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Exchange rate volatility and international trading strategy
Exchange rate volatility and international trading strategy
G~_~NTER FRANICE* Unicersittit Konstnn:,
77.50 Konstatc
I, Germcrn~~
It is usually argued that an increase in exchange rate volatility reduces the volume of international trade as trading firms are risk averse. This paper shows that exporting firms benefit under fairly general conditions from an increase in exchange rate volatility. Firms optimally adjust their export volumes to the level of the exchange rate. Exporting is an option which is exercised if profitable. In addition. the paper presents conditions under which the volume of international trade grows with exchange rate volatility.
Since the collapse of the Bretton Woods agreement in 1973 various papers have investigated the effects of exchange rate volatility on international trade. Many of the early theoretical papers (e.g., Baron, 1976; Clark, 1973; Ethier, 1973) show that an increase in exchange rate volatility will reduce the volume of international trade. The basic argument of these papers can be summarized as follows: the export volume is assumed to be independent of the exchange rate level. The exchange rate risk is the main source of the exporter’s profit risk. If exchange rate volatility increases, then profit risk increases. Since exporters are risk averse and hedging against exchange rate risk is costly or impossible, the increase in profit risk reduces the benefits and therefore the volume of international trade. Hence an increase in exchange rate volatility reduces the volume of international trade. This proposition has been tested frequently (e.g., Hooper and Kohlhagen, 1978; Cushman, 1983, 1986, 1988; Akhtar and Hilton, 1984; Gotur, 198.5; Thursby and Thursby, 1985). Hooper and Kohlhagen (1978) find no significant effects of exchange rate volatility. Subsequent studies found numerous examples of negative effects on bilateral trade flows. Perhaps the strongest support is given by Cushman (1983, 1988) who finds a significant effect in about half of his bilateral trade analyses. But, as Willet (1986, p. S106) points out, the ‘latter results have been ‘I acknowledge the helpful comments of two unknown referees, Wolfgang Buehler, Hans Juerg Buettler, John Chipman, Elhanan Helpman, Horst Siebert, Richard Stapleton, Marti Subrahmanyam, and the members of the Finance Workshop at Tel Aviv University. Any errors are mine. of course. 0261-5606,91/02/0292-16
c
1991 Butterworth-Heinemann
Ltd
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FRANKE
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particularly sensitive to the choice of sample periods and equation specifications.’ While there is some evidence of negative effects from exchange rate volatility, these effects appear to be weak. The purpose of this paper is to analyze the export strategy of a firm in an intertemporal, infinite horizon setting and the effects of exchange rate volatility on exporting. In contrast to the early theoretical papers, an exporter does not sell a constant quantity of his product, irrespective of the exchange rate. He adjusts the export volume to the level of the exchange rate. When the exchange rate surges to high levels, exports increase. When the exchange rate drops below a certain level, exports fall to zero. Whenever the exchange rate permits profitable exports, firms export. Exporting is an option which is exercised if profitable. This export strategy is associated with transaction costs. A firm which starts exporting incurs costs of entering the foreign market. If it stops exporting, then it incurs exit costs. Similar to the value of a stock option, the value of this export strategy depends on exchange rate volatility. It increases with volatility for a firm with a comparative disadvantage in international trade, i.e., a firm which incurs a loss from exporting if the exchange rate equals the parity rate at which internationally traded commodities are equally expensive in the domestic and in the foreign market. For a firm with a comparative disadvantage (=disadvantaged firm), the expected cash flows from exporting grow at a faster rate with exchange rate volatility than the expected entry and exit costs. Therefore the value of exporting grows with exchange rate volatility. This is not generally true for a firm with a comparative advantage (=advantaged firm), i.e., a firm which reaps a profit when the exchange rate equals the parity rate. The next question, addressed in the paper, concerns the impact’of exchange rate volatility on the firm’s expected export volume. The paper presents sufficient conditions for a positive effect of volatility on the expected steady state export volume. Hence, in contrast to the previous theoretical literature, exchange rate volatility may have a positive impact on international trade. This may explain part of the mixed empirical findings. The theoretical analysis of this paper is related to a new strand of literature which analyzes the impact of uncertainty on investment decisions in a framework similar to that of option pricing (Brennan and Schwartz, 1985; Majd and Pindyck, 1987; Pindyck, 1988; Sercu, 1987). A recent paper of Dixit (1989) comes close to this paper. Dixit analyzes a firm’s entry and exit decisions when the output price follows a Wiener process and derives a closed form solution for the optimal strategy. This paper assumes that the real exchange rate being the relevant state variable follows a mean-reverting Ito process. For this setting there is no closed form solution (Dixit, 1989, p. 634). Still the results of this paper are derived analytically. Dixit is not so much concerned about the effects of volatility changes and illustrates these effects only by some numerical examples. In this paper these effects are of central importance and are derived analytically. The paper is organized as follows. Section I defines the economic setting. Section II derives a few characteristics of the optimal export strategy of a firm for a given level of exchange rate volatility. Section III shows the effects of the growth of volatility on the firm’s value and its export volume and aggregates across all firms to derive the effects on international trade volume. Section IV concludes the paper.
I. The economic
setting
This section presents the assumptions of the paper. There exist two countries, the ‘domestic’ and the ‘foreign’ country. In each country there exists a given set of firms which is independent of the exchange rate level and the exchange rate volatility. Firms in both countries produce internationally traded commodities. International trade of these commodities is not hampered by administrative barriers. Each firm sells its products in its home country and exports them whenever the export is expected to be profitable. Consider a firm, located in the ‘domestic’ country. Whenever the firm exports, it earns at date t an incremental real cash flow x,, denominated in domestic currency: (1)
XI= \V,Pr y, -K(Jr,
\1.,),
where \~‘,=real exchange rate at date r (units of domestic currency per unit of foreign currency); p, = real foreign currency-denominated sales price in the foreign market at date t; y, = the firm’s export volume at date f; K( _rI.RJ,)= the firm’s real production cost; this function is assumed to be strictly increasing and convex in the export volume, and increasing and strictly concave in the exchange rate. Part of this cost originates in the foreign country so that this part grows with the exchange rate. As the firm shifts some of the cost generating activities to the domestic country when \v, increases, the cost function is strictly concave in \v,. The model is cast in continuous time so that yI is the instantaneous export volume at date t. Hence X, is the instantaneous cash flow at date t. In the cash flow function, any effects of inflation are neutralized by deflating all prices, costs and cash flows, denominated in domestic currency, by the domestic price index for internationally traded commodities. Similarly, items denominated in foreign currency are deflated by the foreign price index for internationally traded commodities. The exchange rate LVis then defined as the real rate, i.e., the nominal rate adjusted for differential price movements in both countries. The cash flow at date t depends only on the exchange rate at the same date. This appears reasonable if sales contracting and payments are simultaneous, and if sales revenues are converted immediately into domestic currency. Hence no transaction risk exists. In each market the firm faces monopolistic competition (Dixit and Stiglitz, 1977).’ Thus the products of different firms are imperfect substitutes. Given the exchange rate, the demand curve for a firm’s product is downward sloping in each market (monopolistic element). Demand is independent of the decisions of single competing firms, but depends on aggregate competition (competitive element). Let J, denote the aggregate sales volume of all competitors in the foreign market. Then (2) Gp,/Sy,<6p,/6J1
P*= P*(Y,, Y,);
dP,/6Y, < 6P,!~J,
co.
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competitors drastically raise their sales volume. In addition, we assume that, for a given export volume y,, the export price in domestic currency, u’,p,, is convex in w,. For a monopolist, u’,p, is linear in \L’,since then p, depends only on yr, but not on w,. Linearity also exists if F1/ytz~Cr is independent of \v,. In the latter case, pr = pr( y,, c,~,) G fll( y,). This motivates assumption 1. I: For a given export volume L:(,the firm’s marginal export cash flow ds,/6y1 grows with the exchange rate M’,,and the export price in domestic currency. \~‘,p,, is convex in \v,; for every y1> 0 and every K, > 0. From the first part of Assumption 1, it follows that the exporting firm (y,>O) will raise its export volume when the exchange rate grows. Then standard theory of the firm (see, for example, Varian, 1984, p. 46) and the second part of Assumption 1 imply that the exporting firm’s optimal cash flow x,(u;) is increasing and strictly convex. Asswnptiorz
Lemma 1: The firm’s optimal cash flow from exports, _K,(\v,),is increasing and strictly convex in the exchange rate \v,. The following analysis uses the log exchange rate e zz In w. Strict convexity of x(\v) implies, c1fortiori, strict convexity of .u(e). Next, entry and exit costs will be specified. When the firm starts exporting, it has to set up a sales office, inform potential customers, etc. The corresponding entry costs, denominated in domestic currency, are r(a). a is the log exchange rate at which the firm enters and z(u) denotes the function relating the entry cost to the ‘entry rate’n. x(n) is a lump sum payment at the entry date. Time-dependent expenses such as salaries for the sales force are included in the cash flow x,. When the firm stops exports, it exits from the foreign market. Then it has to pay the lump sum T(z) at the exit date.’ T(z) denotes the function relating the exit cost to the log exit exchange rate ;. As part of the entry and exit costs are generated in the foreign country, &/cin >O and tiT/&>O. Therefore, a higher rate motivates the firm to shift entry and exit activities from the foreign country to the domestic country so that the entry and exit costs are strictly concave in the exchange rate \v. But they cannot be positive and concave in the log exchange rate for eE( - x, + zc). At least for small values of e these functions are convex. This motivates Assumption 2. Assumption 2: The entry cost function x(e) and the exit cost function T(e) are increasing, and they are concave for e z e*. e* is the log parity exchange rate for internationally traded commodities. Without loss of generality, e* = 0. I.B.
