Imperfect hedging and export production.

Wahl, Jack E.

I. 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.

II. 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 has a probability distribution that is independent of the foreign exchange rate risk.(3) Note that the mean-zero uncertainty increases the profit risk of the firm by [H.sup.2] var([Mathematical Expression Omitted]). This magnitude is endogenous because of the futures commitment H. This is different from the approach in Zilcha and Broll [25]. In their paper, they investigate the optimal hedging of an exporting firm when there is an additional exogenous risk.

We define hedging in which [Mathematical Expression Omitted] a perfect hedge and hedging in which [Mathematical Expression Omitted] is volatile an imperfect hedge.

In our study, we focus on the impact of the regressibility assumption. This assumption appears in the research on the relation between futures and spot prices and has a long tradition in the economic literature [4] and is also widespread in the risk management literature [10]. At the end of the paper we discuss a more general scheme for the relationship between the risky foreign exchange rate and the risky price of the domestic asset.

The first-order conditions for a maximum in problem (1) become (with [F.sub.K] (K, L) [equivalent to] [Delta]F(K, L)/[Delta]K and [F.sub.L](K, L) [equivalent to] [Delta]F(K, L)/[Delta]L):

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

[Mathematical Expression Omitted].

In the following section we use the conditions (2)-(4) to explore the effects of exchange rate risk and imperfect hedging on the exporting firm's production, hedging policy and welfare.

III. Hedging Policy and Export Production

Let us now demonstrate that uncertainty in the exchange rate and imperfect hedging have real effects on export production. The sign of the correlation determines the sign of the forward position of the exporting firm but in any case the firm's hedging position depends upon the biasedness of the forward market. We claim:

PROPOSITION 1. Consider a competitive risk-averse exporting firm facing exchange rate risk and an imperfect hedging instrument in an unbiased forward market:

1. With positive correlation between the exchange rate and the domestic asset, optimal hedging implies a forward position H [greater than] 0 and an underhedge pF(K, L) [greater than] [Beta]H; with negative correlation optimal hedging implies a forward position H [less than] 0 and also an underhedge pF(K, L) [greater than] [Beta]H.

2. The decision on the optimal level of capital and labor inputs and the optimal hedging decision cannot be separated, i.e., export production depends upon expectations and risk aversion.

Proof (1.) From condition (4) we derive

[Mathematical Expression Omitted].

With the assumption, that the forward market is unbiased, i.e., [Mathematical Expression Omitted], we obtain [Mathematical Expression Omitted] or we can write (due to assumption A.2)

[Mathematical Expression Omitted],

where the profit function is given by

[Mathematical Expression Omitted].

(i.) Positive correlation ([Beta] [greater than] 0): Suppose H [less than or equal to] 0 then [Mathematical Expression Omitted] so that [Mathematical Expression Omitted] by (5) which implies pF(K, L) - [Beta]H [less than or equal to] 0. However this is impossible with [Beta] [greater than] 0. Therefore H [greater than] 0 which implies that [Mathematical Expression Omitted]. Hence from (6) we obtain pF(K, L) - [Beta]H [greater than] 0.

(ii.) Negative correlation ([Beta] [less than] 0): Suppose H [greater than or equal to] 0 then [Mathematical Expression Omitted], so that [Mathematical Expression Omitted] which implies pF(K, L) - [Beta]H [less than or equal to] 0 which is impossible. Therefore H [less than] 0. By equation (6) this implies [Mathematical Expression Omitted]. Hence pF(K, L) - [Beta]H [greater than] 0.

(2.) From equation (2), (3), and (4) we derive

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

which proves that the production decision cannot be separated from expectations and risk behavior of the firm.

Note that the unbiasedness of the forward rate of the domestic asset does not imply a full hedge of export revenue. This will only occur if [Mathematical Expression Omitted] and [Beta] = 1.

In general, the firm reduces its profit risk by engaging in a hedging activity of assets correlated to the foreign exchange rate. In such case the widely discussed separation theorem and the full hedging theorem do not hold.

Conditions (2) and (3) imply the well-known result of the theory of the firm under certainty that capital-intensity K/L is a function of factor-price ratio w/r, only.(4) This is due to the homogeneity assumption of the technology.

COROLLARY. If the production function is linearly homogeneous, then the optimal capital-labor ratio does not depend upon the firm's risk aversion and probability beliefs. Furthermore this ratio is independent of the degree of hedging imperfection.

