt for an over-hedge or an under-hedge, depending on whether the price risk is positively or negatively expectation dependent on the background risk, respectively. When the background risk is multiplicative, both the concept of expectation dependence and the Arrow–Pratt measure of relative risk aversion are called for to determine the firm’s optimal futures position.
Highlights
Since the seminal work of Sandmo (1971), the theory of the competitive firm under price uncertainty has been the subject of considerable research in decision makingI would like to thank Paolo Ghirardato, Frank Riedel, and an anonymous referee for their helpful comments and suggestions
In the general case that the background risk is correlated with the price risk, we show that the concept of expectation dependence (Wright 1987) plays a pivotal role in determining the firm’s optimal futures position
When the background risk is multiplicative, we show that the concept of expectation dependence and the Arrow–Pratt measure of relative risk aversion jointly determine the firm’s optimal futures position
Summary
Since the seminal work of Sandmo (1971), the theory of the competitive firm under price uncertainty has been the subject of considerable research in decision making. Given that the firm’s production decision does not depend on the underlying uncertainty when a futures market exists for hedging purposes, it follows immediately that the separation theorem is robust to the introduction of the correlated background risk. If the price risk is positively expectation dependent on the background risk, the firm finds it optimal to opt for an over-hedge or an under-hedge, depending on whether the firm’s measure of relative risk aversion is everywhere greater or smaller than unity, respectively. If the price risk is negatively expectation dependent on the background risk, the firm finds It optimal to opt for an over-hedge or an under-hedge, depending on whether the firm’s measure of relative risk aversion is everywhere smaller or greater than unity, respectively.
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