International Short-Term Financing and Investment 251

(1 + e_y ⁾ > (1 + L ) 291

(1+f ) 300

(1+f ) 302

International Long-Term Financing, Capital Structure and the Cost of Capital 284

D = Z D, 283

E = Z 283

k = Z 289

International Long-Term Portfolio Investment 296

Rj,t = b 0,j + b 1, jR m,t + b 2,j$t + e j,t (11.40) 311

dL 319

dm ₁ -¹'—-¹'-²'"² --v-^"³ '-³ (1149) 319

+ 2ct(S 1, S ₄)w 4 +1 1 = 0 319

= 2w3 s ² (S3 ) + 2s(S 1 , S3 )w 1 + 2s(S2 , S3 )w2 320

+ 2s(S 3 , S₄ )w 4 +2 1 = 0 = 2w4 s ² (S4 ) + 2s(S 1 , S4 )w 1 + 2s(S 2 , S4 )w2 + 2s(S 3 , S4 )w 4 + 2 2 = 0 320

= w 1 + W2 + w 3 -1 = 0 320

w' = [w 1 w 2 w 3 w 4 2 1 2 2] and 331

Foreign Direct Investment 344

i 341

References 341

i 361

Index 363

which tells us that if the interest differential is equal to the forward spread, then there is no possibility for profitable covered arbitrage. Equation (2.8) implies that the currency offering the higher interest rate must sell at a forward discount and vice versa. This is because if i_x > iy, then f > 0, which means that currency y (offering a lower interest rate) sells at a forward premium, whereas currency x (offering a higher interest rate) sells at a

FIGURE 2.3 The no-arbitrage line in the f - (i_x - iy) space.

forward discount. If, on the other hand, i_x < iy, then f < 0, implying that currency y sells at a discount whereas currency x sells at a premium. That the interest rate differential and the forward spread have similar signs must be a necessary condition for (no-arbitrage) equilibrium, because no investor would want to hold a currency that offers a low interest rate and sells at a discount, whereas everyone would want to hold a currency that offers a high interest rate and sells at a premium. The sufficient condition is that the interest differ­ential and forward spread are equal.

The no-arbitrage condition, as represented by (2.8) can be represented diagrammatically by a 45° line passing through the origin, as shown in Figure 2.3. Any point off the no-arbitrage line represents a profitable arbitrage opera­tion by going short on x and long on y or vice versa, depending on whether the point is above or below the line. Notice that points falling in the second and fourth quadrants represent a more serious violation of the no-arbitrage condi­tion because they imply that the currency with the higher interest rate sells at a premium or vice versa.

A corollary

It can be shown that, in the absence of bid-offer spreads, if there is no covered arbitrage opportunity in one direction, then there is no arbitrage opportunity in the opposite direction. Consider the no-arbitrage condition from x to y (equation 2.2), which may now be written as

^ (1 + iy ) = (1 + ix ) (2.9)

^S(x/y⁾

Since F(x/y) = 1/[F(y/x)] and S(x/y) = 1/[S(y/x)], it follows that which can be manipulated to produce

^^ (1 + ix ) = (1 + iy ) (2.11)

^S(y^/x)

which is the no-arbitrage condition for going from y to x.

2.2 THE NO-ARBITRAGE CONDITION WITH BID-OFFER SPREADS

Let us now reconsider the no-arbitrage condition when there are bid-offer spreads in exchange and interest rates. Remember that an arbitrager in the foreign exchange market buys at the (higher) offer exchange rate and sells at the (lower) bid exchange rate. In the money market, the arbitrager borrows at the (higher) offer interest rate and lends at the (lower) bid interest rate.

Let us first consider arbitrage from x to y in the presence of bid-offer spreads. The operation consists of the following steps:

1. Borrowing one unit of x at the offer interest rate, i_x,_a.

2. Converting the borrowed funds, buying y at the spot offer rate, Sa, obtaining 1/S_a units of y. This amount is invested at the bid interest rate, iy ,b.