Proof. The ratio of marginal productivities is derived from equations (2) and (3) as follows:

[F.sub.L](K, L)/[F.sub.K](K, L) = w/r.

If the production function F(K, L) is homogeneous of degree one, then the ratio of marginal productivities can be expressed as an increasing function of the input ratio K/L. Let [F.sub.L](K, L)/[F.sub.K](K, L) = [T.sup.-1] (K/L). Thus, the inverse function T expresses the input ratio in terms of the factor price ratio w/r as follows:

K/L = T(w/r).

The higher the factor price ratio w/r, the higher is the capital-labor ratio K/L. This implies also that the firm determines the optimal capital-labor ratio regardless of its attitude towards risk and regardless of its beliefs about the distributions of [Mathematical Expression Omitted] and [Mathematical Expression Omitted].

Note the implication of the Corollary that the optimal capital-labor ratio is valid with and without hedging possibilities of the firm regardless of the effectiveness of the hedging device. However, in determining the levels of labor and capital, the firm takes into account its risk aversion level and its expectations. This can be seen from (7) and (8) where [Mathematical Expression Omitted] is an integral part of these conditions.

IV. Welfare and Export Implications

Let us compare the firm's capital and labor input under imperfect hedging with the perfect hedging solution (denoted by [Mathematical Expression Omitted], [Mathematical Expression Omitted]). We can prove:

PROPOSITION 2. With imperfect hedging the firm's labor and capital inputs are lower than in the case of perfect hedging, i.e., [Mathematical Expression Omitted], [Mathematical Expression Omitted]. Therefore exports decrease.

Proof. When the exporting firm can hedge perfectly, i.e., [Mathematical Expression Omitted], with unbiasedness, i.e., [Mathematical Expression Omitted], it follows from (2)-(4) that optimal labor and capital inputs satisfy

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted].

This demonstrates that production and hedging decisions can be separated when there is perfect correlation between the domestic asset's price and the foreign exchange spot rate. With imperfect hedging we obtain from the proof of Proposition 1 that [Mathematical Expression Omitted], which implies from (2) and (3)

[Mathematical Expression Omitted],

[Mathematical Expression Omitted].

Comparing the perfect hedging case equations (9) and (10) with the imperfect hedging case equations (11) and (12), and noting that the maximand in each case is a strictly concave function in K and L, the inequalities of the optimal inputs satisfy the assertion of Proposition 2, if [F.sub.KL] [greater than or equal to] 0.

Now let us compare the imperfect hedging case with the case of no hedging at all. We obtain:

PROPOSITION 3. Regardless of how the firm's optimal level of capital and labor may change, introducing imperfect hedging increases the firm's expected utility. That is, [Mathematical Expression Omitted], where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] denote the firm's random profit with and without imperfect hedging, respectively.

Proof. Since U([Pi]) and F(K, L) are strictly concave, the following inequality holds:

[Mathematical Expression Omitted],

since

F(K, L) - F([K.sub.O], [L.sub.O]) [greater than] [F.sub.K](K, L)(K - [K.sub.O]) + [F.sub.L](K, L)(L - [L.sub.O]).

Hence

[Mathematical Expression Omitted],

from the first-order conditions (2)-(4). This holds independently of the level of capital and labor with imperfect hedging or without any hedging.

Proposition 3 does not give an answer as to whether or not output incenses if imperfect hedging becomes available. In the following we give a sufficient condition for higher capital and labor input if mean-zero uncertainty [Mathematical Expression Omitted] is small. It turns out that the behavior of the Arrow-Pratt measure of absolute risk aversion, [R.sub.A]([Pi]) [equivalent to] -U[double prime]([Pi])/U[prime]([Pi]) [greater than] 0, is essential [3; 20]. We obtain:

PROPOSITION 4. Let the forward market of the domestic asset be unbiased and let the utility function U([Pi]) display constant or decreasing absolute risk aversion. Then, for small mean-zero uncertainty [Mathematical Expression Omitted], optimal level of capital and labor of a linearly homogeneous production function increases, if imperfect hedging becomes available.

