International. chapter. Objectives. Introduction

Transcription

1 chapter International Arbitrage Introduction Arbitrage is generally defined as capitalising on a discrepancy in quoted prices as a result of the violation of an equilibrium (no-arbitrage) condition. The arbitrage process restores equilibrium via changes in the supply of and demand for the underlying commodity, asset or currency. The importance of arbitrage is that no-arbitrage conditions are used for asset pricing, such that the equilibrium price of a financial asset is the price that is consistent with the underlying no-arbitrage condition. In this chapter we consider several kinds of arbitrage involving foreign exchange markets, commodity markets and money markets. Objectives The objectives of this chapter are: To define arbitrage and the no-arbitrage condition. To describe two-point, three-point and multi-point arbitrage in the foreign exchange market. To describe commodity arbitrage. To describe covered interest arbitrage and show how the no-arbitrage condition can be used to determine the forward exchange rate. To describe uncovered arbitrage and introduce the concept of carry trade. To expose some misconceptions of arbitrage.

2 34 international finance: an analytical approach chapter International Arbitrage 35. Two-point arbitrage Also known as spatial arbitrage or locational arbitrage, two-point arbitrage arises when the exchange rate between two currencies assumes two different values in two financial centres at the same time. We will first consider two-point arbitrage without the bid offer spread, then we modify the operation to account for the spread. Two-point arbitrage without the bid offer spread Given two financial centres, A and B, and two currencies, x and y, and assuming (for simplicity) no transaction costs and a zero bid offer spread, arbitrage will be triggered if the following condition is violated: A (x/y) = B (x/y). This condition says that the exchange rate between x and y should be the same in A as in B. If the condition is not satisfied in the sense that the exchange rate between x and y is different in A from its level in B, then the currencies are expensive in one financial centre and cheap in the other. Arbitragers in this case buy one of the currencies where it is cheap and sell it at profit where it is expensive. Consider the case when the condition is violated such that A (x/y) > B (x/y). This violation means that currency y is more expensive in A than in B (or that x is cheaper in A than in B). Let us consider the situation from the perspective of currency y. Arbitragers buy y in B and sell it in A, making profit, p, that is given by p = A (x/y) B (x/y).2 The Reuters Monitor shows the following information about the exchange rate between the Australian dollar and the U dollar (measured in direct quotation in both centres): ydney.7800 (AUD/UD) New York (UD/AUD) To find out whether or not there is an arbitrage opportunity, we have to check whether the no-arbitrage condition is violated. When we invert the exchange rate in New York, we obtain / = Thus, the no-arbitrage condition is violated in the sense that the UD is more expensive in ydney than in New York. Hence, arbitragers buy the U currency in New York at.7400 and sell it in ydney at Profit in Australian dollar per U dollar bought and sold is p = = or 400 points. The effect of arbitrage is to raise the price of the UD in New York and lower it in ydney, until they are equal somewhere between.7800 and uppose that at some stage prior to the restoration of equilibrium, changes in supply and demand cause the exchange rate to fall to.7700 in ydney and rise to.7500 (or in direct quotation) in New York. In this case, profit shrinks to p = = or 200 points. Eventually, the rate falls to.7600 in ydney and rises to the same level ( in direct quotation) in New York. Profit at this stage is p = = 0 which means that arbitrage is not profitable because the no-arbitrage condition is restored.. Figure. The process of arbitrage restores the equilibrium condition via changes in the forces of supply and demand. This is illustrated by Figure., which shows the supply and demand The effect of two-point arbitrage (x/y ) y ( A ) 0 ( A ) ( B ) ( B ) 0 (x/y ) y curves for currency y in financial centres A and B. Initially, the exchange rates in A and B are ( A ) 0 and ( B ) 0 respectively, such that ( A ) 0 > ( B ) 0. As the demand for y increases in B, the exchange rate rises (y appreciates). Conversely, the supply of y increases in A and so the exchange rate falls (y depreciates). This process continues until the exchange rates in both financial centres are equal (that is, until ( A ) = ( B ) ) because this condition eliminates profit and hence the incentive for arbitrage. Two-point arbitrage with the bid offer spread o far we have shown how arbitrage works by assuming that there is no bid offer spread. If this assumption is relaxed, the no-arbitrage condition in this case is given by the equations b,a (x/y) = a,b (x/y).3 a,a (x/y) = b,b (x/y).4 A D y Q y B D y Q y where b,a (x/y) is the bid rate in A, and so on. Let us now see what happens if the equilibrium condition is violated, such that b,a (x/y) > a,b (x/y). In this case the arbitrager can make profit by buying y in B at a,b (x/y) and selling it in A for b,a (x/y). Arbitrage profit is the difference between the selling rate and the buying rate, or b,a (x/y) a,b (x/y).

