Last updated Wed Jan 31, 1996
Special thanks to W Antweiler and J Ries from whom these notes borrow heavily.
An exchange rate is the rate of exchange between two currencies. It is quoted in either of the following forms:
| Term | Description | (E) | (e) |
|---|---|---|---|
| Floating Exchange Rates | |||
| Appreciation | strengthening of a currency in the market |  |  |
| Depreciation | weakening of a currency in the market | | |
| Fixed Exchange rates | |||
| Revaluation | Intentional strengthening of a currency | | |
| Devaluation | intentional weakening of a currency | | |
Foreign currencies are traded on the exchange rate market. There are two important markets:
The spot rate and forward rate are usually not the same. Lets use E notation and denote the spot rate at time [t] as s[t], and the forward rate at time [t] for a contract of length [l] as f[t,l]. Indeed, when
In other words, a positive difference between the forward rate and the spot rate is called a forward premium, and a negative difference is called a forward discount.
We distinguish different two exchange rate regimes:
The real world consists of limited-flexible exchange rates, with colourful names like the "dirty float" and the "crawling peg" :
People who play both sides: Arbitrageurs
Central banks use their foreign exchange reserves to smooth temporary excess demand conditions.
Currency inconvertibility: One method for coping with excess demand for foreign currencies is to restrict purchases of foreign currencies. Usual practice is to allocate licences for purchasing foreign exchange.
The formation of a black market is almost inevitable.
Arbitrage:
Exploiting short-term differences in the exchange rates in two different places. This ensures, as we have seen, market consistency. The profits made from arbitrage can be rationalized as the cost of organizing markets and ensuring the flow of information.
If you spot a difference between two places where a currency is traded, you can buy it in the place where it is cheap and sell it where it is expensive. Along the way you make a profit.
In reality, however, transaction costs matter, and there may be indeed two C$/US$ exchange rates (in, say, Tokyo and Singapore) that are slightly different. Also note the effect of different time zones. When you go across the border from Vancouver to Seattle and back to Vancouver, and change C$ into US$ when you head south, and then change the same amount of cash back from US$ into C$ when you head north again, you'll find that the bank or currency dealer where you have made the change has made a handsome profit.
Some banks and currency dealers engage in such transactions, taking advantage of minute differences in exchange rates. Necessarily, such transactions must be large in order to minimize the relative importance of transaction costs.
What's wrong with this picture?
100 ¥ per US$, 1.25 C$ per US$, 75 ¥ per C$
Triangular Arbitrage===> 80 ¥ perC$.
Suppose you start with 100 C$. Then use them to buy 80 US$. Now buy yen. Finally, take the 8000 yen and buy C$. You end up with 106.67 C$. Then do it again!
Earlier we had seen the importance of arbitrage in exchange rate markets. Not only can arbitrage used in spot transactions, it can also be used in transactions that involve current and future exchange rates.
Consider the example shown in the graph below. Take $1m and invest it in Canada at the going interest rate of 3%, and you obtain $1.03m after a year. However, in Japan the going interest rate is 5%. Would it be more profitable to invest in Japan or in Canada? If the current spot rate is 120 Yen per Dollar, then the initial ¥120m will become ¥126m after one year. How much are ¥126m worth in Canadian dollars in one year? Obviously, that depends on the prevailing exchange rate in one year. If we expect that the exchange rate exceeds 122.33¥/$ (i.e., when the Candian dollar is expected to appreciate strongly vis-à-vis the Yen), we will invest in Canada; if we expect an exchange rate below 122.33¥/$ (i.e., when the Canadian dollar is expected to appreciate mildly or even depreciate vis-à-vis the Yen), we will invest in Japan instead.