Proof. A first-order Taylor expansion of marginal utility around expected profit [Mathematical Expression Omitted], i.e., [Mathematical Expression Omitted] translates condition (2) into

[Mathematical Expression Omitted],

and, by using the profit definition (6) and rearranging terms we get

[Mathematical Expression Omitted],

since [Mathematical Expression Omitted]. Furthermore our Taylor expansion changes condition (4) into

[Mathematical Expression Omitted],

if [Mathematical Expression Omitted]. Again, by using the profit definition (6) we rearrange the expansion result [Mathematical Expression Omitted] to

[Mathematical Expression Omitted].

This leads to

[Mathematical Expression Omitted].

Multiplying this by [Mathematical Expression Omitted] implies(5)

[Beta]H = [[Rho].sup.2]pF(K, L), (14)

where [Mathematical Expression Omitted] is the determination coefficient. Inserting (14) into (13) gives us

[Mathematical Expression Omitted].

Since F(K, L) is homogeneous of degree one, F(K, L)/K = G(L/K) and [F.sub.K](K, L) = G[prime](L/K). For notational simplicity let us introduce

[Mathematical Expression Omitted].

Then we have

[Mathematical Expression Omitted].

Since L/K is independent of the hedging opportunity, be it perfect or imperfect, the LHS of condition (16) is independent of the introduction of an imperfect hedge. Hence, increasing [Beta] and, therefore, [[Rho].sup.2] implies that input factor K increases if absolute risk aversion is nonincreasing. This holds because L/K remains constant and higher input K provides, along with higher input L, higher expected profit [Mathematical Expression Omitted].

V. Concluding Remarks and Discussion

If every financial instrument in the spot market had a futures contract that exactly mirrored its characteristics, futures markets would provide the opportunity to yield a perfect hedge. In the real world hedging must often be accomplished by using futures contracts on different deliverable instruments. Such hedging is called indirect hedging and generally yields an imperfect hedge.

We know from the literature that an international firm facing exchange rate risk eliminates this risk altogether by using a currency forward market, or another financial asset which is perfectly correlated to the spot rate of foreign exchange. In general, the firm can reduce its income risk by engaging in a hedging activity of assets correlated to the foreign exchange rate. In such case the widely discussed separation theorem and the full hedging theorem do not hold.

Our results are: (1.) With imperfect hedging, we obtain an interaction between the optimal export decision and the optimal hedging decision. Hence the separation property is violated. (2.) Although the hedging market is unbiased, imperfect hedging implies an underhedge position. Hence there is no full hedge with an unbiased forward rate. (3.) Regardless of the firm's optimal level of export production, introducing imperfect hedging improves the firm's welfare. (4.) If the mean-zero uncertainty is such that the dispersion of the profit distribution is not "too large," and the domestic asset's forward rate is unbiased, then nonincreasing absolute risk aversion leads to an increase of export production when imperfect hedging becomes available.

Note the following policy and trade implications: exporting firms benefit when hedging devices are offered by governments, for instance, although the hedging instrument may be imperfect. However the effect on international trade is ambiguous because the firm's output for export may decrease or increase.

In our framework the level of hedging depends on [Beta]. Hedging occurs only if there is a regression between the foreign exchange rate and the domestic asset's price. An extension of our analysis can be carried out with a more general scheme than our regressibility assumption.(6) Suppose, for example, a linear marginal utility of profit. Then the optimal hedge ratio H/pF(K, L) is equal to [Mathematical Expression Omitted]. Hence the bivariate probability distribution of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] determines the hedge volume.

The authors wish to thank an anonymous referee for very helpful comments and advice. We also would like to thank Itzhak Zilcha for helpful discussions.

1. The empirical study of Roger [21, Table 1, 16] shows examples of exchange rate volatility for major currencies.

2. The production function F(K, L) displays neoclassical properties. These properties include [F.sub.L] [greater than] 0, [F.sub.K] [greater than] 0, [F.sub.LL] [less than] 0, [F.sub.KK] [less than] 0 and [Mathematical Expression Omitted]. Furthermore we assume [F.sub.KL] [greater than or equal to] 0.

3. For the notion of mean-zero uncertainty, see Kimball [17, 57 and 64].

4. This relationship was extensively investigated by Helpman and Razin [15], Wright [23], and Zilcha and Eldor [24].

5. We set the equality sign in (14), because a portfolio approach, i.e., min [Mathematical Expression Omitted] subject to [Mathematical Expression Omitted], shows that the hedging-output relationship holds for an unbiased futures market.

6. This was pointed out by the referee.

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