Currency risk hedging: Futures vs. forward

Currency risk hedging: Futures vs. forward

Journal of Banking & Finance 22 (1998) 61±81 Currency risk hedging: Futures vs. forward Abraham Lioui 1 Department of Economics, Bar-Ilan Universit...

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Journal of Banking & Finance 22 (1998) 61±81

Currency risk hedging: Futures vs. forward Abraham Lioui

1

Department of Economics, Bar-Ilan University, 52900 Ramat-Gan, Israel Received 14 November 1996; accepted 3 July 1997

Abstract The objective of this paper is to address the issue of choosing between currency forward and currency futures contracts when hedging against currency risk within a stochastic interest rates environment. We compare between the hedging e€ectiveness of the two derivative assets both within a narrow sense (i.e., volatility minimization) and within a wide sense (i.e., risk-return trade-o€). When judging hedging e€ectiveness in the narrow sense, forward and futures contracts give identical results even if they do not have identical prices. When judging hedging e€ectiveness in the wide sense, the choice between the two contracts is determined by the correlation between the domestic and the foreign term structures dynamics. Ó 1998 Elsevier Science B.V. All rights reserved. JEL classi®cation: G11; G13; G15 Keywords: Marking-to-market; Hedging e€ectiveness; Continuous time; Martingale approach

1. Introduction Brealey and Kaplanis (1995) have recently shown that commonly used strategies to hedge against currency risk, such as one-period cash ¯ow hedges and long-term ®xed hedges, may leave the ®rm exposed to foreign exchange risk. A

1

Tel.: 972 3 531 8940; fax: 972 3 535 3180; e-mail: [email protected]

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continuously rebalanced hedge is needed in order to enhance the hedging e€ectiveness of a forward contract. The need to improve upon the above-mentioned strategies is particularly strong when interest rates, either both domestic and foreign or only one of the two, are stochastic, as often found in reality. Their results emphasize the importance of deriving optimal forward strategies for hedgers and speculators who are operating within a continuous time framework and are endowed with a currency risk sensitive non-traded cash position. Work along these lines has been done by Briys and Solnik (1992) and Tong (1996) who also show that the forward strategy, under such circumstances, can be decomposed into minimum-variance hedging components, Merton/ Breeden hedging components and speculative components. An extension of the above results can be found in Glen and Jorion (1993) who included in the investor's strategy not only forward contracts but also primitive assets. They compared between the risk-return performance of globally diversi®ed portfolios with and without forward contracts. They show that inclusion of forward contracts results in statistically signi®cant improvements in the performance of the internationally diversi®ed portfolios. All the above-mentioned research has been carried out on the basis of hedging with forward contracts for which markets are most developed. However, there are also futures markets that o€er hedging opportunities. When interest rates are deterministic and there is no di€erence in price between the two derivative assets, the choice between the two derivatives is of no consequence. The risk-return trade-o€ of a portfolio will be identical when using either futures or forward contracts since the hedger is able to reach a perfect hedge in both cases. However, the relevant framework when considering the above results is that of stochastic interest rates, which are likely to introduce complications to the analysis. Forward and futures contracts are no longer interchangeable due to di€erences between them which are brought about by stochastic interest rates (see Cox et al., 1981; Due and Stanton, 1992). The main di€erence between futures and forward contracts results from the payment schedule. Forward contracts charge gains/losses only when the hedge is lifted, while with futures contracts, gains and losses are continuously marked-to-market in a margin account. The marking-to-market procedure has however been ignored by most authors who do so on the basis of empirical results that show no economically signi®cant di€erence in price between the two derivative assets (see Benninga and Protopapadakis, 1994 2 and the references therein). Lately, Dezhbakhsh (1994) has obtained empirical results that are more compatible with theory and show that the small sample inferences

2

Note that Meulbroek (1992) arrives at a di€erent conclusion than Benninga and Protopapadakis (1994) since she found a signi®cant di€erence, for interest rates derivatives, between the forward and futures prices.

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based on the t-test may be suspect since the di€erence between the prices does not follow a normal distribution. He therefore performs non-parametric distribution-free tests which lead to signi®cant divergence between the two prices. He also stresses the role of marking-to-market as the principal reason for this di€erence. The di€erence in prices as outlined above not only emphasizes the need to return to issues that have been analyzed solely on the basis of forward contracts, but also suggests a di€erence in hedging e€ectiveness that should be investigated. In contrast to recent empirical research that has found signi®cant di€erences in prices, no such di€erence has been found with regard to hedging e€ectiveness (see Herbst et al., 1992 and the references therein). Thus support is lent to initial empirical ®ndings (see Cornell and Reinganum, 1981). However, this issue has still not been comprehensively tackled by theoretical literature. The objective of this paper is to address the issue of choosing between currency forward and currency futures contracts when hedging against currency risk within a stochastic interest rates environment. Our analysis is conducted within the widely used Markovian framework for interest rates. We compare between the hedging e€ectiveness 3 of the two derivative assets both within a narrow sense (i.e., volatility minimization) and within a wide sense (i.e., risk-return trade-o€). Section 2 outlines our framework while Section 3 contains the main results. Some concluding remarks are o€ered in Section 4. Appendix A contains some technical derivations. 2. Arbitrage free international ®nancial market We use a traditional model of a frictionless international ®nancial market where trading takes place continuously over the time interval [0,s]. We consider four sources of uncertainty across the two economies, represented by four independent Brownian motions fZ1 …t†; Z2 …t†; Z3 …t†; Z4 …t†; t 2 ‰0; sŠg on a complete probability space (X, F, Q), where X is the state space, F is the r-algebra representing measurable events, and Q is the historical probability measure. This will allow us to have both speci®c factors and common factors a€ecting each of the domestic term structure, the foreign term structure and the exchange rate. All the processes used below are adapted to the augmented ®ltration generated by the four Brownian motions. This ®ltration which is denoted by F  fFt gt2‰0;sŠ satis®es the usual conditions. 4