Eschnnge
rate mocements
Uncertainty is modeled by a single state variable which is the real exchange rate. The firm operates under a flexible exchange rate regime. The log of the real exchange rate follows a time-homogeneous mean reverting Ito process with stationary, independent increments in the noise: <3>
de,=n(e,-e*)dr+odz(t);
7r
where de, is the instantaneous change of e at date t, TIis a negative constant, the increment n=(t) is normally distributed with expectation 0 and standard deviation J&, the increments n=(t) and &(r) (t#r) are independent, and 0 is the
‘96
E.rcilange rule rokiriiii)
instantaneous standard deviation of e,. The drift of the Ito process is rr(e’,-e*). This term is negative whenever e, > e* and positive whenever er < e*. Therefore the drift moves r, back towards e*. The larger (~1 is, the stronger is the drift. The implications of the Ito process become more transparent by looking at the solution of the process. Let e0 denote the known log exchange rate at date 0. Then according to Wymer (1972), I (1)
exp[n(t -s)] (f:(s).
e,=(e,-e*)exp(rrt)+e*+a I
0
Hence e, is normally distributed with expectation (e, -r*) exp(nt) +r* and variance (exp(2lrt) - 1) ~?/(2rr). Therefore, over time, the expected rate approaches asymptotically the parity rate e* and the variance the upper limit D’/( -2~). Hence, the exchange rate is expected to move back to the parity rate r* in the long run, i.e., internationally traded commodities are expected to be equally expensive in both countries. The actual rate may, of course, deviate strongly from the parity rate. The Ito process is assumed to hold for the log of the exchange rate N’,. This means that: (1) the instantaneous standard deviation of )v, is proportional to the level of wr; (2) Prob (bt’,> 0) = 1; and (3) In 1~~and - In( l/\r,) follow the same Ito process. The last condition means that exporters in the domestic and in the foreign country face the same Ito process. The rationale behind this process is that both international capital movements and international trade determine exchange rate movements. international capital movements may drive the exchange rate a\vay from the parity rate. Sufficiently large differences between these rates create opportunities for international commodity arbitrage. This arbitrage derives the exchange rate back towards the parity level. Hence, mean reversion is caused by this arbitrage. The smaller the obstacles to commodity arbitrage are, the more intensive this arbitrage is and therefore the faster the rate moves back tow,ards the parity level, i.e.. the higher is IzcI.
The firm has an infinite time horizon and operates in a stationary world. Therefore the cash flow function and the entry and exit cost functions are time-invariant. Moreover, the firm faces no constraints nhen it maximizes the benefits from exporting. A risk-neutral firm maximizes the net present value of the expected cash flows from exports. The long-term, risk-free interest rate r is used as a discount rate. The model is applicable, too, in the case of risk aversion. Suppose the firm maximizes its market value. This value equals the net present value of the expected cash flows. using risk-adjusted probability densities instead of probability densities. Hence if the probability densities of the assumed Ito process are risk-adjusted densities, then discounting the expected cash flows by the long-term, risk-free rate yields market values of cash flows. The firm is assumed to follow a stationary bang-bang policy. i.e., the non-exporting firm enters the foreign market whenever the log exchange rate reaches level CIand the exportin, 0 firm exits from the foreign market whenever the log exchange rate drops to level Z, where n and z are constant over time. This
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9 First
entry
First
exit
20
Second entry
Time
FIGURE 1. A stationary
bang-bang
policy is defined by the entry rate cl and the
exit
rate :.
policy is illustrated in Figure 1, assumin, 0 that a> 2~0 and that the firm does not export at date 0. A bang-bang policy is analyzed since an explicit solution to the firm’s optimal policy is not yet known under the previously stated assumptions. No proof exists that a stationary bang-bang policy is the best strategy for mean reverting Ito processes. Such a proof has been provided for Wiener processes by BeneS (1974). But a stationary bang-bang policy appears to be a natural choice for an optimal control policy. Therefore the firm is assumed to follow such a policy. Assumption 3: The exporting firm follo\vs a stationary bang-bang policy. The preceding assumptions do not rule out that a lower entry rate reduces the infinite lifetime expected present value of entry costs. A lower entry rate reduces r(a), but raises the expected frequency of entry and exit. Unless &‘&I is high. the infinite lifetime expected present value of entry costs goes up when the entry rate goes down. Assumption 4: The infinite lifetime expected present value of entry costs goes up when the entry rate goes down. The effects of exchange rate volatility on the firm will be analyzed by taking D as a measure of such volatility. It will be assumed that the parity rate e* and the firm’s cost functions are independent of G. In addition, the discount rate is independent of C. Asswnption 5: The parity exchange rate e *, the firm’s cost functions K(yr, w,). r(a), I?(:) and the discount rate r do not depend on exchange rate volatility G. The preceding assumptions specify a very simple world. These simplifications permit a clear exposition of the basic ideas.
lY8
E.rchangr rate cokatilit~ II.
The optimal
trading strategy
The jirrn i objectisr
//.A.
fitwtiott
This section specifies the firm’s objective function and derives some characteristics of the optimal export policy for a given level of exchange rate volatility. The firm maximizes the expected net present value (NPV) of its incremental cash flows from exports. Let L’t denote the random NPV of the first engagement in the foreign market with the cash flows being discounted to the date of the first entry, O,,. The first engagement starts with the first entry and ends with the first exit from the foreign market. Similarly, C2, F3, . . . denote the NPVs of the second, third,. . . engagement in the foreign market, discounted to the respective dates of entry. The random NPV of all these engagements, discounted to the date of the first entry, is (5)
with (6)
j=1,2,...
dj.j+,=eXp-r[~j+,.,-~j,];
.
Caj+ Is- gj,] is the random time between the jth and the o+ 1)th entry in the foreign market. Therefore, it is the first passage time required by an exchange rate move from the entry rate a to the exit rate : and back to the entry rate ~1.~ The firm maximizes the expectation of equation (5). As the mean reverting Ito process is a strong Markov process, all factors in equation (5) are conditionally independent. Therefore, by the telescope property of the expected value, the expected value of the NPV is NPV=v,+
(7)
i vtfz Rj.j+l. I=2 j= I
Omission of the tilde indicates expected values. c1 = r2 = . . . = v and RI2 = R,, = . . . = R so that
(8)
NpVzv+
f
c,R‘-l=
r=2
Equation
c
implies
c. I-R
(8) defines the firm’s objective function.
(9)
Time-homogeneity
c’ remains to be defined:
-z(a)+CF-I-(-_)R,.
The first component of u is the entry cost r(a). After entry the firm receives the random cash flows _u,(e,) until the exchange rate drops to Z. This cash Bow has to be discounted to the entry date by the discount factor exp( - t-t). The expectation of the discounted cash flows, earned between the entry date t r0 and the exit date, conditional on entry, is (10)
CF=j:
exp(-rt)J:
F(eJa) denotes the conditional probability the conditions e, = CIand e,zVsE[O, t).
xu,(e,)dF(e,Ja)&. distribution
function of e,, subject to
GCINTER
The last drops to z, equals the appropriate distribution
FRANCE
component of r is the discounted the firm incurs the exit cost f(z). first passage time a,,, i.e., the discount factor is exp -rO,,. function of 0,,. Then the expected
(11)
299
exit cost. When the exchange rate Since the lifetime of the engagement time between entry and exit, the Let F(8,,) denote the probability exit cost, conditional on entry, is
I-(Z) a exp( -r-O,,) dF(O,,) s T(z)R,. s0
II.B.