3. The y value of the invested amount at the end of the investment period is (1/Sa)(1 + iy,b).

4. This amount is reconverted into x at the bid forward rate, Fb, to obtain (Fb/Sa)(1 + iy,b).

5. The value of the loan plus interest is (1 + i_x,_a).

The covered margin is given by

_F

P = -b(1 + iy,b ) -(1 + ix,a ) (2.12)

^S

a

Since S_a = Sb(1 + m), where m is the bid-offer spread measured as a percentage of the bid rate, it follows that

F

p = TTT—^¹ + iy,b)-(! + ix,a ) (2.13)

S_b(l+m) ^y

Assuming that the forward spread is equal on both the bid and offer sides, we have

F

= 1 + f (2.14)

^Sb

By substituting equation (2.14) into equation (2.13) we obtain

(1 + f )(1 + iyb )

p = „ s -(1 + ixa ) (2.15)

(1+m) which gives

(1 + f)(1 + i b ) -(1 + m)(1 + ix,a )

p = ^ : (2.16)

(1+m)

Since 1+m»1, and by ignoring the small cross products, an approximate expression for the covered margin can be written as

^p = ⁱy,b ^-ix,a + ^{f -m (2}.¹⁷⁾

It can be demonstrated that the profitability of arbitrage from x to y is smaller in the presence of the bid-offer spreads. Assuming that in the absence of bid-offer spreads, arbitragers act on the basis of the mid-rates, i_x and iy, in which case the covered margin is

^p = ⁱy ^-ix + ^{f (2}.¹⁸)

Since i_x,b < i_x < i_x,_a and iyb < iy < iy,_a, it follows that the covered margin as repre­sented by (12.17) is smaller than the covered margin as represented by (2.18).

Let us now consider arbitrage from y to x. The operation in this case consists of the following steps:

1. Borrowing one unit of y at the offer interest rate, iy,_a.

2. Converting the borrowed funds, selling y at the spot offer rate, obtaining Sb units of x. This amount is invested at the bid interest rate, i_x ,b.

3. The x value of the invested amount at the end of the investment period is

Sb(1 + ix,b).

4. This amount is reconverted at the offer forward rate, F_a, to obtain (Sb/F_a) (1 + ix,b) units of y.

5. The value of the loan plus interest is (1 + iy,_a).

The covered margin is

_S

p = F-(1 + ix,b ) -(1 + iy,a ) (2.19)

^Fa

Since Sb = S_a/(1 + m) and S_a/F_a = 1/(1 + f), it follows that ⁽¹ + ⁱx ,b ⁾

(1 + f)(1+m)

Since (1+f )(1+m) » 1, an approximate expression for arbitrage profit is

^p = ⁱx,b ^-iy,a ^{- f -m (2}.²¹⁾

It can be shown that net profit from arbitrage from y to x is smaller than that resulting from the use of mid-rates.

2.3 DEVIATIONS FROM THE CIP NO-ARBITRAGE CONDITION

Figure 2.4 shows the behaviour of the covered margin (in percentage points) resulting from covered arbitrage involving the US, British and Canadian currencies. It is obvious that the covered margin can deviate from zero, partic­ularly when arbitrage involves the pound against either the Canadian dollar or the US dollar. These deviations imply the availability of profitable arbitrage operations, which begs the question as to why they are not exploited to the extent that reduces the margins to zero. One answer to this question is that these observed deviations may be due to measurement errors. The data used to calculate the covered margins shown in Figure 2.4 are published data, not the data on which transactions could have been conducted. Moreover, they are taken from various sources, which means that they are probably measured or observed at different points in time (for example, mid-day rather than closing).

p = m . m (1 + iya) (2.20)

