Calculating and Interpreting Real Exchange Rates

Why Use a Real Exchange Rate?

A nominal exchange rate tells us how many units of one currency exchange for a unit of another. It does not, by itself, compare the prices of goods in the two countries. To make that comparison, convert the foreign price level into domestic currency and compare it with the domestic price level. The resulting measure is the real exchange rate: a relative price that combines the nominal exchange rate with domestic and foreign prices.

This tutorial uses the convention that $E$ is the nominal exchange rate measured in units of domestic currency per unit of foreign currency. For example, $E=1.20$ means that one unit of foreign currency costs 1.20 units of domestic currency. The price levels $P$ and $P^*$ must be expressed in their respective currencies: $P$ is the domestic price of the domestic goods basket, while $P^*$ is the foreign-currency price of the foreign goods basket.

Definition: With $E$ measured in units of domestic currency per unit of foreign currency, the real exchange rate $q$ is $$ q=\frac{E P^*}{P}. $$ The numerator $E P^*$ is the foreign goods-basket price converted into domestic currency. Thus, $q$ compares the domestic-currency price of the foreign basket with the domestic-currency price of the domestic basket.

If $P$ and $P^*$ are prices of comparable baskets, $q=1$ means that the two baskets have the same price after conversion into a common currency. If $q>1$, the foreign basket costs more than the domestic basket; if $q<1$, it costs less. These comparisons require consistent units and genuinely comparable baskets. When price levels are indexes rather than currency prices, the numerical level of $q$ depends on how those indexes are normalized. In that case, changes in $q$ are often more informative than its level.

Calculating the Real Exchange Rate

The calculation has two parts: first convert the foreign price into domestic currency, then divide by the domestic price. Keeping these steps visible makes the units a useful check. In $E P^*$, foreign-currency units cancel, leaving domestic currency; dividing by $P$, also in domestic currency, leaves a relative price.

Worked Example: Converting a Foreign Basket Price

Suppose a hypothetical domestic basket costs \$120, a comparable foreign basket costs 80 units of foreign currency, and the nominal exchange rate is $E=\$1.20$ per unit of foreign currency. Calculate $q$ and interpret it. State the method and identify the quotation error to avoid.

Method. Since $E$ is domestic currency per unit of foreign currency, multiply the foreign price by $E$, then divide by the domestic price:

E P^*=(\$1.20/\text{foreign unit})(80\text{ foreign units})=\$96.

Now compare this converted price with the domestic basket price:

q=\frac{\$96}{\$120}=0.80.

The foreign basket costs 0.80 times the domestic basket price, or 80% of that price. Equivalently, it is 20% less expensive than the domestic basket on this comparison. The method avoids treating $E$ as foreign currency per domestic currency; the stated units show that multiplication is the correct conversion.

Check. One foreign unit costs \$1.20, so 80 foreign units cost $80\times\$1.20=\$96$. Dividing \$96 by \$120 gives 0.80, as calculated.

When the Quotation Runs the Other Way

Exchange rates are also sometimes quoted as units of foreign currency per unit of domestic currency. Call that quote $R$. It is the reciprocal of $E$, so $E=1/R$. With this opposite quotation, the formula $q=E P^*/P$ still applies, but the quote must first be converted to the convention used in the formula. Alternatively, convert the foreign price directly by dividing it by $R$.

Worked Example: Correcting an Inverted Quotation

A hypothetical domestic basket costs 200 domestic-currency units. A foreign basket costs 150 foreign-currency units. The available exchange-rate quote is $R=0.75$ foreign-currency units per domestic-currency unit. Calculate the real exchange rate. Name the method and the mistake it prevents.

Method. Since one domestic unit buys 0.75 foreign units, the domestic-currency cost of one foreign unit is the reciprocal:

E=\frac{1}{0.75}=\frac{4}{3} \quad\text{domestic units per foreign unit}.

Use the unrounded reciprocal to convert the foreign basket:

E P^*=\frac{4}{3}\times150=200 \quad\text{domestic-currency units}.

Then divide by the domestic basket price:

q=\frac{200}{200}=1.

The two baskets have equal prices after conversion. The method prevents applying a quote in foreign units per domestic unit as though it were in domestic units per foreign unit. It also avoids false precision: if the reciprocal is rounded to $1.3333$, the displayed multiplication gives $1.3333\times150=199.995$, not exactly 200. Using $4/3$ gives exactly 200.

Direct check. At 0.75 foreign units per domestic unit, obtaining 150 foreign units requires $150/0.75=200$ domestic units. Dividing by the domestic price of 200 again gives $q=1$.

Key convention: The formula $q=E P^*/P$ assumes $E$ is quoted as domestic currency per unit of foreign currency. If the quote is foreign currency per domestic currency, invert it first. Do not change the formula without also changing the definition of the exchange-rate quote.

Why the Formula Measures a Relative Price

The formula follows directly from comparing the two basket prices in a common currency. The foreign basket costs $P^*$ units of foreign currency. Each foreign-currency unit costs $E$ domestic-currency units, so the converted foreign price is $E P^*$. The domestic basket already costs $P$ domestic-currency units. The ratio of the first cost to the second is therefore the relative price:

\frac{\text{foreign basket price in domestic currency}} {\text{domestic basket price}} = \frac{E P^*}{P} =q.

This also shows why a change in $q$ cannot be attributed to the nominal exchange rate alone. It can change because $E$ changes, because either price level changes, or because several of these change together. For two dates, labelled 0 and 1, divide the real exchange rate at date 1 by its value at date 0:

\frac{q_1}{q_0} = \frac{E_1}{E_0} \frac{P_1^*}{P_0^*} \frac{P_0}{P_1}.

