Some Definitions

In this paper, the exchange rate will be expressed as the price of a unit of a given foreign currency in terms of local currency. Similarly, the forward exchange rate will be expressed as the unit price in local currency of foreign exchange bought or sold for future delivery. Forward premium (or discount when negative) is to be understood as the discrepancy between the forward and spot exchange rates as a percentage of the spot exchange rate.

In normal circumstances, the forward exchange rate is determined by three main types of operation:

1. Hedging in connection with foreign trade. This type of operation arises out of the normal desire of merchants who trade with foreign countries to insure themselves against the risk of exchange fluctuation affecting their current transactions, which are normally contracted months before the payments or receipts involving foreign currencies become due. The contracted amount of payment or receipt in foreign currency can be transformed immediately into an obligation or a claim fixed in one’s own currency by means of a purchase or sale of forward exchange of the same amount.

2. Speculation. Speculation may be defined in a narrow sense and a wide sense. In the narrow sense, speculation in forward exchange means any sale or purchase which results in the deliberate incurrence of additional risks of exchange uncertainty on the part of the operator—i.e., it increases the net open position (long or short)³ of the operator—with a view to profiting from the discrepancy between the current forward rate and the probable future spot rate that the operator expects to prevail. Speculation in this narrow sense is to be contrasted with “hedging,” which is defined as a sale or purchase of forward exchange calculated to reduce the pre-existing exchange risks of the operator—i.e., it covers (or reduces) his original open position. In this narrow sense, a professional speculator who closes somewhat his excessive open (long or short) position because of a change in his expectations as to future spot rates or a change in the current forward rate must be regarded as a hedger performing a hedging operation. On the other hand, a merchant who chooses to hedge only a part of his exchange commitment, because the current forward rate is too high (or too low) in relation to the future spot rate which he expects, cannot be regarded as speculating at all, even though his decision not to hedge is essentially similar to that of a speculator who takes a chance in incurring an open position.

In a wide sense, speculation may be defined as the deliberate assumption or retention of a net open (long or short) position in foreign exchange upon consideration of the current forward rate and the probable future spot rate which the operator concerned expects to prevail. Under this broad definition, a merchant who fails to hedge to the full extent of his net exchange commitment is, to a certain extent, speculating. This wider definition is the more convenient, since merchants who deliberately leave parts of their commitments unhedged are essentially making the same type of decision as professional speculators who deliberately assume risky commitments (open positions); consequently, the two actions can be analyzed in the same way.

When speculation is defined in this wide sense, however, we must assume that merchants, acting as hedgers, automatically hedge to the full extent against their exchange commitments by purchases or sales of forward exchange unless the exchange risks are already eliminated by other arrangements. If some of them do in fact leave part of their risky commitments unhedged, we shall treat them as if they, this time acting as speculators, have reassumed net open positions by opposite forward transactions, offsetting part of the forward purchases or sales with which they have supposedly hedged themselves against all exchange risks in their original commitments that may arise out of their trade transactions.

3. Covered interest arbitrage. Short-term funds have a natural tendency to flow from one country to another in search of the highest returns. Through the mechanism of the forward exchange market, such search for the highest returns can be divorced of any speculative exchange risk; that is, the person (or institution) that transfers funds from one country to another in search of higher returns on investment can cover his spot exchange transactions by forward transactions in the opposite direction and thus avoid assuming any net open position in foreign exchange. Covered interest arbitrage may thus be defined as an international transfer of spot funds for short-term investment purposes covered by a simultaneous forward transaction of the same amount in the opposite direction. Such an operation leaves the net position and exchange risk of the operator unchanged, which is the essential feature that differentiates it from both hedging and speculation.

Normally, these three types of operation are likely to be performed by different groups of people or institutions; however, not infrequently the same person or institution may perform more than one type of operation at the same time. Commercial hedging (in the automatic sense) is, of course, performed chiefly by merchants, and speculation chiefly by professional or amateur speculators. However, as we have already seen, speculation in the wide sense is practiced to some extent by merchants who decide not to hedge up to the full amount of their commitments. Interest arbitrage is engaged in chiefly by banks and other financial institutions; but, as we shall see later, merchants and speculators may, in a sense, engage in interest arbitrage in connection with their hedging or speculating operations. Furthermore, although banks usually profess that they do not, as a rule, take any speculative open positions in foreign exchange and strive to balance their positions in each currency at the close of each day, available statistics indicate that their net positions (both spot and forward) in each foreign currency are not always zero and, therefore, that they are engaged in some speculative operations according to our definition in the wide sense.

