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11C H A P T E R

InternationalArbitrage

INTRODUCTIONArbitrage 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.

OBJECTIVESThe objectives of this chapter are

To define arbitrage and the

no-arbitrage condition.

To describe two-point, three-p

and multi-point arbitrage in the

foreign exchange market.

To describe commodity arbitra

To describe covered interest

arbitrage and show how the

no-arbitrage condition can beused to determine the forward

exchange rate.

To describe uncovered arbitra

and introduce the concept of

carry trade.

To expose some misconceptio

of arbitrage.

This material is distributed for Marketing Purposes only. No Authorised printing or reproduction permitted. (c)McGraw-Hill Australia

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314 INTERNATIONAL FINANCE: AN ANALYTICAL APPROACH CHAPTER 11 INTERNATIONAL ARI I I I

FIGURE 11.1 The effect of two-point arbitrage

Qy Qy

Dy

Sy

Dy

Sy

S(x/y)S(x/y)

(SA)0(SA)1

(SB)1

(SB)0

A B

EThe Reuters Monitor shows the following information about the exchange rate between the

Australian dollar and the US dollar (measured in direct quotation in both centres):

Sydney 1.7800 (AUD/USD)

New York 0.5747 (USD/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

1/0.5747 = 1.7400. Thus, the no-arbitrage condition is violated in the sense that the USD is

more expensive in Sydney than in New York. Hence, arbitragers buy the US currency in

New York at 1.7400 and sell it in Sydney at 1.7800. Profit in Australian dollar per US dollar

bought and sold is

p= 1.7800 1.7400 = 0.0400

or 400 points. The effect of arbitrage is to raise the price of the USD in New York and lower it

in Sydney, until they are equal somewhere between 1.7800 and 1.7400. Suppose that at some

stage prior to the restoration of equilibrium, changes in supply and demand cause the exchange

rate to fall to 1.7700 in Sydney and rise to 1.7500 (or 0.5714 in direct quotation) in New York. Inthis case, profit shrinks to

p= 1.7700 1.7500 = 0.0200

or 200 points. Eventually, the rate falls to 1.7600 in Sydney and rises to the same level (0.5682

in direct quotation) in New York. Profit at this stage is

p= 1.7600 1.7600 = 0

which means that arbitrage is not profitable because the no-arbitrage condition is restored.

curves for currency y in financial centres A and B. Initially, the exchange rates in A and B

are (SA)0 and (SB)0 respectively, such that (SA)0> (SB)0. As the demand for y increases in B,

the exchange rate rises (yappreciates). Conversely, the supply of y increases inAand so the

exchange rate falls (y depreciates). This process continues until the exchange rates in both

financial centres are equal (that is, until (SA)1= (SB)1) because t his condition eliminates profit

and hence the incentive for arbitrage.

Two-point arbitrage with the bidoffer spread

So far we have shown how arbitrage works by assuming that there is no bidoffer spread. If this

assumption is relaxed, the no-arbitrage condition in this case is given by the equations

Sb,A(x/y) =Sa,B(x/y) 11.3

Sa,A(x/y) = Sb,B(x/y) 11.4

where Sb,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 Sb,A(x/y) > Sa,B(x/y). In this case the arbitrager can make profit by

buyingyin Bat Sa,B(x/y) and selling it inAfor Sb,A(x/y). Arbitrage profit is the difference between

the selling rate and the buying rate, or Sb,A(x/y) Sa,B(x/y).

11.1 Two-point arbitrageAlso known as spatial arbitrageor locational arbitrage, two-point arbitragearises 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 bidoffer spread, then we

modify the operation to account for the spread.

Two-point arbitrage without the bidoffer spread

Given two financial centres, Aand B, and two currencies, xand y, and assuming (for simplic-

ity) no transaction costs and a zero bidoffer spread, arbitrage will be triggered if the following

condition is violated:

SA(x/y)= SB(x/y) 11.1

This condition says that the exchange rate between xandyshould be the same in Aas in B. If

the condition is not satisfied in the sense that the exchange rate betweenxandyis different inAfrom 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 SA(x/y) > SB(x/y). Th is violation

means that currencyyis more expensive inAthan in B(or thatxis cheaper inAthan in B). Let

us consider the situation from the perspective of currency y. Arbitragers buyyin Band sell it

inA, making profit, p, that is g iven by

p= SA(x/y) SB(x/y) 11.2

The process of arbitrage restores the equilibrium condition via changes in the forces of

supply and demand. This is illustrated by Figure 11.1, which shows the supply and demand


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By starting with one unit of currency xand moving clockwise, as in Figure 11.2(a), arbitrage

involves the following steps:

1. Sellingxand buyingyto obtain 1/S(x/y) units of y.

2. Sellingyand buyingzto obtain 1/[S(x/y)S(y/z)] units of z.

3. Sellingz and buyingxto obtain S(x/z)/[S(x/y)S(y/z)] units of x.

Since S(x/y) >S(x/z)/S(y/z), it follows that S(x/z)/S(y/z) S(x/z)/S(y/z), it follows that S(y/z)S(x/y)/S(x/z) > 1. 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 S(x/y) = S(x/z)/S(y/z), then there is no arbitrage opportunity.

If S(x/y) >S(x/z)/S(y/z), then there is a profitable arbitrage opportunity by following the

sequencexzyx.

If S(x/y) S(x/z)/S(y/z). Figure 11.3 shows thatthe buying and selling of currencies result in changes in the forces of supply and demand, as

follows:

1. An increase in the demand for z (the supply of x), and so S(x/z) rises as shown in

Figure 11.3(a).

2. An increase in the demand for y (the supply of z), and so S(y/z) falls as shown in

Figure 11.3(b).

