Currency Cross Rates and Triangular Arbitrage

Economic factors determine the foreign exchange rates of each currency pair, but currency arbitrage ensures that the rates cohere with the rates of all possible combinations of every currency.

Foreign Exchange Rate Determination

Like almost anything else, the value of any currency is determined by supply and demand. The greater the demand relative to supply, the greater the value, and vice versa, even when no foreign exchange is involved. For instance, if a country never expands its money supply, then the available money becomes more valuable as the economy expands. Thus, the price of individual items decreases: deflation.

When the money supply expands faster than the economy, then money becomes less valuable. It takes more money to purchase single items, leading to inflation. Thus, the German mark became almost worthless after World War I, when the government simply printed much more money to pay its bills, or when the United States printed more money to pay for the Vietnam War, leading to significant inflation in the 1970's.

Naturally, in foreign exchange, when currency of a particular country is plentiful, it will have less value against other currencies, and vice versa.

Demand for the currency also affects the exchange rate. If 1 country pays a higher interest rate than another country or has more investment opportunities or a more stable government, then that country's currency will have greater value over the other country's currency.

Sometimes governments, usually through their central banks, will intervene to keep currency pegged to another currency, as China keeps the yuan pegged to the United States dollar (USD), but this is mostly accomplished by manipulating the supply and demand.

But how does supply and demand actually establish the rate?

Purchasing Power Parity (PPP)

The amount of currency in any country that can buy a particular basket of goods and services should equal the amount of another currency that can buy the same basket. This is called purchasing power parity (PPP).

Although purchasing power parity makes sense, it cannot really establish foreign exchange rates, because of the difficulties in equalizing the rates if it should differ from parity.

When the flow of capital or goods and services across international boundaries are not restricted and if the same goods or assets are sold in different markets for different prices, then they can be bought in the cheap market and sold in the more expensive market for a virtually risk-free profit, which is called arbitrage. Arbitrage equalizes prices in different markets to within a narrow range. However, sometimes the expense of transporting and selling the goods in the higher-price market exceeds the price differential.

For instance, if it takes fewer U.S. dollars to buy a basket of goods than Euros in Europe, then how can anyone profit from the difference? Someone might try to buy the basket of goods from the United States and sell it in Europe. However, transportation costs and taxes would reduce or eliminate potential profits significantly. The expenses of buying in 1 country, transporting it to another, then selling it there, creates a wide gap between prices that cannot be closed by arbitrage — at least for most commodities, especially food and energy.

However, arbitrage works very well in currency trades.

Dealers in Currency — Market Makers

Most currency trades are now done online, where time and distance are no barrier. Buying and selling currency is usually done with a market maker in that currency. Many market makers exist for most currencies, especially the major currencies. A market maker may deal in U.S. dollars and Euros, for instance, purchasing and selling both currencies by publishing a bid/ask price for both currencies. If market makers start getting more dollars in exchange for Euros, they will raise the ask price for Euros and lower the bid price for dollars until the orders start equalizing. Otherwise, market makers would soon run out of Euros and be stuck with dollars, unable to continue business since at the established bid/ask price, there would be too few Euros to trade for dollars, which the market is currently demanding. Thus, to stay in business, bid prices for dollars are lowered and ask prices for Euros are increased. This is how supply and demand works.

Each market maker must maintain parity with other market makers in the same currencies — otherwise arbitrageurs would buy, for instance, Euros from the market maker with a lower ask price and sell it to another market maker with a higher bid, and would continue doing so until prices equalized to within transaction costs.

But what about market makers in different currencies?

Currency Cross Rates and Triangular Arbitrage in the FX Spot Market

Cross rates are the exchange rates of 1 currency with other currencies, and those currencies with each other. Cross rates are equalized among all currencies through a process called triangular arbitrage. Below is a table of key cross rates of some major currencies.

All currency quotes are given in pairs, and each pair can be expressed as a fraction. To illustrate, if a Great Britain pound (GBP) is worth 1.98244 USD, then this can be expressed as the fraction 1.98244 USD/1 GBP. We also see from the above table that a Euro is worth 0.68211 GBP, which can be expressed as 0.68211 GBP/1 EUR.

From these 2 fractions, we can calculate what the USD is worth in EUR.

GBP cancels out, resulting in a quote for dollars per Euro. This result is very close to the actual quote, 1.35225, from the table above. Because we are ignoring the bid/ask spread and transaction costs to simplify the math in this example, there is no reason to believe that it would be exact. Also, arbitrage is not a perfect equalizer because the market is not perfectly efficient.

But what if these cross rates didn't equalize. Suppose, for instance, that 1 GBP was exactly equal to 2 USD, with all other cross rates remaining the same. Obviously, the above equation would not hold, but it would present an arbitrage opportunity in the FX spot market, as illustrated below:

This can be illustrated graphically as a self-closing triangle of currency exchanges, so it is called triangular arbitrage.

So, will you profit from triangular arbitrage? Not likely. Many professionals and banks with computers are constantly calculating the cross rates of all currencies. Any inequalities exceeding transaction costs will be quickly closed. But triangular arbitrage does explain how the cross rates of currencies are equalized.