In normal circumstances, these three types of operation constitute practically all the supply and demand for forward exchange. In times of abnormal disturbance, however, the authorities of the countries concerned may intervene directly in the forward market through forward purchases of their own currencies in order to prevent the forward discounts from widening too much—sometimes in addition to restrictions on interest arbitrage or speculation.⁴

Covered Interest Arbitrage

Since the traditional theory of forward exchange as set forth by Keynes in his Tract on Monetary Reform is based mainly upon the possibility of interest arbitrage, we shall now consider this type of operation on the forward exchange market. As pointed out above, short-term funds tend to flow from one center to another (say, from New York to London) if the rate of return in one center (London) is higher than that in the other (New York), after the risk of exchange fluctuations is eliminated by a forward exchange transaction in the opposite direction. This proposition has been most precisely formulated by J. Spraos.⁵ His formulation, with our own notation, is as follows:

SR= spot exchange rate—say, the sterling-dollar rate, quoted in terms of dollars per unit of sterling;

SR	=	spot exchange rate—say, the sterling-dollar rate, quoted in terms of dollars per unit of sterling;
FR	=	three-month forward sterling-dollar rate quoted as for the spot rate;
Ia	=	short-term interest rate in New York per three months;
Ib	=	short-term interest rate in London per three months.

View Table

SR	=	spot exchange rate—say, the sterling-dollar rate, quoted in terms of dollars per unit of sterling;
FR	=	three-month forward sterling-dollar rate quoted as for the spot rate;
Ia	=	short-term interest rate in New York per three months;
Ib	=	short-term interest rate in London per three months.

The proposition can then be stated that short-term funds would tend to flow from New York to London, if

conversely, funds would tend to be transferred from London to New York, if

For example, $100 invested for three months in New York would become $100 × (1+I^a); and if the same sum in dollars is converted into sterling at the current spot rate and invested for three months in London and then converted back into dollars at the current forward rate for sterling for three months’ delivery, it would become $\$100X\frac{FR}{SR}(1+I^b)$ If the latter sum is larger, arbitragers would gain a net profit by a temporary transfer of funds from New York to London; if smaller, the net profit would be gained by a transfer of funds from London to New York. It is then argued that, if the arbitrage funds do not run out, such arbitrage operations (say, transfers of spot funds from New York to London coupled with an equal amount of forward sales of sterling for dollars) would tend to eradicate this inequality through some or all of the following possible effects: raising the spot rate of sterling in terms of dollars (SR); lowering the forward rate (FR); raising the short-term interest rate in New York (I^a) and lowering that in London (I^b). Thus, the equilibrium relationship between the spot and forward exchange rates on the one hand and the interest rates in the two financial centers on the other hand is said to be

If $\begin{array}{c}\frac{FR}{SR}\end{array}$ is expressed as 1 + p, where $p(i\mathrm{.}e\mathrm{.},\frac{FR-SR}{SR})$ is the forward premium (or forward discount, if negative) on sterling as defined above, then equation (3) becomes

If the term pI^b in (4) is considered as of second order of small magnitude and therefore neglected, this equation can be further simplified to

This corresponds to the usual verbal formulation of the so-called interest parity theory of forward exchange, which maintains that the forward premium (or discount) tends to be equal to the interest differential between the two countries concerned.⁶ The currency of the country with a higher interest rate tends to be quoted at a discount on the forward exchange market, compared with its current spot exchange rate. For instance, if the interest rate in London, I^b, is greater than the interest rate in New York, I^a, then p is negative, i.e., forward sterling is at a discount.

This equilibrium condition is not meant to be an exact one. First, the short-term rates of interest in both countries are not uniquely definable entities. There are different kinds of short-term rates in both countries. Second, as pointed out by Keynes at the very beginning, the profit rate on such arbitrage operations must exceed a certain minimum to be sufficient to induce arbitragers to take the trouble of arbitraging—i.e., there is a sort of minimum sensibile on the part of arbitragers, which Keynes put at ½ per cent per annum.⁷ Thus, given a definite interest differential between the two countries, the forward premium (or discount) on sterling could still diverge ½ per cent per annum in either direction from the interest parity without inducing any arbitrage operations to correct the divergence. Lastly, the most important shortcoming of the interest parity theory is the necessary proviso that, even when there is no official restriction of short-term capital movements, the theory is applicable only so long as “arbitrage funds do not run out,”⁸ whereas in fact, as Keynes pointed out, “the floating capital normally available, and ready to move from center to center for the purpose of taking advantage of moderate arbitrage profits between spot and forward exchange, is by no means unlimited in amount, and is not always adequate to the market’s requirements.”⁹