Economists have repeatedly tested CIP to find out if there are profitable arbitrage opportunities. Two important studies of the empirical validity of CIP were conducted in the 1980s by Taylor (1987b, 1989). What was different about these studies was that they were not based on published data, which have some measurement errors. There are certain requirements for proper testing of CIP, including the following: (i) observations on exchange and interest rates must be recorded at the same point in time; (ii) they must include the bid-offer spreads and transaction costs; and (iii) they must represent the data on which arbitragers take decisions. Obviously, these three requirements are not satis­fied by published data, which had been used to test CIP. Taylor (1987b) over­came all of these problems by collecting the data himself from dealers operating in the London foreign exchange market. He found no deviations from CIP (that is, the absence of profitable covered arbitrage opportunities) or zero covered margins. In his subsequent paper, however, he concluded that small but potentially exploitable opportunities of profitable arbitrage emerged occasionally during periods of turbulence in the foreign exchange

(a) USD/GBP

(b) USD/CAD

(c) GBP/CAD

FIGURE 2.4 The covered margin (percentage points).

market, but not during periods of tranquillity. He also found that the degree of reduction in the size and persistence of arbitrage increased with the passage of time. Furthermore, he established the notion of the term structure of arbitrage opportunities, indicating that profitable arbitrage opportunities are positively related to the length of the forward contract (the shorter the maturity, the smaller the profitable arbitrage opportunity). One explanation that Taylor presented for observed deviations from covered interest parity is the size and extent of credit limits. If banks impose restrictions on the amounts, maturities and the counterparties they deal with, this will operate as a liquidity constraint on covered arbitrage, in which case profitable opportunities would arise. In another study, Committeri et al. (1993) cast doubt on Taylor's results on the following grounds. First, his analysis was based on data collected in a specific segment of the Eurocurrency market. Second, the data used in his first study did not cover a long period of time. Third, the data did not represent those that the dealers were actually prepared to deal on. By doing their own analysis, they found no profitable arbitrage opportunities.

Apart from measurement errors, some other factors lead to the distortion of the simple no-arbitrage condition given by (2.8), leading to the belief that there may be a profitable arbitrage operation. The fact of the matter is that there are certain factors that affect the simple no-arbitrage condition just like the case with two-currency arbitrage. These factors will be considered in turn.

Transaction costs

Deviations from CIP have been attributed to transaction costs, which are repre­sented by the bid-offer spread of exchange rates (the cost of transacting in the foreign exchange market), the bid-offer spread of interest rates (the cost of trans­acting in the money market) and brokerage fees. Keynes (1923) asserted that the covered margin must exceed some minimum amount before arbitrage becomes profitable, estimating it to be half a percentage point. Branson (1969) put forward two reasons why such a minimum should exist: (i) each transaction in financial markets requires a payment of brokerage fees, and (ii) banks may require their foreign exchange departments to earn a higher yield than that of their domestic departments.

The effect of transaction costs is to create a band around the CIP no-arbi­trage line within which arbitrage is not profitable, as shown in Figure 2.5. Points falling off the CIP line but within the band indicate that although arbi­trage is possible it is not profitable, since the covered margin is not sufficiently large to offset transaction costs. Points falling outside the band indicate profit­able arbitrage opportunities, because the covered margin is large enough to cover transaction costs and leave out some profit.

To be more precise, recall that the equation of the no-arbitrage line in Figure 2.5 is i_x - iy = f. Without transaction costs, arbitrage from x to y is profitable if i_x - iy < f, whereas arbitrage from y to x is profitable if i_x - iy > f. Let transac­tion costs (measured in percentage points) be t. The equations of the lines

representing the lower and upper limits of the band are i_x - iy = f + t and i_x - iy = f - t respectively. Arbitrage from x to y is not profitable within the lower part of the band because i_x - iy >f - t. Similarly, arbitrage from y to x is not profitable within the upper part of the band if i_x - iy <f + t. Arbitrage from x to y is profit­able if i_x - iy < f - t, in which case the covered margin is i_x - iy + f - t. Arbitrage from y to x is profitable if i_x - iy > f + t, in which case the covered margin is i_x - iy - f - t. Table 2.1 summarises these possibilities.

