Inflation Risk, Hedging, and Exports

Transcription

1 Review of Development Economics, 5(3), , 2001 Inflation Risk, Hedging, and Exports Harald L. Battermann and Udo Broll* Abstract This paper analyzes optimal production and hedging decisions of a risk-averse exporting firm in a developing country. The firm cares about real profits, since the spot exchange rate and the domestic price level are uncertain. It is demonstrated that a separation property holds although there are two sources of risk and only one hedging instrument exists. The authors examine the optimal risk management of the firm. In contrast to most hedging models, the real risk premium is important for the optimal hedging strategy. 1. Introduction In recent years developing economies and transition economies have developed market institutions for coping with uncertainty such as price, exchange rate, and interest rate risks. Additionally, new technologies for financial risk management are becoming more important for development economies, especially derivative instruments such as currency and commodity futures (Powell, 1989; Fry, 1992; Besley, 1995; Lessard, 1995). Establishing a risk sharing market in an economy has important implications for international trade. One important finding of the risk management literature is the well-known separation theorem. This implies that the production decision depends only on futures prices and the cost function of production; i.e., the exporting firm makes its optimal output decision as in the certainty case. The contemporary literature on production and hedging has derived this separation property (Danthine, 1978; Holthausen, 1979; Aradhyula and Kwan Choi, 1993; Broll et al., 1995; Battermann et al., 2000) by assuming that the decision-maker considers nominal risky profits. This assumption can be motivated by having the domestic price level fixed such that nominal risky profits are perfectly correlated with real risky profits. However, in reality the price level is not necessarily stable, especially in developing economies and economies in transition (Kolodko et al., 1992; OECD, 2000). Therefore, a firm located in a high-inflation country should not neglect real risks since the purchasing power of nominal profits may vary substantially. The main objective of this paper is to discuss the case where the owner of the firm faces two sources of risk, namely currency exchange rate risk and inflation risk; that is, we generalize the standard approach by focusing on real profits instead of nominal profits. We assume that a currency forward market exists, so that foreign proceeds can be hedged directly via a currency forward market. On the other hand, the decisionmaker can use forward contracts to reduce indirectly the risk of purchasing power *Battermann: Department of Economics, University of Saarland, D Saarbrücken, Germany. Tel: ; Fax: ; h.battermann@mx.uni-saarland.de. Broll: Department of Economics, University of Saarland, D Saarbrücken, Germany, Tel: ; Fax: ; u.broll@mx.uni-saarland.de. We would like to thank anonymous referees and Bernhard Eckwert for helpful comments. An earlier version of this paper was presented at the Annual Meeting of the American Risk and Insurance Association in Boston, on August The authors would like to thank also the participants of the conference for their remarks., 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA

2 356 Harald L. Battermann and Udo Broll induced by inflation. Does this approach imply that (i) the separation property is still valid, and (ii) that a nominal full hedge is optimal? First, we analyze the optimal export production decision. We demonstrate that the separation theorem is valid, although the firm cares about real profits. Second, we study the optimal risk policy. In contrast to the nominal approach (Benninga et al., 1983; Lence, 1995; Broll and Wahl, 1998) it can be shown that full hedging is optimal, if the real risk premium vanishes (Sibert, 1989; Hakkio and Sibert, 1995) and a type of the purchasing power parity holds. In section 2 we present the model and prove that the separation theorem still holds if a firm cares about real profits. Section 3 focuses on the hedging policy of the firm under different market conditions. The last section summarizes the main results. Proofs of the propositions in the text are collected in an Appendix. 2. The Optimal Export Decision In order to explain the decision problem of a risk-averse competitive exporting firm, let us review the basic model used in the traditional literature on hedging (Kawai and Zilcha, 1986). This model of exporting and hedging extends over two periods of time. At time 0 the production decision has to be made and the forward market opens. The firm delivers the exporting good X to a foreign country at time 1. Let the price of the commodity in the foreign country be Q and assume that it is fixed. At time 0, nominal profits P are uncertain owing to an uncertain spot exchange rate S, where random variables are denoted by tildes. The firm can hedge risky profits by selling or buying a quantity H of contracts on the currency forward market. Let F be the price of a currency forward contract which falls due at time 1. Then risky nominal profits are given by ( ) P= SQX - C( X)+ H F -S, where C(X) denotes the cost function of production, which has the properties C(0) = 0, C (0) < QF, C (X) > 0, C (X) > 0 for all X > 0. As mentioned before, as long as the domestic price level P d is certain, production and hedging decisions can be analyzed by taking nominal profits into account since nominal and real profits are perfectly correlated. However, our decision problem contains two sources of risk to an exporting firm: the domestic inflation risk P d and the currency exchange risk S. Therefore, we focus on real profits P r = P / P d. Expected utility of real profits is ( ( )) EU ( ( P ))= EU S ( QX-H) P + HF P - C( X) P, r d d d where U is a twice continuously differentiable von Neumann Morgenstern utility function with U >0, U <0. Since the firm is risk-averse we obtain necessary and sufficient conditions for the optimum X*, H* due to the strict concavity of the maximand: EU ( ( P *) SQ - C ( X* ) P 0, r EU ( ( P *) F- S P 0. r ( ) d)= ( ) d)= (1) (2)