This proviso to the interest parity theory is often made, e.g., by Spraos, in a way that implies that the theory holds up to the point where arbitrage funds are about to run out, but ceases to hold beyond that point. In reality, the availability of arbitrage funds does not stop abruptly at any given point. What is more likely is that arbitragers (chiefly banks with overseas operations) will generally, after a certain point, become increasingly reluctant to transfer their spot liquid resources from the domestic center to any particular foreign center, say London, if the arbitrage operations they are induced to perform are persistently in the form of spot purchases against forward sales of, e.g., sterling. Similarly, after a certain point, they will become increasingly reluctant to transfer their spot liquid resources from their foreign centers of operations back to their home market through persistent arbitrage operations in the opposite direction. This follows from the fact that, for their regular business operations, banks and other financial institutions (or any firms with overseas operations) must have command over certain amounts of spot liquid funds in every major overseas financial center; mere forward claims would not serve the purpose. That is to say, spot liquid assets yield some intangible returns of convenience or liquidity in addition to their interest yields. Banks (and other financial institutions) would normally be expected to distribute their command over spot liquid assets in various financial centers in such a way that the marginal yields of interest cum liquidity (convenience) net of exchange risks of liquid assets would be approximately equal between different financial centers. As a bank, say in New York, is induced to arbitrage on the forward exchange market in the direction of spot buying of sterling against forward sale of sterling of an equal amount, its spot liquid assets in New York are reduced and those in London increased by that amount, though with the insurance that this amount of sterling can be transferred back into dollars at a fixed rate on a future date. As a result, the marginal yield of convenience of its funds in New York would be increased, while that of its funds in London would be decreased. To induce it to make further arbitrage operations in the same direction, i.e., to purchase more spot sterling against forward sales, the margin of profit represented by the gap between interest parity and the current forward premium (or discount) would have to increase. Thus the equilibrium condition for each arbitrager (this condition has to be stated for each arbitrager, for the marginal convenience yields of liquid assets in various centers are likely to be different for different arbitragers) should be revised to

where ${\rho }_i^a$ is the subjective marginal convenience (or liquidity) yield to the individual arbitrager, i, of his net short-term liquid assets in New York, and ${\rho }_i^b$ that of his net short-term liquid assets in London. Obviously, ${\rho }_i^a$ and ${\rho }_i^b$ are decreasing functions of his net short-term liquid assets in New York and in London, respectively. Since an arbitrager (qua arbitrager, who under our definition does not assume any exchange risk) will always match his purchase of forward exchange (sterling in the present case) with an equal amount of sales of spot sterling exchange, his spot liquid assets in New York would be increased by the amount of his current net purchases (purchases minus sales) of forward sterling exchange less his past contracts of purchases (minus sales) that fall due currently and which may be regarded as swapped against new contracts. By the same token, his net liquid assets in London would be reduced by the same amount. If we make the simplifying assumption that all forward exchange contracts are for three months’ (90 days’) delivery, then the short-term assets of a typical arbitrager in New York may be represented as

where $\begin{array}{c}{S}_{it}^a\end{array}$ denotes the total net short-term assets of the arbitrager, i in New York on day t; D_it is his net purchase of forward exchange on day t, which may be negative if he sells more than he buys on that day; and $\begin{array}{c}{A}_i^a\end{array}$ represents his other net short-term assets in New York unrelated to his foreign exchange dealings. Similarly, his net short-term assets in London may be represented as

where $\begin{array}{c}{S}_{it}^b\end{array}$ denotes his total net short-term assets in London on day t, and $\begin{array}{c}{A}_i^b\end{array}$ his other net short-term assets in London unrelated to his foreign exchange dealings. D_it is defined in the same way, but, for the reason explained above, it takes an opposite sign here.

If the liquidity (convenience yield) functions of his net short-term assets in New York and London may be approximated by the following linear equations

where ${\rho }_i^{\prime a}$ and ${\rho }_i^{\prime b}$ are positive coefficients and α_i and β_i are positive constants, then it can be readily shown that the demand for (or supply of, if the demand turns out to be negative) forward sterling exchange on the part of the arbitrager concerned is a decreasing function of the forward premium minus the excess of the domestic interest rate (New York rate) over the foreign interest rate (London rate). By substituting (8) and (9) in (5), we get

or

where p_t is the forward premium on sterling. Given the forward contracts that the arbitrager had already entered into in the past, his current demand for forward sterling exchange, D_it, is a decreasing function of $[p_t-(I_t^a-I_t^b)]$.

D_it can, of course, be negative as well as positive, that is to say, the arbitrager may be induced to arbitrage in the direction of selling forward against spot buying as well as in the direction of buying forward against spot selling. It is quite obvious from (10) that, if his outstanding previous forward contracts are predominantly forward purchases of sterling, i.e., if $\underset{t-89}{\overset{t-1}{\mathrm{\Sigma }}}{D}_{ir}$, is a large positive magnitude, he would be more inclined to arbitrage in the direction of selling forward against spot buying of sterling, or more reluctant to arbitrage in the direction of buying forward against spot selling of sterling.

Since $p_t=\frac{{FR}_t-{SR}_t}{{SR}_t}$ and, for an individual arbitrager, SR_t as well as $I_t^a$ and $I_t^b$ may be regarded as given (unless he has an oligopolistic or oligopsonistic influence on the spot exchange rate and the interest rates in the two centers), equation (10), therefore, also gives his demand for forward exchange as a decreasing function of the current forward rate FR_t.