Political risk

Another explanation for deviations from CIP is political risk, which involves the uncertainty that while the funds are invested abroad they may be frozen, become inconvertible or be confiscated. In a less extreme case they may face new or higher taxes. Aliber (1973) argues that the comparability criterion, which is critical for choosing the money market assets used to test CIP, requires assets to be identical in terms of political risk. Accordingly, he argues that while Eurocurrency assets satisfy the comparability criterion, domestic assets do not because they are issued under different political jurisdictions.

In diagrammatic terms, political risk creates a band, as described by Figure 2.6, because arbitragers require a risk premium and hence some minimum

| f INTERNATIONAL FINANCIAL OPERATIONS TABLE 2.1 The effect of transaction costs.

x ® y

y ® x

Without transaction costs No arbitrage Profitable arbitrage Covered margin

= f < f + f

ⁱx ^{- i}y = ^{f i}x ^- 'y > ^{f i}x ^- 'y ^{- f}

^y

x- ⁱy * ^f-1 x ^{- i}y < ^f-1 y^{- i}x + ^f-1

'x ^{- i}y ^{£ f} + f

ix - iy > f + t

'x ^- 'y ^{- f-1}

With transaction costs No arbitrage Profitable arbitrage Covered margin

covered margin. This band, however, does not have to be of equal width on either side of the CIP line. For example, if investors from country y view country x as being politically more risky than investors from country x view country y, then there will be a larger political risk premium on x-denominated securities, and the band will be wider to the left of the no-arbitrage line. Moosa (1996a) attributes deviations from CIP between Australia and New Zealand in the period immediately following the abolition of capital controls in the mid-1980s to political risk, in the sense that investors on both sides of the Tasman Sea could have been worried about the possibility of either government reimposing capital controls.

Let us examine Figure 2.6 with some scrutiny. Suppose that y-based inves­tors require a risk premium of p_x to invest in x-denominated assets, whereas x- based investors require a risk premium of py to invest in y-denominated assets such that p_x > py . The equations of the lines defining the lower and upper limits of the band created by political risk are i_x -iy = f -py and i_x -iy = f + p_x . Arbitrage from x to y is not initiated within the lower part of the band because i_x -iy > f -py . Similarly, arbitrage from y to x is not initiated within the upper part of the band because i_x -iy < f + p_x . Arbitrage from x to y is initiated when i_x -iy < f -Py . Notice that the covered margins are equal in both cases because the risk premium is not an actual cost or a source of revenue. Therefore, it only determines the no-arbitrage zone. Table 2.2 summarises the situation.

Tax differentials

Levi (1977) explains deviations from the no-arbitrage condition in terms of differences in tax rates. If tax rates on interest income (t _n ) and foreign exchange or capital gains (tg) are different (t_n ^ tg ), the no-arbitrage line will no longer be a 45° line. Rather, it will have the equation i_x -iy = 6f, where

1 -1 _g

q= ^ (2.22)

1 -t n

If the capital gains tax rate is higher than the income tax rate, the line will be flatter than the 45° line, as shown in Figure 2.7(a). Otherwise, it will be steeper as shown in Figure 2.7(b). Hence, points falling on the line i_x -iy = Of (and hence off the line i_x - iy = f) represent deviations from the no-arbitrage condi­tion in the absence, but not in the presence, of taxes. Table 2.3 shows all of the possibilities.