3

See Howard and d'Antonio (1984, 1987). The r-algebra contains the events whose probability with respect to Q is null (see Karatzas and Shreve, 1991, p. 89). 4

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As usual in the Martingale approach to the term structure, we characterize the domestic and foreign term structures by specifying the evolution of the instantaneous forward interest rates. H1. The domestic instantaneous forward interest rate solves the following stochastic di€erential equation: 5 dfd …t; T † ˆ ld …t; T ; fd …t; T †† dt ‡ md1 dZ1 …t† ‡ md2 dZ2 …t†

…1†

for all x 2 X; t 6 T ; T 2 ‰0; sŠ. ld …t; T ; fd …t; T †† is the drift term that satis®es the usual conditions 6 such that Eq. (1) has a unique solution and md1 and md2 are strictly positive constants. The speci®cation Eq. (1) of the domestic term structure is supported by a general equilibrium model in which two uncorrelated state variables drive the economy. 7 From a partial equilibrium point of view, the two factors term structure models are usually motivated by the presence of a long term factor and a short term factor driving the term structure. We retain the assumption that the instantaneous volatilities parameters (md1 and md2 ) are time invariant to avoid heavy notations. Note that had we assumed time-dependent (deterministic) volatility parameters, the results would not have been a€ected. Of course, conducting our analysis for stochastic volatilities will be a natural extension of our paper as discussed in Section 4 of this paper. The price at each time t of a domestic discount bond maturing at time si is 8 s 9 < Zi = …2† Pd …t; si † ˆ exp ÿ fd …t; T † dT : : ; t

Using Eq. (1), the domestic discount bond price dynamics is as follows: dPd …t; si † ˆ ‰bd …t; si † ‡ rd …t†Š dt ÿ md1 …si ÿ t† dZ1 …t† ÿ md2 …si ÿ t† dZ2 …t†; Pd …t; si † …3† where bd …t; si † is the instantaneous risk premium which could be found by applying Ito's lemma to Eq. (2). 8 We assume a similar structure for the foreign economy. 5 The suggested model for the instantaneous forward rate allows this economic variable to take negative values. Unfortunately, this is one of the few polar cases in Financial Economics which allows for explicit results. Nevertheless, Amin and Morton (1994), when testing contingent claims implications of alternative term structure models, show that the Gaussian models do better than lognormal ones. For a lucid discussion of this issue, see Subrahmanyam (1996). 6 See conditions C.1, p. 80 and C.2, p. 81 of Heath et al. (1992). 7 See Longsta€ and Schwartz (1992). 8 To save space and notation, we do not specify this process. Note that for the remaining, there is no need to specify the process since it is well known that it will not be present in the prices of the contingent claims.

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H2. The foreign economy instantaneous forward interest rate solves the following stochastic di€erential equation: dff …t; T † ˆ lf …t; T ; ff …t; T †† dt ‡ mf2 dZ2 …t† ‡ mf3 dZ3 …t†

…4†

for all x 2 X; t 6 T ; T 2 ‰0; sŠ. lf …t; T ; ff …t; T †† is the drift term that satis®es the usual conditions 9 such that Eq. (4) has a unique solution and mf2 and mf3 are strictly positive constants. The dynamics in Eqs. (1) and (4) incorporate one common factor (Z2 ) which accounts for the instantaneous correlation between the term structures of the two economies. Note that in each economy there is a speci®c factor (Z1 for the domestic term structure and Z3 for the foreign term structure) driving the term structure in addition to the common factor. The price of a foreign discount bond at each time t is 8 s 9 < Zi = …5† Pf …t; si † ˆ exp ÿ ff …t; T † dT : ; t

and its dynamics is as follows: dPf …t; si † ˆ ‰bf …t; si † ‡ rf …t†Š dt ÿ mf2 …si ÿ t† dZ2 …t† ÿ mf3 …si ÿ t† dZ3 …t†; …6† Pf …t; si † where bf …t; si † is the instantaneous risk premium associated with the foreign discount bond. The link between the two economies is guaranteed by the spot exchange rate. It is assumed to evolve as follows. H3. The spot exchange rate (in units of the domestic currency) solves the following stochastic di€erential equation: dS …t† ˆ lS …t; S …t†† dt ‡ mS1 dZ1 …t† ‡ mS2 dZ2 …t† ‡ mS3 dZ3 …t† S …t † ‡ mS4 dZ4 …t†;

…7† 10

where lS (t,s(t)) is the drift term that satis®es the usual conditions such that Eq. (8) has a unique solution; mS: is a strictly positive constant. The speci®cation in Eq. (7) allows the exchange rate to be driven by a source of uncertainty (Z4 ) which does not a€ect the two economies under consideration, in addition to the sources of uncertainty a€ecting the two economies (Z1 , Z2 , Z3 ). This allows us to account for exogenous shocks a€ecting the exchange rate coming from the interdependence of the two economies under consideration with other countries. 9

See conditions C.1, p. 80 and C.2, p. 81 of Heath et al. (1992). See conditions C.1, p. 80 and C.2, p. 81 of Heath et al. (1992).

10

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From a domestic investor's perspective, one needs to express all the foreign asset prices in terms of the domestic currency. This is done by multiplying the price of a foreign asset by the spot exchange rate. We will denote by P^f …t; si † the price of a foreign discount bond in units of the domestic currency. Therefore, we have P^f …t; si † ˆ Pf …t; si †S …t†:

…8†

Applying Ito's lemma to Eq. (8) and using Eqs. (6) and (7), one gets the dynamics of the foreign discount bond in units of the domestic currency, namely, dP^f …t; si † ˆ b^f …t; si † dt ‡ mS1 dZ1 …t† ‡ ‰mS2 ÿ mf2 …si ÿ t†Š dZ2 …t† P^f …t; si † …9† ‡ ‰mS3 ÿ mf3 …si ÿ t†Š dZ3 …t† ‡ mS4 dZ4 …t†; ^ where bf …t; si †  bf …t; si † ‡ rf …t† ‡ lS …t; s…t†† ÿ mS2 mf2 …si ÿ t† ÿ mS3 mf3 …si ÿ t†: We assume the international ®nancial market to be complete and arbitrage free. All the portfolio strategies to be considered in Section 3 are assumed to be admissible portfolio strategies. 11 3. The main results Having set out the framework for analysis we can now proceed with the main objective of the paper. First, we derive the forward contract price and the corresponding futures settlement price and then we set the hedger's problem when confronted with a commitment to pay or receive a certain quantity of the currency. Thirdly, we solve his problem in two separate cases, one when futures contracts are traded and the other when forward contracts are traded. Lastly, we perform a comparison between these two strategies to determine which of the two derivative contracts is preferable from a hedging e€ectiveness point of view. 3.1. Forward price and futures settlement price Within the framework outlined above, the arbitrage free price of a currency forward contract maturing at time s1 , denoted by G…t; s1 †, is G…t; s1 † ˆ

P^f …t; s1 † Pf …t; s1 † ˆ S …t † Pd …t; s1 † Pd …t; s1 †

…10†

11 For an explicit de®nition of admissible strategies in our framework, see Amin and Jarrow (1991).

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while the arbitrage free settlement price of the corresponding futures contract is 12 ( ÿ  …s1 ÿ t†3 ‡ …vdl vS1 H …t; s1 † ˆ G…t; s1 † exp v2dl ‡ v2d2 3 ) …s1 ÿ t†2 …s1 ÿ t†3 ‡vd2 vS2 † ÿ vd2 vf 2 : …11† 2 3 As expected we arrive at a di€erence between the futures settlement price and the forward contract price. Note that a deterministic domestic term structure leads to identical prices. This is due to the fact that the margin account yields or charges are in terms of the domestic interest rate. To further investigate the di€erence in prices, let us write Eq. (11) as follows: ( 2 …s1 ÿ t † P^f …t; s1 † exp q…t; s1 † ‡ …md1 mS1 H …t; s1 † ˆ 3 Pd …t; s1 † ) …s1 ÿ t †2 ; …110 † ‡md2 mS2 † 6 where q…t; s1 †  ‰md1 …mS1 ‡ md1 …s1 ÿ t†† ‡ md2 …mS2 ‡ …md2 ÿ mf2 †…s1 ÿ t††Š: One possible interpretation of this di€erence in prices could be as follows. When q…t; s1 † is positive, the futures contract settlement price is greater than the forward price. Therefore, a buyer of a futures contract is willing to pay more than for a forward contract and a seller will ask for a higher price. To understand this, ®rst, note that q…t; s1 † is the instantaneous covariance between the ¯uctuations of the domestic term structure and the local ¯uctuations of the futures settlement price. Therefore, q…t; s1 † > 0 means that an increase in (domestic) interest rates is followed by an increase in the futures settlement price. A buyer of futures contracts could then invest the margin he receives at a higher interest rate. Similarly, a seller of futures contracts will ®nance the margin that he must pay at a higher (domestic) interest rate. Because the seller is adversely a€ected by the (domestic) interest rates ¯uctuations, the settlement price he will set for the futures contract will be higher than the price for a forward contract. The buyer, who gains from interest rates ¯uctuations, is willing to pay more for the futures contract than for the forward contract.

12

While deriving Eq. (10) is trivial this is not the case for the futures settlement price. Our framework is similar to the one assumed by Amin and Jarrow (1991), namely the completeness of the primitive assets market, and therefore one can recover Eq. (11) using the results of these authors. Nevertheless, a complete proof of Eq. (11) is given in Appendix A for self-countenance of the paper.

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Assume now that q…t; s1 † is suciently 13 negative such that the futures settlement price is less than the forward price. Then the futures ¯uctuations are opposite to the ¯uctuations of the domestic term structure. An increase in interest rates implies a decrease in the futures settlement price. A seller of futures contracts will then receive a margin that he can invest at a higher interest rate. A buyer pays a margin that he must ®nance at a higher interest rate. Therefore, the buyer who is adversely a€ected by the ¯uctuations of the domestic term structure is willing to buy the futures contract only at a price which is lower than that of the corresponding forward contract. The above analysis is consistent with the way the forward price and the futures settlement price di€er. An inspection of Eq. (110 ) shows that the forward price and the futures settlement prices dynamics will have the same instantaneous volatilities while their drifts will di€er. Had they had di€erent instantaneous volatilities, we would not have been able to say anything (in general) concerning the relationship between the futures settlement price and the forward price. 3.2. The hedger's problem The hedger has a commitment to p units (he pays when p < 0 and he receives when p > 0) of the foreign currency at time s1 < s which is his hedging horizon. The hedger can hedge against currency risk by using either forward or futures contracts. Let bH (t) denote the number of futures contracts held by the hedger at time t for the purpose of hedging his non-traded position. The futures contract positions are marked-to-market in a domestic interest rate bearing account. If we denote by X(t) the value of the margin account at time t, then 8 t 9 Zt
s

The hedger's wealth at each time t, denoted WH …t†, is as follows: WH …t† ˆ pS …t† ‡ X …t†:

…13†

Despite being allowed to trade the riskless asset (i.e., the domestic money market account), we do not include it in his portfolio because the corresponding position is indeterminate. To see why this is the case, let us recall the solution to the hedging problem in the case of deterministic interest rates (domestic and foreign). One computes the volatility and the return of the hedged portfolio which is constituted of the non-traded position, the margin account and a

13

Assume for example that jq…t; s1 †j > …md1 mS1 ‡ md2 mS2 †…s1 ÿ t†2 =2:

