Some Notes on Arbitrage

Arbitrage is a concept that comes up in finance and economics a lot. As I understand the concept, it requires that a market exist for at least three items (say, tomatoes, lettuce, and cucumbers). Suppose a head of lettuce costs two tomatoes, and one cucumber. In that case, we can assume that the cucumber is also worth two tomatoes. But what if it's actually 1.5 tomatoes? In that case, you want to exchange cucumbers for lettuce, then exchange the lettuce for tomatoes, and then "buy" cucumbers for all those tomatoes. If you start out with 1 cucumber, then you'll wind up with 1.333 cucumbers, then 1.777, then 2.370, then 3.161....without doing anything other than cart around a burgeoning pile of vegetables.

That's pretty much the whole concept. What makes it tricky is finding comparable items. Arbitrage opportunities involving things that belong to identical categories (like foreign exchange) will be rare.

Precious Metals Arbitrage

In the 19th-20th century, while currencies were officially pegged to gold, they often oscillated within a range of ±0.4-0.9% of the actual mint parity ratio (ratio of the two currencies respective gold content). These were known as the gold points: if the spot rate for a foreign currency exceeded the gold points, it was financially worthwhile to redeem the local currency in gold, and physically transport the currency abroad (or, conversely, import it from abroad). The problem was that the gold points had to be calculated by experts, and varied over time.


If the spot price of a currency remained within its gold point spread, then it made more sense to remit payments abroad using bills of exchange. But there were different types of bills of exchange, and their creditworthiness varied. Therefore, one could locate spreads between different bills and form an independent estimate of what they ought to be. For example, the most reliable bill of exchange for 19th century remittances was the council bill (for exchanges between India and the UK). This was basically government paper issued by the government of India for financing its debt with the Bank of England (Palgrave, 1894, p.442). Another form was commercial paper, which was very frequently discounted abroad (Houston, 1917, p.1235). A seasoned arbitrageur could take advantage of different discount rates for different bills of exchange, whether domestically or internationally.


In cases where different gold/silver standards are in effect, or where a country has multiple metals in use, arbitrage can have serious consequences. An obvious example is when a currency's currency moves well outside the gold points. For example, a country might suffer a minor regional panic, which triggers other banks to restrict loans so as to restore their reserves. In the Usonian banking system of the 19th century, this could mean country banks withdrawing interest-earning reserves deposited in city banks. Fears of a more general collapse of the banking system could lead to a scramble to convert US dollars to UK pounds, driving the pound upward against the dollar. Instead of the mint parity ratio of $4.8667 per pound, this surged to $5.11 per pound. In such a case, the exchange rate has shifted by only 5% in favor of the UK pound, but this would be massively more than the gold point. Gold in the form of a sovereign (a gold £1 coin) would be worth more than gold in the form of an eagle (a gold $10 coin), so of course gold would physically flow out of the USA to London, where it would be minted into sovereigns. Notice this would usually occur at precisely the worst possible time, whether for England or for the USA.

Note also that the problem would not be insufficient gold content in the eagle relative to the sovereign; it would be panic about the "specie liquidity" of Usonian banks, and hence the currency they issued (i.e., 19th century nationally chartered banks in the USA could issue currency but it had to be backed by some quantity of specie). In order to cover the cost of operations, gold in a "country bank" (an outlying bank) not located in a designated reserve city was partly relocated to a reserve or central reserve city, where it was usually used as the reserves of that bank's currency issue as well as earning interest. Both banks might well be solvent, and capable of meeting lawful obligations, but they would require some time in order to do so. Runs rendered them illiquid, but not insolvent.


A problem peculiar to the USA was the political demand for "free coinage" of silver, in which people with silver in non-coined form (including silver ore) could bring it to a mint, have it assayed, and receive coins equal to the value of the silver. This was never actually implemented, but the US Treasury was obligated after 1890 to buy 4 million ounces per month (124.4 metric tonnes) of silver and issue coins from it. During the period that the USA was on the gold coin standard, this was a serious problem.


The official ratio of gold to silver was 16:1; one ounce of gold was supposed to be worth 16 ounces of silver. In fact, it was 20:1 in '90 and had been 22:1. It reached 26:1 during the 1893-95 Depression, and would rise much further. But silver coins were legal tender at the 16:1 ratio, which meant they were overvalued relative to gold. If the Treasury did not buy silver, this was moot; but during the period of the Sherman Silver Purchase Act, this led to a massive outflow of gold.² The reason was that silver could be sold to the US mint for legal tender US dollars, and the US dollars could be redeemed for gold worth about 37% more than the silver used initially. This led to an enormous outflow of gold that nearly caused the collapse of the Usonian financial system.