E[t] (1+i*) / E[t+1] = (1+i)
That can be rearranged a bit:
E[t+1]/E[t] = (1+i*)/(1+i)
Now subtract "1" on each side:
(E[t+1]-E[t])/E[t] = (1+i*-1-i)/(1+i) = (i*-i)/(1+i)
Let's call the expression on the left side the expected appreciation of our domestic currency, and name it (E^). If (1+i) is close to one, we can simplify the expression above and write:
E^ = i* - i
In other words, if the foreign interest rate exceeds the domestic interest rate, the domestic currency is expected to appreciate. The equation above describes the case of uncovered interest rate parity. When we substitute the spot rate s[t] for E and the forward rate f[t,1] for E[t+1], the equation describes covered interest rate parity. For covered interest rate parity, if the foreign interest rate exceeds the domestic interest rate, there will be a forward premium.
It would seem to be an easy task since all we have to do is open the business pages and look up the forward rate. Three problems. Forward markets only exist for a small number of countries. They are often short to medium run (360 days is usually about the longest) so not of much use for long-run forecasting. And they are lousy predictors.
If exchange rate markets make best use of the available information, the forward rate should predict future spot rates. However, in reality the forward rate is a poor predictor of the spot rate. To demonstrate this, we look at monthly exchange rates between the US dollar and the Canadian dollar between 1990 and 1995. The horizontal axis shows change predicted by the 90-day forward rate. Negative values indicate an expected depreciation of the Canadian dollar, positive values indicate an expected appreciation of the Canadian dollar.The vertical axis shows the actual change in the Canadian dollar in this 90-day period. Again, positive values indicate a realized appreciation, and negative values indicate a realized depreciation of the Canadian dollar vis-à-vis the US dollar.The thick diagonal line indicates a perfect forecast. If the forward rate was a good predictor, the dots should be scattered around that line. Obviosuly they are not.
We can make two observations:
How can we predict exchange rates? The oldest predictor of exchange rates is based on the observation that one can carry out "physical" arbitrage. This is known as the law of one price that applies when goods (or services)
For example, consider buying a PC either in Vancouver or Seattle, and assume that visiting Seattle does not take any extra time or extra cost. The PC costs C$1,620 in Canada and US$1,200 in Seattle. Where will you buy the PC? When the exchange rate exceeds 0.74US$/C$ (i.e, the Canadian dollar is strong) you will buy in Seattle, but when the exchange rate is below 0.74US$/C$ (i.e., the Canadian dollar is weak), you will buy in Vancouver. What happens when there is such a discrepancy in prices? If buying in Seattle was cheaper, everyone would want to buy in Seattle instead of Vancouver. This would increase the demand in Seattle and raise the price there. (At the same time, Vancouverites changing their loonies for greenbacks would weaken the Canadian dollar.) Falling demand in Vancouver would lower prices there. In the end, prices will settle on a level that is compatible with the prevailing exchange rate.
We can turn the law of one price into a theory of real exchange rates. Let us use the symbol R for the real exchange rate, and E for the nominal exchange rate. Furthermore, P and P* denote the domestic and foreign price level, respectively. Then we can write
R=E(P/P*)
Note that the real exchange rate X is an index number, i.e., it does not have a physical dimension. Since E is measured in, eg., ¥/$, and the price level are measure in $ and ¥, the dimensions will cancel out.
Consider the following "types" of purchasing power parity (PPP):
Take the expression X=EP/P* and take logarithms. This operation yields:
log(X)=log(E)+log(P)-log(P*)
Now differentiate with respect to time [note that dlog(x)/dt=(dx/dt)/x=x^] and put a (^) behind a variable to denote that this is a rate of change. Then:
X^ = E^ + P^ - (P*)^
PPP in rates of change says that X^=0. Hence:
E^ = (P*)^ - P^
The rate of change of the price level is the inflation rate. Therefore, P^ and (P*)^ are the inflation rates at home and abroad. Now we can conclude that if the inflation in the foreign country exceeds the inflation at home, we expect our own currency to appreciate. Consider Canada and Italy. Inflation in Canada is 2%, and inflation in Italy runs at 9%. This implies that the Canadian dollar can be expected to appreciate vis-à-vis the Italian lira by 9%-2%=7% over the next year (e.g., increase from 1200Lit/$ to 1284Lit/$).