The effects of entry and e.rit costs
In order to gain insight into the optimal bang-bang policy, the effects of entry and exit costs will be investigated. First suppose that these costs approach zero. Then the optimal policy (a+,~+) would be a++2 and z+-$, i.e., the optimal entry rate a+ and the optimal exit rate zf approach the break-even rate 6. 6 is defined by x(e) CO for e c6 and x(e) >O for e> P. The firm exports whenever e,>2, and never exports when e, ~6. This policy implies infinitely many entries and exits since, for adz, the expected first passage time from a to z and that from z to a approach zero. The zero entry and exit cost case is a good starting point for illustrating the optimal bang-bang policy. The firm is not passively exposed to exchange rate changes. Instead it actively pursues a policy of reaping positive and avoiding negative cash flows from exporting. Hence exporting represents an option on positive cash flows. Next, it will be shown that a+ >D>z+ 1s . implied by positive entry and exit costs. Suppose, to the contrary, e 6. Finally, suppose that the net present value of an engagement, U, is negative for every finite (a, z) with a > 2 > z. Then the firm cannot benefit from exporting so that the optimal policy is a+ + CL,.These results are summarized in Proposition 1. Proposition I:
(a) Ifentry and exit costs approach zero, the optimal entry and exit rates approach the break-even rate 6. (b) Positive entry and exit costs imply that the optimal entry rate a+ exceeds the break-even rate g and the optimal exit rate cf is smaller than the break-even rate 2.
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300
(c)
The firm never enters the foreign market ((I* -+ x ) if the net present value of an engagement in the foreign market, c. is negative for every policy (n,:) with LZ>P>I.
III.
The effects of an increase in exchange
/11..4.
The
rate volatilit>
ctrlue qf’ r.vpor[irq
The literature argues that an increase in exchange rate volatility reduces the expected utility of risk av’erse international traders because their risk increases. This paper shows that this statement is generally not true. In order to keep the analysis tractable, competition will be assumed to be independent of exchange rate volatility. Competition is measured by the aggregate sales volume function of competitors, _F,(r,). With constant competition, this function is independent of exchange rate volatility. Then the sales price function ~,(~~,l;(r,)) is independent of exchange rate volatility. Hence, whenever the firm exports, its optimal sales volume J~(L~~) is independent of exchange rate volatility since, by Assumption 5, the firm’s production cost is independent of G. This implies that the firm’s optima! cash How function s,(e() is independent of G, too. Lerl~rrrr -7: Suppose that competition does not depend on exchange rate volatility. Then, whenever the firm exports, its optima! sales volume _~,,(e,)and cash flow s,fc’,) are independent of exchange rate volatility. Lemma Z is essential for the proof that the firm’s net present value of exporting grows with exchange rate volatility under fairly genera! conditions. Define an increase in exchange rate volatility such that G is replaced by S t with S3 1. Correspondingly. the date O-log exchange rate co is assumed to change from e’oo for S= 1 to pol for S> I. en, -r*=S(e,,,-e*).
(12)
If f~‘,~denotes a realization of e, for S= 1, then r,,, the realization with the same probability density as r,,, is given by e,,-e*=S(e,,-c*);
(13)
of r, for S> 1
Vr.
This follows from equation (4). Intuitively speaking, the increase in exchange rate volatility is similar to a mean preserving spread plus a change in the mean.’ Since explicit solutions for an optimal bang-bang policy are not known, a mechanical adjustment policy to a growth of exchange rate volatility will be analyzed. If such a policy raises the net present value of exporting, then optima! adjustment does it even more. The mechanical adjustment raises the distances (n-e”) and (I--e*) by the factor S. More precisely, let CI~and z. denote the optima! entry and exit rates for S = 1 and 111 and z1 denote the mechanically adjusted rates for S> 1. Then (14’)
0, -e*=S(n,-e*),
(14”)
zi -e*rS(z,-e*).
This mechanical because
adjustment Prob(:,
is called a probability-invariant
adjustment Vt.
(= PIA)
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The following proposition states that firms with a comparative disadvantage in international trade (.u(e*)GO) always benefit from a growth in exchange rate volatility. The same is true of firms with a comparative advantage (.u(e*)>O) if a, 6 e*. Proposition 2: Assume that exchange rate volatility grows, but competition remains the same. Then a probability-invariant adjustment raises the net present value of exporting. This need not be true for an advantaged firm with ~z~>e*. Proposition 2 is proved in the appendix. A firm enters and exits from the foreign market if the present value of the cash flow x exceeds the present value of entry and exit costs. Hence the firm benefits from an increase in exchange rate volatility if the present value of the cash flows grows faster than that of entry and exit costs. The latters’ present value is lowered by a PIA if z0r*=O, then LI,>LI~ and, perhaps, I~>:~. As entry and exit costs are concave in the positive log entry and exit rates, the elasticity of their present value with respect to the volatility is smaller than 1. given a PIA. As the cash flow function x(e) is strictly convex in the log exchange rate, the present value of the cash flow is raised by a PIA. Therefore the firm benefits from an increase in exchange rate volatility if the present value of entry and exit costs is not raised by a PIA (a0 < r*). If this present value grows, howev-er, then its elasticity is less than 1. Hence a sufficient condition for growth of the firm’s net present value is that the elasticity of the present value of the cash flow .Kwith respect to the volatility is greater than 1. This is always true for a disadvantaged firm since x(e*)
The ejfects on the firm’s expected export colrme
Next, the effects of an increase in exchange rate volatility on the firm’s expected export volume will be analyzed. Although Proposition 2 does not address this issue, an interpretation of this proposition points to the possibility that a growth in volatility raises the expected export volume. According to Proposition 2, for a disadvantaged firm the relative magnitude of transaction costs declines when volatility increases. Therefore, a growth of volatility reduces frictions in international trading created by transaction costs, and thus supports growth of international trade of disadvantaged firms.
302
E.rchange
rutc rolatilit?
A strict proof cannot be based on a PIA, but has to be based on the firm’s optimal adjustment. Proposition 3 characterizes this for a disadvantaged firm. This firm has been shown to benefit from an increase in exchange rate volatility. Advantaged firms may suffer and stop exporting forever. Proposition 3: Assume that exchange rate volatility increases, but competition remains the same. Then an increase in exchange rate volatility implies that the optimal entry and exit rates u: and z: are smaller than the entry and exit rates cur and =I under a probability-invariant adjustment if the comparative disadvantage I.u(e*)l is sufficiently large. Proposition 3 states that a sufficiently disadvantaged firm reacts on a volatility increase by expanding its export activities. Under a probability-invariant adjustment the probability that the firm exports at a future date remains the same. Under optima1 adjustment, however, the firm enters earlier (a: 0 with e* z 0. Proposirion 4: Assume that exchange rate volatility grows, but competition remains the same. Consider a firm for which optima1 adjustment entails ~7: ,<(I, and 2: <<=,. In addition, assume that the firm’s optimal export volume is
(1%
y(e) = maxf0; c(e) +s(e)j ,
with c(e) being an increasing convex function and s(e) being an increasing function which is symmetric with respect to the point (s(e*); e*). Then the firm’s steady state expected export volume grows with exchange rate volatility. Proposition 4 shows that, in contrast to the previous literature, the firm’s expected export volume may grow with exchange rate volatility. On the aggregate level, if the conditions of Proposition 4 hold for all firms, then the aggregate steady state expected export volume grows with exchange rate volatility. Since the assumptions of the analysis are chosen symmetrically for both countries, the steady state expected volume of international trade grows with exchange rate volatility if the conditions of Proposition 4 hold for all firms in the domestic and in the foreign country. Hence it is not surprising when empirical studies on international trade and exchange rate volatility do not find unambiguous results. The results of this paper can be interpreted such that they appear to be consistent with a general equilibrium. This interpretation is based on violations ofthe ‘law of one price’and the corresponding international commodity arbitrage. Deviations of the exchange rate from its mean are generated by international capital movements. These deviations create differences between the domestic and the foreign price index for internationally traded commodities which are not offset by the exchange rate. Hence, prices of the internationally traded
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commodities violate the ‘law of one price.’ The price differences create arbitrage opportunities for international trade. This arbitrage reduces price differences although it cannot eliminate price differences completely because of transaction costs. An increase in exchange rate volatility increases the potential price differences. Therefore, more scope for profitable commodity arbitrage through international trade exists. Hence the expected volume of international trade may grow depending on how strongly entry and exit costs vary with entry and exit rates and on firms’ comparative advantage or disadvantage in international trade. This interpretation does not suggest that welfare grows with exchange rate volatility. Although the owners of some international trade firms may benefit, consumers may lose. Also firms producin g only for the domestic market may lose since higher exchange rate volatility induces higher volatility in imports and hence in their production volume and, therefore, higher adjustment costs (Willet, 1956). IV.