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riskless asset position. Since interest rates are deterministic, a perfect hedge of the non-traded position is possible and the futures position is determined such that the volatility of the hedged portfolio is zero. However, since the portfolio is riskless, it should yield the riskless interest rate. Therefore, one must set the return on the portfolio to be equal to the riskless rate and then solve for the riskless asset's position. Interest rates are stochastic, therefore our investor is able only to achieve a minimum-variance hedge and therefore the return on the hedged portfolio will di€er from the spot interest rate. Finally, had we included a position in a riskless asset, we would not have been able to determine it since one cannot say anything concerning the return on the hedged portfolio. Turning now to an investor who uses forward contracts to hedge against currency risk, the investor's wealth at each time t is as follows: Zt WG …t† ˆ pS …t† ‡

Pd …s; s1 †bG …s† dG…s; s1 †;

…14†

0

where bG …t† is the number of forward contracts held by the hedger in his portfolio at time t. The second term on the right-hand side of Eq. (14) represents the present value of the gains and losses on the forward positions. The hedger's problem is to construct the optimal derivatives strategy so as to minimize the instantaneous volatility of his wealth. As a result he would have made a decision as to whether he prefers forward or futures contracts. 3.3. The solutions to the hedger's problem The forward and the futures strategies, respectively, are as follows: r

bG …t† ˆ ÿp

S …t † m…t; s1 † mS ; Pd …t; s1 †G…t; s1 † m…t; s1 †r m…t; s1 †

bH …t† ˆ ÿp

S …t† m…t; s1 † mS ; H …t; s1 † m…t; s1 †r m…t; s1 †

where

14

0

…15†

r

mS1 ‡ md1 …s1 ÿ t†

…16† 0

1

B m ÿ m …s ÿ t † ‡ m …s ÿ t † C f2 1 d2 1 B S2 C m…t; s1 †  B C @ A mS3 ÿ mf3 …s1 ÿ t†

and

mS1

1

Bm C B S2 C mS  B C: @ mS3 A

mS4 mS4 Eqs. (15) and (16) give the number of forward contracts and futures contracts, respectively, held by the investor at each time t. To get the number of contracts 14

Both strategies are derived in Appendix A.

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exchanged between t and t ‡ dt it suces to di€erentiate Eqs. (15) and (16) using Ito's lemma. These two strategies call for the following remarks. (i) The two strategies have an identical structure which comprises the size of the non-traded position multiplied by a price adjusting factor …1=Pf …t; s1 † or S…t†=H …t; s1 †† and by a volatility adjusting factor …m…t; s1 †r mS =m…t; s1 †r m…t; s1 ††. The volatility adjusting factor is the usual instantaneous minimum-variance hedge ratio of the ¯uctuations of the exchange rate using the derivative contracts. It is important to emphasize that the futures contract and the forward contract strategies share the same minimum-variance hedge ratio component because stochastic interest rates a€ect only the drift and not the volatility of the futures contract, which remains equal to the volatility of the forward contract (see the preceding paragraph for the economic intuition of this result). The minimum-variance hedging factor adds a time dimension to the two strategies. However, unlike previous research, the source of this time dimension is not the volatility of the underlying asset (Kroner and Sultan, 1993), but rather the stochastic interest rates that generate a need to continuously rebalance. The price adjusting factors are also a source of a time dimension that strengthen the need to continuously rebalance. (ii) An interesting characteristic of the two strategies is that the hedger will not necessarily choose a derivative contract's position which is opposite in sign to the non-traded position. To see this, it suces to remark that 4 X r m2Si ‡ ‰…mS1 md1 ‡ mS2 md2 † ÿ …mS2 mf2 ‡ mS3 mf3 †Š…s1 ÿ t†: …17† m…t; s1 † mS …t† ˆ iˆ1

A sucient condition for the investor to take a position in the derivative assets market which is opposite in sign to the non-traded position is that ‰…mS1 md1 ‡ mS2 md2 † ÿ …mS2 mf2 ‡ mS3 mf3 †Š > 0. This condition can be given an intuitive explanation since ‰…mS1 md1 ‡ mS2 md2 † ÿ …mS2 mf2 ‡ mS3 mf3 †Š is the di€erence between (a) the instantaneous covariance between the exchange rate dynamics and the domestic term structure dynamics and (b) the instantaneous covariance between the exchange rate dynamics and the foreign term structure dynamics. 3.4. Futures vs. forwards Now that we have derived the hedging strategies chosen by the investor when hedging with forward or futures contracts, we are able to compare between the two strategies and decide which of the two contracts is preferable. The framework of our analysis dictates a di€erence in price between the two derivative assets, which plays an important role in our analysis. The ®rst result is as follows.

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3.4.1. Rule 1 When judging hedging e€ectiveness in the narrow sense, i.e., measuring the volatility of a hedged portfolio, forward and futures contracts give identical results. This ®nding can be explained by the fact that the instantaneous volatility of the forward contract is equal to that of the futures contract as explained in Section 3.1. Our results ®ll the gap left by previous theoretic research that has not addressed the issue of hedging e€ectiveness. Furthermore, our conclusion is compatible with empirical ®ndings in the ®eld of hedging e€ectiveness (see Herbst et al., 1992). However, the above result is not sucient for a comprehensive analysis, because the hedger's portfolio is not risk-free, thus we are compelled to measure hedging e€ectiveness in a wider sense, i.e., the risk-return trade-o€. For such an analysis we need only compare between the drifts of the two hedged portfolios, due to the equality in instantaneous volatility. When comparing between the two drifts we obtain the following result: 15 r

lH …t† ÿ lG …t† ˆ rd …t†X …t† ‡

m…t; s1 † mS

…18† q…t; s1 †…s1 ÿ t†pS …t†; m…t; s1 †r m…t; s1 † where lH …t† is the instantaneous drift of the hedger's portfolio when futures are used, lG …t† is the instantaneous drift of the hedger's portfolio when forward contracts are used. The ®rst term on the right-hand side of Eq. (18) depicts the accrued interest on the margin account due to marking-to-market of the hedger's position. The second term on the right-hand side of Eq. (18) re¯ects the contribution of interest rates risk to the di€erence in the drifts. This is so because it is a function (through q( )) of the di€erence in the drifts between the futures settlement price dynamic and the forward price dynamic. The intuition behind the second term on the right-hand side of Eq. (18) is best captured by rewriting Eq. (18) using Eq. (16) as follows: lH …t† ÿ lG …t† ˆ rd …t†X …t† ÿ bH …t†H …t; s1 †q…t; s1 †…s1 ÿ t†:

…180 †

The second term on the right-hand side of Eq. (180 ) can now be interpreted as the ``premium'' paid or received depending on the sign of the futures position and on the di€erence between the forward price and the futures settlement price. An immediate result is, that in general one cannot determine which of the two derivatives is preferable when the domestic interest rates are stochastic. The reason for this is that outcome depends on the signs taken on by bH …t† and q…t; s1 †. The following table gives all the possible outcomes for di€erent parameters of the model.