The aggregate demand on the part of arbitragers as a whole is obtained by totaling the individual demand functions for all the arbitragers:

The essential characteristics of the aggregate demand function for forward exchange on the part of arbitragers as a whole remain unchanged, compared with the demand function for an individual arbitrager. The aggregate demand, $\underset{i}{\mathrm{\Sigma }}{D}_{it}$ is still a decreasing function of $[p_t-(I_t^a-I_t^b)]$. Again, if the aggregate outstanding forward contracts, which arbitragers as a whole have entered into in the past, are predominantly forward purchases of sterling, i.e., if $\underset{i}{\mathrm{\Sigma }}\underset{t-89}{\overset{t-1}{\mathrm{\Sigma }}}{D}_{\mathit{ir}}$, is a large positive magnitude, the arbitragers would, on the whole, be more inclined to arbitrage in the direction of selling forward and buying spot, or more reluctant to arbitrage in the direction of buying forward and selling spot.

There is, however, one important point of difference: for arbitragers as a whole, $S{R}_t,I_t^a,\mathrm{and}{I}_t^b$ may not be given but may be functions of the aggregate demand, even though individually none of them may be powerful enough to affect these variables. As pointed out above, an arbitrager, qua arbitrager, would match an increase in his forward position by a corresponding decrease in his spot position (or vice versa). An increase in his forward position implies an excess of his current net purchase of forward exchange over the previous forward purchase contracts (net of previous forward sale contracts) that fall due on the current day. If we make the simplifying assumption that all forward contracts are for 90 days’ delivery, then the current change in his net forward position is

The corresponding adjustment in his spot position which he has to make to avoid a net open position (spot and forward) is, therefore, –(D_it-D_it-90). If his current net purchase of forward sterling exceeds his previous net forward purchases due for settlement on the current day, i.e., if (D_it-D_it-90) >0, he would sell spot sterling of an equal amount, and vice versa if (D_it-D_it-90)< 0. Therefore, the aggregate net purchase (or sale) of spot sterling by arbitragers as a whole on account of their arbitrage operations is equal to $-(\underset{i}{\mathrm{\Sigma }}{D}_{it}-\underset{i}{\mathrm{\Sigma }}{D}_{it-90})$. If this is a positive magnitude, it indicates an aggregate net purchase; if negative, it indicates an aggregate net sale.

Unless SR and I^a are rigidly pegged by the monetary authorities, they tend to be pushed up by the aggregate net purchase of spot exchange by arbitragers as a whole. On the other hand, I^b tend to be lowered by the transfer of spot funds to London by arbitragers as a whole. That is

and

Hence, given $\underset{i}{\mathrm{\Sigma }}{D}_{it-90}$ as a predetermined quantity,

and

We shall leave the determination of the spot rate of exchange for later discussion, as we shall see that, unless it is pegged, it must be determined jointly with the forward exchange rate. We shall also refrain from broadening our model to encompass all the variables and equations that are relevant for the determination of the interest rates both at home and in the foreign center concerned. We shall merely assume that the domestic and foreign short-term interest rates may be approximated by the following linear equations:

and

where $I_t^{a*}\mathrm{and}{I}_t^{b*}$ are determined by factors extraneous to our limited model and I_’a and I_’b are positive coefficients representing, respectively, $-\frac{{\partial I}_t^a}{\partial (\underset{i}{\mathrm{\Sigma }}{D}_{it}-\underset{i}{\mathrm{\Sigma }}{D}_{it-90})}$ and $\frac{{\partial I}_t^b}{\partial (\underset{i}{\mathrm{\Sigma }}{D}_{it}-\underset{i}{\mathrm{\Sigma }}{D}_{it-90})}$. Substituting (13) and (14) in (11) and writing $p_t=\frac{{FR}_t-{SR}_t}{{SR}_t}$, we get

where $\varphi =\underset{i}{\mathrm{\Sigma }}\frac{1}{\left(({\rho }_i^{\prime a}+{\rho }_i^{\prime b})\right)}/1+\underset{i}{\mathrm{\Sigma }}\frac{(I^{\prime a}+I^{\prime b})}{({\rho }_i^{\prime a}+{\rho }_i^{\prime b})}$. It is obvious that the greater the effects of arbitrage operations upon the domestic and foreign interest rates, the smaller (numerically) the elasticity of arbitragers’ demand for forward exchange.

Thus the analysis of the behavior of covered interest arbitragers at best provides us merely with a demand (or supply) curve for forward exchange on the part of arbitragers. To assume that such a demand function on the part of only one sector of the dealers in the forward exchange market would by itself determine the forward exchange rate is to assume by implication that such a demand curve is perfectly elastic over a sufficiently wide range to overwhelm all other demands and supplies. We have just seen, however, that this demand is normally less than perfectly elastic either as a function of the current forward rate or as a function of the divergence of the forward premium (or discount) from the interest differential.