Conclusion
The literature argues that an increase in exchange rate volatility reduces the volume of international trade since firms are risk averse. The empirical evidence is mixed. This paper considers international trade as an option for firms to profit from price differences of internationally traded commodities. Depending on the exchange rate level, this option is sometimes exercised. Firms are assumed to follow a stationary bang-bang policy of entry and exit. Such a policy appears to be a natural one since the optimal policy is not yet known. Given a bang-bang policy, firms with a comparative disadvantage in international trade benefit from an increase in exchange rate volatility since their expected cash flows from exporting grow at a higher rate than their entry and exit costs. This need not be true for firms with a comparative of a firm grows with exchange
advantage. Whether the expected export rate volatility depends on the optimal
volume export
volume being a function of the exchange rate, and on the optimal adjustment of entry and exit rates. The paper presents sufficient conditions such that the expected export volume of a firm with a comparative disadvantage grows with exchange rate volatility. If these conditions prevail for all firms, then the expected volume of international trade grows with exchange rate volatility. This paper does not deal explicitly with risk aversion. Although the probability densities of the Ito process may be defined as risk-adjusted, this approach to risk is highly restrictive. Further research should incorporate risk aversion in a more general manner. At the same time, hedging opportunities have to be taken into consideration. In addition, the assumptions of an infinite time horizon and a stationary world need be relaxed to investigate more realistic strategies of international trade firms. Appendix
Proof of Proposition 2
(a) First, consider the effects of a PIA on probability densities and on the discount factors. When 0 is replaced by S G with .S> 1, then the date O-exchange rate e,, is replaced by e,,t according to equation (12). Hence the probability density of e,,, defined by equation
E.uchnnye role cohrilirj
304
(13) for S> 1, is the same as that of e,, for S = 1. A PIA. defined by equations (14”). then implies for the conditional probability distribution functions (16)
F(r,,ln,;
S> l)~F(e,,lcl,)=F(r,,lcl,.
(14’) and
S= l)=f(e,&,).
Similarly, the probability distribution functions of the first passage times (flj+,.,-Hi,) and 0,: remain the same under a PIA. Therefore the discount factors R and R= remain the same under a PIA. (b) Second, consider the effect of a PLA on the .VPZ’. As R does not change, the 1VPV (equation (S)) grows if and only if L‘grows. Let tlr.‘dS denote the change in L’if volatility grows by rr t/S and the firm adjusts by a PIA. Then equation (9) yields tfc (1X((Z) _=____ _ RT “1:) r/S dS ---++
dg dS
+ $;
dCF dS ’
dr,ldS > 0 iff dCF, (1.5> dyldS or iff
07)
dy s I/ dCF S _____>1_.1. (1.7 CF r1.S g CF
For S = I, the firm engages in the foreign market iff !VPV> 0. .&‘PV> 0 iff y/CF < I. Hence inequality (17) holds if the elasticity of CF with respect to S is at least as great as the elasticity of 9 with respect to S. First consider the latter elasticity. dy/dS e* so that a, > a,, and, perhaps. -_, >ce. Then concavity of x(n) and T(Z) for a>e* and -_>e* implies together with x(r*)>O and T(e*)>O that the elasticity of y with respect to S is smaller than 1. Therefore in order to prove Proposition 2, it remains to be shown that (dCF/dS)(S/CF)k 1. CF is defined in equation (10). Decompose CF=CF(x(e)) into two components CF(.u(e)-x(e*))+CF(x(e*)). If x(e) were linear, then CF(s(e)-s(e*)) would be strictly proportional to S under a PIA. As x(e) is strictly convex, CF(.u(e) - .u(r*) grows at a faster rate than S. CF(u(e*)) is independent of S. If .u(e*) 1. Hence dNP V/dS > 0. dN VP/dS 0 and if n,, > e*. This proves Proposition 2. Proof of’ Proposition 3
It suffices to prove Proposition 3 for a marginal growth of volatility. u0 and z0 denote the optimal entry and exit rates before a marginal growth of volatility (S= 1). 0, and I, denote the mechanically adjusted entry and exit rates. a; and I; the optimally adjusted entry and exit rates. (a) First, it will be shown that sign(n: -0,) and sign(-_; --_,) can be inferred from 6NPV:6a and dNPl’;dz. An optimal bang-bang policy requires 6!VPViiin= 6,VPV/d:=O. For S= 1 and (a,,:,) these conditions hold. Now consider a marginal increase in S and a PIA. If, after this adjustment, GNPV/da>[ [
d(l -R)-’ &I
_ r($5
--
(i[CF,( 1-R)] + ___-. &I (jr
GBNTER
FRANKE
305
For notational simplicity, ‘X2 Y’ is defined as follows: ‘X is proportional to Y if the firm reacts on a marginal increase in exchange rate volatility by a PIA.’ Without loss of generality, e* E 0. Then a PIA implies that the scale e,, -e* = r,, is replaced by the scale Se,,. Hence I&J@., .)/& : I/S with f being the probability density of the first passage of time 0.). . Thus -6( 1 - R)- ‘/&I z l/S and -Sk/&r z l/S. As R and R, are independent of a PIA (see proof of Proposition 2) Rz is also independent. This implies for 6NPV/cTa: As .x(O)< 0, CI,,> 0. Hence n, > ao. As %((I)is increasing and concave for (12 0, a PIA raises - [rfr,l&] (I- R)- ’ which is always nonpositive. In addition, r(O) > 0 and concavity imply Z(N, )
D = 6[ - (I -R)- ‘f(e,ltr)]:ba is nonnegative since raising the entry rate strictly reduces the probability density of exporting for eE[-_,(l]. Dz l/S. Strict convexity of .v,(e,) and .x(O)d 0 imply .x,(e,, ) > Ss,(e’,,,). Hence S[CF/( I - R )], &r declines when the firm reacts on a marginal volatility increase by a PIA. This decline is the stronger, the smaller z(0) is. Hence for a sufficiently low s(O), 3NPV/Sa declines. too. As dVPV$a=O for (Q.-_~), 6NPV/Su
(a) First consider a PIA. Let f(e) denote the steady state density of r and h(esje) the conditional probability that the firm actually exports, given P. h(e\-lr)=O for e<:, h(esle)= 1 for e>cl and h(esle) increases for eE(z,Ll). The steady state expected export volume E(y) is li (18)
E(y)=
y(eMexle)
dF(e)
i 1
=
max(O;s(e)+c(e))h(rxle)dF(e) J:
z =
(s(e) + c(e))h(eule) r/F(e), s.
with _-_=max(c,els(ee)+c(e)=O). _ Suppose 220. Then a PIA shifts the probability mass Prob(e>z) to higher exchange rates and-thus to higher export volumes. Therefore E(y) grows. Now suppose r
(s(e) + c(e))h(e.ule) dF(e)
s: +
I; s III
(s(e)+c(e))h(esje)dF(e).
3Y6
E.rchunyr
rule
colurilif~
By the same argument as before, the second integral grows under a PIA. The first integral would remain the same if c(r) and h(rslr) were constants. As s(e). c(p) and h(r.~le) are increasing and c(e) is convex. the first integral grows, too, or remains constant. Hence E(J) grows under a PIX. (b) Now suppose a,* <(I, and :,*
Notes I. 2. 3. 4.
If a firm faces perfect competition in the domestic and in the foreign market. it would sell its products only in one of both markets. For a discussion of exit costs see Siebert (1986). As the mean reverting Ito process does not satisfy a reflection principle, we do not kno\\ an explicit probability density function of the first passage time. For theexpected date r-exchange rate E(e,)equation (13) implies E(r,,)-r*=S[E(e,,)-e*].