3 36 international finance: an analytical approach chapter International Arbitrage 37.2 The exchange rate between the pound and Australian dollar (GBP/AUD) as recorded in ydney and London is as follows: ydney London To make profit the arbitrager will buy the Australian dollar in London at GBP and sell it in ydney at GBP Profit in pounds per Australian dollar is given by p = = or 0 points. Equivalently, profit is made by buying the pound in ydney and selling it in London. The effect of the bid offer spread is to reduce the profitability of arbitrage, since the spread is a transaction cost. If arbitrage is possible at the mid-rates, then we have the following: ydney London The arbitrager in this case buys the Australian dollar in London at GBP and sells it in ydney at GBP Arbitrage profit in this case is or 50 points..2 Three-point and multi-point arbitrage In this section we consider arbitrage involving more than two currencies. We start with arbitrage involving three currencies. Three-point arbitrage Given three currencies (x, y and z) and making the same assumptions as in the case of twopoint arbitrage, three-point arbitrage (also called triangular arbitrage) will be triggered if the following condition is violated: (x/y) = (x/z) } (y/z) In this case, the three exchange rates are equal across financial centres, which precludes the possibility of two-point arbitrage (this is why the exchange rates in Equation (.5) do not have subscripts to indicate the financial centres where they are quoted). This condition tells us that cross exchange rates are consistent in the sense that if we calculate one of them on the basis of the other two, the calculated rate should be identical to the rate that is actually quoted. Two steps are involved in three-point arbitrage: (i) checking whether or not the condition is violated (that is, whether or not the cross rates are consistent); and (ii) determining the profitable sequence. Let us assume that the no-arbitrage condition is violated such that (x/y) > (x/z)/(y/z). The profitable sequence can be determined with the aid of a triangle, placing each one of the three currencies in one of its corners (in no special order), as shown in Figure.2. Determination of the profitable sequence is simple. We start with one unit of any of the three currencies and move clockwise as in Figure.2(a) around the triangle until we end up where we started from, with the same currency. In this case, we end up with less than one unit of the currency we started with, which gives the unprofitable sequence. The profitable sequence will be in an anti-clockwise direction, as in Figure.2(b)..5 Figure.2 Profitable and unprofitable sequences in three-point arbitrage z x (a) Unprofitable sequence By starting with one unit of currency x and moving clockwise, as in Figure.2(a), arbitrage involves the following steps:. elling x and buying y to obtain /(x/y) units of y. 2. elling y and buying z to obtain /[(x/y)(y/z)] units of z. 3. elling z and buying x to obtain (x/z)/[(x/y)(y/z)] units of x. ince (x/y) > (x/z)/(y/z), it follows that (x/z)/(y/z) < (x/y) or that (x/z)/[(x/y)(y/z)] <. Hence, we end up with less than one unit of x. The profitable sequence must, therefore, be in a counter-clockwise direction. This sequence, as represented by Figure.2(b), involves the following steps:. elling x and buying z to obtain /(x/z) units of z. 2. elling z and buying y to obtain (y/z)/(x/z) units of y. 3. elling y and buying x to obtain (y/z)(x/y)/(x/z) units of x. ince (x/y) > (x/z)/(y/z), it follows that (y/z)(x/y)/(x/z) >. Thus, we end up with more than one unit of x, and this must be the profitable sequence. The possibilities for three-point arbitrage can be summarised as follows: If (x/y) = (x/z)/(y/z), then there is no arbitrage opportunity. If (x/y) > (x/z)/(y/z), then there is a profitable arbitrage opportunity by following the sequence x z y x. If (x/y) < (x/z)/(y/z), then there is a profitable arbitrage opportunity by following the sequence x y z x. Arbitrage restores the equilibrium condition via changes in the supply of, and demand for, currencies. Let us trace what happens when (x/y) > (x/z)/(y/z). Figure.3 shows that the buying and selling of currencies result in changes in the forces of supply and demand, as follows:. An increase in the demand for z (the supply of x), and so (x/z) rises as shown in Figure.3(a). 2. An increase in the demand for y (the supply of z), and so (y/z) falls as shown in Figure.3(b). 3. An increase in the demand for x (the supply of y), and so (x/y) falls as shown in Figure.3(c). y z (b) Profitable sequence x y

4 38 international finance: an analytical approach chapter International Arbitrage 39 Figure.3 (x/z) The effect of three-point arbitrage z (y/z) z The following exchange rates are quoted in ydney, Auckland and Hong Kong: (HKD/AUD) (NZD/AUD) (HKD/NZD) To find out whether or not there is a possibility for three-point arbitrage, we have to check the consistency of the cross rates (the validity of the no-arbitrage condition). (HKD/NZD) can be calculated from the other two rates as (HKD/NZD) = }} (HKD/AUD) (NZD/AUD) = } = (a) Dz Qz (b) Dz Qz Hence, the equilibrium condition is violated, implying a possibility for three-point arbitrage. First, try the sequence HKD NZD AUD HKD, starting with one unit of HKD:. ell HKD.0000 for NZD to obtain (/ = 0.279) units of NZD. 2. ell NZD0.279 for AUD to obtain (0.279/.2052 = 0.236) units of AUD. (x/y) 3. ell AUD0.236 for HKD to obtain ( = ) units of HKD. y Obviously, this is not the profitable sequence. Now, try the opposite sequence, starting with one unit of HKD:. ell HKD.0000 for AUD to obtain (/4.548 = ) units of AUD. 2. ell AUD for NZD to obtain ( = 0.290) units of NZD. 3. ell NZD0.290 for HKD to obtain ( =.0392) units of HKD. This is obviously the profitable sequence. If (HKD/NZD) = , then there is no possibility for three-point arbitrage because this rate is consistent with the others. At this stage we have: (c) Dy Qy and (HKD/AUD) }}} (HKD/NZD)(NZD/AUD) = }} =.0000 }}} (NZD/AUD)(HKD/NZD) = }} =.0000 (HKD/AUD) which shows that there is no profitable sequence. REEARCH The profitability of three-point arbitrage We have presented three-point arbitrage as a risk-free operation, because all of the decision variables (three exchange rates) are known at the time the decision is made. However, Kollias and Metaxas argue that three-point arbitrage involves some degree of risk due to the effect of slippages in currency quotes. By using high-frequency, tick-by-tick data on the exchange rates they found that arbitrage opportunities do exist. However, they also found that the exploitation of such opportunities involves a degree of risk that can adversely affect realised returns. C. Kollias and K. Metaxas, How Efficient are FX Markets? Empirical Evidence of Arbitrage Opportunities Using High-Frequency Data, Applied Financial Economics,, 200, pp Multipoint arbitrage Arbitrage involving four, five or more currencies can take place. However, three-point arbitrage is sufficient to establish consistent exchange rates, eliminating the profitability of multi-point arbitrage. In the case of three-point arbitrage involving currencies x, y and z, the no-arbitrage condition may be written as (x/y)(y/z)(z/x) =.6 If four currencies are involved (x, x 2, x 3 and x 4 ), then we have four-point arbitrage, in which case the no-arbitrage condition is