3. An increase in the demand for x (the supply of y), and so S(x/y) falls as shown in

Figure 11.3(c).

EXAMPLE

11.2

The exchange rate between the pound and Australian dollar (GBP/AUD) as recorded in Sydney

and London is as follows:

Sydney 0.37500.3790

London 0.37000.3740

To make profit the arbitrager will buy the Australian dollar in London at GBP0.3740 and sell it

in Sydney at GBP0.3750. Profit in pounds per Australian dollar is given by

p= 0.3750 0.3740 = 0.0010

or 10 points. Equivalently, profit is made by buying the pound in Sydney and selling it in

London.

The effect of the bidoffer 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:

Sydney 0.3770

London 0.3720

The arbitrager in this case buys the Australian dollar in London at GBP0.3720 and sells it inSydney at GBP0.3370. Arbitrage profit in this case is 0.005 or 50 points.

FIGURE 11.2 Profitable and unprofitable sequences in three-point arbitrage

z

(a) Unprofitable sequence

y

x

z

(b) Profitable sequence

y

x

11.2 Three-point and multi-point arbitrageIn this section we consider arbitrage involving more than two currencies. We start with arbitrage

involving three currencies.

Three-point arbitrage

Given three currencies (x, yand z) and making the same assumptions as in the case of two-

point arbitrage, three-point arbitrage(also called triangular arbitrage) will be triggered if the

following condition is violated:

S(x/y) =S(x/z)}S(y/z)

11.5

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 (11.5) 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 condi-

tion is violated (that is, whether or not the cross rates are consistent); and (ii) determiningthe profitable sequence. Let us assume that the no-arbitrage condition is violated such that

S(x/y) > S(x/z)/S(y/z). T he profitable sequence can be determ ined w ith the aid of a tr iangle,

placing each one of the three currencies in one of its corners (in no special order), as shown in

Figure 11.2. Determination of the profitable sequence is simple. We start with one unit of any

of the three currencies and move clockwise as in Figure 11.2(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 11.2(b).


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FIGURE 11.3 The effect of three-point arbitrage

Qz

S(x/z)

Dz

Sz

(a)Qz

S(y/z)

Dz

Sz

(b)

Qy

S(x/y)

Dy

Sy

(c)

W

e have presented three-point arbitrage as a risk-free

operation, because all of the decision variables (threeexchange 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.1

By using high-frequency, tick-by-tick data on the

exchange rates they found that arbitrage opportunitiesdo exist. However, they also found that the exploitation

of such opportunities involves a degree of risk that can

adversely affect realised returns.

RESEARCH

THE PROFITABILITY OF THREE-POINT ARBITRAGE

1C. Kollias and K. Metaxas, How Efficient are FX Markets? Empirical Evidence of Arbitrage Opportunities Using High-Frequency Data,

Applied Financial Economics, 11, 2001, pp. 43544.

The following exchange rates are quoted in Sydney, Auckland and Hong Kong:

S(HKD/AUD) 4.1548

S(NZD/AUD) 1.2052

S(HKD/NZD) 3.5825

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). S(HKD/NZD) can be

calculated from the other two rates as

S(HKD/NZD) =S(HKD/AUD)}}S(NZD/AUD)

=4.1548}1.2052

= 3.4474

Hence, the equilibrium condition is violated, implying a possibility for three-point arbitrage.

First, try the sequence HKD NZDAUD HKD, starting with one unit of HKD:

1. Sell HKD1.0000 for NZD to obtain (1/3.5825 = 0.2791) units of NZD.

2. Sell NZD0.2791 for AUD to obtain (0.2791/1.2052 = 0.2316) units of AUD.

3. Sell AUD0.2316 for HKD to obtain (0.2316 4.1548 = 0.9623) units of HKD.

Obviously, this is not the profitable sequence. Now, try the opposite sequence, starting with one

unit of HKD:

1. Sell HKD1.0000 for AUD to obtain (1/4.1548 = 0.2407) units of AUD.

2. Sell AUD0.2407 for NZD to obtain (0.2407 1.2052 = 0.2901) units of NZD.

3. Sell NZD0.2901 for HKD to obtain (0.2901 3.5825 = 1.0392) units of HKD.

This is obviously the profitable sequence. If S(HKD/NZD) = 3.4474, then there is no possibility

for three-point arbitrage because this rate is consistent with the others. At this stage we have:

S(HKD/AUD)}}}

S(HKD/NZD)S(NZD/AUD)= 4.1548}}

3.4474 1.2052= 1.0000

and

S(NZD/AUD)S(HKD/NZD)}}}

S(HKD/AUD) = 1.2052 3.4474}}

4.1548 = 1.0000

which shows th at there is no profitable sequence.

E

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-pointarbitrage. In the case of three-point arbitrage involving currencies x,yand z, the n o-arbitrage

condition may be written as

S(x/y)S(y/z)S(z/x) = 1 11.6

If four currencies are involved (x1,x2,x3andx4), then we have four-point arbitrage, in which

case the no-arbitrage condition is


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S(x1/x2)S(x2/x3)S(x3/x4)S(x4/x1) = 1 11.7

and when ncurrencies are involved, the no-arbitrage condition is

S(x1/x2)S(x2/x3)S(x3/x4)S(xn1/xn)S(xn/x1) = 1 11.8

EXAMPLE

11.4

Consider the following exchange rates:

S(AUD/USD) 1.8811

S(JPY/USD) 132.68

S(JPY/GBP) 189.24

S(GBP/EUR) 0.6125

S(EUR/AUD) 0.6086

The no-arbitrage condition in this case is

S(AUD/USD)S(USD/JPY)S(JPY/GBP)S(GBP/EUR)S(EUR/AUD) = 1

which gives

1.8811 1}132.68

189.24 0.6125 0.6086 = 1.000

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 11.4. 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).