15

See the derivation of this result in Appendix A.

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3.4.2. Rule 2 q…t; s1 † > 0

q…t; s1 † < 0

m…t; s1 † mS > 0

r

p > 0 ) futures p < 0 ) forward

p > 0 ) forward p < 0 ) futures

r

p > 0 ) forward p < 0 ) futures

p > 0 ) futures p < 0 ) forward

m…t; s1 † mS < 0

The above table describes the cases in which a hedger who has a long position (p > 0, the long-hedger hereafter) will prefer one derivative asset over the other, and the corresponding results for a hedger who has a short position (p < 0, the short-hedger hereafter). Remember that the sign of m…t; s1 †r mS which is the instantaneous correlation between the ¯uctuations of the derivative contracts and the spot exchange rate, determines the sign of the derivative contracts position (b given by Eqs. (15) and (16)) in relation to the sign of the non-traded position. Furthermore, q( ) is the instantaneous correlation between the price dynamics of the derivative contracts and the dynamics of the domestic term structure. 3.4.2.1. Analysis of the top left-hand cell of the table. When m…t; s1 †r mS is positive, a hedger will always choose a position in the derivative contract that is opposite in sign to the non-traded position. Thus, the long-hedger (p > 0) will short the derivative contract while the short-hedger will be long in derivative contract. When q…t; s1 † is positive, then the table shows that the long-hedger chooses to short the futures contract and the short-hedger chooses to be long in the forward contract. The choice between futures and forward contracts can be given the following intuitive explanation. 16 When q…t; s1 † is positive, a long-hedger will always bene®t from an increment in the instantaneous drift of his wealth, i.e., ÿbH …t†H …t; s1 †q…t; s1 †…s1 ÿ t† > 0. The short-hedger is a€ected adversely by

16 At ®rst glance a simpler intuition for this could be that when q(t,s1 ) is positive, the futures settlement price is greater than the forward price and therefore the long-hedger prefers to short the most expensive contract. Similarly, the short-hedger must buy derivative contracts to hedge his non-traded position and therefore prefers to buy the forward contract which is cheaper. However, this kind of explanation is not satisfactory since it does not explain the right-hand column of the table. When q(t,s1 ) is non-positive, there is still a range of values for q(t,s1 ) such that the futures settlement price is greater than the forward price and the long-hedger will nevertheless prefer to hedge using forward contracts!

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a positive q…t; s1 † since ÿbH …t†H …t; s1 †q…t; s1 †…s1 ÿ t† is negative and he will therefore choose the forward contract for hedging against the risk that stems from his non-traded position. The perceptive reader will notice that the explanations so far, as well as the results that appear in Rule 2, relate only to the e€ect that changes in interest rates have on the di€erence between the drifts of the two derivative assets (see the second term on the right-hand side of Eq. (18)). However, we can show that the ®rst term on the right-hand side of Eq. (18), namely, the traditional component which consists of interest accrued in the margin account, will not a€ect our results. To understand why this is true, consider an extremely bad case, from a long-hedger's point of view, in which interest rates are constantly increasing. The long-hedger who is short in the futures contract, will ®nd himself constantly paying a margin. This is so because a positive q( ) implies a positive correlation between changes in the interest rates and changes in the futures settlement price. Thus, any increase in interest rates brings about an increase in the settlement price of the futures contract thereby creating a need to pay a margin. Moreover, the newly created margins must be ®nanced by ever growing interest rates. However, the increase in expense incurred by the long-hedger is likely to be more than o€set by the premium from trading in futures contracts, (ÿbH …t†H …t; s1 †q…t; s1 †…s1 ÿ t††. Note that the premium is a function of the value at time t of the non-traded position (see Eq. (18)) while the interest rate is paid only on margins, i.e., on the changes in the futures settlement prices. Turning now to the short-hedger a similar analysis can be made. Consider the best situation a short-hedger can ®nd himself in, namely, increasing interest rates. Assume that this short-hedger has chosen a long position in futures contracts. The increase in interest rates creates a new margin in his favor, which he can invest at a higher interest rate. Furthermore, this increase in income is unlikely to o€set the adverse e€ect from trading in futures contracts brought about by the negative premium …ÿbH …t†H …t; s1 †q…t; s1 †…s1 ÿ t† is negative). Thus, the short-hedger will prefer to trade forward contracts. The remaining three cells of Rule 2 can be explained in a similar manner. The analysis of these three cells is omitted to avoid repetitions that do not add intuition to the paper. Furthermore, note that from a practical point of view, the currency and the derivative contracts are likely to be very highly correlated, i.e., m…t; s1 †r mS > 0. On the other hand, q…t; s1 † > 0 unless vf2  0. Thus, it appears that the only cell in Rule 2 that is relevant is the top left-hand side one. q…t; s1 † also has an impact on the size of the derivative contract's position, since 17

17

This term is known in the literature as the ``tailing factor''. See Figlewski et al. (1991).