References Akhtar. M.A., and R.S. Hilton, ‘Exchange Rate Uncertainty and International Trade: Some Conceptual Issues and New Estimates for Germany and the U.S.,’ Federal Reserve Bank of New York. Research Paper No. 8403. Ma> 1984. Exchange Rates and the Pricing of Exports.’ Ecotzottrit BARON. D.P., ‘Fluctuating Inquit-J,, September 1976. 14: 425438. BENES, V.E., ‘Gersanov Functionais and Optimal Bang-Bang Laws for Final Value Stochastic Control,’ Srocllclstic Processrs md their Applicariom. April 1974, 2: 127-140. BRENNAS, M., AND E. SCHWARTZ, ‘Evaluating Natural Resource Investments.’ Jownnl q/’ Bu.sitwss. April 19S5. 58: 135-l 57. CLARK. P.B., ‘Uncertainty. Exchange Risk, and the Level of International Trade.’ Westertz Econott~ic Jourtd. September 1973. 1 I : 302-3 13. CUSH~IAS, D.O., ‘The Effects of Real Exchange Rate Risk on International Trade,’ Jowxcd of Intrrnrrtionnl Econottzics. August 1983, 15: 45-63. CUSHSlAN. D.O., ‘Has Exchange Risk Depressed International Trade? The Impact of Third-Country Exchange Risk,’ Jourtd of Zt~retwtrtiotd Motw~. utztl Fitzmce. October 1986, 5: 361-379. CL~SH~IAS. D.O., ‘U.S. Bilateral Trade Flows and Eschange Risk During the Floating Period.’ Jownnl of‘ lntrrtxrriond Ecommics. May 1988. 21: 3 17-330. DIXIT, A., ‘Entry and Exit Decisions Under Uncertainty.’ Jomxd of Poliricd Ecotwt~~.v. June 1989.97: 620-638. Drxrr, A.K.. AND J.E. STIGLITZ, ‘Monopolistic Competition and Optimum Product Diversity.’ American Economic Review. June 1977, 67: 297-308. ETHIER. W.. ‘International Trade and the Forward Exchange Market.’ ,-lmrricmr Economics Rcrieu.. June 1973. 63: 494-503. GOTUR, P., ‘Effects of Exchange Rate Volatility on Trade: Some Further Evidence.’ I.WF .SM#‘ Prprs. September 1985, 32: 475-512. HOOPER. P., AND SW. KOHLHAGES, ‘The Effect of Exchange Rate Uncertainty on the Prices and Volume of International Trade,’ Journal of‘In~ertzn~ionnl Economics. Xovcmber 1978. 8: 483-511. MAID, S.. AND R.S. PIXDYC~~, ‘Time to Build, Option Value, and Investment Decisions.’ Journd o/‘Financial Econonrics, March 1987. 18: 7-27. PINDYCK. R.S., ‘Irreversible Investment, Capacity Choice. and the Value of the Firm.’ .4mericm~ Economic ReGw., December 1988, 78: 969-985. SERCU, P.. ‘Exchange Risk. Exposure, and the Option to Trade.’ University of Leuven. Discussion paper, 1957. Jourttnl q/’ Itutirurionul cud Tlworericui SIEBERT, H.. ‘Closing Costs in Ailing Industries.’ Economics. March 1986. 142: 156-162.
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THURSBY. M.C., AND J.G. THURSBY, ‘The Uncertaintv Effects of Floating Exchange Rates. Empirical Evidence on International Trade Flows. ‘in S.W. Arndt. R.J. Sweeny, and T.D. Willett. eds, E.~chunye Rates. Trade cm/ the U.S. Econom_v, Cambridge. MA: Ballinger. 1985. 1 d edition, New York and London: Norton and Camp. VARIAN. H.. .Cficroeconornic Amlysis, _n 1984. WILLETT, T.D.. ‘Exchange-Rate Volatility, International Trade. and Resource Allocation: A Perspective on Recent Research’, Jownal of Interncrtional ,Lloney md Finmce. March 1986. Supplement, 5: SIOL-Sl12. WYSIER, C.R., ‘Econometric Estimation of Stochastic Differential Equations Systems.’ Ecorlometricrc, May 1972, -lo: 565-577.
G~_~NTER FRANICE* Unicersittit Konstnn:,
77.50 Konstatc
I, Germcrn~~
It is usually argued that an increase in exchange rate volatility reduces the volume of international trade as trading firms are risk averse. This paper shows that exporting firms benefit under fairly general conditions from an increase in exchange rate volatility. Firms optimally adjust their export volumes to the level of the exchange rate. Exporting is an option which is exercised if profitable. In addition. the paper presents conditions under which the volume of international trade grows with exchange rate volatility.
Since the collapse of the Bretton Woods agreement in 1973 various papers have investigated the effects of exchange rate volatility on international trade. Many of the early theoretical papers (e.g., Baron, 1976; Clark, 1973; Ethier, 1973) show that an increase in exchange rate volatility will reduce the volume of international trade. The basic argument of these papers can be summarized as follows: the export volume is assumed to be independent of the exchange rate level. The exchange rate risk is the main source of the exporter’s profit risk. If exchange rate volatility increases, then profit risk increases. Since exporters are risk averse and hedging against exchange rate risk is costly or impossible, the increase in profit risk reduces the benefits and therefore the volume of international trade. Hence an increase in exchange rate volatility reduces the volume of international trade. This proposition has been tested frequently (e.g., Hooper and Kohlhagen, 1978; Cushman, 1983, 1986, 1988; Akhtar and Hilton, 1984; Gotur, 198.5; Thursby and Thursby, 1985). Hooper and Kohlhagen (1978) find no significant effects of exchange rate volatility. Subsequent studies found numerous examples of negative effects on bilateral trade flows. Perhaps the strongest support is given by Cushman (1983, 1988) who finds a significant effect in about half of his bilateral trade analyses. But, as Willet (1986, p. S106) points out, the ‘latter results have been ‘I acknowledge the helpful comments of two unknown referees, Wolfgang Buehler, Hans Juerg Buettler, John Chipman, Elhanan Helpman, Horst Siebert, Richard Stapleton, Marti Subrahmanyam, and the members of the Finance Workshop at Tel Aviv University. Any errors are mine. of course. 0261-5606,91/02/0292-16
c
1991 Butterworth-Heinemann
Ltd
GCJNTER
FRANKE
293
particularly sensitive to the choice of sample periods and equation specifications.’ While there is some evidence of negative effects from exchange rate volatility, these effects appear to be weak. The purpose of this paper is to analyze the export strategy of a firm in an intertemporal, infinite horizon setting and the effects of exchange rate volatility on exporting. In contrast to the early theoretical papers, an exporter does not sell a constant quantity of his product, irrespective of the exchange rate. He adjusts the export volume to the level of the exchange rate. When the exchange rate surges to high levels, exports increase. When the exchange rate drops below a certain level, exports fall to zero. Whenever the exchange rate permits profitable exports, firms export. Exporting is an option which is exercised if profitable. This export strategy is associated with transaction costs. A firm which starts exporting incurs costs of entering the foreign market. If it stops exporting, then it incurs exit costs. Similar to the value of a stock option, the value of this export strategy depends on exchange rate volatility. It increases with volatility for a firm with a comparative disadvantage in international trade, i.e., a firm which incurs a loss from exporting if the exchange rate equals the parity rate at which internationally traded commodities are equally expensive in the domestic and in the foreign market. For a firm with a comparative disadvantage (=disadvantaged firm), the expected cash flows from exporting grow at a faster rate with exchange rate volatility than the expected entry and exit costs. Therefore the value of exporting grows with exchange rate volatility. This is not generally true for a firm with a comparative advantage (=advantaged firm), i.e., a firm which reaps a profit when the exchange rate equals the parity rate. The next question, addressed in the paper, concerns the impact’of exchange rate volatility on the firm’s expected export volume. The paper presents sufficient conditions for a positive effect of volatility on the expected steady state export volume. Hence, in contrast to the previous theoretical literature, exchange rate volatility may have a positive impact on international trade. This may explain part of the mixed empirical findings. The theoretical analysis of this paper is related to a new strand of literature which analyzes the impact of uncertainty on investment decisions in a framework similar to that of option pricing (Brennan and Schwartz, 1985; Majd and Pindyck, 1987; Pindyck, 1988; Sercu, 1987). A recent paper of Dixit (1989) comes close to this paper. Dixit analyzes a firm’s entry and exit decisions when the output price follows a Wiener process and derives a closed form solution for the optimal strategy. This paper assumes that the real exchange rate being the relevant state variable follows a mean-reverting Ito process. For this setting there is no closed form solution (Dixit, 1989, p. 634). Still the results of this paper are derived analytically. Dixit is not so much concerned about the effects of volatility changes and illustrates these effects only by some numerical examples. In this paper these effects are of central importance and are derived analytically. The paper is organized as follows. Section I defines the economic setting. Section II derives a few characteristics of the optimal export strategy of a firm for a given level of exchange rate volatility. Section III shows the effects of the growth of volatility on the firm’s value and its export volume and aggregates across all firms to derive the effects on international trade volume. Section IV concludes the paper.