CLOCKWISE

T RAN SA CT ION END C UR RE NC Y NU MBE R OF U NI TS

AUD USD USD 0.5316

USD JPY JPY 70.53

JPY GBP GBP 0.3727

GBP EUR EUR 0.6085

EUR AUD AUD 1.0000

ANTICLOCKWISE

AUD EUR EUR 0.6086

EUR GBP GBP 0.3728

GBP JPY JPY 70.55

JPY USD USD 0.5317

USD AUD AUD 1.0000

FIGURE 11.4 Five-point arbitrage

GBP

EUR

AUD

USD

JPY

(a) Clockwise

AUD

JPGBP

EUR

(b) Anti-clockwise

1Commodity arbitrageThe 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

arbitrageis 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

Pi= SPi* 11.9

wherePiis the domestic price of commodity i,Pi*is its foreign price and Sis the exchange rate

expressed as the number of units of the domestic currency per one unit of the foreign currency.

Thus, SPi*is the domestic currency equivalent of the foreign price of the commodity. Likewise,

Pi/Sis the foreign currency equivalent of the domestic currency price of the commodity.

When arbitragers buy a commod ity 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 anda decline in its price in the market where it is expensive until profit is eliminated and the no-

arbitrage condition is restored.

Figure 11.5 shows how commodity arbitrage works, starting from a disequilibrium posi-

tion described by the inequalityPi> SPi*. Initially, the domestic currency prices of commodity i

abroad and at home are SPi0*andPi0respectively. 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


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The price of commodity iin Australia is AUD100 and the exchange rate (AUD/USD) is 1.80.

The LOP implies that the equilibrium US price is USD56, because at this price the equilibriumcondition represented by equation (11.9) is not violated, implying no possibility for profitable

arbitrage. This is because the selling price and the buying price measured in the same cur-

rency (Piand SPi*respectively) are equal, which produces zero arbitrage profit. If the US price is

USD50, thenPi SPi*> 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 States), paying USD50 (or AUD90), and selling it in Australia at AUD100.

Net arbitrage profit is then given by Pi SPi*= AUD10 per unit of the commodity.

EXAMPLE

11.5

FIGURE 11.5 The effect of commodity arbitrage

Qi Qi

D0 D0

D1

S1

S0

S0

SPiPi

Pi0

Pi1

*1

SPi*1

SPi*0

(a) Domestic market (b) Foreign market

SPi1*and the latter reaches Pi1, which are equa l. 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. Several 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 imagi ne buyi ng a Big Mac in Melbour ne 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 1990s, 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 Sheffield and Newcastle.

Since 1986, The Economist magazine has used theprice of a homogenous product, the Big Mac, to showthat 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 US dollars and

compared. The exchange rate compatibl

US dollar) is subsequently calculated by d

a Big Mac in any country by the price in

The deviation of the actual rate from

rate is calculated and used to indicate

overvaluation or undervaluation of the do

is present when the actual rate is

no-arbitrage rate, and vice versa).IN

PRACTICE

USING THE LOP FOR CURRENCY VALUATION

1

While the LOP ty pically 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 indifferent currencies. In this case, the LOP can be written as

P= SP* 11.10

which is the same as equation (11.9) except that it is written in terms of the prices of baskets of

commodities,PandP*, and not the prices of individual commodities,PiandPi

*. If PandP

*are

taken to be the general price levels at home and abroad, then Equation (11.10) may be taken to

represent purchasing power parity, which we studied in Chapter 4.

Covered interest arbitrageCovered interest arbitrageis triggered by the violation of the covered interest parity(CIP)

condition, which describes the equilibrium relation between the spot exchange rate, the for-

ward 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 repre-

sented by CIP.

The CIP conditionConsider 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 for-

eign currency to buy foreign assets, earning the foreign interest rate, i*. Since 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


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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(1 + i) =K}S(1 + i

*)F 11.11

By expressing the condition in terms of one unit of the domestic currency, we obtain

(1 + i) =F}S(1 + i) 11.12

This condition (CIP) tells us that gross domestic return is equal to gross covered foreign return.

The left-hand side of equation (11.12) represents gross domestic return: it is gross because it

includes the amount invested (one unit of the domestic currency) and the interest earned, i.

SinceF/S= 1 +f, wherefis the forward spread, it follows that

(1 + i) = (1 +f)(1 + i*) 11.13

By simplifying equation (11.13), ignoring the term i*f, we obtain the approximate CIP condition

i i*=f 11.14

which tells us that in equilibrium the interest differential must be equal to the forward spread.

Equation (11.14) 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 cur-

rency (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 bidoffer 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 pro-

ceeds 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 11.7.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 domes-

tic 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).

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 posi-

tion 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.

Suppose 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 11.6).

When the inves tor chooses the domestic investment, t he invested capital is compounded at

the domestic interest rate, and the investor ends up with the initial capital plus interest income,

that is,K(1 + i). If the investor chooses the foreign investment, she converts the initial capital to

foreign currency at the current spot exchange rate, obtainingK/Sunits of the foreign currency,

where Sis measured as domestic currency units per one unit of the foreign cur rency. IfK/Sworth

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/S)(1 + i*). The domestic currency value of th is investment is obtained by conver ting th is

amount into the domestic currency at the forward rate, F, to obtain F(K/S)(1 + i*).