TABLE 2.2 The effect of political risk.

x ® y y ® x

Without political risk
No arbitrage	ix- iy = f	ix - iy = f
Arbitrage initiated	ix- iy < f	ix - iy > f
Covered margin	iy - ix + f	ix - iy - f
With political risk
No arbitrage	ix-iy ^ f -Py	ix - iy £ f + Px
Arbitrage initiated	l'x - l'y < f - Py	ix -iy > f + Px
Covered margin	iy- ix + f	ix - iy - f

TABLE 2.3 The effect of tax differentials.

x ® y y ® x
Without tax differentials (0 = 1)
No arbitrage	x	-/y= f	x- iy = f
Profitable arbitrage	x	-iy < f	x- iy > f
Covered margin	y	-ix + f	x - iy - f
With tax differentials (0 * 1)
No arbitrage	x	-iy= Of	x- iy= Of
Profitable arbitrage	x	-iy< Of	x- iy > Of
Covered margin i	y "	-ix + Of	x- iy- Of

TABLE 2.4 The combined effect of transaction costs, political risk and tax differen­tials (0 > 1).

x ® y			y ® x
Without the effect
No arbitrage i	x	-V=f		x - iy = f
Profitable arbitrage i	x	iy < f		x - iy > f
Covered margin i	y -	~ix + f		x - iy- f
With the effect
No arbitrage i	x	-iy > Of -1	- Py	x -iy < Of + t + Px
Profitable arbitrage i	x	-iy < Of -1	" Py	x -iy > Of + t + Px
Covered margin i	y -	-ix + Of -1		x- iy- Of -1

Combining the three factors

It is possible to combine the three factors mentioned so far: transaction costs, political risk and tax differentials. The effect of this combination of factors is represented by Figure 2.8 when 0 > 1. In this case, the no-arbitrage line is steeper than the 45° line with an equation i_x -iy = 0f. The effect of transaction costs and political risk is to create a band around this line. The unequal band width is due to differences in the risk premia, as we are assuming that invest­ment in country x is more risky as viewed by investors from country y. The no- arbitrage line in the absence of these factors is the 45° line with the equation i_x - iy = f. Table 2.4 illustrates all of the possibilities.

Liquidity differences

Liquidity differences may also cause deviations from the no-arbitrage condi­tion. The liquidity of an asset can be judged by how quickly and cheaply it can be converted into cash. The more uncertainty there is concerning future needs

(a) 6 <1

(b) 6 >1

and alternative sources of short-term financing, the higher will be the premium that should be received before choosing to invest in the non­base currency, y. This again implies the existence of a band around the no- arbitrage line, and this band may have different widths on the two sides of the line.

Other factors

Some other factors have been suggested to explain deviations from the CIP no-arbitrage condition because they hinder the movement of arbitrage

FIGURE 2.8 Covered arbitrage in the presence of transaction costs, political risk and tax differentials.

funds when the possibility for arbitrage arises. One of these factors is the exis­tence of inelastic (or less than perfectly elastic) supply and demand for arbitrage funds (Pippenger, 1978), which amounts to a violation of one of the basic assumptions of CIP. A related factor is capital market imperfections (Prachowny, 1970), which is not important at the present time given the increasing degree of market perfection. Another factor is capital controls, a factor that has lost importance because of the worldwide tendency to abolish these controls and implement financial deregulation.

These factors prevent the forces of supply and demand from moving to the extent required to restore the equilibrium condition. We have seen from Figure 2.2 that equilibrium is restored when arbitrage causes some changes that take interest and exchange rates to the levels required to achieve the no- arbitrage condition. Let us denote the initial levels of the variables i_x,o, iy,o, So and Fo. Consider now the situation illustrated by Figure 2.9 assuming that the levels of the variables required to restore equilibrium are i_x,2 (> i_x,l), iy,2 (< iy,l), S2 (> Si) and F2 (< Fi). If these factors prevent the forces of supply and demand

FIGURE 2.9 The effect of covered arbitrage in the presence of inelastic supply and demand and similar factors.

from achieving the levels of the variables consistent with the no-arbitrage condition, then deviations from this condition will persist. Figure 2.9 shows that changes in supply and demand can only achieve the levels i_x,i, iy,i, Si and Fi. In terms of the CIP diagram represented by Figure 2.10, changes in the forces of supply and demand lead to a fall in the forward spread from fo to fi and to a rise in the interest differential from (i_x - iy)o to (i_x - iy)i, which is inade­quate to restore the no-arbitrage condition. In other words, arbitrage causes a move towards the no-arbitrage line from A to B. To achieve equilibrium, however, a further move from B to C is required, as indicated by the dotted arrow. Because of these factors, changes in supply and demand reduce, but do not eliminate, deviations from the no-arbitrage condition.