I. The economic
setting
This section presents the assumptions of the paper. There exist two countries, the ‘domestic’ and the ‘foreign’ country. In each country there exists a given set of firms which is independent of the exchange rate level and the exchange rate volatility. Firms in both countries produce internationally traded commodities. International trade of these commodities is not hampered by administrative barriers. Each firm sells its products in its home country and exports them whenever the export is expected to be profitable. Consider a firm, located in the ‘domestic’ country. Whenever the firm exports, it earns at date t an incremental real cash flow x,, denominated in domestic currency: (1)
XI= \V,Pr y, -K(Jr,
\1.,),
where \~‘,=real exchange rate at date r (units of domestic currency per unit of foreign currency); p, = real foreign currency-denominated sales price in the foreign market at date t; y, = the firm’s export volume at date f; K( _rI.RJ,)= the firm’s real production cost; this function is assumed to be strictly increasing and convex in the export volume, and increasing and strictly concave in the exchange rate. Part of this cost originates in the foreign country so that this part grows with the exchange rate. As the firm shifts some of the cost generating activities to the domestic country when \v, increases, the cost function is strictly concave in \v,. The model is cast in continuous time so that yI is the instantaneous export volume at date t. Hence X, is the instantaneous cash flow at date t. In the cash flow function, any effects of inflation are neutralized by deflating all prices, costs and cash flows, denominated in domestic currency, by the domestic price index for internationally traded commodities. Similarly, items denominated in foreign currency are deflated by the foreign price index for internationally traded commodities. The exchange rate LVis then defined as the real rate, i.e., the nominal rate adjusted for differential price movements in both countries. The cash flow at date t depends only on the exchange rate at the same date. This appears reasonable if sales contracting and payments are simultaneous, and if sales revenues are converted immediately into domestic currency. Hence no transaction risk exists. In each market the firm faces monopolistic competition (Dixit and Stiglitz, 1977).’ Thus the products of different firms are imperfect substitutes. Given the exchange rate, the demand curve for a firm’s product is downward sloping in each market (monopolistic element). Demand is independent of the decisions of single competing firms, but depends on aggregate competition (competitive element). Let J, denote the aggregate sales volume of all competitors in the foreign market. Then (2) Gp,/Sy,<6p,/6J1
P*= P*(Y,, Y,);
dP,/6Y, < 6P,!~J,
co.
GRUNTER FRANKE
295
competitors drastically raise their sales volume. In addition, we assume that, for a given export volume y,, the export price in domestic currency, u’,p,, is convex in w,. For a monopolist, u’,p, is linear in \L’,since then p, depends only on yr, but not on w,. Linearity also exists if F1/ytz~Cr is independent of \v,. In the latter case, pr = pr( y,, c,~,) G fll( y,). This motivates assumption 1. I: For a given export volume L:(,the firm’s marginal export cash flow ds,/6y1 grows with the exchange rate M’,,and the export price in domestic currency. \~‘,p,, is convex in \v,; for every y1> 0 and every K, > 0. From the first part of Assumption 1, it follows that the exporting firm (y,>O) will raise its export volume when the exchange rate grows. Then standard theory of the firm (see, for example, Varian, 1984, p. 46) and the second part of Assumption 1 imply that the exporting firm’s optimal cash flow x,(u;) is increasing and strictly convex. Asswnptiorz
Lemma 1: The firm’s optimal cash flow from exports, _K,(\v,),is increasing and strictly convex in the exchange rate \v,. The following analysis uses the log exchange rate e zz In w. Strict convexity of x(\v) implies, c1fortiori, strict convexity of .u(e). Next, entry and exit costs will be specified. When the firm starts exporting, it has to set up a sales office, inform potential customers, etc. The corresponding entry costs, denominated in domestic currency, are r(a). a is the log exchange rate at which the firm enters and z(u) denotes the function relating the entry cost to the ‘entry rate’n. x(n) is a lump sum payment at the entry date. Time-dependent expenses such as salaries for the sales force are included in the cash flow x,. When the firm stops exports, it exits from the foreign market. Then it has to pay the lump sum T(z) at the exit date.’ T(z) denotes the function relating the exit cost to the log exit exchange rate ;. As part of the entry and exit costs are generated in the foreign country, &/cin >O and tiT/&>O. Therefore, a higher rate motivates the firm to shift entry and exit activities from the foreign country to the domestic country so that the entry and exit costs are strictly concave in the exchange rate \v. But they cannot be positive and concave in the log exchange rate for eE( - x, + zc). At least for small values of e these functions are convex. This motivates Assumption 2. Assumption 2: The entry cost function x(e) and the exit cost function T(e) are increasing, and they are concave for e z e*. e* is the log parity exchange rate for internationally traded commodities. Without loss of generality, e* = 0. I.B.
Eschnnge
rate mocements
Uncertainty is modeled by a single state variable which is the real exchange rate. The firm operates under a flexible exchange rate regime. The log of the real exchange rate follows a time-homogeneous mean reverting Ito process with stationary, independent increments in the noise: <3>
de,=n(e,-e*)dr+odz(t);
7r
where de, is the instantaneous change of e at date t, TIis a negative constant, the increment n=(t) is normally distributed with expectation 0 and standard deviation J&, the increments n=(t) and &(r) (t#r) are independent, and 0 is the
‘96
E.rcilange rule rokiriiii)
instantaneous standard deviation of e,. The drift of the Ito process is rr(e’,-e*). This term is negative whenever e, > e* and positive whenever er < e*. Therefore the drift moves r, back towards e*. The larger (~1 is, the stronger is the drift. The implications of the Ito process become more transparent by looking at the solution of the process. Let e0 denote the known log exchange rate at date 0. Then according to Wymer (1972), I (1)
exp[n(t -s)] (f:(s).
e,=(e,-e*)exp(rrt)+e*+a I
0
Hence e, is normally distributed with expectation (e, -r*) exp(nt) +r* and variance (exp(2lrt) - 1) ~?/(2rr). Therefore, over time, the expected rate approaches asymptotically the parity rate e* and the variance the upper limit D’/( -2~). Hence, the exchange rate is expected to move back to the parity rate r* in the long run, i.e., internationally traded commodities are expected to be equally expensive in both countries. The actual rate may, of course, deviate strongly from the parity rate. The Ito process is assumed to hold for the log of the exchange rate N’,. This means that: (1) the instantaneous standard deviation of )v, is proportional to the level of wr; (2) Prob (bt’,> 0) = 1; and (3) In 1~~and - In( l/\r,) follow the same Ito process. The last condition means that exporters in the domestic and in the foreign country face the same Ito process. The rationale behind this process is that both international capital movements and international trade determine exchange rate movements. international capital movements may drive the exchange rate a\vay from the parity rate. Sufficiently large differences between these rates create opportunities for international commodity arbitrage. This arbitrage derives the exchange rate back towards the parity level. Hence, mean reversion is caused by this arbitrage. The smaller the obstacles to commodity arbitrage are, the more intensive this arbitrage is and therefore the faster the rate moves back tow,ards the parity level, i.e.. the higher is IzcI.
The firm has an infinite time horizon and operates in a stationary world. Therefore the cash flow function and the entry and exit cost functions are time-invariant. Moreover, the firm faces no constraints nhen it maximizes the benefits from exporting. A risk-neutral firm maximizes the net present value of the expected cash flows from exports. The long-term, risk-free interest rate r is used as a discount rate. The model is applicable, too, in the case of risk aversion. Suppose the firm maximizes its market value. This value equals the net present value of the expected cash flows. using risk-adjusted probability densities instead of probability densities. Hence if the probability densities of the assumed Ito process are risk-adjusted densities, then discounting the expected cash flows by the long-term, risk-free rate yields market values of cash flows. The firm is assumed to follow a stationary bang-bang policy. i.e., the non-exporting firm enters the foreign market whenever the log exchange rate reaches level CIand the exportin, 0 firm exits from the foreign market whenever the log exchange rate drops to level Z, where n and z are constant over time. This
G&TER FRAME
197
9 First
entry
First
exit
20
Second entry
Time
FIGURE 1. A stationary
bang-bang
policy is defined by the entry rate cl and the
exit
rate :.
policy is illustrated in Figure 1, assumin, 0 that a> 2~0 and that the firm does not export at date 0. A bang-bang policy is analyzed since an explicit solution to the firm’s optimal policy is not yet known under the previously stated assumptions. No proof exists that a stationary bang-bang policy is the best strategy for mean reverting Ito processes. Such a proof has been provided for Wiener processes by BeneS (1974). But a stationary bang-bang policy appears to be a natural choice for an optimal control policy. Therefore the firm is assumed to follow such a policy. Assumption 3: The exporting firm follo\vs a stationary bang-bang policy. The preceding assumptions do not rule out that a lower entry rate reduces the infinite lifetime expected present value of entry costs. A lower entry rate reduces r(a), but raises the expected frequency of entry and exit. Unless &‘&I is high. the infinite lifetime expected present value of entry costs goes up when the entry rate goes down. Assumption 4: The infinite lifetime expected present value of entry costs goes up when the entry rate goes down. The effects of exchange rate volatility on the firm will be analyzed by taking D as a measure of such volatility. It will be assumed that the parity rate e* and the firm’s cost functions are independent of G. In addition, the discount rate is independent of C. Asswnption 5: The parity exchange rate e *, the firm’s cost functions K(yr, w,). r(a), I?(:) and the discount rate r do not depend on exchange rate volatility G. The preceding assumptions specify a very simple world. These simplifications permit a clear exposition of the basic ideas.
lY8
E.rchangr rate cokatilit~ II.