FIGURE 11.6 Return on domestic and foreign investment (with covered position)

Converting at

spot rate

Investor

(K)

Foreign

investment

Investing in

foreign assets

Reconverting at

forward rate

K

S

K

S(1 + i*)

KF

S(1 + i*) K(1 + i)

Domestic

investment


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This amount is reconverted into domestic currency at the forward rate,F, to obtain (F/S)(1 + i*)

domestic currency units.

The value of the loan plus interest is (1 + 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}S(1 + i) (1 + i*) 11.15

or approximately

p= i* i+f 11.16

Equation (11.16) tells us that the covered margin on arbitrage from the domestic to a foreign cur-

rency consists of the interest rate differential (foreign less domestic) and the forward spread.

Arbitrage fr om a foreign to the domestic currency con sists of the follow ing 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, S, obtaining Sdomestic currencyunits. 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

S(1 + i).

This amount is reconverted into the foreign currency at the forward rate,F, to obtain (S/F)

(1 + i) foreign currency units.

The value of the loan plus interest is (1 + i*) foreign currency units.

The covered margin is again the difference between the domestic currency value of the

proceeds and loan repayment, which gives

p=S}

F(1 + i) (1 + i

*) 11.17

or approximately

p= i i*f 11.18

Equation (11.18) 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

The no-arbitrage condition is obtained when the covered margin is zero. By substituting p= 0

in equation (11.15) or (11.17), we obtain

}

F = SF1 + i}1 + i

*G 11.19

where}

F is the particular value of the forward rate that is consistent with the no-arbitrage condi-

tion, which we may call the interest parity forward rate. If CIP holds then}

F =F.

Suppose 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

Arbitrage, however, does not have to involve the domestic currency, as two foreign currenciesmay provide a profitable arbitrage opportunity.

Arbitrage from t he domestic to a foreign c urrency consi sts of the followi ng 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, S, obtaining 1/Sforeign 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

(1/S)(1 + i*).

FIGURE 11.7 Covered interest arbitrage without bidoffer spreads

Borrowing

domestic

currency

Converting

at spot rate

Investing at

foreign rate

1 unit

1

S

Loan

repayment

Loan

repayment

Borrowing

foreign

currency

Converting

at spot rate

Reconverting at

forward rate

Reconverting at

forward rate

Investing at

domestic rate

Covered margin Covered margin

1 + i*

Domestic Foreign Foreign Domestic

1

S(1 + i*)

F

S(1 + i*)

F

S(1 + i*) (1 + i)

S

F(1 + i) (1 + i*)

S

F(1 + i)

S(1 + i)

1 + i

S

1 unit


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Covered arbitrage with bidoffer spreads

To reconsider covered arbitrage in the presence of bidoffer 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 r ate of the market-maker. Covered arbitrage in

the presence of bidoffer spreads is illustrated in Figure 11.8.

Arbitrage fr om the domestic currenc y to a foreign c urrency consi sts of the followi ng steps:

The arbitrager borrows domestic currency funds at the domestic offer interest rate, ia.

Borrowed funds are converted into the foreign currency at the spot offer rate, Sa, obtaining

1/Saforeign currency units. This amount is invested at the foreign bid interest rate, ib*.


(1/Sa)(1 + ib*).

This amount is reconverted into the domestic currency at the bid forward rate,Fb, to obtain

(Fb/Sa)(1 + ib*) domestic currency units.

The value of the loan plus interest per unit of the domestic currency is (1 + ia).

The covered margin in this case is

p=Fb}Sa(1 + i

b*) (1 + ia) 11.20

SinceFb/Sa= (1 +f)/(1 + m), where fis the forward spread and mis the bidoffer spread, it fol-

lows that

p= ib* ia+f m 11.21

By comparing Equations (11.16) and (11.21), we can see that the covered margin is lower if we

allow for the bidoffer spreads. This is simply because bidoffer spreads are transaction costs.

Arbitrage fr om a foreign cur rency to the domestic cu rrency consists of the following steps:

The arbitrager borrows foreign currency funds at the foreign offer interest rate, ia*.

Borrowed funds are converted into the domestic currency at the spot bid rate, Sb, obtaining

Sbdomestic currency units. This amount is invested at the domestic bid interest rate, ib.

open his manual to search for a formula that gives him an expression for the forward rate. This

formula would look like Equation (11.19). Why would the banker use this formula to calculate

the forward rate? Simply 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.

EXAMPLE

11.6

Suppose 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) 2.6500

Spot rate (AUD/GBP) 2.7500

One-year AUD interest rate 8%

One-year GBP interest rate 4%

You observe immediately that the pound is s elling 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 arbi-

trage, starting with arbitrage from the pound to the Australian dollar:

Borrow GBP1000 (or any other amount).

Convert the pound spot at 2.75 to obtain AUD2750 (1000 2.75).

Invest the AUD amount at 8 per cent for one year. At the end of the year, you will have

AUD2970 (2750 (1 + 0.08)).

Reconvert the AUD proceeds at the forward rate to pounds to obtain GBP1120.8

(2970/2.65).

The loan repayment that you have to make is GBP1040 (1000 (1 + 0.04)).

Net arbitrage profit is GBP80.8 (= 1120.8 1040).

Notice that this profit is made without bearing any risk, since all of the decision variables (includ-

ing 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 AUD1000 (or any other amount).

Convert the Australian dollar spot at 2.75 to obtain GBP363.6 (1000/2.75).

Invest the GBP amount at 4 per cent for one year. At the end of the year, you will have

GBP378.1.

Reconvert the GBP proceeds at the forward rate to Australian dollars to obtain AUD1002

(3781 2.65).

The loan repayment that you have to make is AUD1080 (1000 (1 + 0.08)).