5 320 international finance: an analytical approach chapter International Arbitrage 32 (x /x 2 )(x 2 /x 3 )(x 3 /x 4 )(x 4 /x ) =.7 and when n currencies are involved, the no-arbitrage condition is Figure.4 Five-point arbitrage (x /x 2 )(x 2 /x 3 )(x 3 /x 4 ) (x n /x n )(x n /x ) =.8 AUD AUD.4 Consider the following exchange rates: (AUD/UD).88 (JPY/UD) EUR UD EUR UD (JPY/GBP) (GBP/EUR) (EUR/AUD) The no-arbitrage condition in this case is (AUD/UD)(UD/JPY)(JPY/GBP)(GBP/EUR)(EUR/AUD) = which gives.88 } = Because the no-arbitrage condition is not violated there is no possibility for profitable five-point arbitrage. We can actually check that this is the case by working out the process step by step, starting with one AUD and moving as shown in Figure.4. In this case, we have a pentagon rather than a triangle. The results of the calculations are displayed in the following table, which shows that starting with one Australian dollar we end up with one Australian dollar no matter which direction we move (try the same exercise by starting with one pound). You may want to check for yourself that there is no possibility for three-point arbitrage either, using all possible currency combinations (taking three currencies at a time). GBP (a) Clockwise Commodity arbitrage JPY GBP (b) Anti-clockwise The no-arbitrage condition in the case of commodity arbitrage is the law of one price (LOP), which stipulates that, in the absence of frictions such as shipping costs and tariffs, the price of a commodity expressed in a common currency is the same in every country. Commodity arbitrage is conducted by buying a commodity in a market where it is cheap and selling it in a market where it is more expensive. The LOP can be written as JPY.3 Clockwise Transaction End currency Number of units AUD UD UD UD JPY JPY JPY GBP GBP GBP EUR EUR EUR AUD AUD.0000 Anticlockwise AUD EUR EUR EUR GBP GBP GBP JPY JPY JPY UD UD UD AUD AUD.0000 P i = P * i.9 where P i is the domestic price of commodity i, P * i is its foreign price and is the exchange rate expressed as the number of units of the domestic currency per one unit of the foreign currency. Thus, P * i is the domestic currency equivalent of the foreign price of the commodity. Likewise, P i / is the foreign currency equivalent of the domestic currency price of the commodity. When arbitragers buy a commodity in a market where it is cheap and sell it where it is more expensive they make profit as the difference between the selling price and the buying price. This activity leads to a rise in the price of the commodity in the market where it is cheap and a decline in its price in the market where it is expensive until profit is eliminated and the noarbitrage condition is restored. Figure.5 shows how commodity arbitrage works, starting from a disequilibrium position described by the inequality P i > P * i. Initially, the domestic currency prices of commodity i abroad and at home are P * i0 and P i0 respectively. Arbitragers, then, buy the commodity where it is cheap (in the foreign market), leading to an increase in demand and a shift in the demand curve. They sell the commodity in the domestic market, leading to an increase in supply. Thus, the price rises in the foreign market and falls in the domestic market, until the former reaches

6 322 international finance: an analytical approach chapter International Arbitrage 323 Figure.5 The effect of commodity arbitrage P i P i0 P i 0 P i * P i * 0 IN PRACTICE Using the LOP for currency valuation ince 986, The Economist magazine has used the price of a homogenous product, the Big Mac, to show that there are cross-border differences in prices (when measured in the same currency) and to use these prices to calculate the level of exchange rates compatible with the no-arbitrage condition. The idea is very simple. Big Mac prices are recorded in a number of countries, then converted into U dollars and compared. The exchange rate compatible with the LOP (per U dollar) is subsequently calculated by dividing the price of a Big Mac in any country by the price in the United tates. The deviation of the actual rate from the no-arbitrage rate is calculated and used to indicate the extent of the overvaluation or undervaluation of the dollar (overvaluation is present when the actual rate is higher than the no-arbitrage rate, and vice versa). P i * 0 (a) Domestic market D 0 D 0 D Q i (b) Foreign market Q i While the LOP typically applies to the prices of individual commodities (such as a Big Mac), there is no reason why it cannot be applied to baskets of goods whose prices are measured in different currencies. In this case, the LOP can be written as.5 P i * and the latter reaches P i, which are equal. At this point, arbitrage profit is eliminated and the equilibrium condition is restored. In reality, however, commodity arbitrage is not as effective as to bring prices into equality and substantial cross-border differences in prices exist. everal reasons can be presented to explain deviations from the LOP, including transportation costs, differences in taste and differences in quality. Remember that for the LOP to work, we must consider exactly similar products in the absence of transportation costs. But even these conditions may not be adequate. Just imagine buying a Big Mac in Melbourne and selling it in New York: by the time it gets there no one would want to buy it. There are, however, real episodes of commodity arbitrage. In the early 990s, for example, quantitative restrictions on the imports of alcoholic beverages to the United Kingdom from France were relaxed in the spirit of the European single market. Given that beer was cheaper to buy in France, the English found it profitable to go across the Channel, buy a vanload of French beer and sell it at profit in England. French beer was sold by individuals as far north as heffield and Newcastle. The price of commodity i in Australia is AUD00 and the exchange rate (AUD/UD) is.80. The LOP implies that the equilibrium U price is UD56, because at this price the equilibrium condition represented by equation (.9) is not violated, implying no possibility for profitable arbitrage. This is because the selling price and the buying price measured in the same currency (P i and P * i respectively) are equal, which produces zero arbitrage profit. If the U price is UD50, then P i P * i > 0, implying that the no-arbitrage condition is violated, and that there is a possibility of profitable arbitrage. Arbitragers make profit by buying the commodity where