Why Integrated Exchange-Rate Problems Need a Consistent Convention
An exchange-rate calculation can be numerically correct and still answer the wrong question if the quotation is reversed or the direction of appreciation is misread. Integrated problems often combine several steps: convert a quotation, measure a nominal change, adjust for inflation, calculate a trade price, and compare returns across currencies. The safest method is to define the exchange-rate convention first and carry it through every step.
Throughout this tutorial, let \(E\) denote the number of units of the home currency required to buy one unit of the foreign currency. For example, if the home currency is the U.S. dollar and \(E=1.20\) dollars per euro, one euro costs $1.20. Under this convention, a rise in \(E\) means the foreign currency has become more expensive in home currency; equivalently, the home currency has depreciated against the foreign currency.
A percentage change must be calculated relative to the initial value. If \(E_0\) changes to \(E_1\), its percentage change is \((E_1-E_0)/E_0\). The reciprocal quotation does not generally have a percentage change equal in size and opposite in sign. Real exchange rates add price levels to the comparison: they measure relative prices, not just currency prices.
Worked Example: Quotation, Inflation, and Trade Prices
Worked Example: Quotation, Inflation, and Trade Prices
A U.S. importer observes the exchange rate rise from \(E_0=\$1.20\) per euro to \(E_1=\$1.26\) per euro. Over the same period, U.S. prices rise by 4% and euro-area prices rise by 2%. A machine costs 800 euros, while a U.S.-made component has a fixed U.S. price of $500. Find the nominal exchange-rate change, the exact real exchange-rate change, and the currency-price effects on both goods.
Method. Keep the quotation in dollars per euro. Calculate the nominal change using the initial exchange rate, then adjust the exchange-rate factor for the two price-level changes. For the trade prices, convert each fixed sticker price using the relevant exchange rate.
Nominal change. The dollar-per-euro quotation changes by \[ \frac{1.26-1.20}{1.20}=\frac{0.06}{1.20}=0.05=5\%. \] The euro has appreciated by 5% in dollar terms, and the dollar has depreciated against the euro. The reciprocal quotation changes from \(1/1.20=0.833333\) euros per dollar to \(1/1.26=0.793651\) euros per dollar. Its percentage change is \[ \frac{0.793651-0.833333}{0.833333}\approx -0.047619=-4.7619\%. \] The reciprocal quotation falls, but its percentage change is not \(-5\%\).
Real change. Define the real exchange rate as \(q=EP^*/P\), where \(P^*\) is the foreign price level and \(P\) is the home price level. With this definition, a rise in \(q\) is a real depreciation of the home currency: the foreign basket becomes more expensive relative to the home basket. The exact change in \(q\) is \[ \frac{q_1}{q_0}=\frac{E_1}{E_0}\frac{P_1^*/P_0^*}{P_1/P_0} =\frac{1.05(1.02)}{1.04} =\frac{1.071}{1.04} =1.0298077. \] Thus \(q\) rises by \(1.0298077-1=0.0298077\), or 2.9808%. The dollar has depreciated in real terms under this definition.
Import price. The 800-euro machine costs \(800(1.20)=\$960\) initially and \(800(1.26)=\$1{,}008\) after the exchange-rate change. Its dollar price rises by \((1{,}008-960)/960=5\%\), assuming its euro price stays fixed.
Export price abroad. The $500 component costs \(500/1.20=416.6667\) euros initially and \(500/1.26=396.8254\) euros after the change. Its euro price falls by approximately 4.7619%. The fall differs from the 5% rise in the dollar-per-euro quotation because the converted price uses the reciprocal.
Check. A euro costs more dollars after the change, so the fixed-euro import must cost more dollars. A dollar buys fewer euros, so the fixed-dollar export costs fewer euros. Both directions agree with the quotation.
From Nominal to Real Exchange-Rate Changes
The real exchange rate \(q=EP^*/P\) compares the home-currency price of a foreign basket, \(EP^*\), with the price of a home basket, \(P\). This convention is useful for integrated calculations because every factor has a clear interpretation: \(E\) converts foreign prices into home currency, \(P^*\) is the foreign price level, and division by \(P\) expresses the result relative to home prices.
Here \(g_E\) is the percentage change in \(E\), \(\pi^*\) is foreign inflation, and \(\pi\) is home inflation. This is an exact multiplicative relation, not an approximation. For small rates, the real exchange-rate change is approximately \(g_E+\pi^*-\pi\), but the exact formula is preferable when a question supplies finite percentage changes. This measure does not, by itself, determine how much consumers or firms change their purchases; that response depends on other factors, including demand and price adjustment.