Speculation in Forward Exchange

We shall next analyze the speculative demand for (or supply of) forward exchange. We shall adopt the wider definition under which we mean by speculation in foreign exchange any deliberate assumption of a net open position (an excess of uncovered claims over liabilities, or the other way around) in a foreign currency. Such an open position can, of course, be created through dealings either in forward exchange or in spot exchange—in other words, speculation can be either in the forward market or in the spot market. We shall, however, first tentatively assume that all speculation in foreign exchange is in forward exchange, as this is indeed the most convenient way of speculating in foreign exchange. This assumption will not impair the generality of the analysis, since it will presently be shown that speculation in spot exchange can always be treated as a combination of speculation in forward exchange and an interest arbitrage operation such as that described above.

Suppose that a speculator with no initial net commitments (or net position) in foreign exchange, and hence with no exchange risk to begin with, is confronted with the current forward rate. If he believes that the future spot rate (90 days from the current day) is likely to be higher than the current forward rate, he would purchase forward exchange, and thus assume a long position, until the marginal risk of extending his long position by an extra unit of foreign exchange is only just compensated by the margin by which his expected future spot rate exceeds the current forward rate. That is, his equilibrium position can be represented as

where ER_it is, in the expectation of the speculator concerned (denoted here by the subscript j), the most probable future spot rate 90 days from the current day, and r_it is the marginal risk of extending his long position in foreign exchange.

The expected future spot rate of any speculator is never a definite rate, but a range of possible rates with different degrees of probability. However, as long as a speculator’s expectation as to the future spot rate is normally distributed, it can always be uniquely represented by the most likely expected rate (or the mean expected rate) and the variance (or its square root, the standard deviation) of the probability distribution of his expectation. Thus whenever we speak of the expected rate of a speculator, we always mean the most likely future spot rate in his expectation, or his modal estimate of the future rate.

Obviously, r_jt is an increasing function of the speculator’s current purchase of forward exchange, D_jt, and the variance of the probability distribution of his current expectation, σ_it. Thus

Conversely, a speculator with no initial net position would sell forward exchange, and thus assume a short position, if he believed that the future spot rate was likely to be lower than the current forward rate, until the marginal risk of extending his short position by an extra unit of foreign exchange was only just compensated by the margin by which the current forward rate exceeded his expected future spot rate. The marginal risk for such a speculative seller of forward exchange is obviously an increasing function of his current sale of forward exchange, –D_kt (according to our convention, sales of forward exchange are to be treated as negative purchases), and the standard deviation of the probability distribution of his current expectation, σ_kt. That is

which, put in the same form as (16’) above, gives

Comparison between (16’) and (17’) suggests that the equilibrium conditions for a speculator who purchases forward exchange and for one who sells are essentially the same. When the left-hand side of the equation is written as ER_jt—FR_t, the marginal risk on the right-hand side takes positive or negative sign according to whether the speculator concerned is a purchaser or a seller of forward exchange, i.e., according to whether D_jt is positive or negative. The standard deviation of his expectation, σ_jt, which is always a positive magnitude, affects only the magnitude of his marginal risk but not its sign. Thus the speculator concerned will be a buyer or a seller of forward exchange according to whether his expected rate is higher or lower than the current forward rate.

The problem becomes much more complicated, however, when the assumption is waived that the speculator has no initial net position, and hence no initial exchange risk, before he makes any purchase or sale on the current market. If he has already accumulated a large long position from his previous purchases of forward exchange, the excess of his current expected rate over the current forward rate has to be quite considerable to induce him to make any further purchase of forward exchange. Indeed, he may become a seller of forward exchange to cover his pre-existing long position, when the excess of his expected rate over the forward rate is reduced even though it is still positive. Conversely, if he already has a large short position from his previous sales of forward exchange, his current expected rate would have to fall considerably short of the current forward rate to induce him to make further sales of forward exchange; and if the gap is so reduced that he decides to close his initial overextended (in the light of the new situation) short position, he may become a purchaser of forward exchange even though his expected rate still falls short of the current forward rate. The difficulty of this problem lies in the fact that his initial short or long position is an aggregate of his previous purchases and sales which have been contracted during the past 89 days and which therefore have days to maturity varying from 1 to 89. Purchases or sales that were contracted yesterday, though possibly at a different forward rate, have almost the same effect upon his willingness to make further purchases or sales today as purchases or sales concluded today, whereas purchases or sales contracts that are to mature in a day’s time are already written off as realized profits or losses almost as definitely as contracts that have already matured.¹⁰

It appears, therefore, that past contracts of purchase or sale would affect the speculator’s willingness or reluctance to take on more commitments in different degrees, varying in direct relation to the number of days the contracts have to run before maturity. If the effects of past forward commitments upon the speculator’s current risk position are assumed to diminish exponentially, and his current marginal risk to be a linear homogeneous function of his current and past commitments,¹¹ his marginal risk may be represented as

where γ_j is a positive coefficient, λ_j a positive fraction measuring the diminishing influences of past commitments such that ${\lambda }_j^{90}$ is practically zero, and D_jt, D_jt-1, etc., are the speculator’s net purchases of forward exchange on the days indicated by the second subscripts. Each of these D_j’s may be either positive or negative.