it is cheap (the United tates), paying UD50 (or AUD90), and selling it in Australia at AUD00. Net arbitrage profit is then given by P i P * i = AUD0 per unit of the commodity. P = P *.0 which is the same as equation (.9) except that it is written in terms of the prices of baskets of commodities, P and P *, and not the prices of individual commodities, P i and P * i. If P and P * are taken to be the general price levels at home and abroad, then Equation (.0) may be taken to represent purchasing power parity, which we studied in Chapter 4. Covered interest arbitrage Covered interest arbitrage is triggered by the violation of the covered interest parity (CIP) condition, which describes the equilibrium relation between the spot exchange rate, the forward exchange rate, domestic interest rates and foreign interest rates. In essence, CIP is an application of the law of one price to financial markets, postulating that, when foreign exchange risk is covered in the forward market, the rate of return on a domestic asset must be equal to that on a foreign asset with similar characteristics. If this is not the case, then covered interest arbitrage is set in motion and continues until the resulting changes in the forces of supply and demand (for the underlying assets) lead to a restoration of the no-arbitrage condition represented by CIP. The CIP condition Consider an investor who has initial capital, K, and faces two alternatives: (i) domestic investment, whereby the investor buys domestic assets, earning the domestic interest rate, i; and (ii) foreign investment, whereby the investor converts the domestic currency into foreign currency to buy foreign assets, earning the foreign interest rate, i *. ince the domestic investment does not involve currency conversion, it does not involve foreign exchange risk (the risk arising from changes in the spot exchange rate). On the other hand, the foreign investment produces exposure to foreign exchange risk, but this exposure can be covered by.4

7 324 international finance: an analytical approach chapter International Arbitrage 325 Figure.6 selling the foreign currency (buying the domestic currency) forward. Foreign exchange risk is eliminated because the forward exchange rate is known in advance, although it is used to settle transactions involving delivery of the currencies some time in the future. Thus, the investor knows in advance the domestic currency value of her foreign investment. If the position is not covered in the forward market, the investor has to wait until maturity and apply the spot exchange rate prevailing then to determine the domestic currency value of the foreign investment. uppose that we are considering a one-period investment starting with the acquisition of a financial asset (for example, a deposit) and ending with the maturity of this asset (Figure.6). When the investor chooses the domestic investment, the invested capital is compounded at the domestic interest rate, and the investor ends up with the initial capital plus interest income, that is, K ( + i). If the investor chooses the foreign investment, she converts the initial capital to foreign currency at the current spot exchange rate, obtaining K/ units of the foreign currency, where is measured as domestic currency units per one unit of the foreign currency. If K/ worth of the foreign currency is invested in foreign assets, this capital is compounded for one period at the foreign interest rate, such that the foreign currency value of the investment on maturity is (K/)( + i * ). The domestic currency value of this investment is obtained by converting this amount into the domestic currency at the forward rate, F, to obtain F (K/)( + i * ). Return on domestic and foreign investment (with covered position) Converting at spot rate Investing in foreign assets forward rate Foreign investment K K ( + i*) Investor (K) Domestic investment KF ( + i*) K( + i) Let us assume that there are no restrictions on the movement of capital and that there are no transaction costs. We also assume that agents are risk-neutral, in the sense that they are indifferent between holding domestic and foreign assets if these assets offer equal returns. The equilibrium condition that precludes the possibility of profitable arbitrage is that the two investments must be equally profitable, in the sense that they provide the same domestic currency amount of capital plus interest. Hence K( + i) = K } ( + i* )F. By expressing the condition in terms of one unit of the domestic currency, we obtain ( + i) = } F ( + i).2 This condition (CIP) tells us that gross domestic return is equal to gross covered foreign return. The left-hand side of equation (.2) represents gross domestic return: it is gross because it includes the amount invested (one unit of the domestic currency) and the interest earned, i. ince F/ = + f, where f is the forward spread, it follows that ( + i) = ( + f)( + i * ).3 By simplifying equation (.3), ignoring the term i * f, we obtain the approximate CIP condition i i * = f.4 which tells us that in equilibrium the interest differential must be equal to the forward spread. Equation (.4) implies that the currency offering the higher interest rate must sell at a forward discount, and vice versa. This is because if i > i *, then f > 0, which means that the foreign currency (offering a lower interest rate) sells at a forward premium whereas the domestic currency (offering a higher interest rate) sells at a forward discount. If, on the other hand, i < i *, then f < 0, implying that the foreign currency sells at a discount while the domestic currency sells at a premium. Covered arbitrage without bid offer spreads Covered interest arbitrage consists of going short on (borrowing) one currency and long on (investing in) another currency, while covering the long position via a forward contract (selling the currency forward). Upon the maturity of the investment (and the forward contract) the proceeds are converted at the forward rate and used to repay the loan (covering the short position). The difference between the proceeds from the investment and the loan repayment (principal plus interest) is net arbitrage profit, or the covered margin. For arbitrage to be profitable the covered margin must be positive. This process is illustrated in Figure.7. Depending on the configuration of exchange and interest rates, an arbitrager may choose to arbitrage from the domestic to a foreign currency (taking a short position on the domestic currency and a long position on the foreign currency) or vice versa. The choice depends on which sequence produces profit or positive covered margin. For a given configuration of exchange and interest rates, if arbitrage is profitable in one direction it must produce a loss in the opposite direction. In the following descriptions the spot and forward exchange rates are measured in direct quotation as the price of one foreign currency unit (domestic/foreign).