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A. Lioui / Journal of Banking & Finance 22 (1998) 61±81

bH …t† ˆ Pd …t; s1 † exp bG …t†

(

2

…s1 ÿ t† …s1 ÿ t† q…t; s1 † ÿ …vd1 vS1 ‡ vd2 vS2 † ÿ 3 6

2

) : …19†

When q…t; s1 † > 0, the futures position is always smaller than the forward position. To understand why this is the case it suces to rewrite Eqs. (15) and (16) as follows: bG …t†G…t; s1 † ˆ ÿp

5

S…t† v…t; s1 † vS ; Pd …t; s1 † v…t; s1 †5 v…t; s1 †

bH …t†H …t; s1 † ˆ ÿpS…t†

5

v…t; s1 † vS : 5 v…t; s1 † v…t; s1 †

…150 † …160 †

Therefore, the value of the forward position is always greater than the value of the futures position at each time t due to the presence of the discount factor in Eq. (150 ). Moreover, when q…t; s1 † > 0, the futures settlement price is greater than the forward price. Consequently, the hedger must always hold less futures contracts than forward contracts. Note that when interest rates are deterministic, it is well known that the marking-to-market of the futures position requires a continuous rebalancing of the futures position in order to achieve a perfect hedge. The ratio of the futures position to the forward position is called the tailing factor. A similar result is found in Eq. (19), i.e., the discount factor. In the presence of stochastic interest rates, a second factor is added which is brought about by the di€erence in prices between the two contracts. Note that q…t; s1 † > 0 is only a sucient condition but this case is the one which is the most likely to prevail in the international ®nancial markets. 4. Concluding remarks The main message of this paper is that currency forward contracts and currency futures contracts are not interchangeable when interest rate risk exists. Therefore, conclusions obtained in the ®eld of International Hedging and Synthetic International Diversi®cation are sensitive to the nature of the derivative contract included in the portfolios. For example, it is not clear that the results of Glen and Jorion (1993) will still hold when using futures contracts instead of forward contracts. The dynamic hedging strategy derived by Briys and Solnik (1992) and used by Tong (1996) is also likely to be a€ected by the marking-tomarket procedure. One important question from a practical point of view is to what extent the introduction of frictions in the international ®nancial market (such as transac-

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tion costs) could change our results. Fortunately, our results are likely to be robust even if one takes into account the presence of transaction costs. The reason for this is simply because strategies Eqs. (15) and (16) need a continuous rebalancing of the hedger's portfolio. Moreover, the futures position is always smaller than the forward positions. Therefore, transactions costs, should they be taken into account, would probably not change the results in the table. Our results are sensitive to the dynamic of the domestic term structure postulated a priori. While the assumption of deterministic instantaneous volatility of the domestic term structure has been made for convenience, there is no doubt that a more realistic framework should account for the possibility of stochastic volatility. Moreover, because interest rates are the main tool of monetary policies, a more realistic modeling of the term structure dynamics should include the possibility of jumps in the dynamic of the domestic interest rates. However, such properties, although important, would have made our analysis almost entirely untractable since it is well known that in the case of jumps, for example, even explicit discount bond prices are hard to obtain (see El-Jahel et al., 1996). Our results could be extended in several other directions. While only the case of a hedger has been addressed in this paper, the e€ect on the behavior of a speculator (expected utility maximizer) could also be examined. Di€erences in the welfare reached in the case of forward trading and futures contracts trading are likely to exist since the two contracts generate di€erent opportunity sets. However the task of comparing between the welfare levels could be dicult since the speculator has access to an incomplete market in both cases. 18 Several currencies are subject to a realignment risk since they belong to a target zone. Dumas et al. (1995) analyzed the e€ect of this risk on currency option pricing. It is likely that, due to stochastic interest rates and the marking-tomarket procedure, the futures settlement price will incorporate a premium to compensate for this risk. How this risk a€ects the hedging strategy and the hedging e€ectiveness of both contracts is an important question both from a theoretical and an empirical point of view. The presence of the Siegel Paradox in currency options pricing had been widely studied (see Bardhan, 1995). The marking-to-market procedure could be a source of a Siegel Paradox in pricing futures contracts since the margin account bears the domestic spot rate.

18

The paper by Lioui and Poncet (1996) is an attempt to apply the martingale approach to address the issue of intertemporal hedging in complete markets. Such an analysis could be extended to the suggested problem based on the results of He and Pearson (1991) and Karatzas et al. (1991).

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Acknowledgements I would like to thank Rafael Eldor, Miriam Krausz and especially Patrice Poncet for very insightful discussions and comments. Two anonymous referees provided extensive comments and suggestions which greatly improved the presentation of the paper. All remaining errors are mine.

Appendix A A.1. Derivation of Eqs. (10) and (11) The derivative contracts represent the right to get one unit of the foreign currency at the contract's maturity date, namely s1 . Therefore, it is straightforward to show that the forward contract price in units of domestic currency, denoted G…t; s1 †, is P^f …t; s1 † Pf …t; s1 † ˆ S…t†: …A:1† Pd …t; s1 † Pd …t; s1 † The international ®nancial market is complete and arbitrage free. This implies that there exists a (unique) probability measure associated to the numera~ such that the dynamics of the domestic ire Bd ( ), equivalent to Q denoted Q, discount bond and the foreign discount bond price processes could be written as follows: G…t; s1 † ˆ

dPd …t; s† ˆ rd …t† dt ÿ vd1 …s ÿ t† dZ~1 …t† ÿ vd2 …s ÿ t† dZ~2 …t†; Pd …t; s† dP^f …t; s† ˆ rd …t† dt ‡ vs1 dZ~1 …t† ‡ ‰vS2 ÿ vf2 …s ÿ t†Š dZ~2 …t† P^f …t; s†

…A:2†

…A:3†

‡ ‰vS3 ÿ vf3 …s ÿ t†Š dZ~3 …t† ‡ vS4 dZ~4 …t†;

where Z~1 …t†; Z~2 …t† Z~3 …t†; Z~4 …t† are standard Brownian motions with respect to ~ Moreover, following Heath et al. (1992), 19 one obtains that Q.  t …A:4† fd …t; T † ˆ fd …0; T † ‡ …v2d1 ‡ v2d2 †t T ÿ ‡ vd1 Z~1 …t† ‡ vd2 Z~2 …t† 2 and then rd …t† ˆ fd …t; t† ˆ fd …0; t† ‡ …v2d1 ‡ v2d2 †