3 INFLATION RISK, HEDGING, AND EXPORTS 357 From equations (1) and (2) we obtain the following proposition. Proposition 1 (Production rule). Given an international firm in a developing country as described above, if a currency forward market exists, then the firm s optimal production rule is given by max X FQX - C(X). Proposition 1 shows that the production decision does not depend on the hedging decision, since the production rule is independent of the optimal hedging position H*. The production decision is also independent of the domestic inflation risk, the spot exchange risk, and the degree of risk aversion; i.e., the separation property holds although the firm cares about real profits. Does this result imply that the firm fully hedges real profits, if the nominal risk premium vanishes? 3. Hedging against Inflation Risk In this section we examine the optimal hedging policy of the firm. We investigate under which market conditions the firm fully hedges real profits. We say a real (nominal) full hedge is a hedge policy such that real (nominal) profits are certain. Currency Forward Markets and Hedging Since the optimal export decision X* is given by the production rule (Proposition 1) we can analyze the hedging decison by maxeu ( ( P ( H) )), H r ( H)= S ( QX* -H) P + HF P - C( X* ) P. P r d d d (3) Maximizing expected utility leads to the following first-order condition: EU ( ( P r( H * ))( F- S ) P d)= 0. (4) Using the covariance identity for random variables, we can transform equation (4) into E( ( S - F) P d)= cov( U ( P r( H* )),( F - S ) P d) EU ( P r( H* )). (5) Thus, if the real risk premium, E(( S - F)/ P d) (Sibert, 1989; Hakkio and Sibert, 1995; Obstfeld and Rogoff, 1996) is zero then the decision-maker chooses a hedge policy such that the covariance term vanishes. Before analyzing the optimal risk policy of the firm, let us introduce the following assumption. Assumption 1. Suppose the uncertain spot exchange rate is given by S = l P d /P f, l > 0, where P f denotes the certain foreign price level; i.e., a type of purchasing power parity holds. With Assumption 1, real profits can be decomposed into a certain and an uncertain part:

4 358 Harald L. Battermann and Udo Broll ( H)= S ( QX* -H) P + ( HF - C( X* )) P. P r d d The first part of real profits is certain due to our Assumption 1. The second term is uncertain in general. However, choosing a hedge position equal to total domestic cost measured in units of foreign currency would imply that future real income is certain. The following proposition states that this hedge position is optimal as well. Proposition 2 (Real full hedging rule). Consider an exporting firm producing under exchange rate risk and inflation risk as described above. The firm fully hedges real profits if and only if the real risk premium vanishes and Assumption 1 holds; i.e., H* = C(X*)/F. Proposition 2 demonstrates under which market conditions the firm fully hedges real profits. Of course, this result heavily relies on our assumption concerning the exchange rate. Our assumption postulates a perfect correlation between the domestic price level and the spot exchange rate. 1 Assumption 1 may be an acceptable approximation for the link between prices and exchange rates in high-inflation economies. 2 One implication of Proposition 2 is that the optimal risk policy depends on the sign of the real risk premium. 3 A positive real risk premium implies an underhedge (H* < C(X*)/F). An overhedge (H* > C(X*)/F) is optimal when the real risk premium is negative. Corollary (Real risk premium). Consider an exporting firm as described above. If Assumption 1 holds, then (a) the firm underhedges if the real risk premium is positive, and (b) the firm overhedges if the real risk premium is negative. Forward-Futures Markets and Hedging So far we have assumed that the foreign commodity price Q is certain. In the following we extend the model of the previous section by assuming that the commodity price Q is uncertain. The firm has access to a currency forward market as in the previous section. In addition a commodity futures market exists at time 0. Real profits are given by ( ) + ( - ) = SQX P - C( X) P + H F -S P Y Q Q S P, P r d d d f d where Y denotes the quantity of the commodity which is sold in the forward market, and Q f is the commodity futures price, which is known at time 0. Now let us show under which conditions the firm fully hedges real profits. The firm chooses its choice variables X, Y, and H in a way that maximizes expected utility of real income. Thus the first-order conditions are EU ( ( P *) SQ - C ( X* ) P 0, r ( ) d))= EU ( ( P *) r ( F-S ) P d))= 0, ( ) )= EU ( ( P *) Q - QS P 0. r f d (6) (7) (8)