This exact relationship is useful for interpreting movements. A higher $E$, with prices fixed, raises $q$: foreign currency costs more in domestic currency, so the converted foreign basket becomes more expensive relative to the domestic basket. A higher foreign price level also raises $q$, all else equal. A higher domestic price level lowers $q$, all else equal. For finite percentage changes, the factors multiply; the percentage changes should not simply be added unless an approximation is explicitly intended.

Worked Example: Separating Exchange-Rate and Price-Level Changes

At date 0, suppose $E_0=1.10$ domestic units per foreign unit, $P_0^*=100$ foreign-currency units, and $P_0=100$ domestic-currency units. At date 1, $E_1=1.155$, $P_1^*=104$, and $P_1=105$. Calculate both real exchange rates and the percentage change in $q$. Use the exact ratio method, and avoid adding the component percentage changes as if that were exact.

Method. Calculate the level at date 0:

q_0=\frac{1.10\times100}{100}=1.10.

At date 1, the converted foreign price is $1.155\times104=120.12$ domestic-currency units. Therefore:

q_1=\frac{1.155\times104}{105} =\frac{120.12}{105} =1.144.

The percentage change from date 0 to date 1 is based on the ratio of the two levels:

\frac{q_1-q_0}{q_0}\times100\% = \frac{1.144-1.10}{1.10}\times100\% = \frac{0.044}{1.10}\times100\% =4\%.

The exact decomposition confirms the result. The nominal rate rises by 5%, the foreign price level rises by 4%, and the domestic price level rises by 5%:

\frac{q_1}{q_0} = 1.05\times1.04\times\frac{1}{1.05} =1.04.

Thus $q$ rises by exactly 4%. The foreign basket becomes 4% more expensive relative to the domestic basket. The common mistake is to add $5\%+4\%-5\%$ and treat that procedure as generally exact. It happens to give 4% here because the two 5% factors cancel; for other values, the exact multiplicative calculation is required.

Check. $1.10\times1.04=1.144$, which matches the directly calculated date-1 value.

Reading Changes Carefully

A real exchange-rate change describes a change in a relative goods price, not just a movement in a currency quotation. Under the convention used here, a rise in $q$ means the foreign basket has become more expensive relative to the domestic basket. A fall means it has become less expensive relative to the domestic basket. The cause can be an exchange-rate movement, a difference in price-level movements, or both.

When comparing dates, keep the order of the ratios consistent: date 1 divided by date 0. Also keep the same exchange-rate quotation and the same price indexes over time. If a price index is rebased between dates, that change in presentation must not be mistaken for an economic change in prices. A rebase multiplies index values by a scaling factor; with consistent treatment across the dates, the measured growth rate of that price index remains unchanged.

A real exchange rate is a relative-price measure, not a direct record of how many goods are traded or how much a country exports. The calculation alone does not establish the cause of a movement or its effect on trade. It tells us how the converted foreign goods-basket price compares with the domestic goods-basket price under the stated definitions.

Common Exam Traps

Using the wrong direction of the quote. Check the units on $E$. Domestic currency per foreign unit means multiply $E P^*$; foreign currency per domestic unit must be inverted first.
Reversing the price ratio. With this convention, $q$ is the converted foreign price divided by the domestic price. Writing $P/(E P^*)$ calculates the reciprocal measure.
Comparing currencies without conversion. The raw values $P^*$ and $P$ are in different currencies, so comparing them directly has no price interpretation.
Adding percentage changes as an exact rule. Use $q_1/q_0=(E_1/E_0)(P_1^*/P_0^*)(P_0/P_1)$ for an exact finite-change calculation.
Overinterpreting an index level. If price indexes have different base conventions, a level such as $q=0.8$ need not imply that an actual foreign basket costs 80% as much. Confirm the index construction before making that claim.

Key takeaway: With $E$ in domestic currency per foreign unit, calculate $q=E P^*/P$. It compares the converted foreign price level with the domestic price level; a change in $q$ is a change in that relative price.

Check Your Understanding

For each problem, use the stated quotation convention and show the conversion before interpreting the result.

A hypothetical foreign basket costs 64 foreign-currency units, the domestic basket costs 100 domestic-currency units, and $E=1.25$ domestic units per foreign unit. Find $q$ and state which basket costs less. (Answer: $q=0.8000$; the foreign basket costs less.)
The domestic basket costs 150 domestic units, the foreign basket costs 120 foreign units, and the quote is 0.80 foreign units per domestic unit. Find $q$. (Answer: $q=1.0000$.)
At date 0, $E_0=2.00$, $P_0^*=50$, and $P_0=100$. At date 1, $E_1=2.10$, $P_1^*=55$, and $P_1=105$. Find the percentage change in $q$. (Answer: $q_0=1.0000$, $q_1=1.1000$; $q$ increases by 10%.)
A foreign basket costs 100 foreign units and the domestic basket costs 240 domestic units. If $q=0.9000$, find the exchange rate $E$ in domestic units per foreign unit. (Answer: $E=2.16$ domestic units per foreign unit.)
Between two dates, $E$ rises by 3%, $P^*$ rises by 6%, and $P$ rises by 4%. Using the exact multiplicative relationship, find the percentage change in $q$. (Answer: $1.03\times1.06/1.04=1.0498077$, so $q$ increases by approximately 4.98%.)

What You'll Learn