8 326 international finance: an analytical approach chapter International Arbitrage 327 Figure.7 Covered interest arbitrage without bid offer spreads Domestic Foreign Foreign Domestic Borrowing domestic currency unit unit Borrowing foreign currency This amount is reconverted into domestic currency at the forward rate, F, to obtain (F/)( + i * ) domestic currency units. The value of the loan plus interest is ( + i) domestic currency units. The covered margin, p, is the difference between the domestic currency value of the proceeds and loan repayment, which gives p = F } ( + i) ( + i* ).5 or approximately p = i * i + f.6 Converting at spot rate Investing at foreign rate forward rate ( + i*) Loan repayment Loan repayment ( + i) Converting at spot rate Investing at domestic rate forward rate Equation (.6) tells us that the covered margin on arbitrage from the domestic to a foreign currency consists of the interest rate differential (foreign less domestic) and the forward spread. Arbitrage from a foreign to the domestic currency consists of the following steps: The arbitrager borrows foreign currency funds at the foreign interest rate, i *. For simplicity we again assume that the amount borrowed is one foreign currency unit. Borrowed funds are converted at the spot exchange rate,, obtaining domestic currency units. This amount is invested at the domestic interest rate, i. The domestic currency value of the invested amount at the end of the investment period is ( + i). This amount is reconverted into the foreign currency at the forward rate, F, to obtain (/F) ( + i) foreign currency units. The value of the loan plus interest is ( + i*) foreign currency units. The covered margin is again the difference between the domestic currency value of the proceeds and loan repayment, which gives F ( + i*) + i + i* ( + i) F p = } F ( + i) ( + i* ).7 or approximately Covered margin F ( + i*) ( + i) Covered margin ( + i) ( + i*) F p = i i * f.8 Equation (.8) tells us that the covered margin on arbitrage from the foreign to the domestic currency consists of the interest rate differential (domestic less foreign) and the negative of the forward spread. The interest parity forward rate Arbitrage, however, does not have to involve the domestic currency, as two foreign currencies may provide a profitable arbitrage opportunity. Arbitrage from the domestic to a foreign currency consists of the following steps: The arbitrager borrows domestic currency funds at the domestic interest rate, i. For simplicity we assume that the amount borrowed is one domestic currency unit. Borrowed funds are converted at the spot exchange rate,, obtaining / foreign currency units. This amount is invested at the foreign interest rate, i *. The foreign currency value of the invested amount at the end of the investment period is (/)( + i * ). The no-arbitrage condition is obtained when the covered margin is zero. By substituting p = 0 in equation (.5) or (.7), we obtain } F = F + i } + i * G.9 where } F is the particular value of the forward rate that is consistent with the no-arbitrage condition, which we may call the interest parity forward rate. If CIP holds then } F = F. uppose that you approached your banker, requesting a quote for the forward rate between the domestic currency and a foreign currency, perhaps because you want to buy the foreign currency forward to cover future payables. The banker may not know what CIP is, but he will

9 328 international finance: an analytical approach chapter International Arbitrage open his manual to search for a formula that gives him an expression for the forward rate. This formula would look like Equation (.9). Why would the banker use this formula to calculate the forward rate? imply because if the banker chose any other forward rate, you can simply make (riskless) profit out of your banker by indulging in covered arbitrage. The following example explains the situation. uppose that you asked your banker to quote a one-year forward rate on the pound, which he does, giving you the following information: One-year forward rate (AUD/GBP) pot rate (AUD/GBP) One-year AUD interest rate 8% One-year GBP interest rate 4% You observe immediately that the pound is selling at a forward discount because the forward rate is lower than the spot rate. Let us see what happens if you try to indulge in covered arbitrage, starting with arbitrage from the pound to the Australian dollar: Borrow GBP000 (or any other amount). Convert the pound spot at 2.75 to obtain AUD2750 ( ). Invest the AUD amount at 8 per cent for one year. At the end of the year, you will have AUD2970 (2750 ( )). Reconvert the AUD proceeds at the forward rate to pounds to obtain GBP20.8 (2970/2.65). The loan repayment that you have to make is GBP040 (000 ( )). Net arbitrage profit is GBP80.8 (= ). Notice that this profit is made without bearing any risk, since all of the decision variables (including the forward rate) are known at the time you decided to indulge in this operation. Now, let us see what happens if instead you decided to indulge in arbitrage from the Australian dollar to the pound: Borrow AUD000 (or any other amount). Convert the Australian dollar spot at 2.75 to obtain GBP363.6 (000/2.75). Invest the GBP amount at 4 per cent for one year. At the end of the year, you will have GBP378.. Reconvert the GBP proceeds at the forward rate to Australian dollars to obtain AUD002 ( ). The loan repayment that you have to make is AUD080 (000 ( )). Net arbitrage loss is AUD78.0 (= ). In this case you make a loss. Now assume that the banker quoted a forward rate of If you indulge in arbitrage from the pound to the Australian dollar at this forward rate, you will (after reconversion) obtain GBP040 (2970/2.8558), in which case your arbitrage profit is zero. If you go from the Australian dollar to the pound you obtain AUD080 ( ). Again, your profit is zero. Your banker will always quote you this rate so that you will not make profit out of him. This rate is calculated from equation (.9) as 2.75 F.08 }.04 G = You make profit if the forward rate is 2.65 because this rate is not consistent with the no- arbitrage condition (but is). If the quoted forward rate is not consistent with the no-arbitrage condition you will make profit in one direction and loss in the other (exactly as in the case of two-point and three-point arbitrage). How do you know which way to go? Very simply by calculating the covered margin, which must be positive for arbitrage to be profitable. We have to be very careful about the deannualisation of interest rates when these calculations are carried out. In this example we did not deannualise interest rates because we used a time horizon of one year. If, on