Net arbitrage loss is AUD78.0 (= 1080 1002).In this case you make a loss. Now assume that the banker quoted a forward rate of 2.8558. If you

indulge in arbitrage from the pound to the Australian dollar at this forward rate, you will (after

reconversion) obtain GBP1040 (2970/2.8558), in which case your arbitrage profit is zero. If you

go from the Australian dollar to the pound you obtain AUD1080 (378.1 2.8558). 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 (11.19) as

2.75F1.08}1.04G= 2.8558You make profit if the forward rate is 2.65 because this rate is not consistent with the no-arbitrage

condition (but 2.8558 is). If the quoted forward rate is not consistent with the no-arbitrage condi-

tion 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 caref ul about the deannualisationof interest rates when these calcula-

tions 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 dean-

nualise interest rates by dividing by (12/N) where Nis the time horizon in months.

Ex


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Sb(1 + ib).

This amount is reconverted into the foreign currency at the offer forward rate,Fa, to obtain

(Sb/Fa)(1 + ib) foreign currency units.

The value of the loan plus interest is (1 + ia*) foreign currency units.

The covered margin in this case is

p=Sb}Fa

(1 + ib) (1 + ia*) 11.22

FIGURE 11.8 Covered interest arbitrage with bidoffer spreads

Borrowing

domestic

currency

Converting at

spot offer rate

Investing at

foreign bid rate

1 unit

1

Sa

Loan

repayment

Loan

repayment

Borrowing

foreign

currency

Converting at

spot bid rate

Reconverting at

forward bid rate

Reconverting at

forward offer rate

Investing at

domestic bid rate

Covered margin Covered margin


Fb

Sa

1 + ia

Sb

1 unit

Sb(1 + ib)1

Sa

(1 + i*)b

(1 + i*)bSb

Fa

(1 + ib)(1 + i*)a

Fb

Sa

(1 + i*) (1 + ia)bSb

Fa

(1 + ib) (1 + ia*)

EYou request your banker to quote a one-year forward rate on the pound, w hich he does. The

following information is available:

One-year forward rate (AUD/GBP) 2.64502.6550

Spot rate (AUD/GBP) 2.74502.7550

One-year AUD interest rate 7.758.25

One-year GBP interest rate 3.754.25

Consider arbitrage from the pound to the Australian dollar:

Borrow GBP1000 (or any other amount).

Convert the pound spot at 2.7450 to obtain AUD2745 (10002.7450).

Invest the AUD amount at 7.75 per cent for one year. At the end of the year, you will have

AUD2958 (2745 (1 + 0.0775)).

Reconvert the AUD proceeds at the offer forward rate into pounds to obtain GBP1114

(2958/2.6550).

The loan repayment that you have to make is GBP1042.5 (1000 (1 + 0.0425)).

Net arbitrage profit is GBP71.50 (= 1114 1042.50).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 AUD1000 (or any other amount).

Convert the Australian dollar spot at 2.7550 to obtain GBP363 (1000/2.7550).

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 (377 2.6450).

The loan repayment that you have to make is AUD1082.50 (1000 (1 + 0.0825)).

Net arbitrage loss is AUD85.5 (= 997 1082.50).

The loss incurred in this case is greater than that incurred in the previous case.

1

Since Sb/Fa= 1/[(1 + m)(1 +f)], it follows that

p= ib ia*f m 11.23

which again shows that the covered margin would be lower if the bidoffer spreads are

allowed for.

Uncovered interest arbitrageUncovered arbitrageis triggered by the violation of the uncovered interest parity(UIP) con-

dition. 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


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investor chooses foreign investment, he or she will convert the initial capital into foreign cur-

rency at the current spot exchange rate, S, obtaini ngK/Sunits of the foreign currency, where

Sis measured as domestic currency units per one unit of the foreign currency. If K/Sworth of

the foreign currency is invested in foreign assets, this capital is compounded for one period at

the foreign interest rate, i*, such that t he foreign cur rency value of t he investment on matur ity

is (K/S)(1 + i*). The expec ted domestic currency v alue of this in vestment is obtained by recon-

verting th is amount into the domestic c urrency at the expected spot rate, Se, to S

e(K/S)(1 + i

*).

The two alternatives are described in Figure 11.9.

Again, we assume that there are no restrictions on the movement of capital and no trans-

action 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

K(1 + i) =K}S(1 + i

*)S

e 11.24

or

1 + i=Se}S(1 + i

*) 11.25

which says that gross domestic return must be equal to gross foreign uncovered return.

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 arbi-trage 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 altern atives: ( 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 i n 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 dom-

estic 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 1. 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(1 + i)

where iis the interest earned by holding the domestic asset between time 0 and time 1. If the

CIP has two important practical business implicationspertaining to two activities: hedging and short-terminvestment/financing.The first implication is that if CIP holds then there isno 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.

IN

PRACTICE

SOME PRACTICAL BUSINESS IMPLICATIONS OF CIP

FIGURE 11.9 Return on domestic and foreign investment (with uncovered position)

Converting at

current spot rate

Investor

(K)

Foreign

investment

Investing in

foreign assets

Reconverting at

expected spot rate

K

S

K

S

(1 + i*)

KSe

S(1 + i*) K(1 + i)

Domestic

investment


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The value of the loan plus interest is (1 + i) domestic currency units.

The uncovered margin,p

, is the d ifference between the domestic currency value of theproceeds and loan repayment, which gives

p=S1}S0

(1 + i*) (1 + i) 11.28

or approximately

p= i i*+ S

. 11.29

Since Se/S = 1 + S

e, w here S

e is the expected percentage change in the exchange rate, it

follows that

1 + i= (1 + Se)(1 + i

*) 11.26

Equation (11.26) can be used to derive an approximate UIP condition by ignoring the term i*Se

on the assumption that it is too small. The approximate condition is

i i*= S

e 11.27

Equation (11.27) implies that the currency offering the higher interest rate must be expected to

depreciate, and vice versa. This is because if i> i*, then S

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

Se< 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 inves-

tor 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.