It may be shown¹² then that

This function for the marginal risk of the speculator indicates that his current marginal risk is not only an increasing function of his current purchases of forward exchange (sales are treated as negative purchases), but that the risk freshly incurred by current purchases has to be added algebraically (i.e., allowing for signs) to a remnant of his marginal risk of the day before. And the marginal risk of the day before would be positive or negative according to whether his weighted cumulative position on that day is long (positive) or short (negative).

Substituting (19) in (16), we get

and hence

Equation (21) may be regarded as a linear approximation of the demand (or supply) function for forward exchange on the part of a speculator regardless of whether he is initially in a long or a short position.

The aggregate demand (supply) function of speculators as a whole is the aggregation of a set of equations like (21), i.e.,

where ${\overline{ER}}_t$, the weighted average expected rate of all speculators, is defined as $\mathrm{\Sigma }\frac{1}{{\gamma }_j}{ER}_{jt}/\mathrm{\Sigma }\frac{1}{{\gamma }_j}$.

The essential characteristics of the aggregate speculative demand function remain the same as the demand functions of the individual speculators; the aggregate speculative demand (or supply) is an increasing (decreasing) function of the excess of the weighted average of the expected rates of all the speculators over the current forward rate and a decreasing (increasing) function of their aggregate weighted cumulative long positions.

Equation (22) gives the aggregate speculative demand for forward exchange as a definite function of the current forward rate only when the weighted average expected rate of all the speculators is either given independently of the current forward rate by extraneous factors or is a known function of the current forward rate (as well as of other extraneous factors). We shall not be able to go into the details of the dynamics of the formation of exchange rate expectation here. Suffice it only to point out that if the average current expected rate is influenced by the current forward rate so that it is adjusted upward when the current forward rate increases, the aggregate speculative demand for forward exchange would become much less elastic and might even become an increasing function of the current forward rate. Write

where ${\overline{ER}}_t^{*}$ is given by extraneous factors and η represents the influence of FR_t upon ${\overline{ER}}_t$. Substitution of this equation in (22) produces

If (η - 1) > 0, the aggregate speculative demand is an increasing function of the current forward rate. If η = 1, the aggregate speculative demand is entirely determined independently of the current forward rate by extraneous factors and lagged variables. Normally, however, the current forward rate is unlikely to have a considerable direct influence upon the average expected rate of all the speculators, so that the aggregate speculative demand can be expected to be a decreasing function of the current forward rate.

It has been pointed out above that a speculator in foreign exchange may speculate on the spot market as well as on the forward market. Speculative purchases of spot exchange, however, unlike those of forward exchange, cost the speculator interest charges and the sacrifice of liquidity on the local currency funds that he must raise in order to buy the foreign exchange. On the other hand, the spot foreign exchange he purchased can earn him interest and convenience abroad until the time he decides to part with it. The converse is true for speculative sales of spot exchange.

The equilibrium condition for a spot speculator would be

where SR_t is the current spot rate, r_it the marginal risk of extending his current long position in foreign exchange, $I_t^a$ and $I_t^b$, the current short-term rates of interest at home and in the foreign center, and t and ${{\rho }_j^a}_t$ and ${{\rho }_j^b}_t$ the marginal subjective yield of convenience (or liquidity) to the speculator concerned of funds at home and in the foreign center, respectively. Since the possibilities of speculating on the spot market and on the forward market are both open to the speculator concerned, this equilibrium condition must exist side by side with equation (16), i.e., ER_jt - FR_t = r_jt, if he indulges himself in both forms of speculation. The marginal risk of extending his current long (or short) position by spot or forward dealings must be the same, since the risk of loss involved in a wrong expectation is the same. That is, r_jt in (23) and (16) must be the same.

When the left-hand and right-hand sides of (16) are subtracted from the corresponding sides of (23), the result is

which is identical in form to the equilibrium condition of a typical interest arbitrager stated in equation (5) above. Thus a speculator who speculates on the spot exchange market may, in fact, be regarded as acting implicitly in the combined capacity of an interest arbitrager and a forward exchange speculator. If he speculates by buying spot foreign exchange (instead of buying forward exchange), what he is doing is in fact equivalent to acting first as an interest arbitrager in buying spot exchange against a sale of an equal amount of forward exchange and then as a forward speculator in buying the forward exchange from himself (in the former capacity) in the expectation that the future spot rate will rise higher than the current forward rate and yield him a pure speculative gain. If he speculates by selling spot exchange (instead of forward exchange), he is in fact acting first as an interest arbitrager in selling spot exchange against an equal amount of forward purchase, and then as a forward speculator in selling the forward exchange to himself in the former capacity in the expectation that the future spot rate will turn out to be lower than the current forward rate.

Thus the tentative assumption stated above, that all speculation in foreign exchange is carried on in the forward market, does not impair the generality of our analysis as long as speculators who in effect speculate in the spot market are treated according to their implicit dual capacity, namely, first as interest arbitragers and then as speculators in forward exchange.