the other hand, we used a horizon of six months, deannualised interest rates on the two currencies would be 4 and 2 per cent respectively. In general, we deannualise interest rates by dividing by (2/N) where N is the time horizon in months. Covered arbitrage with bid offer spreads To reconsider covered arbitrage in the presence of bid offer spreads in both exchange and interest rates we have to remember that a price-taker in the foreign exchange market (like our arbitrager) buys at the (higher) offer exchange rate and sells at the (lower) bid exchange rate of the market-maker (the banker). A price-taker in the money market borrows at the (higher) offer interest rate and lends at the (lower) bid interest rate of the market-maker. Covered arbitrage in the presence of bid offer spreads is illustrated in Figure.8. Arbitrage from the domestic currency to a foreign currency consists of the following steps: The arbitrager borrows domestic currency funds at the domestic offer interest rate, i a. Borrowed funds are converted into the foreign currency at the spot offer rate, a, obtaining / a foreign currency units. This amount is invested at the foreign bid interest rate, i * b. The foreign currency value of the invested amount at the end of the investment period is (/ a )( + i * b ). This amount is reconverted into the domestic currency at the bid forward rate, F b, to obtain (F b / a )( + i * b ) domestic currency units. The value of the loan plus interest per unit of the domestic currency is ( + i a ). The covered margin in this case is p = F b } ( + i * b ) ( + i a ).20 a ince F b / a = ( + f )/( + m), where f is the forward spread and m is the bid offer spread, it follows that p = i * b i a + f m.2 By comparing Equations (.6) and (.2), we can see that the covered margin is lower if we allow for the bid offer spreads. This is simply because bid offer spreads are transaction costs. Arbitrage from a foreign currency to the domestic currency consists of the following steps: The arbitrager borrows foreign currency funds at the foreign offer interest rate, i * a. Borrowed funds are converted into the domestic currency at the spot bid rate, b, obtaining b domestic currency units. This amount is invested at the domestic bid interest rate, i b. Example.6 continued

10 330 international finance: an analytical approach chapter International Arbitrage 33 Figure.8 Covered interest arbitrage with bid offer spreads Domestic Foreign Foreign Domestic Borrowing domestic currency Borrowing foreign currency You request your banker to quote a one-year forward rate on the pound, which he does. The following information is available: One-year forward rate (AUD/GBP) pot rate (AUD/GBP) One-year AUD interest rate One-year GBP interest rate Converting at spot offer rate Investing at foreign bid rate forward bid rate unit a ( + i*) b a F b a ( + i*) b Loan repayment + i a Loan repayment ( + i*) a unit b b ( + i b ) Converting at spot bid rate Investing at domestic bid rate forward offer rate b F a ( + i b ) Consider arbitrage from the pound to the Australian dollar: Borrow GBP000 (or any other amount). Convert the pound spot at to obtain AUD2745 ( ). Invest the AUD amount at 7.75 per cent for one year. At the end of the year, you will have AUD2958 (2745 ( )). Reconvert the AUD proceeds at the offer forward rate into pounds to obtain GBP4 (2958/2.6550). The loan repayment that you have to make is GBP042.5 (000 ( )). Net arbitrage profit is GBP7.50 (= ). which is less than was obtained in the previous example. Now, let us see what happens if instead you indulge in arbitrage from the Australian dollar to the pound: Borrow AUD000 (or any other amount). Convert the Australian dollar spot at to obtain GBP363 (000/2.7550). Invest the GBP amount at 3.75 per cent for one year. At the end of the year, you will have GBP377. Reconvert the GBP proceeds at the bid forward rate into Australian dollar to obtain AUD997 ( ). The loan repayment that you have to make is AUD (000 ( )). Net arbitrage loss is AUD85.5 (= ). The loss incurred in this case is greater than that incurred in the previous case. Covered margin Covered margin F b ( + i*) ( b + i a ) a b F a ( + i b ) ( + i a *) ince b /F a = /[( + m)( + f )], it follows that p = i b i * a f m.23 which again shows that the covered margin would be lower if the bid offer spreads are allowed for. The domestic currency value of the invested amount at the end of the investment period is b ( + i b ). This amount is reconverted into the foreign currency at the offer forward rate, F a, to obtain ( b /F a )( + i b ) foreign currency units. The value of the loan plus interest is ( + i a * ) foreign currency units. The covered margin in this case is p = b } ( + i F b ) ( + i * a ).22 a Uncovered interest arbitrage Uncovered arbitrage is triggered by the violation of the uncovered interest parity (UIP) condition. It is described as uncovered because, unlike covered arbitrage, the long currency position is not covered in the forward market but rather left uncovered or open. This means that the proceeds of an investment in foreign currency assets are reconverted into the domestic currency (or vice versa) at the spot exchange rate prevailing on the maturity date of the investment rather than at the forward rate determined in advance. Thus, foreign exchange risk is present, which.5

11 332 international finance: an analytical approach chapter International Arbitrage 333 IN PRACTICE ome practical business implications of CIP CIP has two important practical business implications pertaining to two activities: hedging and short-term investment/financing. The first implication is that if CIP holds then there is no difference between the effectiveness of forward hedging and money market hedging (borrowing and lending in the money market). This is because money market hedging creates a synthetic forward contract with an implicit forward rate that, under CIP, is equal to the forward rate quoted in the market. In this case, money market and forward market hedging produce identical results in terms of the domestic currency values of payables and receivables under the two modes of hedging. The second implication pertains to financing and investment decisions. It implies that there is no difference between financing or investing in the domestic currency and in a foreign currency while covering the position in the forward market. This is because the two modes of financing/investment give exactly the same cost of funding/rate of return if CIP holds. investor chooses foreign investment, he or she will convert the initial capital into foreign currency at the current spot exchange rate,, obtaining K/ units of the foreign currency, where