- The nominal exchange rate e is the price in foreign currency of one unit of a domestic currency.
- The real exchange rate (RER) is defined as , where Pf is the foreign price level and P the domestic price level. P and Pf must have the same arbitrary value in some chosen base year. Hence in the base year, RER = e.
The RER is only a theoretical ideal. In practice, there are many foreign currencies and price level values to take into consideration. Correspondingly, the model calculations become increasingly more complex. Furthermore, the model is based on purchasing power parity (PPP), which implies a constant RER. The empirical determination of a constant RER value could never be realised, due to limitations on data collection. PPP would imply that the RER is the rate at which an organization can trade goods and services of one economy (e.g. country) for those of another. For example, if the price of a good increases 10% in the UK, and the Japanese currency simultaneously appreciates 10% against the UK currency, then the price of the good remains constant for someone in Japan. The people in the UK, however, would still have to deal with the 10% increase in domestic prices. It is also worth mentioning that government-enacted tariffs can affect the actual rate of exchange, helping to reduce price pressures. PPP appears to hold only in the long term (3–5 years) when prices eventually correct towards parity.
More recent approaches in modelling the RER employ a set of macroeconomic variables, such as relative productivity and the real interest rate differential.
Bilateral vs. effective exchange rate
Bilateral exchange rate involves a currency pair, while effective exchange rate is weighted average of a basket of foreign currencies, and it can be viewed as an overall measure of the country's external competitiveness. A nominal effective exchange rate (NEER) is weighted with the inverse of the asymptotic trade weights. A real effective exchange rate (REER) adjust NEER by appropriate foreign price level and deflates by the home country price level. Compared to NEER, a GDP weighted effective exchange rate might be more appropriate considering the global investment phenomenon.
Uncovered interest rate parity
Uncovered interest rate parity (UIRP) states that an appreciation or depreciation of one currency against another currency might be neutralized by a change in the interest rate differential. If US interest rates increase while Japanese interest rates remain unchanged then the US dollar should depreciate against the Japanese yen by an amount that prevents arbitrage. The future exchange rate is reflected into the forward exchange rate stated today. In our example, the forward exchange rate of the dollar is said to be at a discount because it buys fewer Japanese yen in the forward rate than it does in the spot rate. The yen is said to be at a premium.
UIRP showed no proof of working after 1990s. Contrary to the theory, currencies with high interest rates characteristically appreciated rather than depreciated on the reward of the containment of inflation and a higher-yielding currency.
The uncovered interest rate parity postulates that
The equality assumes that the risk premium is zero, which is the case if investors are risk-neutral. If investors are not risk-neutral then the forward rate (F + 1) can differ from the expected future spot rate (E[S + 1]), and covered and uncovered interest rate parities cannot both hold.
The uncovered parity is not directly testable in the absence of market expectations of future exchange rates. Moreover, the above rather simple demonstration assumes no transaction cost, equal default risk over foreign and domestic currency denominated assets, perfect capital flow and no simultaneity induced by monetary authorities. Note also that it is possible to construct the UIP condition in real terms, which is more plausible.
Uncovered interest parity example
An example for the uncovered interest parity condition: Consider an initial situation, where interest rates in the home country (e.g. U.S.) and a foreign country (e.g. Japan) are equal. Except for exchange rate risk, investing in the US or Japan would yield the same return. If the dollar depreciates against the yen, an investment in Japan would become more profitable than a US-investment - in other words, for the same amount of yen, more dollars can be purchased. By investing in Japan and converting back to the dollar at the favorable exchange rate, the return from the investment in Japan, in the dollar terms, is higher than the return from the direct investment in the US. In order to persuade an investor to invest in the US nonetheless, the dollar interest rate would have to be higher than the yen interest rate by an amount equal to the devaluation (a 20% depreciation of the dollar implies a 20% rise in the dollar interest rate).
Technically however, a 20% depreciation in the dollar only results in an approximate rise of 20% in U.S. interest rates. The exact form is as follows: Change in spot rate (Yen/Dollar) equals the dollar interest rate minus the yen interest rate, with this expression being divided by one plus the yen interest rate.
Uncovered vs. covered interest parity example
Let's assume you wanted to pay for something in Yen in a month's time. There are several ways to do this.
- (a) Buy Yen forward 30 days to lock in the exchange rate. Then you may invest in dollars for 30 days until you must convert dollars to Yen in a month. This is called covering because you now have covered yourself and have no exchange rate risk.