Worked Example: Real Competitiveness and an Import Invoice
Worked Example: Real Competitiveness and an Import Invoice
A U.S. firm uses a British component priced at 400 pounds. The quotation, in dollars per pound, rises from $1.25 to $1.30. Over the same period, U.S. inflation is 3% and U.K. inflation is 5%. A U.S.-made product sells for a fixed $900 price. Calculate the exact real exchange-rate change, the new dollar cost of the component, and the product's price in pounds before and after the exchange-rate change.
Nominal change. Here \(E_1/E_0=1.30/1.25=1.04\), so the dollar-per-pound quotation rises by 4%. The dollar depreciates nominally against the pound.
Real change. With home prices in the United States and foreign prices in the United Kingdom, \[ \frac{q_1}{q_0} =\frac{1.04(1.05)}{1.03} =\frac{1.092}{1.03} =1.0601942. \] The real exchange rate rises by 6.0194%, indicating a real depreciation of the dollar under the stated definition.
Import invoice. The component costs \(400(1.25)=\$500\) initially. At the new exchange rate it costs \(400(1.30)=\$520\). The cost rises by \((520-500)/500=4\%\), consistent with the change in dollars per pound.
Export price in pounds. Initially, the $900 product costs \(900/1.25=720\) pounds. At the new rate it costs \(900/1.30=692.3077\) pounds. The pound price changes by \[ \frac{692.3077-720}{720}\approx -0.038462=-3.8462\%. \] The pound price falls because each pound now exchanges for more dollars.
Check. The exchange-rate conversion uses pounds times dollars per pound for the import, and dollars divided by dollars per pound for the export. The units cancel to dollars and pounds, respectively.
Covered Interest Parity: Derivation and Application
Covered interest parity (CIP) compares a home-currency investment with a foreign-currency investment whose future conversion back to home currency is fixed by a forward contract. Let \(i_h\) and \(i_f\) be effective interest rates over the same investment horizon in the home and foreign countries. Let \(F\) be the forward exchange rate in home currency per unit of foreign currency, quoted for that horizon. Assume investors can borrow and lend at the stated rates, exchange and forward transactions are available at the stated rates, and there are no transaction costs, taxes, capital controls, or default risk.
Why the formula holds. Start with one unit of home currency. Investing it at home produces \(1+i_h\) home currency at the end of the horizon. Alternatively, converting it at the spot rate \(E\) produces \(1/E\) units of foreign currency. Investing abroad produces \((1+i_f)/E\) foreign currency. A forward contract converts that amount back at \(F\) home currency per foreign unit, giving \(F(1+i_f)/E\) home currency. With covered returns and no risk, no-arbitrage requires these final amounts to be equal. Rearranging gives the displayed parity formula. If a market forward rate differs from the parity rate, a covered arbitrage may be possible, subject to the assumptions and trading costs.
Worked Example: Testing a Six-Month Forward Quote
The spot rate is \(E=\$1.10\) per euro. Over the next six months, the U.S. investment return is 1.8% and the euro investment return is 0.9%. A dealer quotes a six-month forward rate of $1.12 per euro. With no transaction costs or other frictions, determine the parity forward rate and the covered return advantage on an initial $1,000,000 if the investor uses the quoted forward rate instead.
Method. Use both six-month returns, rather than annual rates, in the parity equation. Then compare the two six-month final dollar amounts using the same initial investment.
Parity rate. \[ F_{\mathrm{CIP}}=1.10\frac{1.018}{1.009} =1.10\left(\frac{1.018}{1.009}\right) \approx \$1.1098117\text{ per euro}. \] The quoted forward rate of $1.12 is above the parity rate.
Home investment. Investing $1,000,000 in the United States produces \[ \$1{,}000{,}000(1.018)=\$1{,}018{,}000. \]
Covered foreign investment. Convert the initial dollars at spot: \[ \$1{,}000{,}000/\$1.10\text{ per euro} =909{,}090.9091\text{ euros}. \] After the euro investment return, the amount is \[ 909{,}090.9091(1.009)=917{,}272.7273\text{ euros}. \] Sell those euros forward at $1.12 per euro: \[ 917{,}272.7273(1.12)=\$1{,}027{,}345.45. \] The covered foreign route therefore yields \(\$1{,}027{,}345.45-\$1{,}018{,}000=\$9{,}345.45\) more than the home investment over six months.
Check. The euro amount is converted into dollars by multiplying by the forward quote, since the quote is dollars per euro. The result is a guaranteed comparison only under the stated assumptions and if the quoted transactions are actually available.
Uncovered Interest Parity: Expected, Not Guaranteed, Exchange Rates
Uncovered interest parity (UIP) uses the expected future spot rate \(E^e\), rather than a forward rate that fixes the conversion. Under the simplified assumptions of risk-neutral investors, comparable assets, and no risk premium or other wedge, it equates expected returns:
The foreign investment's home-currency return depends on the future spot rate, which is not known when the investment is made. Consequently, UIP is a relationship involving an expected return, not a guaranteed arbitrage condition. Risk premia, transaction costs, and other market conditions can cause observed outcomes or expectations to differ from the simple relation. Rates and exchange-rate expectations must refer to the same horizon.
