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4.2 EXCHANGE RATE EQUILIBRIUM
With FX volume around $2 trillion per day, you can bet your bottom dollar (euro, or yuan) that there are plenty of arbitrageurs looking for opportunities such as these. Dealers are just as vigilant in ensuring that their bid and offer quotes overlap those of other FX dealers. If a bank’s bid or offer quotes drift outside of the band defined by other dealers’ quotes, it quickly finds itself inundated with buy (sell) orders for its low-priced (high-priced) currencies. Even if banks’ quoted rates do not allow arbitrage, banks offering the lowest offer (or highest bid) prices in a currency will attract the bulk of customer purchases (sales) in that currency.
The Long and the Short of It
A long position refers to a currency purchase.
A long position is synonymous with ownership of an asset. Ashort position means the holder of the position has sold the asset with the intention of buying it back at a later time. Long positions benefit if the price of the asset goes up, whereas short positions benefit if the price of the asset goes down. For example, a bank is in a long euro position and a short dollar position when, on balance, it has purchased euros and sold dollars. Conversely, a bank is short euros and long dollars when it has sold euros and purchased dollars. Currency balances must be netted out; if a bank has bought 100 million and sold 120 million in two separate transactions, then its net position is short 20 million. Banks try to minimize their net exposures, because currency dealers operating with large imbalances risk big gains or losses if new information arrives and currency values unexpectedly change.
Cross Rates and Triangular Arbitrage
An exchange rate that does not involve the domestic currency is called a cross exchange rate,or simply across rate.Financial newspapers such asThe Wall Street Journaland theLondon Financial Timespublish bilateral exchange rates in a cross- rate table like the one in Figure 4.3. Cross-rate tables report bid-ask midpoints, so these rates do not represent prices that actually can be traded in the market.
Suppose you are given bilateral exchange rates for currencies d, e, and f. The no-arbitrage condition fortriangular arbitragein the currency markets is
Sd/eSe/fSf/d=1 (4.3)
Equation 4.3 can be stated in its reciprocal form (Sd/eSe/fSf/d)−1=Se/dSf/eSd/f=1. Again, remember to follow Rule #1 and keep track of your currencies. If this condition does not hold within the bounds of transaction costs, then triangular arbitrage provides an opportunity for a riskless profit.
Triangular arbitrage ensures that cross rates are in equilibrium.
Currency BRL GBP CNY EUR INR JPY CHF USD
Brazilian real BRL 1 1.1804 0.2914 2.3633 0.0347 0.0239 1.9415 1.8357
British pound GBP 0.8471 1 0.1021 0.8279 0.0122 0.0084 0.6801 0.6430
Chinese new yuan CNY 3.4317 9.7965 1 8.1101 0.1191 0.0819 6.6627 6.2996
Euro area EUR 0.4231 1.2079 0.1233 1 0.0147 0.0101 0.8215 0.7768
Indian rupee INR 28.806 82.234 8.3942 68.078 1 0.6877 55.928 52.880
Japanese yen JPY 41.890 119.58 12.207 98.998 1.4542 1 81.330 76.898
Swiss franc CHF 0.5151 1.4703 0.1501 1.2172 0.0179 0.0123 1 0.9455
U.S. dollar USD 0.5448 1.5551 0.1587 1.2874 0.0189 0.0130 1.0576 1
FIGURE 4.3 Currency Cross Rates.
Source:www.federalreserve.gov (January 2012). Exchange rates in this table are
left-over-top; that is, the left-hand currency divided by one unit of the currency at the top of the table. For example, the dollar-per-yen spot rate S$/¥=$0.0130/¥ indicates one yen is worth $0.0130. Similarly, the yen-per-dollar spot rate S¥/$=¥76.898/$ indicates one dollar is worth 76.898 yen. The relation between the two exchange rates is S$/¥=1/S¥/$.
An Example of Triangular Arbitrage Suppose S$/¥ = $0.0130/¥ and S¥/SFr =
¥81.330/SFr as in Figure 4.3. However, rather than the equilibrium rate of SSFr/$ = SFr0.9455/$ as in Figure 4.3, suppose you can buy dollars (in the denominator of the quote) at a bargain price of SSFr/$ =SFr0.9400/$. The product of the spot rates is less than 1
S$/¥S¥/SFrSSFr/$ =($0.0130/¥)(¥81.330/SFr)(SFr0.9400/$)=0.9939<1 and SSFr/$is too low relative to the cross-currency equilibrium in Figure 4.3. There is an arbitrage opportunity here, so long as transaction costs are not too high.