Uncovered interest arbitrage without bidoffer spreads

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 11.10) with-

out 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 1 to be the

time at which the investment matures and the short position is covered.

Arbitrage from t he domestic to a foreign c urrency consi sts of the followi ng 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,S0, obtaining 1/S0foreign currency

units. This amount is invested at the foreign interest rate, i*.


(1/S0)(1 + i*).

This amount is reconverted into the domestic currency at the spot exchange rate prevailing

at time 1, S1, to obtain (S1/S0)(1 + i*) domestic currency units.

The current exchange rate between the Australian and US dollars is 1.80 (AUD/USD) and the

three-month interest rates on the Australian and US currencies are 6 and 4 per cent p.a. respec-tively. 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}

4= 0.5

The US dollar should appreciate by 0.5 per cent. The level of the exchange rate three months

from now should be

1.005 1.80 = 1.809

EXAMPLE

11.8

FIGURE 11.10 Uncovered interest arbitrage without bidoffer spreads

Borrowing

domestic

currency

Converting

at spot rate

Investing at

foreign rate

1 unit

Loan

repayment

Loan

repayment

Borrowing

foreign

currency

Converting

at spot rate

Reconverting at

spot rate

Reconverting at

spot rate

Investing at

domestic rate

Uncovered margin Uncovered margin

1 + i*

1

S0

1

S0

(1 + i*)

S1

S0(1 + i*)

S1

S0(1 + i*) (1 + i)

S0

S1(1 + i) (1 + i*)

S0

S1(1 + i)

S0(1 + i)

1 + i

S0

1 unit



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where Sis the percentage change in the exchange rate between 0 and 1. Equation (11.29) 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 c urrency consi sts of the followi ng 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,S0, obtaining S0domestic currency

units. This amount is invested at the domestic interest rate, i.


S0(1 + i).

This amount is reconverted into the foreign currency at the spot rate, S1, to obtain

(S0/S1)(1 + i) foreign currency units.

The value of the loan plus interest is (1 + i*) foreign currency units.

The uncovered margin is the difference between the domestic currency value of the pro-

ceeds and loan repayment, which gives

Carry trade is essentially another name for uncoveredarbitrage, which may be a better name, since it isa 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

Sharpe 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.1They find that carry trade produces higher Sharpe

ratios than that of the Standard and Poors 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 Swiss franc asfunding currencies, pointing out that carry trade is pursued

when the interest differential is wide enough to compensate

traders for the underlying foreign exchange risk.2They 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 Sharpe ratio, though it remains higher than

those obtainable from share markets. Like Gyntelberg and

Remolona (2007), Hottori and Shin (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 t he period 19962006.4

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 Sharpe ratios than

those associated with share market investment, several

reasons are presented for why carry trade is not as lucrativeas it may appear.

RESEARCH

THE PROFITABILITY OF CARRY TRADE

1C. Burnside, M. Eichenbaum, I. Kleshcelski and S. Rebelo, The Returns to Currency Speculation, NBER Working Papers, No 12489,

2006.2J. Gyntelberg and E. M. Remolona, Risk in Carry Trades: A Look at Target Currencies in Asia and the Pacific, BIS Quarterly Review,

December, 2007, pp. 7382.3M. Hottori and H. S. Shin, The Broad Yen Carry Trade, Bank of Japan, Institute for Monetary and Economic Studies, Discussion Paper

No. 2007-E-19, 2007.4I. A. Moosa, Risk and Return in Carry Trade, Journal of Financial Transformation, 22, 2008, pp. 813.

EThe one-year interest rates on the Australian dollar and the US dollar are 4 and 7 per cent

respectively. The current exchange rate (AUD/USD) is 1.80. An investor is considering uncovered

arbitrage by taking a short position on (borrowing) the Australian dollar and a long position

on (lending) the US dollar. Since the interest rate differential is 3 per cent, this investor will

make profit (ignoring transaction costs) as long as the US dollar does not depreciate against the

Australian dollar by more than 3 per cent. The following table shows the rate of return ( uncov-

ered margin) for various levels of the exchange rate prevailing on the maturity of the investment

for arbitrage in both directions.

S0 S1 AUD USD USD AUD

1.80 1.85 5.8 5.8

1.80 1.80 3.0 3.0

1.80 1.75 0.2 0.2

1.80 1.70 2.6 2.6

1.80 1.65 5.3 5.3

Suppose now that the Australian dollar interest rate rose to 9 per cent while the US dollar

interest rate remained unchanged. The investor will be willing to do the same only if he or

she expects the US dollar to appreciate by more than 2 per cent. The following table shows the

uncovered margin for various levels of the final exchange rate:


1.80 1.95 0.8 0.7

1.80 1.90 2.0 2.0

1.80 1.85 4.8 4.8

1.80 1.80 7.5 7.6

1.80 1.75 10.3 10.3

p=S0}S1

(1 + i) (1 + i*) 11.30

or approximately

p= i i* S 11.31

Equation (11.31) 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 nega-

tive of the percentage change in the exchange rate.

Uncovered arbitrage with bidoffer spreads

Let us now reconsider uncovered arbitrage by allowing for bidoffer spreads in both exchange

and interest rates, as illustrated in Figure 11.11. Remember that a price-taker in the foreign


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340 INTERNATIONAL FINANCE: AN ANALYTICAL APPROACH CHAPTER 11 INTERNATIONAL AR340 INTERNATIONAL FINANCE: AN ANALYTICAL APPROACH

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.