FIGURE 2.10 The effect of covered arbitrage in the presence of inelastic supply and demand and similar factors (the CIP diagram).

2.4 COMBINING COVERED ARBITRAGE WITH THREE- CURRENCY ARBITRAGE

Three-currency arbitrage is normally viewed as the activity that maintains the consistency of the cross spot exchange rates (the no-arbitrage condition). There is no reason why the same is not valid for the forward market. It is important, however, to bear in mind that consistency of the cross spot rates does not neces­sarily imply consistency of the cross forward rates. It will be shown later that consistency of the cross spot rates is a necessary but not sufficient condition for the consistency of the cross forward rates. This means that the absence of profit­able three-currency arbitrage opportunities from the spot market does not neces­sarily imply the absence of such opportunities from the forward market.

In what follows, it is shown that covered interest arbitrage and three- currency arbitrage in the spot market can maintain the consistency of the cross forward rates, eliminating three-currency arbitrage opportunities from the forward market. Put differently, the objective is to show that the effect of three-currency arbitrage in the forward market is equivalent to the combined effect of covered interest arbitrage (which involves four markets) and three- currency arbitrage in the spot market.

To simplify the exposition we assume that there are no bid-offer spreads and that transactions are conducted on the basis of the mid rates. Assume that there are three currencies (x, y and z) and three exchange rates. Let currency z be the numeraire. Consistency of the cross forward rates (the no-arbitrage condition in the forward market) requires the following condition to be satisfied:

F(x/y) =

(2.23)

F(x/z) F( y/z)

where F(x/y) is the forward exchange rate between x and y, and so on. This no- arbitrage condition is maintained by three-currency arbitrage in the forward market. It can be shown that the condition (2.23) can be established by a combination of covered interest arbitrage and three-currency arbitrage in the spot market.

From the previous discussion, we have

1 + ix 1 + i_z

(2.24)

F( x/z) = S( x/z)

Likewise, the condition can be written for y and z and for x and y as

(2.25)

"1 + i_y

^ly

F( y/z) = S( y/z)

1+ix 1 + i_y

(2.26)

^{F( x/}y⁾ = ^{S( x/}y⁾

Since three-currency arbitrage in the spot market implies that S(x/y) = S(x/z)/S(y/ z), it follows that

1 + ix 1 + k

(2.27)

S(x/z)

F(x/y) =

S( y/z)

(2.28)

By multiplying and dividing the right hand side of equation (2.27) by 1 + i_z, we obtain

S(x/z)	1 + x	1 + iz
S( y/z)	1 + iy	1 + iz _

F( x/y) = -

By rearranging we obtain

(2.29)

S(x/z)[(1 + i_x )/(1 + i_z )] F(x/z)

F(x/y) =

S(y/z)[(1 + iy )/(1 + iz )] F(y/z)

which is the no-arbitrage condition in the forward market. Thus, profitable three-currency arbitrage opportunities will not be present in the spot market if there are no profitable opportunities for three-currency arbitrage in the spot market and no profitable covered arbitrage.