Interest Arbitrage

After 1897, exchange rates for different national currencies were very rare; in that year, Japan, Chile, and Korea went on the gold standard, and gold point spreads were in decline. This state of affairs continued until the outbreak of World War I (1914), when the international gold standard was suspended. The restoration of gold convertibility in some countries only increased volatility with respect to those that did not; and the Interwar gold standard was short-lived. During the postwar period, most countries were on a gold exchange standard with the US dollar as the anchor.

Under a gold exchange standard, gold points no longer existed because currencies were not convertible to gold, but only to the dollar; and the cost of transporting US dollars internationally was nil. Speculation in currencies occurred within a narrow band when there was evidence that a revaluation or devaluation would occur (i.e., a change in the fixed exchange rate under the Bretton Woods regime). Under the gold exchange standard, national governments were compelled to use monetary policy to ensure they had adequate reserves against foreign liabilities, rather than respond to recessions.

With exchange rates that move within wider bands, or rates that float, currency zones have their own rates of interest. A speculator from A can sell bonds domestically and use the proceeds to buy bonds in B.¹ If the interest rate on b (the currency of B) is higher than the rate on a, then the speculator makes a profit on the spread.

Supposing a rises against b. If this happens, then payments on the b loan will probably be worth less than payments on the a loan, even though the speculator knew the nominal value of both at the time of the transaction. The speculator could potentially make huge losses for a small mistake. If a declines against b, the payoffs could be large; but assuming constant relative risk aversion, the speculator is probably going to think twice about taking such a risk without some form of hedging (like buying futures contracts for a to cover the risks of a change in the exchange rate).

Covered Interest Arbitrage

This is an especially important form of arbitrage, for reasons explained below. In covered interest arbitrage, one tries to capitalize on the differences between exchange rates and between interest rates at the same time; and one uses derivatives to eliminate the risks.

For a borrower in country A, country B may well represent better venues for raising money since its local rates are lower. But add to that the risk that the exchange rate may fall, and those rates may not look so good. Our borrower could

1. sell a bond in B and buy call options for b to match repayments in the future
2. sell a bond in A and buy put options for a in units of b to match repayments in the future.

In the first case, the borrower guarantees the payments for the loan will not cost more in the future, since she can buy b for a price guaranteed not to exceed the strike price on the option. In the second case, she raises the possibility of paying off a domestic loan with a foreign currency that is rising against a. If b does not rise against a, she doesn't have to exercise the put, and can pay it off in a.

Needless to say, covering one's risk is expensive: options aren't free. And in this case, the borrower has essentially bid up the price of derivatives to eliminate the arbitrage opportunity. Economists tend to assume that markets will find such opportunities and eliminate them, although accurately establishing risk premia is still open to interpretation.

Importance

In the case of interest rate arbitrage, we used the example of a speculator selling bonds domestically in order to buy them abroad. The proceeds from the foreign bond pay for the domestic one. Assuming the speculator does nothing to cover the risks, then the great danger of huge losses would strongly discourage such arbitrage. The real profit-making opportunity could exist, but it would not be utilized, and therefore the benefits of arbitrage would be greatly reduced.³ Instead of eliminating persistent windfalls to one winner, arbitrage can only contain them.

Hedging reduces the risk but also reduces the potential profits; indeed, the exact return is known the day the investment is made. Options are more expensive, but leave a (remote) possibility of a surprising upside windfall for the arbitrageur. Also, covering risk allows the possibility for the arbitrageur to sell his positions as a new derivative.

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NOTES

1. Supposing A uses a currency called "a," and so on for B and C. For the borrower in A, a rise in the exchange rate means that b and c now cost more units of a. For an outsider, it would look as though the a is falling. If exchange rates rise, then the actual interest rate for the borrower is higher than if the borrower lived in the country from which she was borrowing money. A good introduction to the subject is "Assessing Exchange Rate Risk," BNET (date unknown).
2. For the Sherman Silver Purchase Act, see Wesley Clair Mitchell, Business cycles, Burt Franklin (1913/1970), Table 72, p.285; or see Graph 45, p.284. Notice the sharp spike in gold exports after '91,
3. The benefits of arbitrage are that economic rents are eliminated.

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SOURCES 🙵 ADDITIONAL READING

Frank K. Houston, "Commercial Paper: Its Uses Part I," The Financier Magazine, Vol.CIX (12 May 1917)

Jeff Madura, International financial management, Abridged 9th edition, Cengage Learning (2009)

R.H.I. Palgrave, Dictionary of Political Economy, Vol. 1, Macmillan (1894); "Exchange between Great Britain and British India," p.776; Vol. 3, "Silver, as Standard," p.395