is measured as domestic currency units per one unit of the foreign currency. If K/ worth of the foreign currency is invested in foreign assets, this capital is compounded for one period at the foreign interest rate, i *, such that the foreign currency value of the investment on maturity is (K/)( + i * ). The expected domestic currency value of this investment is obtained by reconverting this amount into the domestic currency at the expected spot rate, e, to e (K/)( + i * ). The two alternatives are described in Figure.9. Again, we assume that there are no restrictions on the movement of capital and no transaction costs. Assume also that traders are risk-neutral, in the sense that they are indifferent between holding domestic and foreign assets if these assets offer the same (expected) return. The equilibrium condition that precludes the possibility of profitable arbitrage is that the two investments must be equally attractive, offering the same return. Hence means that it is more like speculation than arbitrage. This is why another name for this activity is carry trade. There are, however, reasons why this activity is called arbitrage. The term uncovered arbitrage is used so that it can be the counterpart of covered arbitrage. Moreover, if arbitragers firmly believed in their exchange rate expectations, then they would behave as if there was no foreign exchange risk. And if the operation involves a pair of currencies with a fixed or highly stable exchange rate, then foreign exchange risk will be absent or minimal, in which case the term arbitrage is appropriate. Then there is the view that arbitrage is not really a risk-free operation, but this is a controversial proposition that we will put aside. Deriving the UIP condition The UIP condition can be derived by combining CIP with the unbiased efficiency condition, which stipulates (in its strongest form) that the future spot rate is equal to the current forward rate. Thus, we can obtain UIP if the forward rate (forward spread) in CIP is replaced with the expected spot rate (expected percentage change in the spot rate). This difference in specification reflects the difference between covered arbitrage, which is based on the forward rate (or forward spread), and uncovered arbitrage, which is based on the expected spot rate (or expected change in the spot rate). Alternatively, UIP can be derived directly as follows. Consider an investor with initial capital, K, who is facing two alternatives: (i) domestic investment whereby the investor buys domestic assets, earning the domestic interest rate, i; and (ii) foreign investment whereby the investor converts the domestic currency into foreign currency to buy foreign assets, earning the foreign interest rate, i *. The foreign investment produces exposure to foreign exchange risk, which in this case is not covered, as the investor leaves the position open. Foreign exchange risk is present because, unlike the forward rate, the spot exchange rate used to reconvert the proceeds of foreign investment into the domestic currency is not known in advance (that is, prior to the maturity of the investment). In this case, the investor acts upon the spot rate expected to prevail on the maturity date of the investment, not knowing in advance the domestic currency value of his or her foreign investment on maturity. Consider a one-period investment starting with the acquisition of a financial asset (for example, a bank deposit) at time 0 and ending with the maturity of this asset at time. When the investor chooses domestic investment, the invested capital is compounded at the domestic interest rate, and the investor ends up with the initial capital plus interest, which is K ( + i) where i is the interest earned by holding the domestic asset between time 0 and time. If the or K( + i) = K } ( + i* ) e.24 + i = e } ( + i* ).25 which says that gross domestic return must be equal to gross foreign uncovered return. Figure.9 Return on domestic and foreign investment (with uncovered position) Converting at current spot rate Investing in foreign assets expected spot rate Foreign investment K K ( + i *) Investor (K) Domestic investment K e ( + i *) K( + i )

12 334 international finance: an analytical approach chapter International Arbitrage 335 ince e / = + Ṡ e, where Ṡ e is the expected percentage change in the exchange rate, it follows that + i = ( + Ṡ e )( + i * ).26 Figure.0 Uncovered interest arbitrage without bid offer spreads Domestic Foreign Foreign Domestic Equation (.26) can be used to derive an approximate UIP condition by ignoring the term i * Ṡ e on the assumption that it is too small. The approximate condition is i i * = Ṡ e.27 Borrowing domestic currency Borrowing foreign currency Equation (.27) implies that the currency offering the higher interest rate must be expected to depreciate, and vice versa. This is because if i > i *, then Ṡ e > 0, which means that the foreign currency (offering a lower interest rate) is expected to appreciate, while the domestic currency (offering a higher interest rate) is expected to depreciate. If, on the other hand, i < i *, then Ṡ e < 0, implying that the foreign currency is expected to depreciate, while the domestic currency is expected to appreciate. This must be a necessary condition for equilibrium because no investor wants to hold a currency that offers a low interest rate and is expected to depreciate, while everyone wants to hold a currency that offers a high interest rate and is expected to appreciate. Converting at spot rate unit 0 unit 0 Converting at spot rate Uncovered interest arbitrage without bid offer spreads Investing at foreign rate Loan repayment Loan repayment Investing at domestic rate Uncovered interest arbitrage consists of taking a short position on (that is, borrowing) a currency, and a corresponding long position on (that is, investing in) another currency (Figure.0) without covering the long position. One of the two currencies may be the domestic currency and the other a foreign currency (although this is not necessarily the case). We will illustrate uncovered arbitrage by taking time 0 to be the time at which the operation is initiated and time to be the time at which the investment matures and the short position is covered. Arbitrage from the domestic to a foreign currency consists of the following steps: The arbitrager borrows domestic currency funds at the domestic interest rate, i. For simplicity we assume that the amount borrowed is one domestic currency unit. Borrowed funds are converted at the spot exchange rate, 0, obtaining / 0 foreign currency units. This amount is invested at the foreign interest rate, i *. The foreign currency value of the invested amount