- (b) Convert spot to Yen today. Invest in a Japanese bond (in Yen) for 30 days (or otherwise loan out Yen for 30 days) then pay your Yen obligation. Under this model, you are sure of the interest you will earn, so you may convert fewer dollars to Yen today, since the Yen will grow via interest. Notice how you have still covered your exchange risk, because you have simply converted to Yen immediately.
- (c) You could also invest the money in dollars and change it for Yen in a month.
According to the interest rate parity, you should get the same number of Yen in all methods. Methods (a) and (b) are covered while (c) is uncovered.
- In method (a) the higher (lower) interest rate in the US is offset by the forward discount (premium).
- In method (b) The higher (lower) interest rate in Japan is offset by the loss (gain) from converting spot instead of using a forward.
- Method (c) is uncovered, however, according to interest rate parity, the spot exchange rate in 30 days should become the same as the 30 day forward rate. Obviously there is exchange risk because you must see if this actually happens.
General Rules: If the forward rate is lower than what the interest rate parity indicates, the appropriate strategy would be: borrow Yen, convert to dollars at the spot rate, and lend dollars.
If the forward rate is higher than what interest rate parity indicates, the appropriate strategy would be: borrow dollars, convert to Yen at the spot rate, and lend the Yen.Covered interest rate parity
Covered interest parity
(also called interest parity condition) means that the following equation holds:
where:
- is the domestic interest rate implied by debt of a given maturity;
- ic is the interest rate in the foreign country for debt of the same maturity;
- S is the spot exchange rate, expressed as the price in domestic currency ($) of one unit of the foreign currency c, i.e. $/c;
- F is the forward exchange rate implied by a forward contract maturing at the same time as the domestic and foreign debt underlying and ic. F is expressed in the same units as S, namely $/c.
Taking natural logs of both sides of the interest parity condition yields:
where all interest rates are now the continuously compounded equivalents. ln(F/S) is the forward premium, the percentage difference between the forward rate and the spot rate. Covered interest parity states that the difference between domestic and foreign interest rates equals the forward premium. When , the forward price of the foreign currency will be below the spot price. Conversely, if , the forward price of the foreign currency will exceed the spot price.
Covered interest parity assumes that debt instruments denominated in domestic and foreign currency are freely traded internationally (absence of capital controls), and have similar risk. If the parity condition does not hold, there exists an arbitrage opportunity. (see covered interest arbitrage and an example below).
The interest parity condition may also be expressed as:
The following common approximation is valid when S is not too volatile:
In short, assume that
- This would imply that one dollar invested in the US <>
The following rudimentary example demonstrates covered interest rate arbitrage (CIA). Consider the interest rate parity (IRP) equation,
Assume:
- the 12-month interest rate in US is 5%, per annum
- the 12-month interest rate in UK is 8%, per annum
- the current spot exchange rate is 1.5 $/£
- the forward exchange rate implied by a forward contract maturing 12 months in the future is 1.5 $/£.
Clearly, the UK has a higher interest rate than the US. Thus the basic idea of covered interest arbitrage is to borrow in the country with lower interest rate and invest in the country with higher interest rate. All else being equal this would help you make money riskless. Thus,
-
- Per the LHS of the interest rate parity equation above, a dollar invested in the US at the end of the 12-month period will be,
- $1 · (1 + 5%) = $1.05
- Per the RHS of the interest rate parity equation above, a dollar invested in the UK (after conversion into £ and back into $ at the end of 12-months) at the end of the 12-month period will be,
- $1 · (1.5/1.5)(1 + 8%) = $1.08
Thus one could carry out a covered interest rate (CIA) arbitrage as follows,
- Borrow $1 from the US bank at 5% interest rate.
- Convert $ into £ at current spot rate of 1.5$/£ giving 0.67£
- Invest the 0.67£ in the UK for the 12 month period
- Purchase a forward contract on the 1.5$/£ (i.e. cover your position against exchange rate fluctuations)
At the end of 12-months
- 0.67£ becomes 0.67£(1 + 8%) = 0.72£
- Convert the 0.72£ back to $ at 1.5$/£, giving $1.08
- Pay off the initially borrowed amount of $1 to the US bank with 5% interest, i.e $1.05
The resulting arbitrage profit is $1.08 − $1.05 = $0.03 or 3 cents per dollar.
Obviously, arbitrage opportunities of this magnitude would vanish very quickly.
In the above example, some combination of the following would occur to reestablish Covered Interest Parity and extinguish the arbitrage opportunity:
- US interest rates will go up
- Forward exchange rates will go down
- Spot exchange rates will go up
- UK interest rates will go down
