Imperfect hedging and export production.

• Author: Broll, Udo

1. Introduction

   Firms engaged in international operations are highly interested in developing ways to protect themselves from exchange rate risk. The incentive for risk management comes from the enormous volatility of the floating foreign exchange rates.(1) Our study shows that an exporting firm can benefit from hedging exchange rate risks even when no perfect hedge is possible. Since in reality, not every currency is traded in a futures market [7, Chap. 15], the exporting firm uses futures contracts with other underlying assets whose spot prices are highly correlated with the foreign exchange spot rate. In the real world hedging must often be accomplished by using futures contracts on different deliverable instruments. Such hedging may result in imperfect hedging as shown by Anderson and Danthine [1], Eaker and Grant [11], Dellas and Zilberfarb [9], Broll, Wahl and Zilcha [6].

   It has been shown in recent publications [12; 8; 15; 13; 23; 16; 2; 14; 5; 18; 22] that an international firm facing exchange rate risk can eliminate this risk altogether if it can use a currency forward market, another financial asset or a portfolio of assets which is perfectly correlated to the exchange rate. In the absence of such markets, the firm can reduce its income risk by engaging in a hedging activity of assets correlated to the foreign exchange.

   Recent studies of firm behavior under exchange rate uncertainty examine the influence of futures markets on the export and hedging decision. These papers derive two major theorems: One is the "separation theorem" which states that, when futures markets exist, the firm's export production decision is determined solely by technology and input-output prices, including the futures prices. This result holds if the gain from the futures contract is perfectly correlated with export revenue. The other theorem is the "full hedging theorem" which asserts that with unbiased futures markets, the firm completely avoids exchange rate risk by entering into optimum futures contracts.

   However, many spot assets are not delivered in any futures market, nor are there bank forward contracts available. Hence, firms must cross hedge, which means hedge in a futures contract delivering a different asset. In this case hedging must be accomplished by using existing futures contracts that involve similar price fluctuations with the cash market instrument being hedged. These matches of the futures contract to the cash instrument are known as imperfect hedges. An example of an imperfect hedge is the use of T-bill futures contracts to hedge a commitment in another money market instrument.

   The aim of our study is to examine the role of imperfect hedging on the firm's export and hedging policy. Imperfect hedging is a method our firm can use to manage foreign currency risk because there is no futures or forward market in the currency. Imperfect hedging expands the opportunity set of hedging alternatives. Our research provides some insights into the output and welfare implications of imperfect hedging.

   The paper is organized as follows. In section II, the model of an exporting firm is presented. The main results are derived in section III, where we examine the impact of imperfect hedging of exchange rate risk on the exporting firm's decision making. We show that imperfect hedging violates both the separation theorem and the full hedging theorem. Nonetheless, introducing an imperfect hedging device increases the welfare of the firm though the effect on production is ambiguous. In section IV we derive conditions under which output increases if hedging is imperfect. Section V has a discussion of possible extensions and conclusions.

2. The Model

   Consider a competitive risk-averse exporting firm facing a random exchange rate [Mathematical Expression Omitted]. The firm's production function F(K, L) depends on capital K and labor L.(2) The factor rentals are denoted by r and w for capital and labor, respectively. The firm cannot hedge its foreign currency risk directly in a given futures market. However, there is a forward market for some domestic financial asset correlated to the exchange currency which can be entered by the firm. Therefore, there exists an indirect, but imperfect hedging device.

   The firm has access to the futures market when the production decision takes place. It can sell (or buy) forwards at a volume of H at a competitively given futures domestic price [g.sub.f]. With a von Neumann-Morgenstern utility function U, where positive marginal utility is decreasing the decision problem of the exporting firm becomes

   [Mathematical Expression Omitted],

   where the firm's random profit

   [Mathematical Expression Omitted].

   Here, [Mathematical Expression Omitted] denotes the random spot value of the domestic asset correlated to the exchange rate. We now make two assumptions:

   (A.1) Unbiasedness: We assume that the futures market is unbiased, i.e., [Mathematical Expression Omitted].

   (A.2) Regressibility: We assume that [Mathematical Expression Omitted] is a linear function of [Mathematical Expression Omitted] with noise, i.e., [Mathematical Expression Omitted] where [Beta] [not equal to] 0 and the mean-zero uncertainty [Mathematical Expression Omitted] is independent of [Mathematical Expression Omitted].

   We define mean-zero uncertainty [Mathematical Expression Omitted] as an additional risk with expected value of zero, which...