Arbitrage is less profitable when we allow for bidoffer spreads, because these spreads

represent transaction costs.

The uncovered margin in this case is

p=Sb0}Sa1

(1 + ib) (1 + i*a) 11.34

Since Sb0/Sa1= 1/[(1 + S.b)(1 + m)], it follows that

p= ib i*a S

.b m 11.35

EXAMPLE

11.10

Consider the previous example with bidoffer spreads. The exchange rates are as shown in the

table, whereas the interest rates are 3.754.25 and 6.757.25. The uncovered margin should

now be as in the table.


1.79501.8050 1 .94501.9550 10.8 12.0

1.79501.8050 1.89501.9050 7.8 9.5

1.79501.8050 1.84501.8550 4.9 6.9

1.79501.8050 1.79501.8050 1.9 4.1

1.79501.8050 1.74501.7550 1.1 1.1

SUMMARY Arbitrage is generally defined as capitalising on a discrepancy in quoted prices as a result of

the violation of an equilibrium (no-arbitrage) condition.

Two-point arbitrage arises when the exchange rate between two currencies assumes two

different values in two financial centres at the same time.

Three-point arbitrage is triggered by the violation of a no-arbitrage condition, which is the

consistency of the cross rates.

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.

Multi-point arbitrage involves four, five or more currencies. Three-point arbitrage is sufficient

to establish consistent exchange rates, eliminating the profitability of multi-point arbitrage.

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.

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.

arbitrage 313

carry trade 332

commodity arbitrage 321

covered interest arbitrage 323

covered interest parity 323

covered margin 325

deannualisation 329

interest parity forward rate 327

law of one price (LOP) 321

locational arbitrage 314

multi-point arbitrage 319

net arbitrage profit 325

no-arbitrage condition 313

risk-neutral 325

spatial arbitrage 314

three-point arbitrage

triangular arbitrage 3

two-point arbitrage 3

uncovered arbitrage 3

uncovered interest par

uncovered margin 335

1. What are the no-arbitrage conditions for (i) two-point arbitrage, (ii) three-point arbitrage, and (iii)

arbitrage?

2. Explain how the equilibrium condition implied by the LOP is maintained and restored when it is viol

3. What are the practical business implications of CIP?

4. Why is risk neutrality an important assumption for deriving the CIP condition?

5. Why is covered interest arbitrage covered?

6. What is the interest parity forward rate?

7. In equilibrium, the currency offering a lower interest rate must sell at a forward premium, while the

a higher interest rate must sell at a forward discount. Why?

8. Explain how covered arbitrage restores the CIP equilibrium condition when it is violated.

9. What is the covered margin?

10.What is the effect of the presence of bidoffer spreads in interest and exchange rates on the CIP eq

condition and on the profitability of covered interest arbitrage?

11.Why i s uncovered interest arbitrage uncovered?

12.What is the connection between CIP and UIP?

13.The difference between the specification of CIP and UIP reflects the difference between covered and

arbitrage. Explain.14.Uncovered interest parity tells us that a currency that offers a higher interest rate is expected to de

possible to reconcile this proposition with the prediction of the supply and demand model pertainin

interest rate on the exchange rate?

KEY TERMS

REVIEW QUESTIONS


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6. You are given the following information:

Spot exchange rate (AUD/EUR) 1.60

One-year forward rate (AUD/EUR) 1.62

One-year interest rate on the Australian dollar 8.5%

One-year interest rate on the euro 6.5%

(a) Is there any violation of CIP?

(b) Calculate the covered margin (going short on the AUD).

(c) Calculate the interest parity forward rate and compare it with the actual forward rate.

(d) Calculate the forward spread and compare it with the interest differential.

(e) What would arbitragers do?

(f) If arbitrage is initiated, suggest some values for the interest and exchange rates after it has stop

equilibrium has been reached.


Spot exchange rate (AUD/CHF) 1.1500

Three-month forward rate (AUD/CHF) 1.1585

Australian three-month interest rate 10.5% p.a.

Swiss three-month interest rate 6.5% p.a.


(b) Calculate the covered margin (going short on the AUD).

(c) Calculate the interest parity forward rate and compare it with the actual forward rate.

(d) Calculate the forward spread and compare it with the interest differential.



Spot exchange rate (CAD/GBP) 2.42

Six-month forward rate (CAD/GBP) 2.46

Canadian six-month interest rate 8% p.a.

UK six-month interest rate 10% p.a.


(b) Calculate the covered margin from a Canadian perspective (going short on the CAD).

(c) Calculate the interest parity forward rate in direct quotation from a Canadian perspective and co

actual forward rate.

(d) Calculate the forward spread and compare it with the interest differential from a Canadian persp


(f) Redo all the calculations from a UK perspective (going short on the GBP).

9. You are given the following information: Spot exchange rate (AUD/EUR) 1.59501.6050

One-year forward rate (AUD/EUR) 1.61501.6250

One-year interest rate on the Australian dollar 8.258.75

One-year interest rate on the euro 6.256.75

(a) Calculate the covered margin (going short on the AUD).

(b) What would arbitragers do?

(c) Compare the results with those obtained by solving Problem 6.

1. The following exchange rates are quoted in Sydney and London at the same time:

Sydney (AUD/GBP) 2.56

London (GBP/AUD) 0.35

(a) Is there a possibility for two-point arbitrage?

(b) If so, what will arbitragers do?

(c) What is the profit earned from arbitrage?