In fact, speculators in foreign exchange will speculate in spot exchange only if they find it cheaper to perform the implicit interest arbitrage themselves rather than to leave it to the professional arbitragers, i.e., banks. In other words, unless they find that they can carry the speculative stock of spot exchange at a lower net interest and liquidity cost than the banks, they will prefer to concentrate on speculative operations in the forward market only.

Hedging in Connection with Foreign Trade

Under the wide definition for speculation that has been adopted in this paper, a merchant who takes a chance in not covering his commitments in foreign exchange with a forward purchase or sale is treated as a speculator; therefore, it will be assumed that all foreign trade transactions contracted for future payments automatically lead to corresponding amounts of purchases or sales of forward exchange. A current import contract would automatically give rise to a demand for forward exchange (on the part of the domestic importer if the contracted payment is stipulated in foreign currency, or on the part of the foreign exporter if the payment is stipulated in domestic currency). A current export contract would automatically give rise to a supply of forward exchange (on the part of the domestic exporter if the payment is stipulated in foreign currency, or on the part of the foreign importer if the payment is stipulated in domestic currency).

It is true that, by immediate purchases or sales of spot exchange importers and exporters can equally get rid of the risks connected with the exchange rate uncertainty at the time of payment. Thus, for instance, an importer could pay for his order either in advance or so many days after receiving the documents of shipment. As Spraos has pointed out,¹³ however, hedging by purchase or sale of spot exchange is really equivalent to a combination of hedging by means of forward exchange and an interest arbitrage operation by the hedger himself. This can be demonstrated more precisely in the following way. The cost of eliminating the exchange risk of an importer by means of a forward purchase may be represented as ER_s - FR, where ER_s is the expected future spot rate of the importer concerned. On the other hand, the cost of eliminating the exchange risk by an immediate purchase of spot exchange may be represented as ${ER}_s-SR[1+(I^a+{\rho }_s^a)-(I^b+{\rho }_s^b\left)\right]$ where I^a is the domestic rate of interest that the importer has to pay to raise the funds for the immediate purchase of spot exchange; ${\rho }_s^a$, is the sacrifice of liquidity at the margin for thus diverting the use of his funds or borrowing power; and $(I^b+{\rho }_s^b)$ may be regarded either as the discount he can obtain from his trading partner for advance payment—in which I^b and ${\rho }_s^b$ represent the interest rate in the foreign center and the marginal convenience his partner can receive from the funds—or as the interest and convenience he can obtain himself by putting the exchange funds aside in the foreign center pending the future date of payment.

It may then be observed that hedging by advance purchase of spot exchange really implies an interest arbitrage operation by the hedger himself; for

That is, hedging by advance purchase of spot exchange is equivalent to acting first as an interest arbitrager in buying spot exchange against a sale of an equal amount of forward exchange and then as a hedger in buying the forward exchange supplied by himself in the former capacity. If the hedger concerned attains his equilibrium as an implicit interest arbitrager, so that, in conformity to (5) above,

or

then

i.e., the two ways of hedging would be indifferent to him.

Mutatis mutandis, the analysis is the same for exporters who hedge by advance sale of spot exchange (including the discount of drafts on their trading partners or their partners’ banks).

Thus it may be assumed that all currently contracted imports and exports that need hedging (which means practically all import and export contracts that are not directly financed by foreign long-term investment) automatically constitute demands for and supplies of forward exchange on the forward market, as long as those importers and exporters who hedge their commitments by advance purchases or sales of spot exchange are treated as both hedgers in the forward exchange market and implicit interest arbitragers.

It is sometimes pointed out that, apart from international trade in goods and services, foreign investments and the holdings of internationally traded commodities may give rise to demands for hedging by forward exchange. However, as described above, hedging of short-term foreign investments by means of forward exchange is simply a component part of covered interest arbitrage. The demands for (or supplies of) forward exchange arising from such operations have already been classified as arbitrager’s demands (or supplies) (see section, Covered Interest Arbitrage, above). As for long-term foreign investments, the exchange risks involved therein cannot be adequately hedged by means of forward exchange. An exchange contract due in three months is an inappropriate hedge for a ten-year bond. Insofar as holdings of internationally traded commodities are concerned, the facilities of hedging are much more adequately provided by the futures markets in those commodities.

Therefore, the assumption will be made here that the demand and supply of forward exchange for hedging purposes arise almost entirely from contracts of imports and exports of goods and services, and that other sources of demand or supply for hedging purposes either are already accounted for as arbitragers’ demands or are negligible.

Equilibrium of the Forward and Spot Exchange Markets

The forward rate of exchange, like all other prices on a free market, will be determined at the level where demand equals supply. If supply is treated as negative demand, as it was above, then the forward rate will be determined at the level where aggregate net demand (or excess demand) for forward exchange on the part of all dealers in the forward market is zero. That is to say, if it is assumed that there is no government intervention, the equilibrium condition of the forward exchange market is

where $\underset{i}{\mathrm{\Sigma }}{D}_{it}$ is the aggregate net demand for forward exchange on the part of interest arbitragers (including the implicit arbitragers who operate as speculators or hedgers at the same time); $\underset{j}{\mathrm{\Sigma }}{D}_{jt}$ is the aggregate net demand on the part of speculators (including those merchants who deliberately leave parts of their foreign exchange commitments unhedged); M_t measures import contracts currently entered into; M^*_t is the part of current import contracts directly financed by long-term foreign investments; X^*_t represents export contracts currently entered into; and X^*_t is the part of current export contracts directly financed by long-term investment abroad by domestic residents.

