CURRENCY ARBITRAGE
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 defini
tion 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 opera
tion 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 opera
tion. 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 elimi
nated under certain conditions (for example, by choosing a pair of curren
cies 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-arbi- trage condition, which is known as covered interest parity (CIP), is main
tained. If the counterpart to covered interest parity in this case is uncovered
interest parity (UIP), then it is convenient to call the mechanism that main
tains 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 arbi
trage from x to y. The operation consists of the following steps:
1. At time t, the arbitrager borrows one unit of x at ix for a period extending between t and t + 1.
2. The amount borrowed is converted at St, obtaining 1/St units of y. This amount is then invested at iy.
3. At t + 1, the y value of the investment is (1/St)(1 + iy).
4. The x currency value of the investment, converted at the spot rate prevailing at t+ 1 is (St+1/St)(1 + iy).
5. At t+ 1 the loan matures, and the amount (1 + ix) has to be repaid.
Net profit, or the uncovered margin, is given by p=St+ 1
(1 +iy ) -(1+ix ) (2.38)
St
Hence the no-arbitrage condition is St+ 1
(1 +iy ) =(1 +ix ) (2.39)
St
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
St+ 1
(1+iy ) -1 =i (2.40)
St x
Another specification of the no-arbitrage condition can be obtained by manipulating equation (2.39) to obtain
=Ft (2.41)
St+ 1
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 possi
bility 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 arbi
trage from x to y, the uncovered margin is given by p=Sb ,t+ 1
(1+iy,b ) -(1 +ix,a ) (2.43)
Sa,t or
p=ix,a -i y,b + S -m (2.44)
If, on the other hand, arbitrage runs from yto x, then p Sb ,t
(1+ix,b ) -(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 St+1(S) replacing Ft(f).
The empirical validity of UIP
For UIP to be valid, indicating the absence of uncovered arbitrage opportuni
ties, the exchange rate between tand 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 uncov
ered 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
–15 –10 –5 0 5 10 15 20
Mar-78 Mar-82 Mar-86 Mar-90 Mar-94 Mar-98 Mar-02
(b) USD/CAD
–6 –4 –2 0 2 4 6
Mar-78 Mar-82 Mar-86 Mar-90 Mar-94 Mar-98 Mar-02
(c) GBP/CAD
–15 –10 –5 0 5 10 15 20
Mar-78 Mar-82 Mar-86 Mar-90 Mar-94 Mar-98 Mar-02
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.