5 INFLATION RISK, HEDGING, AND EXPORTS 359 Conditions 6, 7, and 8 in combination with Assumption 1 lead to the next proposition. Proposition 3 (Real full double hedging rule). Suppose the real risk premium is zero and the commodity futures market is unbiased; i.e., E( Q - Q f ) = 0. If Assumption 1 holds, then the optimal hedging rule is given by H* = C(X*)/F and Y* = X*, where X* is determined by the production rule max X FQ f X - C(X); i.e., FQ f = C (X*). Under the conditions of Proposition 3, any risk-averse decision-maker fully hedges real profits, 4 i.e., a real full double hedging policy is optimal. This finding is related to the nominal full double hedging theorem for nominal profits derived by Kawai and Zilcha (1986). This theorem states that the optimal forward-futures contracting is a complete nominal full double hedge (i.e., H* = Q f X*, Y* = X*) if the currency forward market is unbiased (nominal risk premium vanishes) and the forward-futures markets are jointly unbiased (i.e., E( S - F) = 0 and E( S Q - FQ f ) = 0). The main difference is that the real risk premium rather than the nominal risk premium must vanish in order to generate a full hedge policy. 4. Concluding Remarks Since the volatility of exchange rates and commodity prices seems more likely to rise than fall in coming years, the demand for risk management by developing countries and economies in transition is large, and it will continue to grow (Lessard, 1995). Our study analyzes optimal production and hedging decisions of a risk-averse exporting firm in a developing country under uncertainty. The decision-maker cares about real profits, since the firm faces inflation risk in addition to currency exchange rate risk and commodity price risk. Contrary to most hedging models we demonstrate that the real risk premium and not the nominal risk premium is important for the optimal risk management, if a currency forward market exists. Our study shows that (i) the separation theorem is still valid, if a firm cares about real profits, and (ii) the firm fully hedges real profits if and only if the real risk premium is zero and a type of purchasing power parity holds. This is of potential interest to policymakers. Exporting firms benefit from hedging, although the hedging instrument is not perfect; the output effect is that hedging opportunities promote national and international trade. Appendix Proof of Proposition 1 Equations (1) and (2) can be written as EU ( ( P *) SQ P ) = EU ( ( P *) C ( X* ) P ), r d r d ( ) = ( ) EU ( P *) S P EU( P *) F P. r d r d Combining both equations we get FQ = C (X*).

6 360 Harald L. Battermann and Udo Broll Proof of Proposition 2 ( fi ) (a) If H* is a full hedge, then future real profit is certain. Equation (4) implies that the real risk premium must vanish. (b) Suppose H* is a full hedge; i.e., there exists a constant K such that real profits equal K: P* = SQX ( *-H*) P + HF * P - C( X* ) P = K, K>, d d d 0 (A1) where X* is determined by the production rule. From equation (A1) we conclude that Assumption 1 must hold. ( ) If the real risk premium vanishes, and if a type of purchasing power parity holds, then H* = C(X*)/F follows from equation (4). Proof of the Corollary Part (a): We sketch the main idea of the proof. If it is possible to show that the function f(h): = de(u( P r(h)))/dh is negative at point H 1 = C(X*)/F, then H* (the optimal hedge policy) must be less than C(X*)/F since the function f(h) is strictly decreasing (f (H) < 0) due to the assumption that the decision-maker is risk-averse. The function f(h) can be written as fh ( )= EU ( ( P r( H) )) E( ( F- S ) P d)+ cov ( U ( P r( H) ), ( F- S ) P d). (A2) Now let us show that f(h 1 ) < 0 holds. Since cov (U (P r (H 1 )), (F - S )/ P d) is zero, we get f(h 1 ) < 0 (see equation (A2)). Thus H* must be less than C(X*)/F. This proves the claim. Part (b) of the corollary can be proved in the same way. Proof of Propositon 3 Deriving the optimal production rule The first-order conditions are equivalent to EU ( ( P *) SQ P ) = EU ( ( P *) C ( X* ) P ), r d r d EU ( ( P *) S P ) = EU ( ( P *) F P ), r d r d EU ( ( P *) r SQ P d) = EU ( ( P *) r QfS P d). (A3) (A4) (A5) Replacing the left side of (A5) with the right side of (A3) we get EU ( ( P *) r C ( X* ) P d) = EU ( ( P *) r QfS P d). (A6) Combining (A4) and (A6) we get FQ f = C (X*). Deriving the optimal hedging decision From the first-order conditions (7) and (8) it follows that (X*, H*, Y*) is a solution of the optimization problem, since real profits are certain.

