Purchasing Power Parity
and International Commodity Arbitrage
Foreign exchange refers to two different things. The first is currency claims expressed in the equivalent value in foreign money. The second is actual transactions involving the conversion of money of one country into that of another.
Foreign exchange is necessary because different countries have different monetary units. One country’s currency typically cannot be used in another country. The determination of the price at which the currency of one country will be or should be exchanged for that of another country is the basis of this and many other essays and studies.
Foreign exchange is a commodity, and its price fluctuates based on supply and demand, like any commodity. This is not the place for a complete discussion of supply and demand as relates to foreign exchange, but for our purposes, we will assume that supply of and demand for a country’s currency moves along with the supply of or demand for that country’s products or the products of its trading partners. For example, if one country buys many more goods from its neighbor than its neighbor buys from it, the balance of payments at the end of the year will cause its neighbor’s currency to be in great demand, thereby driving its price up.
What in fact sets the exchange ratio between two currencies? Obviously supply and demand, but what causes supply and demand to set exchange rates at appropriate levels? With this question we begin the next section.
What is Purchasing Power Parity?
Perhaps the single most well known concept in foreign exchange theory is that of Purchasing Power Parity (PPP). The basic idea of PPP is that currencies represent purchasing power over goods and services. Either the exchange rate or price levels adjust to keep purchasing power constant. For example, say a particular basket of goods sells for $2000 in America and 1000 GBP in Great Britain. According to PPP, the exchange rate of dollars to pounds should be 2:1. If it were not 2:1; if, for example only 1.5 dollars was needed to purchase 1 pound, an arbitrageur would buy the basket of goods for 1000 pounds ($1500) and resell them in America for $2000. He would continue to do this until currency traders realized that they were being underpaid for their pounds and started to charge two dollars each for them, or Americans realized that they were paying too much for these goods and became willing to pay only $1500 for them, or some combination of the two. Supply and demand forces acting both on Great British pounds and this basket of goods would set the price based on the fact that the same amount of money should buy the same amount of goods anywhere in the world (PPP). The increased demand
for pounds (in order to purchase these goods) will push its price up relative to the dollar, and the increased supply of this basket of goods will push its price down, to the point where PPP is achieved.
Expressed mathematically, that point is:
P(i,t) = S(t) x P*(i,t), where
S(t) = the current exchange rate (the domestic price of foreign
P(i,t) = the current domestic currency price of commodity i
P*(i,t) = the current foreign currency price of commodity i
At its simplest formulation, PPP is also called the law of one price (LOP). This formulation contains several caveats. First, LOP assumes that there are no transaction costs involved in buying a commodity in one market and selling it in another. Obviously, in a situation involving substantial transaction costs relative to the cost of the commodity in question, LOP will be empirically meaningless. Second, for this formulation to hold true, there must no barriers to trade. This would include prohibitions, tariffs, taxes, and quotas. Lastly, and perhaps most obviously, is that we must be comparing homogeneous goods.
The absolute version of PPP hypothesizes a similar relationship, with the exception that it uses price levels instead of specific commodity prices. Mathematically, the absolute version of PPP can be expressed thus:
P(t) = S(t) x P*(t),where
P(t) = the domestic price level in domestic currency
P*(t) = the foreign price level in foreign currency
In reality though, price levels are certainly not calculated with any regularity. The cost and time to compute it would be prohibitive. Instead, countries typically calculate a variety of price indexes. Therefore, people do not make absolute PPP calculations, but rather relative ones, involving price index ratios instead. Relative PPP calculations also compensate for the fact that two economies may not have the same composition of goods.
Mathematically, the absolute version of PPP can be expressed thus:
P(t+T) = S(t+T) x P*(t+T),where
P(t) = an index of a subset of goods and services in domestic
P*(t) = an index of a subset of goods and services in foreign currency
t = some date
t + T = some later date
Relative PPP can also be modified to include the effects of disparate rates of inflation on the exchange rate. That is, since P(t+T) = 1+  , where  is the domestic rate of inflation and P*(t+T) = 1+ * where * is the foreign exchange rate, we can algebraically alter our previous equation to:
S(t+T) = 1+
This latest version of our equation says that the proportional appreciation or depreciation of the foreign currency depends on whether inflation is higher in, respectively, the domestic or the foreign country.
The Real Exchange Rate
Often however, exchange rates do not move within the framework of PPP. The ‘real exchange rate’ is the exchange rate when PPP does not hold.
PPP calculations are used extensively when developing international trade and monetary policy. Central banks use them to establish par values for their currencies, and arbitrageurs use them to help determine when market exchange rates are too high or low (i.e. when currencies are overvalued or undervalued). Most users of PPP calculations assume that the real exchange rate should return to levels dictated by PPP.
In reality though, exchange levels clearly do not stabilize at PPP dictated levels. Foreign exchange does exist. While some of that is clearly due to products
that lay outside our discussion of PPP, such as goods that are not available in some area, obviously not all of it is. For foreign trade to occur, there has to be a price differential high enough at the destination sufficient to cover shipping, tariffs, financing, insurance, and any other costs involved in transferring goods from one location to another. Not only must price differentials exist, but they also must remain for trade to persist.
We will define the real exchange rate, in terms of price levels as:
R(t) = P(t)/(S(t)P*(t))
And in terms of price indexes as follows:
R(t+T) = P(t+T)/P(t)
Obviously, when absolute and relative PPP holds true for whichever scenario is under discussion, the real exchange (R(t)) will be equal to one.
Despite our previous discussions, however, these exchange differentials are consistent with both purchasing power parity and with the absence of commodity arbitrage opportunities. These price differentials cover the additional costs mentioned above and allow the importer/exporter to reap a nominal profit.
For this reason, empirical studies have established that there exist large and continuing deviations of the real exchange rate from LOP. It seems that these deviations embody the cost of trade, and are not arbitrage opportunities. Empirical
tests have generally supported this. One positive to this, in terms of the usefulness of LOP is that current deviations from the LOP can be used to forecast future deviations, since these would not be expected to substantially change.
In conclusion, though the simple law of one price formulation of purchasing power parity we started with obviously does not hold true when the requirements for that theory are stripped away, as in the real world, the concept that the same amount of money should purchase the same amount everywhere does have validity. Any of us who have traveled between countries knows that some goods are more expensive, and some are less expensive in other countries. When we factor in all the variables discussed above, this begins to make sense. It is not logical to pay $1 to ship something from Haiti to America if the price differential is only $0.50. This also obviously does not constitute an arbitrage opportunity.