Since the amount that importers would actually have to pay for their foreign exchange obligations, including the premium for exchange risks, and the amount that exporters would actually receive in domestic currency for their sales proceeds abroad, net of the premium for exchange risks, are governed by the forward rate, current imports and export contracts are primarily functions of the current forward rate rather than of the current spot rate; viz.,

Provided that the spot exchange rate is given, e.g., pegged by the monetary authorities, equations (15), (22’), (24), (25), and (26) would be sufficient to determine the current forward rate together with the aggregate demand for forward exchange on the part of all arbitragers (implicit as well as explicit); the aggregate demand of forward exchange on the part of speculators; current import contracts; and current export contracts.

The determination of the forward rate is illustrated by Diagram 1.

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In this diagram, $(M_t-M_t^{*})$ is represented by a rather inelastic downward sloping line, and $(X_t-X_t^{*})$ by a rather inelastic upward sloping line. The aggregate speculators’ demand for forward exchange, $\underset{j}{\mathrm{\Sigma }}{D}_{jt}$ is drawn as a rather elastic downward line with reference to a fixed point on the vertical axis representing the average expected rate of all the speculators determined by extraneous factors, i.e., ${\overline{ER}}_t^{*}$. This line cuts the vertical axis at a point below ${\overline{ER}}_t^{*}$ indicating that when FR_t equals ${\overline{ER}}_t^{*}$, speculators’ demand would be negative (i.e., $\underset{j}{\mathrm{\Sigma }}\frac{\eta }{{\gamma }_j}{\overline{ER}}_t^{*}-\underset{j}{\mathrm{\Sigma }}\frac{{\lambda }_j}{{\gamma }_j}{r}_{jt-1}$ happens to be negative.¹⁴ The arbitragers’ demand curve for forward exchange is drawn with reference to a given spot rate, SRt. Since arbitragers’ demand is a function of the percentage gap between the forward rate and the spot rate (rather than a function of the absolute level of the former), the position of this curve would obviously shift with any change in the spot rate. If the spot rate is pegged, as is assumed for the time being, the forward rate is determined as follows: Add the horizontal distances between the speculators’ demand curve $\left(\underset{j}{\mathrm{\Sigma }}{D}_{jt}\right)$ and the vertical axis to the corresponding horizontal distances between $(M_t-M_t^{*})$ and the axis to form the $(M_t-M_t^{*}+\underset{j}{\mathrm{\Sigma }}{D}_{jt})$ curve. The equilibrium forward rate (given the spot rate, SR_t) will then be the rate at which the excess of the demand of importers and speculators over the supply of exporters, CD in Diagram 1, is exactly equal to the arbitragers’ supply (negative demand), AB; that is, $(M_t-M_t^{*})+\underset{j}{\mathrm{\Sigma }}{D}_{jt}-(X_t-X_t^{*})=-\underset{j}{\mathrm{\Sigma }}{D}_{jt}$, as indicated by (24).

If the spot exchange rate is not pegged by the monetary authorities but must be regarded as an endogenous variable dependent upon, inter alia, arbitragers’ purchases or sales of spot exchange, then the system of equations outlined above is not determinate until one more equation is added, viz., that for equilibrium of the spot exchange market. In other words, the forward rate is indeterminate unless the spot rate is determined at the same time.

For the sake of simplification, assume that all import and export contracts uniformly stipulate for payments in 90 days, to coincide with the length of forward exchange contracts; then the equilibrium of the spot exchange market (i.e., equality between the supply of and demand for spot exchange) may be written as

where $(M_{t-90}-M_{t-90}^{*})$ denotes import orders, not directly financed by foreign long-term investment at home, that are currently due for payments; $(X_{t-90}-X_{t-90}^{*})$ denotes export orders (excluding those directly financed by domestic long-term investment abroad) currently due for receipt of payments; L_t represents net current long-term capital outflow (or inflow when negative) and, therefore, $L_t-(X_t^{*}-M_t^{*})$ denotes the change in net current long-term capital movement not used directly to finance exports or imports; $(\underset{i}{\mathrm{\Sigma }}{D}_{it}-\underset{i}{\mathrm{\Sigma }}{D}_{it-90})$, as already explained above, represents the net sales (or purchases when negative) of spot exchange by arbitragers to cover the net increases (or decreases when negative) in their forward positions; and G_t is the net sale (purchase when negative) of spot exchange by the monetary authorities of either of the two countries concerned¹⁵ for stabilization purposes.