Suppose you start with $1 million and simultaneously make the following transactions in a ‘‘round turn’’ (i.e., buying and then selling each currency in turn):
Buy ¥ with $($1,000,000)/($0.0130/¥)=¥76,923,077 Buy SFr with ¥ (¥76,923,077)/(¥81.330/SFr)=SFr945,814 Buy $ with SFr (SFr945,814)/(SFr0.9400/$)=$1,006,185
There is no net investment if you execute these trades simultaneously. So long as your credit is good and your counterparties are trustworthy, each cash outflow in a given currency is covered by an offsetting cash inflow in that same currency. With no net investment and no time delay between trades, you have no money at risk.
And, you’ll have captured an arbitrage profit of $6,185.
Suppose you go the wrong way on your round turn and start by purchasing Swiss francs.
Buy SFr with $ ($1,000,000)(SFr0.9400/$)=SFr940,000 Buy ¥ with SFr (SFr940,000)(¥81.330/SFr)=¥76,450,200 Buy $ with ¥ (¥76,450,200)($0.0130/¥)=$993,853
Oops! In this case, you’ve locked in an arbitrage lossof $6,147. How can you tell which direction to go on your round turn? If you start with dollars, do you first convert them to Japanese yen or to Swiss francs?
APPLICATION Significant Digits and Rounding Error
The result of a foreign exchange calculation (indeed, any calculation) is only as precise as the least precise value in the calculation. Consider the following three spot exchange rates:
S$/¥ =$0.0130/¥ S¥/SFr=¥81.330/SFr SSFr/$=SFr0.9400/$ Suppose the spot rate SSFr/$=SFr0.9400/$ is precise. S$/¥=$0.0130/¥ is then the least precise of these values because it is quoted with only three significant digits. As a consequence, any calculation based on these values will have rounding error in the third digit. An arbitrage profit based on a $1,000,000 initial transaction as in the example below is accurate only to about the nearest
$10,000. In the example, triangular arbitrage yields a profit of $6,185, plus or minus a few thousand dollars. The precision implied by the seemingly precise answer of $6,185 is spurious unless you can trade at the exact prices in the quotes.
As a general rule, it is best to retain as many significant digits as possible in your calculations. The result of any calculation is only as accurate as your inputs.
Which Way Do You Go? The no-arbitrage condition is Sd/eSe/fSf/d=1. If Sd/eSe/fSf/d<1, then at least one of these exchange rates should increase as triangular arbitrage forces these rates back toward equilibrium. This suggests the winning arbitrage strategy should be to buy the currency in the denominator of each spot rate with the currency in the numerator.
Conversely, if Sd/eSe/fSf/d >1 then at least one of the rates Sd/e, Se/f, or Sf/d must fall to achieve parity. In this case, you want to sell the high-priced currency in the denominator of each spot rate for the low-priced currency in the numerator.
Here’s the rule for determining which currencies to buy and sell in triangular arbitrage.
■ If Sd/eSe/fSf/d<1, then Sd/e, Se/f, and Sf/dare too low relative to equilibrium.
■ Buy the currencies in the denominators with the currencies in the numerators.
■ If Sd/eSe/fSf/d>1, then Sd/e, Se/f, and Sf/dare too high relative to equilibrium.
■ Sell the currencies in the denominators for the currencies in the numerators.
In our example, S$/¥S¥/SFrSSFr/$ =0.9939<1 at the disequilibrium rate SSFr/$ = SFr0.9400/$. One or more of these exchange rates must rise to return to equilibrium, so you should buy the currency in the denominator of each spot rate with the currency in the numerator. You should (1) buy yen with dollars at S$/¥, (2) buy francs with yen at S¥/SFr, and (3) buy dollars with francs at SSFr/$. In this example, triangular arbitrage is worth doing so long as transaction costs on the round turn are less than (1−S$/¥S¥/SFrSSFr/$)≈0.61 percent of the transaction amount, or about $6,100.
Here’s a complementary way of viewing the example. The inequality S$/¥S¥/SFrSSFr/$ =0.9939<1 can be restated in its reciprocal form (S$/¥S¥/SFrSSFr/$)−1
= (0.9939)−1, or S¥/$SSFr/¥S$/SFr ≈1.0061 > 1. The product of the exchange rates is greater than one, so you should (1) sell dollars for yen, (2) sell yen for francs, and (3) sell francs for dollars. Of course, whenever you sell the currency in the denominator you are simultaneously buying the currency in the numerator. Viewed in this way, the two inequalities for determining ‘‘which way to go’’ are equivalent.
No matter which inequality you use, in our example you want to buy yen with dollars (sell dollars for yen), buy francs with yen (sell yen for francs), and buy dollars with francs (sell francs for dollars).
In actuality, all three exchange rates (as well as any related bilateral exchange rates) are likely to change as financial market arbitrage forces these prices toward equilibrium. Cross-rate tables must be internally consistent within the bounds of transaction costs to preclude arbitrage opportunities. Interbank currency markets for large transactions between major banks are highly competitive, and the no-arbitrage condition ensures that currency cross rates are in equilibrium at all times.