The optimal
trading strategy
The jirrn i objectisr
//.A.
fitwtiott
This section specifies the firm’s objective function and derives some characteristics of the optimal export policy for a given level of exchange rate volatility. The firm maximizes the expected net present value (NPV) of its incremental cash flows from exports. Let L’t denote the random NPV of the first engagement in the foreign market with the cash flows being discounted to the date of the first entry, O,,. The first engagement starts with the first entry and ends with the first exit from the foreign market. Similarly, C2, F3, . . . denote the NPVs of the second, third,. . . engagement in the foreign market, discounted to the respective dates of entry. The random NPV of all these engagements, discounted to the date of the first entry, is (5)
with (6)
j=1,2,...
dj.j+,=eXp-r[~j+,.,-~j,];
.
Caj+ Is- gj,] is the random time between the jth and the o+ 1)th entry in the foreign market. Therefore, it is the first passage time required by an exchange rate move from the entry rate a to the exit rate : and back to the entry rate ~1.~ The firm maximizes the expectation of equation (5). As the mean reverting Ito process is a strong Markov process, all factors in equation (5) are conditionally independent. Therefore, by the telescope property of the expected value, the expected value of the NPV is NPV=v,+
(7)
i vtfz Rj.j+l. I=2 j= I
Omission of the tilde indicates expected values. c1 = r2 = . . . = v and RI2 = R,, = . . . = R so that
(8)
NpVzv+
f
c,R‘-l=
r=2
Equation
c
implies
c. I-R
(8) defines the firm’s objective function.
(9)
Time-homogeneity
c’ remains to be defined:
-z(a)+CF-I-(-_)R,.
The first component of u is the entry cost r(a). After entry the firm receives the random cash flows _u,(e,) until the exchange rate drops to Z. This cash Bow has to be discounted to the entry date by the discount factor exp( - t-t). The expectation of the discounted cash flows, earned between the entry date t r0 and the exit date, conditional on entry, is (10)
CF=j:
exp(-rt)J:
F(eJa) denotes the conditional probability the conditions e, = CIand e,zVsE[O, t).
xu,(e,)dF(e,Ja)&. distribution
function of e,, subject to
GCINTER
The last drops to z, equals the appropriate distribution
FRANCE
component of r is the discounted the firm incurs the exit cost f(z). first passage time a,,, i.e., the discount factor is exp -rO,,. function of 0,,. Then the expected
(11)
299
exit cost. When the exchange rate Since the lifetime of the engagement time between entry and exit, the Let F(8,,) denote the probability exit cost, conditional on entry, is
I-(Z) a exp( -r-O,,) dF(O,,) s T(z)R,. s0
II.B.
The effects of entry and e.rit costs
In order to gain insight into the optimal bang-bang policy, the effects of entry and exit costs will be investigated. First suppose that these costs approach zero. Then the optimal policy (a+,~+) would be a++2 and z+-$, i.e., the optimal entry rate a+ and the optimal exit rate zf approach the break-even rate 6. 6 is defined by x(e) CO for e c6 and x(e) >O for e> P. The firm exports whenever e,>2, and never exports when e, ~6. This policy implies infinitely many entries and exits since, for adz, the expected first passage time from a to z and that from z to a approach zero. The zero entry and exit cost case is a good starting point for illustrating the optimal bang-bang policy. The firm is not passively exposed to exchange rate changes. Instead it actively pursues a policy of reaping positive and avoiding negative cash flows from exporting. Hence exporting represents an option on positive cash flows. Next, it will be shown that a+ >D>z+ 1s . implied by positive entry and exit costs. Suppose, to the contrary, e
(a) Ifentry and exit costs approach zero, the optimal entry and exit rates approach the break-even rate 6. (b) Positive entry and exit costs imply that the optimal entry rate a+ exceeds the break-even rate g and the optimal exit rate cf is smaller than the break-even rate 2.
E.uci~r~nye rule rolnrilir!.
300
(c)
The firm never enters the foreign market ((I* -+ x ) if the net present value of an engagement in the foreign market, c. is negative for every policy (n,:) with LZ>P>I.
III.
The effects of an increase in exchange
/11..4.
The
rate volatilit>
ctrlue qf’ r.vpor[irq
The literature argues that an increase in exchange rate volatility reduces the expected utility of risk av’erse international traders because their risk increases. This paper shows that this statement is generally not true. In order to keep the analysis tractable, competition will be assumed to be independent of exchange rate volatility. Competition is measured by the aggregate sales volume function of competitors, _F,(r,). With constant competition, this function is independent of exchange rate volatility. Then the sales price function ~,(~~,l;(r,)) is independent of exchange rate volatility. Hence, whenever the firm exports, its optimal sales volume J~(L~~) is independent of exchange rate volatility since, by Assumption 5, the firm’s production cost is independent of G. This implies that the firm’s optima! cash How function s,(e() is independent of G, too. Lerl~rrrr -7: Suppose that competition does not depend on exchange rate volatility. Then, whenever the firm exports, its optima! sales volume _~,,(e,)and cash flow s,fc’,) are independent of exchange rate volatility. Lemma Z is essential for the proof that the firm’s net present value of exporting grows with exchange rate volatility under fairly genera! conditions. Define an increase in exchange rate volatility such that G is replaced by S t with S3 1. Correspondingly. the date O-log exchange rate co is assumed to change from e’oo for S= 1 to pol for S> I. en, -r*=S(e,,,-e*).
(12)
If f~‘,~denotes a realization of e, for S= 1, then r,,, the realization with the same probability density as r,,, is given by e,,-e*=S(e,,-c*);
(13)
of r, for S> 1
Vr.
This follows from equation (4). Intuitively speaking, the increase in exchange rate volatility is similar to a mean preserving spread plus a change in the mean.’ Since explicit solutions for an optimal bang-bang policy are not known, a mechanical adjustment policy to a growth of exchange rate volatility will be analyzed. If such a policy raises the net present value of exporting, then optima! adjustment does it even more. The mechanical adjustment raises the distances (n-e”) and (I--e*) by the factor S. More precisely, let CI~and z. denote the optima! entry and exit rates for S = 1 and 111 and z1 denote the mechanically adjusted rates for S> 1. Then (14’)
0, -e*=S(n,-e*),
(14”)
zi -e*rS(z,-e*).
This mechanical because
adjustment Prob(:,
is called a probability-invariant
adjustment Vt.
(= PIA)
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The following proposition states that firms with a comparative disadvantage in international trade (.u(e*)GO) always benefit from a growth in exchange rate volatility. The same is true of firms with a comparative advantage (.u(e*)>O) if a, 6 e*. Proposition 2: Assume that exchange rate volatility grows, but competition remains the same. Then a probability-invariant adjustment raises the net present value of exporting. This need not be true for an advantaged firm with ~z~>e*. Proposition 2 is proved in the appendix. A firm enters and exits from the foreign market if the present value of the cash flow x exceeds the present value of entry and exit costs. Hence the firm benefits from an increase in exchange rate volatility if the present value of the cash flows grows faster than that of entry and exit costs. The latters’ present value is lowered by a PIA if z0
The ejfects on the firm’s expected export colrme
Next, the effects of an increase in exchange rate volatility on the firm’s expected export volume will be analyzed. Although Proposition 2 does not address this issue, an interpretation of this proposition points to the possibility that a growth in volatility raises the expected export volume. According to Proposition 2, for a disadvantaged firm the relative magnitude of transaction costs declines when volatility increases. Therefore, a growth of volatility reduces frictions in international trading created by transaction costs, and thus supports growth of international trade of disadvantaged firms.
302
E.rchange
rutc rolatilit?