The violation of equation (2.23) may be due to one or all of the following: (i) inconsistency of the spot exchange rates; (ii) violation of CIP between x and 2 (2.24); (iii) violation of CIP between y and 2 (2.25); and (iv) violation of CIP between x and y (2.26). Let us assume that equation (2.23) is violated such that

(2.30)

F(x/2)

F( x/y ⁾ >

F( y/z)

This violation is due to one, some or all of the following violations of the no- arbitrage conditions:

S(x/z)

^{S( x/}y⁾ >

(2.31)

%/Z)

1+x

1 + K

S( ^x/y⁾

(2.32)

>^F(x/y⁾

1+ix

1 + iz

1 + iy 1 + i_z

S(x/z)

>F(x/z)

S( y/z)

< F( y/z)

It can be demonstrated that if the forward rates are inconsistent only because the spot rates are inconsistent, then the same profit can be obtained from three-currency arbitrage in the spot market or in the forward market. Consider the violation represented by equation (2.31). If this is the case, then profit can be obtained by following the sequence 2 ® y ® x ® 2. For each unit of 2, profit realised from three-currency arbitrage in the spot market, p _s, is

S⁽y^/z)S^(x/y⁾

p _s = 1

(2.36)

^s S( x/z)

It is easy to show that the same profit can be realised by indulging in three- currency arbitrage in the forward market. In this case, profit, p f, is

_p F(y/z)F( x/y) .

(2.35)

p f = -1

^f F(x/z)

Since there are no covered arbitrage opportunities in any of the currency pairs, it follows that

S(y/z)[(1 + iy )/(1 + iz )]S(x/y) [(1 + ix )/(1 + iy )] P f = -1

S^(x/z)[(1 + ⁱx ^)/(1 + ⁱz ^)] (_{2 37})

S⁽ y^{/z)S( x/}y⁾

-1 = p s

S(x/z)

Likewise, it can be shown that profit realised from covered interest arbitrage is identical to the profit realised from three-currency arbitrage for any violation of CIP.

2.5 UNCOVERED INTEREST ARBITRAGE

Uncovered interest arbitrage is similar to covered arbitrage, except that it does not involve the forward market. The long position is not covered by converting the underlying currency proceeds at the forward rate that is known in advance, but rather at the spot rate prevailing on the maturity of the long position, which is not known in advance. If the currency underlying the long position does not depreciate against the currency underlying the short position by more than the interest rate differential, then profit will be made.

Obviously, this operation involves foreign exchange risk, because it is based on a decision variable that is not known in advance: the expected spot rate on the maturity of the long position. So, this operation does not satisfy the defini­tion of arbitrage that it is a riskless operation, and hence it should be classified as speculation. In general, there is nothing wrong with classifying this opera­tion as speculation, but there are some reasons why it may also be classified as arbitrage, including the following:

1. As we are going to see, it is a misconception that arbitrage is a riskless opera­tion. Even in the simple case of two-currency arbitrage, some risk may be involved.

2. We will also find out that foreign exchange risk can be minimised or elimi­nated under certain conditions (for example, by choosing a pair of curren­cies with a relatively stable exchange rate). In this case, the operation boils down to borrowing a low-interest currency and investing in a high-interest currency, making profit out of the interest rate differential. The forward cover would be replaced by a natural hedge from the choice of a currency pair with a stable exchange rate.

3. If, and this is a big if, the arbitrager believes in the accuracy of her forecasts with a high confidence level, then she will be in a position to calculate the arbitrage profit in advance with a high degree of confidence.

4. As we have seen, covered arbitrage is the mechanism whereby the no-arbi­trage condition, which is known as covered interest parity (CIP), is main­tained. If the counterpart to covered interest parity in this case is uncovered

interest parity (UIP), then it is convenient to call the mechanism that main­tains this condition uncovered arbitrage.

5. We will also find out that foreign exchange risk can be eliminated in some special situations. One such situation is when a short (long) position is taken on a currency that is pegged to a basket, while a corresponding long (short) position is taken on the components of the basket in such a way that it reflects the weights of the basket components. In this case, foreign exchange risk would disappear, and the arbitrager would know in advance how much profit will be made on the interest rate differentials.

