Covered interest arbitrage is an operation that is conducted in four markets involving two currencies: (i) the spot foreign exchange market, (ii) the forward foreign exchange market, (iii) the money market in currency x, and (iv) the money market in currency y. The objective is to make profit by going short on one currency and long on the other, while covering the long position in the forward market. When it matures, the long position is unwound and the proceeds are converted into the other currency at the forward rate agreed upon in advance.
The proceeds are then used to meet the obligations arising from the short posi
tion, and any left over would then represent net arbitrage profit. Notice that the operation is risk-free in the sense that the decision variables are known at the time when the transaction is initiated.
Let Sand Fbe the spot and forward exchange rates between currencies x and ymeasured as S(x/y) and F(x/y). Also let ix and iy be the interest rates on x and y respectively, such that the maturities of the assets and liabilities under
lying ix and iy (for example, deposits and loans) are identical to the maturity of the forward contract. We will assume a two-period model where t is the present time at which the operation is initiated and t+1 is the future when the long position, short position and the forward contract mature. Whether the arbitrager goes short on x and long on y or the other way round depends on the configuration of interest and exchange rates. A covered arbitrage opera
tion by going short on xand long on y(x®y) consists of the following steps:
1. At time t, the arbitrager borrows one unit of xat ix for a period extending between t and t + 1, when the forward contract matures.
2. The amount borrowed is converted at S, obtaining 1/S units of y. This amount is then invested at iy.
3. At t+ 1, the value of the investment is (1/S)(1 + iy) units of y.
1 9
4. The x currency value of the investment converted at the forward rate is (F/
S)(1 + iy).
5. At t+ 1 the loan matures, and the amount (1 + ix) has to be repaid.
The net profit arising from this operation, which is also called the covered margin, is given by
p=F (1+iy ) -(1 +ix ) (2.1)
S
Hence the no-arbitrage condition is
F (1 +iy ) =(1 +ix ) (2.2)
S
The equality of the gross return on yand the cost of borrowing x(principal plus interest) after covering the foreign exchange risk by selling yforward, as represented by (2.2), is called covered interest parity (CIP). This relationship is an application of the law of one price to financial markets (identical financial assets should produce identical returns after covering the foreign exchange risk).
Figure 2.1 shows the no-arbitrage condition represented by equation (2.2).
The no-arbitrage line is represented by a 45° line passing through the origin.
Any point above the line represents profitable arbitrage by going short on x and long on y(x®y), whereas points below the line represent profitable arbi
trage by going short on y and long on x (y®x).
F ( 1 + i y) S
x Æ y
¨
) (
) x
y i i
S i
F + = +
y Æ x Profitable arbitrage Profitable arbitrage
No-arbitrage line 1
(
1 ( + ix ) FIGURE 2.1 The no-arbitrage condition implied by CIP.
Suppose now that the interest rate and exchange rate configuration is such that the no-arbitrage condition is violated, as represented by a point above the no-arbitrage line. In this case, arbitrage will lead to changes in the forces of supply and demand as illustrated in Figure 2.2. The following will happen:
1. Demand declines in the money market for x-denominated assets, leading to a rise in ix.
2. Demand rises in the money market for y-denominated assets, leading to a decline in iy.
3. The demand for currency y increases in the spot market, leading to a rise in the spot exchange rate, S.
4. The supply of currency yrises in the forward market, leading to a decline in the forward exchange rate, F.
These changes combined lead to a decline in the covered return on yand an increase in the cost of borrowing x. When they are equal, the covered margin is
Supply Demand
Supply i x
Demand
i y
Money market ( x) Money market ( y)
( / ) F x y
S x y ( / )
Supply Supply
Demand Demand
Spot market Forward market
FIGURE 2.2 The effect of covered arbitrage.
equal to zero, and the no-arbitrage condition is re-established. Arbitrage comes to an end, as the new configuration of interest and exchange rates is represented by a point falling on the no-arbitrage line.
Other forms of the no-arbitrage condition
The no-arbitrage condition can be expressed differently by manipulating equation (2.2). First of all we could rewrite this equation in terms of the net amounts, by subtracting 1 from both sides of the equation, to obtain
F (1 +iy ) - =1 i x (2.3)
S
Another specification of the no-arbitrage condition can be obtained by deriving the value of the forward rate consistent with CIP, the so-called equi
librium or interest parity forward rate, from equation (2.2). In order to distin
guish between the actual forward rate (which prevails whether or not CIP holds) and the equilibrium rate, the latter is denoted F. Thus the CIP no-arbi- trage condition may be written as
F F where
= (2.4)
é1 ù F=S +ix
(2.5) êêë ú
1 +iyúû
which means that the interest parity forward rate, as represented by equation (2.5), is calculated by adjusting the spot rate for a factor reflecting the interest rate differential. Since
F
S= +1 f (2.6)
where fis the forward spread, it follows that
(1 + f)(1 + iy) = 1 + ix (2.7)
which can be manipulated to obtain an approximate but useful expression for the no-arbitrage condition by ignoring the (small) term iyf. The approximate expression is
ix – iy = f (2.8)
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 ix > 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
ix -iy
¨
f i i x - y=
y Æ x
y Æ x y Æ x
x Æ y
¨
x Æ y
x Æ y
f No-arbitrage line
No-arbitrage line
FIGURE 2.3 The no-arbitrage line in the f – (ix – iy) space.
forward discount. If, on the other hand, ix < 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 differ
ential 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 opera
tion by going short on xand long on yor 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 condi
tion 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
F x y
S x y( / ) i y ix
( / ) ( 1 + ) ( = 1+ ) (2.9)
Since F(x/y) = 1/[F(y/x)] and S(x/y) = 1/[S(y/x)], it follows that S y x
F y x( / ) iy ix
( / ) ( 1 + ) ( = 1+ ) (2.10)
which can be manipulated to produce F y x
S y x( / ) i x iy
( / ) ( 1 + ) ( = 1+ ) (2.11)
which is the no-arbitrage condition for going from yto x.