at the end of the investment period is (/ 0 )( + i * ). This amount is reconverted into the domestic currency at the spot exchange rate prevailing at time,, to obtain ( / 0 )( + i * ) domestic currency units. spot rate ( + i*) 0 0 ( + i*) Uncovered margin 0 ( + i*) ( + i) + i + i* Uncovered margin 0 ( + i) ( + i*) 0 ( + i) 0 ( + i) spot rate.8 The current exchange rate between the Australian and U dollars is.80 (AUD/UD) and the three-month interest rates on the Australian and U currencies are 6 and 4 per cent p.a. respectively. If UIP holds, the level of the exchange rate expected to prevail three months from now can be calculated from the (deannualised) interest rate differential as follows: i i * = } 6 4 } 4 = The U dollar should appreciate by 0.5 per cent. The level of the exchange rate three months from now should be =.809 The value of the loan plus interest is ( + i) domestic currency units. The uncovered margin, p, is the difference between the domestic currency value of the proceeds and loan repayment, which gives p = } 0 ( + i * ) ( + i).28 or approximately p = i i * + Ṡ.29

13 336 international finance: an analytical approach chapter International Arbitrage 337 REEARCH where Ṡ is the percentage change in the exchange rate between 0 and. Equation (.29) tells us that the uncovered margin on arbitrage from the domestic to a foreign currency consists of the interest rate differential and the percentage change in the exchange rate. Arbitrage from a foreign to the domestic currency consists of the following steps: The arbitrager borrows foreign currency funds at the foreign interest rate, i *. For simplicity we again assume that the amount borrowed is one foreign currency unit. Borrowed funds are converted at the spot exchange rate, 0, obtaining 0 domestic currency units. This amount is invested at the domestic interest rate, i. The domestic currency value of the invested amount at the end of the investment period is 0 ( + i). This amount is reconverted into the foreign currency at the spot rate,, to obtain ( 0 / )( + i) foreign currency units. The value of the loan plus interest is ( + i*) foreign currency units. The uncovered margin is the difference between the domestic currency value of the proceeds and loan repayment, which gives The profitability of carry trade arry trade is essentially another name for uncovered Carbitrage, which may be a better name, since it is a risky operation. This operation became very popular as the interest rate on the yen declined to near zero in the recent past, and this is why it came to be known as yen carry trade where the yen is described as being the funding currency and a high-interest currency, such as the Australian dollar, is known as the target currency. On the profitability of carry trade, Burnside et al. conclude that although the operation produces very large harpe ratios (which is a measure of return relative to risk), the amount of money produced by carry trade is rather small because of transaction costs and price pressure limits. They find that carry trade produces higher harpe ratios than that of the tandard and Poor s 500 index even after taking into account transaction costs. However, they point out that the pay-off in terms of the sums of money obtainable from carry trade is relatively small. In their study of the profitability of carry trade, Gyntelberg and Romolona use the yen and wiss franc as funding currencies, pointing out that carry trade is pursued when the interest differential is wide enough to compensate traders for the underlying foreign exchange risk. 2 They find evidence supporting the view that downside risk is an important feature of carry trade and that using measures of downside risk (as opposed to the standard deviation) reduces the harpe ratio, though it remains higher than those obtainable from share markets. Like Gyntelberg and Remolona (2007), Hottori and hin (2007) find evidence indicating that volumes of carry trade involving the yen are high when interest differentials against the yen are high. 3 Moosa used currency combinations involving two funding currencies and three target currencies to analyse the profitability of carry trade over the period The results show that carry trade can be profitable over a long period of time but it is also highly risky because an adverse movement in the underlying exchange rate could wipe out the carry trader once and for all. Although it may appear that carry trade produces higher harpe ratios than those associated with share market investment, several reasons are presented for why carry trade is not as lucrative as it may appear. C. Burnside, M. Eichenbaum, I. Kleshcelski and. Rebelo, The Returns to Currency peculation, NBER Working Papers, No 2489, J. Gyntelberg and E. M. Remolona, Risk in Carry Trades: A Look at Target Currencies in Asia and the Pacific, BI Quarterly Review, December, 2007, pp M. Hottori and H.. hin, The Broad Yen Carry Trade, Bank of Japan, Institute for Monetary and Economic tudies, Discussion Paper No E-9, I. A. Moosa, Risk and Return in Carry Trade, Journal of Financial Transformation, 22, 2008, pp The one-year interest rates on the Australian dollar and the U dollar are 4 and 7 per cent respectively. The current exchange rate (AUD/UD) is.80. An investor is considering uncovered arbitrage by taking a short position on (borrowing) the Australian dollar and a long position on (lending) the U dollar. ince the interest rate differential is 3 per cent, this investor will make profit (ignoring transaction costs) as long as the U dollar does not depreciate against the Australian dollar by more than 3 per cent. The following table shows the rate of return (uncovered margin) for various levels of the exchange rate prevailing on the maturity of the investment for arbitrage in both directions. 0 AUD UD UD AUD uppose now that the Australian dollar interest rate rose to 9 per cent while the U dollar interest rate remained unchanged. The investor will be willing to do the same only if he or she expects the U dollar to appreciate by more than 2 per cent. The following table shows the uncovered margin for various levels of the final exchange rate: 0 AUD UD UD AUD p = 0 } ( + i) ( + i * ).30 or approximately p = i i * Ṡ.3 Equation (.3) tells us that the uncovered margin on arbitrage from the foreign to the domestic currency consists of the interest rate differential (measured the other way round) and the negative of the percentage change in the exchange rate. Uncovered arbitrage with bid offer spreads Let us now reconsider uncovered arbitrage by allowing for bid offer spreads in both exchange and interest rates, as illustrated in Figure.. Remember that a price-taker in the foreign.9