Worked Example: Interpreting a UIP-Implied Exchange Rate
The spot rate is \(E=\$1.20\) per euro. Over a one-year horizon, the U.S. interest rate is 5% and the euro interest rate is 2%. Under the simplified UIP assumptions, calculate the expected exchange rate in dollars per euro and interpret the implied movement.
Method. Use the one-year rates in the UIP relation and compare the implied expected rate with the current rate.
Expected rate. \[ E^e=1.20\frac{1.05}{1.02} =1.20(1.0294118) \approx \$1.2352941\text{ per euro}. \]
Implied percentage change. \[ \frac{1.2352941-1.20}{1.20} \approx 0.0294118=2.9412\%. \] The expected dollar-per-euro rate rises by approximately 2.9412%. Thus, in this convention, UIP implies an expected euro appreciation and expected dollar depreciation.
Check. The home interest rate is higher than the foreign rate. In the simplified UIP calculation, that difference is offset by an expected rise in the dollar cost of a euro. The future exchange rate is expected, not locked in, so this does not establish a risk-free profit.
A Reliable Sequence and Common Exam Traps
A linked problem is easier to manage when each calculation has a defined role. The following sequence keeps units, signs, and assumptions visible:
Record whether \(E\) is home currency per foreign unit or its reciprocal. State what a rise in \(E\) means.
Use the initial quotation as the denominator. If the question asks for the reciprocal quote, invert both endpoint rates before calculating its percentage change.
Apply the exact real-exchange-rate relation using the home and foreign price changes in the same period.
Multiply foreign-currency prices by \(E\) to obtain home currency; divide home-currency prices by \(E\) to obtain foreign currency.
Use a forward rate for covered parity and an expected future spot rate for uncovered parity. Match interest-rate and exchange-rate horizons.
Common mistake: using the quote in the wrong direction. If \(E=\$1.10\) per euro, multiplying $500 by 1.10 to find its euro price is incorrect: the units become dollars squared per euro. The correction is \(500/1.10\approx454.5455\) euros. Conversely, a 300-euro invoice costs \(300(1.10)=\$330\).
Common mistake: treating a reciprocal percentage change as the negative of the original one. If \(E\) rises from 1.20 to 1.26, the change is \(+5\%\). The reciprocal falls from 0.833333 to 0.793651, a change of \(-4.7619\%\), not \(-5\%\). Invert the rates first; do not merely change the sign.
Common mistake: using the nominal rate change as the real change. With 4% nominal depreciation, 2% foreign inflation, and 4% home inflation, the exact real change is \(1.04(1.02)/1.04-1=2\%\), not 4%. Inflation adjustment matters because a currency movement alone does not measure the relative change in basket prices.
Common mistake: treating UIP as arbitrage. A future spot rate is uncertain. A parity calculation based on an expected rate is not a guaranteed final payoff. Covered parity, by contrast, uses a forward contract to lock in the conversion under the stated assumptions.
Check Your Understanding
For each problem, keep the quotation convention and time horizon explicit. Answers are rounded only where needed.
- A quotation of $0.92 per Swiss franc rises to $0.966 per franc. What is the percentage change in this quotation, and what is the percentage change in its reciprocal, Swiss francs per dollar? (Answer: The dollar-per-franc quote rises 5.000%; the reciprocal falls approximately 4.7619%.)
- With the home country as the United States, \(E\) in Australian dollars is not used: instead, use \(E=\$0.68\) per Australian dollar, which rises by 5%. U.S. inflation is 2% and Australian inflation is 4%. Under \(q=EP^*/P\), what is the exact percentage change in the real exchange rate? (Answer: Approximately a 7.0588% increase, a real depreciation of the U.S. dollar under this definition.)
- A U.S. business buys a product priced at 750 pounds. The exchange rate is $1.24 per pound and then rises by 3%. What is the initial dollar cost and the new dollar cost, assuming the pound price is unchanged? (Answer: Initial cost $930.00; new cost $957.90.)
- The spot rate is $1.25 per unit of foreign currency. Over the same six-month horizon, the home return is 2% and the foreign return is 1%. Under covered interest parity, what is the six-month forward rate? (Answer: Approximately $1.262376 per foreign-currency unit.)
- The spot rate is $0.80 per Canadian dollar. The one-year home interest rate is 4% and the Canadian interest rate is 6%. Under simplified UIP, what expected future spot rate is implied, and what percentage change does it represent? (Answer: Approximately $0.784906 per Canadian dollar, a 1.8868% decrease in the quotation.)