7 INFLATION RISK, HEDGING, AND EXPORTS 361 References Adam-Müller, Axel F. A., Hedging Price Risk when Real Wealth Matters, Journal of International Money and Finance 19 (2000): Aradhyula, Sathees V. and E. Kwan Choi, Production, Hedging, and Speculative Decisions with Options and Futures Markets, American Journal of Agricultural Economics 75 (1993): Battermann, Harald L. and Udo Broll, Real Income Risk and Hedging in Transition Economies, in J. Hölscher (ed.), Financial Turbulence and Capital Markets in Transition Countries, London: Macmillan (2000): Battermann, Harald L., Michael Braulke, Udo Broll, and Jörg Schimmelpfennig, The Preferred Hedge Instrument, Economics Letters 66 (2000): Benninga, Simon, Raphael Eldor, and Itzhak Zilcha, Optimal Hedging in the Futures Market under Price Uncertainty, Economics Letters 13 (1983): Besley, Timothy, Nonmarket Institutions for Credit and Risk Sharing in Low-Income Countries, Journal of Economic Perspectives 9 (1995): Broll, Udo and Jack E. Wahl, Missing Risk Sharing Markets and the Benefits of Cross-Hedging in Developing Countries, Journal of Development Economics 55 (1998): Broll, Udo, Jack E. Wahl, and Itzhak Zilcha, Indirect Hedging of Exchange Rate Risk, Journal of International Money and Finance 14 (1995): Danthine, Jean-Pierre, Information, Futures Prices, and Stabilizing Speculation, Journal of Economic Theory 17 (1978): Fry, Maxwell J., Money, Interest, and Banking in Economic Development, Baltimore: Johns Hopkins University Press (1992). Hakkio, Craig S. and Anne Sibert, The Foreign Risk Premium: Is it Real? Journal of Money, Credit, and Banking 27 (1995): Holthausen, Duncan M., Hedging and the Competitive Firm under Price Uncertainty, American Economic Review 69 (1979): Kawai, Masahiro and Itzhak Zilcha, International Trade with Forward-Futures Markets under Exchange Rate and Price Uncertainty, Journal of International Economics 20 (1986): Kolodko, Grzegorz W., Danuta Gotz-Kozierkiewicz, and Elzbieta Skrzeszewska-Paczek, Hyperinflation and Stabilization in Postsocialist Economies, Boston: Kluwer Academic (1992). Lence, Sergio H., On the Optimal Hedge under Unbiased Futures Prices, Economics Letters 47 (1995): Lessard, Donald R., Financial Risk Management for Developing Countries: A Policy Overview, Journal of Applied Corporate Finance 8 (1995):4 18. Obstfeld, Maurice and Kenneth Rogoff, Foundations of International Economics, Cambridge, MA: MIT Press (1996). OECD, The Baltic States, OECD Economic Surveys (2000). Powell, Andrew, The Management of Risk in Developing Country Finance, Oxford Review of Economic Policy 5 (1989): Rivera-Batiz, Francisco L., and Luis A. Rivera-Batiz, International Finance and Open Economy Macroeconomics, 2nd edn, New York: Macmillan (1994). Sibert, Anne, The Risk Premium in the Foreign Exchange Market, Journal of Money, Credit, and Banking 21 (1989): Notes 1. The case where a type of purchasing power parity does not necessarily hold is discussed in Adam-Müller (2000) and in Battermann and Broll (2000). 2. For example, in 1989 and January 1990 we observe a strong relationship between the monthly rate of depreciation of the new cruzado (Brazil s currency at that time) and Brazil s rate of inflation at the same time (see Table 1).

8 362 Harald L. Battermann and Udo Broll Table 1. The Positive Relationship Between Brazilian Inflation and New Cruzado Depreciation Versus the Dollar Rate of Brazilian new cruzado inflation rate depreciation Year Month (%) (%) 1990 January December November October September Inflation rate = (P d (t) - P d (t - 1))/P d (t - 1) and rate of depreciation = (S(t) - S(t - 1))/S(t - 1); where P d (t - 1) is the consumer price index at time t - 1, P d (t) is the consumer price index at time t, S(t - 1) is the exchange rate at the end of month t - 1, and S(t) is the exchange rate at the end of month t. Source: Wharton Econometric Forecasting Associates Group, Latin America Monthly Economic Report (February 1990), in Rivera-Batiz and Rivera-Batiz (1994). 3. Let us note that in an equilibrium model the real risk premium does not necessarily vanish (Sibert, 1989). 4. Note that the production rule does not depend on Assumption 1.