The mechanism of uncovered arbitrage

Let us now see what happens when the arbitrager indulges in uncovered arbi­trage from x to y. The operation consists of the following steps:

1. At time t, the arbitrager borrows one unit of x at i_x for a period extending between t and t + 1.

2. The amount borrowed is converted at S_t, obtaining 1/S_t units of y. This amount is then invested at i_y.

3. At t + 1, the y value of the investment is (1/S_t)(1 + iy).

4. The x currency value of the investment, converted at the spot rate prevailing at t + 1 is (S_t+x/S_t)(1 + iy).

5. At t + 1 the loan matures, and the amount (1 + i_x) has to be repaid.

Net profit, or the uncovered margin, is given by

S

t+ 1 ^St

p = -

(1 + iy ) - (1 + ix ) (2.38)

Hence the no-arbitrage condition is

-(1 + iy ) = (1 + ix ) (2.39)

which is one version of the uncovered interest parity condition. The UIP no- arbitrage condition can be expressed differently by manipulating equation (2.39). First of all we could rewrite equation (2.39) in terms of net returns and

costs, by subtracting 1 from both sides of the equation, as

_S

_J±1 (1 + iy ) -1 = ix (2.40)

^St

Another specification of the no-arbitrage condition can be obtained by manipulating equation (2.39) to obtain

^St+ 1 = ^Ft (².⁴¹)

^St+ 1 ^St

where Ft is the interest parity forward rate defined in equation (2.5). Likewise, we can derive the following expression for the UIP condition:

ix -iy= S (2.42)

where S is the percentage change in the exchange rate between t and t + 1. Equation (2.42) tells us that if the (expected) percentage change in the exchange rate is equal to the interest rate differential, then there is no possi­bility of indulging in profitable arbitrage. This is because if the arbitrager wants to go long on the high-interest currency, any gains from the interest rate differential will be offset by the deprecation of the currency. Otherwise, if the arbitrager wants to go long on the currency that is expected to appreciate to make profit out of foreign exchange gains, then this gain will be offset by the interest rate differential in favour of the other currency. No profit will be made in either case.

Deriving an expression for the uncovered margin in the presence of bid-offer spreads is similar to that under covered arbitrage. In the case of arbi­trage from x to y, the uncovered margin is given by

S

P = -S+¹ (1 + iyb ) -(1 + ix,a ) (2.43)

^pa,t

or

^p = ⁱx,a ^-iy,b + ^{S-m (2}.⁴⁴⁾ If, on the other hand, arbitrage runs from y to x, then

^Sb t

P (1 + ix) -(1 + iy,a ) (2.45)

^Sa,t+ 1

or

^p = ⁱx,b ^-iy,a ^{- S-m (2}.⁴⁶⁾

As we can see, the UIP condition is similar to the CIP condition with S_t+i(S) replacing Ft(f).

The empirical validity of UIP

For UIP to be valid, indicating the absence of uncovered arbitrage opportuni­ties, the exchange rate between t and t + 1 must change by a percentage that is equal (or at least related) to the interest rate differential at time t. Formal tests of UIP have produced mixed results that are highly sensitive to the model specification. If we observe historical data, however, we can see that uncov­ered arbitrage would have produced some significantly positive or negative uncovered margins, as shown in Figure 2.11. It is obvious that the uncovered margin is greater in currency pairs involving the pound because of the relative stability of the exchange rate between the US and Canadian currencies and the narrower interest rate differential.

(a) USD/GBP

(b) USD/CAD

(c) GBP/CAD

FIGURE 2.11 The uncovered margin (percentage points).

Notice that the uncovered margins move within wider ranges than the corresponding covered margins shown in Figure 2.4. This is a manifestation of the motto "there is no such thing as a free lunch", in the sense that if you want to enjoy the possibility of the high return offered by uncovered arbitrage you must be prepared to endure the high risk associated with this operation.

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CHAPTER 3

Other Kinds of Arbitrage and Some Extensions