A strict proof cannot be based on a PIA, but has to be based on the firm’s optimal adjustment. Proposition 3 characterizes this for a disadvantaged firm. This firm has been shown to benefit from an increase in exchange rate volatility. Advantaged firms may suffer and stop exporting forever. Proposition 3: Assume that exchange rate volatility increases, but competition remains the same. Then an increase in exchange rate volatility implies that the optimal entry and exit rates u: and z: are smaller than the entry and exit rates cur and =I under a probability-invariant adjustment if the comparative disadvantage I.u(e*)l is sufficiently large. Proposition 3 states that a sufficiently disadvantaged firm reacts on a volatility increase by expanding its export activities. Under a probability-invariant adjustment the probability that the firm exports at a future date remains the same. Under optima1 adjustment, however, the firm enters earlier (a:
(1%
y(e) = maxf0; c(e) +s(e)j ,
with c(e) being an increasing convex function and s(e) being an increasing function which is symmetric with respect to the point (s(e*); e*). Then the firm’s steady state expected export volume grows with exchange rate volatility. Proposition 4 shows that, in contrast to the previous literature, the firm’s expected export volume may grow with exchange rate volatility. On the aggregate level, if the conditions of Proposition 4 hold for all firms, then the aggregate steady state expected export volume grows with exchange rate volatility. Since the assumptions of the analysis are chosen symmetrically for both countries, the steady state expected volume of international trade grows with exchange rate volatility if the conditions of Proposition 4 hold for all firms in the domestic and in the foreign country. Hence it is not surprising when empirical studies on international trade and exchange rate volatility do not find unambiguous results. The results of this paper can be interpreted such that they appear to be consistent with a general equilibrium. This interpretation is based on violations ofthe ‘law of one price’and the corresponding international commodity arbitrage. Deviations of the exchange rate from its mean are generated by international capital movements. These deviations create differences between the domestic and the foreign price index for internationally traded commodities which are not offset by the exchange rate. Hence, prices of the internationally traded
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commodities violate the ‘law of one price.’ The price differences create arbitrage opportunities for international trade. This arbitrage reduces price differences although it cannot eliminate price differences completely because of transaction costs. An increase in exchange rate volatility increases the potential price differences. Therefore, more scope for profitable commodity arbitrage through international trade exists. Hence the expected volume of international trade may grow depending on how strongly entry and exit costs vary with entry and exit rates and on firms’ comparative advantage or disadvantage in international trade. This interpretation does not suggest that welfare grows with exchange rate volatility. Although the owners of some international trade firms may benefit, consumers may lose. Also firms producin g only for the domestic market may lose since higher exchange rate volatility induces higher volatility in imports and hence in their production volume and, therefore, higher adjustment costs (Willet, 1956). IV.
Conclusion
The literature argues that an increase in exchange rate volatility reduces the volume of international trade since firms are risk averse. The empirical evidence is mixed. This paper considers international trade as an option for firms to profit from price differences of internationally traded commodities. Depending on the exchange rate level, this option is sometimes exercised. Firms are assumed to follow a stationary bang-bang policy of entry and exit. Such a policy appears to be a natural one since the optimal policy is not yet known. Given a bang-bang policy, firms with a comparative disadvantage in international trade benefit from an increase in exchange rate volatility since their expected cash flows from exporting grow at a higher rate than their entry and exit costs. This need not be true for firms with a comparative of a firm grows with exchange
advantage. Whether the expected export rate volatility depends on the optimal
volume export
volume being a function of the exchange rate, and on the optimal adjustment of entry and exit rates. The paper presents sufficient conditions such that the expected export volume of a firm with a comparative disadvantage grows with exchange rate volatility. If these conditions prevail for all firms, then the expected volume of international trade grows with exchange rate volatility. This paper does not deal explicitly with risk aversion. Although the probability densities of the Ito process may be defined as risk-adjusted, this approach to risk is highly restrictive. Further research should incorporate risk aversion in a more general manner. At the same time, hedging opportunities have to be taken into consideration. In addition, the assumptions of an infinite time horizon and a stationary world need be relaxed to investigate more realistic strategies of international trade firms. Appendix
Proof of Proposition 2
(a) First, consider the effects of a PIA on probability densities and on the discount factors. When 0 is replaced by S G with .S> 1, then the date O-exchange rate e,, is replaced by e,,t according to equation (12). Hence the probability density of e,,, defined by equation
E.uchnnye role cohrilirj
304
(13) for S> 1, is the same as that of e,, for S = 1. A PIA. defined by equations (14”). then implies for the conditional probability distribution functions (16)
F(r,,ln,;
S> l)~F(e,,lcl,)=F(r,,lcl,.
(14’) and
S= l)=f(e,&,).
Similarly, the probability distribution functions of the first passage times (flj+,.,-Hi,) and 0,: remain the same under a PIA. Therefore the discount factors R and R= remain the same under a PIA. (b) Second, consider the effect of a PLA on the .VPZ’. As R does not change, the 1VPV (equation (S)) grows if and only if L‘grows. Let tlr.‘dS denote the change in L’if volatility grows by rr t/S and the firm adjusts by a PIA. Then equation (9) yields tfc (1X((Z) _=____ _ RT “1:) r/S dS ---++
dg dS
+ $;
dCF dS ’
dr,ldS > 0 iff dCF, (1.5> dyldS or iff
07)
dy s I/ dCF S _____>1_.1. (1.7 CF r1.S g CF
For S = I, the firm engages in the foreign market iff !VPV> 0. .&‘PV> 0 iff y/CF < I. Hence inequality (17) holds if the elasticity of CF with respect to S is at least as great as the elasticity of 9 with respect to S. First consider the latter elasticity. dy/dS
It suffices to prove Proposition 3 for a marginal growth of volatility. u0 and z0 denote the optimal entry and exit rates before a marginal growth of volatility (S= 1). 0, and I, denote the mechanically adjusted entry and exit rates. a; and I; the optimally adjusted entry and exit rates. (a) First, it will be shown that sign(n: -0,) and sign(-_; --_,) can be inferred from 6NPV:6a and dNPl’;dz. An optimal bang-bang policy requires 6!VPViiin= 6,VPV/d:=O. For S= 1 and (a,,:,) these conditions hold. Now consider a marginal increase in S and a PIA. If, after this adjustment, GNPV/da>[
d(l -R)-’ &I
_ r($5
--
(i[CF,( 1-R)] + ___-. &I (jr
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For notational simplicity, ‘X2 Y’ is defined as follows: ‘X is proportional to Y if the firm reacts on a marginal increase in exchange rate volatility by a PIA.’ Without loss of generality, e* E 0. Then a PIA implies that the scale e,, -e* = r,, is replaced by the scale Se,,. Hence I&J@., .)/& : I/S with f being the probability density of the first passage of time 0.). . Thus -6( 1 - R)- ‘/&I z l/S and -Sk/&r z l/S. As R and R, are independent of a PIA (see proof of Proposition 2) Rz is also independent. This implies for 6NPV/cTa: As .x(O)< 0, CI,,> 0. Hence n, > ao. As %((I)is increasing and concave for (12 0, a PIA raises - [rfr,l&] (I- R)- ’ which is always nonpositive. In addition, r(O) > 0 and concavity imply Z(N, )
D = 6[ - (I -R)- ‘f(e,ltr)]:ba is nonnegative since raising the entry rate strictly reduces the probability density of exporting for eE[-_,(l]. Dz l/S. Strict convexity of .v,(e,) and .x(O)d 0 imply .x,(e,, ) > Ss,(e’,,,). Hence S[CF/( I - R )], &r declines when the firm reacts on a marginal volatility increase by a PIA. This decline is the stronger, the smaller z(0) is. Hence for a sufficiently low s(O), 3NPV/Sa declines. too. As dVPV$a=O for (Q.-_~), 6NPV/Su
(a) First consider a PIA. Let f(e) denote the steady state density of r and h(esje) the conditional probability that the firm actually exports, given P. h(e\-lr)=O for e<:, h(esle)= 1 for e>cl and h(esle) increases for eE(z,Ll). The steady state expected export volume E(y) is li (18)
E(y)=
y(eMexle)
dF(e)
i 1
=
max(O;s(e)+c(e))h(rxle)dF(e) J:
z =
(s(e) + c(e))h(eule) r/F(e), s.
with _-_=max(c,els(ee)+c(e)=O). _ Suppose 220. Then a PIA shifts the probability mass Prob(e>z) to higher exchange rates and-thus to higher export volumes. Therefore E(y) grows. Now suppose r
(s(e) + c(e))h(e.ule) dF(e)
s: +
I; s III
(s(e)+c(e))h(esje)dF(e).
3Y6
E.rchunyr
rule
colurilif~
By the same argument as before, the second integral grows under a PIA. The first integral would remain the same if c(r) and h(rslr) were constants. As s(e). c(p) and h(r.~le) are increasing and c(e) is convex. the first integral grows, too, or remains constant. Hence E(J) grows under a PIX. (b) Now suppose a,* <(I, and :,*
Notes I. 2. 3. 4.
If a firm faces perfect competition in the domestic and in the foreign market. it would sell its products only in one of both markets. For a discussion of exit costs see Siebert (1986). As the mean reverting Ito process does not satisfy a reflection principle, we do not kno\\ an explicit probability density function of the first passage time. For theexpected date r-exchange rate E(e,)equation (13) implies E(r,,)-r*=S[E(e,,)-e*].
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