19

Proposition 3, p. 86, of Heath et al. (1992).

t2 ‡ vd1 Z~1 …t† ‡ vd2 Z~2 …t†: 2

…A:5†

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77

From Due and Stanton (1992), the futures settlement R s price at time t is equal to the price at time t of a cash ¯ow S…s1 † exp t 1 rd …u† du at the instant s1 . Therefore, if H(t,s1 ) denotes the futures settlement price, it is such that  R s1   rd …u† du H …t; s1 † Q~ S…s1 † exp 1 …A:6† ˆE Ft Bd …s1 † Bd …t† and ~

H …t; s1 † ˆ EQ ‰S…s1 †jFt Š:

…A:7†

Using Eq. (8), this can also be written as follows: ~

H …t; s1 † ˆ EQ ‰P^f …s1 ; s1 †jFt Š:

…A:8†

Applying Ito's lemma to Ln P^f …t; s1 †, one obtains 8 t 
Zt ‡ 0

9 =

~ n…u† dZ…u† ; ; 5

…A:9†

where n…t†  …vS1 ; vS2 ÿ vf2 …s1 ÿ t†; vS3 ÿ vf3 …s1 ÿ t†; vS4 †5 . Then, 8s 
9 Zs1 = 5 ~ : ‡ n…u† dZ…u† ;

…A:10†

t

Using Eqs. (A.4) and (A.5), one can show that Zs1

Zs1 rd …u† du ˆ

t

fd …t; T † dT ‡ …v2d1 ‡ v2d2 †

t

Zs1 ‡ vd1 t

…s1 ÿ t† 6

…s1 ÿ u† dZ~1 …u† ‡ vd2

Zs1

3

…s1 ÿ u† dZ~2 …u†:

…A:11†

t

Substituting for Eq. (A.11) in Eq. (A.10) and using Eq. (2), Eq. (A.10) becomes

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8 Zs1 3 < ^f …t1 ; s1 † 1 P 5 2 2 …s1 ÿ t† ^ exp …vd1 ‡ vd2 † ÿ n…u† n…u† du Pf …t1 ; s1 † ˆ : 6 Pd …t; s1 † 2 t 9 Zs1 = 5 ~ ; …A:12† ‡ …n…u† ‡ v…u†† ‡ dZ…u† ; t

5

where v…t†  …vd1 …s1 ÿ t† vd2 …s1 ÿ t† 0 0† . The futures settlement price is thus 8 2 <ÿ ^  …s1 ÿ t†3 Pf …t; s1 † ~  EQ 4 exp m2d1 ‡ m2d2 H …t; s1 † ˆ : 6 Pd …t; s1 † 9 3 Zs1 Zs1 = 1 r r ~ …A:13† n…u† n…u† du ‡ …n…u† ‡ v…u†† dZ …u† Ft 5: ÿ ; 2 t

t

With respect to the ®ltration generated by the Brownian motion Z( ), the stochastic integral in the exponential term on the right-hand side of Eq. (A.13) is independent of this ®ltration (thanks to Novikov's criterion), and is distributed as a Gaussian random variable. Result Eq. (11) follows. A.2. Derivation of Eqs. (15) and (16) Applying Ito's lemma to Eqs. (10) and (11), one obtains the dynamics of the forward contract price and the futures contract settlement price as follows: dG…t; s1 † r ˆ lG …t; s1 † dt ‡ m…t; s1 † dZ~…t† G…t; s1 †

…A:14†

and dH …t; s1 † ˆ ‰lG …t; s1 † ÿ q…t; s1 †…s1 ÿ t†Š dt ‡ m…t; s1 †r dZ~…t†: H …t; s1 † Now applying Ito's lemma to Eq. (14) we get dWG …t† ˆ p dS …t† ‡ Pd …t; s1 †bG …t† dG…t; s1 †:

…A:15†

…A:16†

Substituting for Eqs. (7) and (A.14) in Eq. (A.16), one obtains dWG …t† ˆ lG …t† dt ‡ ‰pS …t†mS ‡ Pd …t; s1 †bG …t†G…t; s1 †m…t; s1 †Š dZ~…t†; r

…A:17†

where lG …t†  pS…t†lS …t† ‡ Pd …t; s1 †bG …t†G…t; s1 †lG …t; s1 †:

…A:18†

The instantaneous variance of this wealth is ‰pS …t†mS ‡ Pd …t; s1 †bG …t†G…t; s1 †m…t; s1 †Šr ‰pS …t†mS ‡Pd …t; s1 †bG …t†G…t; s1 †m…t; s1 †Š

…A:19†

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79

and Eq. (15) follows. To derive the dynamics of Eq. (13), apply Ito's lemma to Eqs. (12) and (13) to get: dX …t† ˆ rd …t†X …t† dt ‡ bH …t† dH …t; s1 †;

…A:20†

dWH …t† ˆ p dS …t† ‡ rd …t†X …t† dt ‡ bH …t† dH …t; s1 †:

…A:21†

Substituting for Eqs. (7), (A.15) and (A.20) in Eq. (A.21), one gets dWH …t† ˆ lH …t† dt ‡ ‰pS …t†mS ‡ bH …t†H …t; s1 †m…t; s1 †Š dZ~…t†; r

…A:22†

where lH …t†  pS …t†lS …t† ‡ rd …t†X …t† ‡ bH …t†H …t; s1 †‰lG …t; s1 † ‡q…t; s1 †…s1 ÿ t†Š:

…A:23†

The instantaneous variance is ‰pS …t†mS ‡ bH …t†H …t; s1 †m…t; s1 †Šr ‰pS …t†mS ‡ bH …t†H …t; s1 †m…t; s1 †Š