2. The following exchange rates are quoted simultaneously in Sydney, Frankfurt and Zurich:

AUD/EUR 1.6400

CHF/AUD 0.8700

CHF/EUR 1.4600


(b) Is there a possibility for three-point arbitrage?

(c) If so, what is the profitable sequence?

(d) What is the profit earned from arbitrage?

(e) How do the three exchange rates change as a result of arbitrage?

(f) What is the value of the CHF/EUR exchange rate that eliminates the possibility for profitable arbitrage?

3. The following exchange rates are quoted in Sydney and London at the same time:

Sydney (AUD/GBP) 2.55752.5625

London (GBP/AUD) 0.34750.3525


(b) If so, what will arbitragers do?

(c) What is the profit earned from arbitrage?

(d) Compare the results with those obtained from Problem 1 above.

4. The following exchange rates are quoted:

JPY/AUD 67.16

GBP/AUD 0.3484

CHF/AUD 0.8012

CAD/AUD 0.8711

(a) Calculate all possible cross rates.

(b) Using the calculated cross rates, show that there is no opportunity for three-point, four-point or five-point

arbitrage.

(c) If the cross rates were 10 per cent higher than those obtained in (a) above, show that there are opportunities forprofitable three-point, four-point or five-point arbitrage.

5. The price of a commodity in New Zealand is NZD10, while the price of the same commodity in Australia is AUD6. The

current exchange rate (NZD/AUD) is 1.15.

(a) Is there a violation of the LOP?

(b) If so, what will happen?

(c) What is the Australian dollar price compatible with the LOP at the current exchange rate?

(d) At the current Australian dollar price, what is the exchange rate compatible with the LOP?

PROBLEMS


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13.You are given the following information:

Spot exchange rate (USD/GBP) 1.46

Spot exchange rate (USD/CAD) 0.64

US one-year interest rate 6%

UK one-year interest rate 8%

Canadian one-year interest rate 10%

(a) Calculate the one-year forward rate between the Canadian dollar and the UK pound (CAD/

GBP) by adjusting the spot rate for the interest rate differential.

(b) Calculate the same forward rate as a cross rate. Do you obtain the same answer? Why or

why not?

14.The current AUD/EUR exchange rate is 1.60, the Australian three-month interest rate is 8.5

per cent p.a. and the three-month interest rate on the euro is 6.5 per cent p.a. Where will the

exchange rate be in three months time if UIP holds?

15.Reconsider Problem 14 by assuming that the exchange rate in three months time turned out

to be 1.68. Calculate the uncovered margins obtained by going short on the Australian dollar

and long on the euro, a nd vice versa.

16.The following information is available:

Spot exchange rate (CAD/GBP) 2.32

Canadian six-month interest rate 8% p.a.

UK six-month interest rate 10% p.a.

Calculate the uncovered margin obtained by going short on the Canadian dollar if the

exchange rate assumes the following values in six months: (a) 2.25, (b) 2.28, (c) 2.32,

(d) 2.35 and (e) 2.38. Do the same by going short on t he pound.

17.The following information is available:

Spot exchange rate (CAD/GBP) 2.31502.3250

Canadian six-month interest rate 7.758.25 p.a.

UK six-month interest rate 9.7510.25 p.a.

Calculate the uncovered margin by going short on the Canadian dollar if the exchange rate

assumes the following values in six months: (a) 2.24752.2525, (b) 2.27752.2825, (c)

2.31752.3225, (d) 2.34752.3525 and (e) 2.37752.3825. Do the same by going short on

the pound.


Spot exchange rate (AUD/CHF) 1.14501.1550

Three-month forward rate (AUD/CHF) 1.15351.1635

Australian three-month interest rate 10.2510.75 p.a.

Swiss three-month interest rate 6.256.75 p.a.

(a) Calculate the covered margin (going short on the AUD).

(b) What would arbitragers do?

(c) Compare the results with those obtained by solving Problem 7.


Spot exchange rate (CAD/GBP) 2.41502.4250

Six-month forward rate (CAD/GBP) 2.45502.4650

Canadian six-month interest rate 7.758.25 p.a.

UK six-month interest rate 9.7510.25 p.a.

(a) Calculate the covered margin from a Canadian perspective (going short on the CAD).

(b) Calculate the covered margin from a UK perspective (going short on the GBP). (c) What would arbitragers do?

(d) Compare the results with those obtained by solving Problem 8.

12.The table below shows a set of data consisting of 15 observations on the spot and three-month forward rates

between the Australian dollar and the Canadian dollar, as well as the Australian and Canadian three-month interest

rates. On the basis of this data set, you are required to do the following (all calculations are to be carried out from

an Australian perspective):

(a) Calculate the interest parity forward rate and plot it against the actual forward rate.

(b) Calculate the percentage deviation of the actual forward rate from the interest parity forward rate and plot it.

(c) Calculate the covered margin and plot it.

O BS ER VAT IO N S PO TAUD/CAD

FORWARDAUD/CAD

AUSTRALIANINTEREST

CANADIANINTEREST

1 1.0643 1.0692 4.84 4.59

2 1.1000 1.1037 4.93 4.87

3 1.1003 1.0904 4.85 4.91

4 1.0672 1.0520 4.62 4.66

5 1.0449 1.0440 4.66 4.63

6 1.0170 1.0188 4.70 4.56

7 1.0447 1.0398 4.78 4.66

8 1.0555 1.0538 5.08 4.85

9 1.1365 1.1339 5.78 5.27

10 1.1325 1.1295 5.87 5.53

11 1.2259 1.2163 6.41 5.56

12 1.1942 1.1969 6.03 5.49

13 1.3066 1.3165 4.98 4.58

14 1.2918 1.2941 4.88 4.30

15 1.2890 1.2838 4.30 3.05


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