…A:24†

and Eq. (16) follows. A.3. Proof of Rule 1 Using bG …t† given in Eqs. (15) and (A.17), one can write the hedger's wealth volatility as follows: pS …t†mS ‡ Pd …t; s1 †bG …t†G…t; s1 †m…t; s1 † r

ˆ pS …t†mS ÿ pS …t†

m…t; s1 † mS

m…t; s1 †: …A:25† r m…t; s1 † m…t; s1 † Similarly, using Eqs. (16) and (A.22), the hedger's wealth volatility when futures are used is pS …t†mS ‡ bH …t†H …t; s1 †m…t; s1 † r

ˆ pS …t†mS ÿ pS …t†

m…t; s1 † mS r

m…t; s1 † m…t; s1 † which yields the desired result.

m…t; s1 †

…A:26†

A.4. Proof of Rule 2 Using Eqs. (A.18) and (15), the drift of the hedger's wealth when forward contracts are used as hedging instruments is lG …t† ˆ pS …t†lS …t† ‡ Pd …t; s1 †bG …t†G…t; s1 †lG …t; s1 † r

ˆ pS …t†lS …t† ÿ pS …t†

m…t; s1 † mS r

m…t; s1 † m…t; s1 †

lG …t; s1 †

…A:27†

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and using Eqs. (A.23) and (16), one shows that lH …t† ˆ pS …t†lS …t† ‡ rd …t†X …t† ‡ bH …t†H …t; s1 †‰lG …t; s1 † ÿ q…t; s1 †…s1 ÿ t†Š r

ˆ pS …t†lS …t† ‡ rd …t†X …t† ÿ pS …t†

m…t; s1 † mS r

m…t; s1 † m…t; s1 †

ÿq…t; s1 †…s1 ÿ t†Š:

‰lG …t; s1 † …A:28†

By comparing Eqs. (A.27) and (A.28) one gets Eq. (18).

References Amin, K., Jarrow, R., 1991. Pricing foreign currency options under stochastic interest rates. Journal of International Money and Finance 10, 310±329. Amin, K., Morton, A., 1994. Implied volatility functions in arbitrage free term structure models. Journal of Financial Economics 35, 141±180. Bakshi, G., Chen, Z., 1996. Equilibrium valuation of foreign exchange claims. Journal of Finance 52, 799±826. Bardhan, I., 1995. Exchange rate schocks, currency options and the Siegel Paradox. Journal of International Money and Finance 14, 441±458. Benninga, S., Protopapadakis, A., 1994. Forward and Futures prices with markovian interest rates processes. Journal of Business 67, 401±421. Brealey, R., Kaplanis, E., 1995. Discrete exchange rate hedging strategies. Journal of Banking and Finance 19, 765±784. Briys, E., Solnik, B., 1992. Optimal currency hedge ratios and interest rates risk. Journal of International Money and Finance 11, 431±445. Cornell, B., Reinganum, M., 1981. Forward and futures prices: evidence from the foreign exchange market. Journal of Finance 36, 1035±1045. Cox, J., Ingersoll, J., Ross, S., 1981. The relation between forward and futures prices. Journal of Financial Economics 9, 321±346. Dezhbakhsh, H., 1994. Foreign exchange forward and futures prices: are they equal. Journal of Financial and Quantitative Analysis 29, 75±87. Due, D., Stanton, R., 1992. Pricing Continuously resettled contingent claim. Journal of Economic Dynamics and Control 16, 561±573. Dumas, B., Jennergren, P., Naslund, B., 1995. Realignment risk and currency option pricing in target zones. European Economic Review 39, 1523±1544. El-Jahel, L., Lindberg, H., Perraudin, W., 1996. Yield curves with jump short rates, The Institute for Financial Research (Working Paper). Figlewski, S., Landskroner, Y., Silber, W., 1991. Tailing the hedge: Why and How. Journal of Futures Markets 11, 201±212. Glen, J., Jorion, P., 1993. Currency hedging for international portfolios. Journal of Finance XLVIII, 1865±1886. He, H., Pearson, N., 1991. Consumption and portfolio policies with incomplete markets and short sale constraints. Journal of Economic Theory 54, 259±304. Heath, D., Jarrow, R., Morton, A., 1992. Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60, 77±105. Herbst, A., Swanson, P., Caples, S., 1992. A redetermination of hedging strategies using foreign currency futures contracts and forward markets. Journal of Futures Markets 12, 93±104.

A. Lioui / Journal of Banking & Finance 22 (1998) 61±81

81

Howard, C., d'Antonio, L., 1984. A risk-return measure of hedging e€ectiveness. Journal of Financial and Quantitative Analysis 19, 101±112. Howard, C., d'Antonio, L., 1987. A risk-return measure of hedging e€ectiveness: a reply. Journal of Financial and Quantitative Analysis 22, 377±381. Karatzas, I., Lehoczky, J., Shreve, S., Xu, G., 1991. Martingale and duality methods for utility maximization in an incomplete market. SIAM Journal Control and Optimization 29, 702±730. Karatzas, I., Shreve, S., 1991. Brownian motion and stochastic calculus, 2nd ed. Springer, Berlin. Kroner, K., Sultan, J., 1993. Time-varying distributions and dynamic hedging with foreign currency futures. Journal of Financial and Quantitative Analysis 28, 535±551. Lioui, A., Poncet, P., 1996. Optimal hedging in a dynamic futures market with a non-negative constraint on wealth. Journal of Economic Dynamics and Control 20, 1101±1113. Longsta€, F., Schwartz, E., 1992. Interest Rate Volatility and the term structure: A Two ± Factor General equilibrium model. Journal of Finance XLVII, 1259±1282. Meulbroek, L., 1992. A comparison of forward and futures prices of an interest rate sensitive ®nancial asset. Journal of Finance XLVII, 381±396. Subrahmanyam, M., 1996. The term structure of Interest rates: Alternative Paradigms and Implications for Financial Risk Management. Geneva Papers on Risk and Insurance Theory 21, 7±29. Tong, W., 1996. An examination of dynamic hedging. Journal of International Money and Finance 15, 19±35.