Testing Purchasing Power Parity in Practice

Why Test Purchasing Power Parity?

Purchasing power parity (PPP) gives a benchmark for comparing currencies through the prices of goods and services. Testing PPP in practice means asking whether observed exchange rates and price data line up with that benchmark. The answer depends on which version of PPP is being examined: absolute PPP compares exchange-rate levels with price levels, while relative PPP compares exchange-rate changes with inflation differences.

For consistency, this tutorial quotes the exchange rate \(E\) as units of home currency per one unit of foreign currency. Let \(P\) be the home-currency price of a specified basket and \(P^*\) the foreign-currency price of a comparable basket. A rise in \(E\) means that one unit of foreign currency costs more home currency.

Definition: Absolute PPP holds for a specified comparable basket when the observed exchange rate equals the ratio of its home and foreign prices: \(E=P/P^*\). The absolute-PPP benchmark is \(E^{\mathrm{PPP}}=P/P^*\). Relative PPP is a hypothesis about changes: over a period, the observed exchange-rate ratio equals the ratio of the home and foreign gross inflation rates, \(E_1/E_0=(1+\pi)/(1+\pi^*)\), where \(\pi\) and \(\pi^*\) are inflation rates over that same period.

The two tests answer different questions. Absolute PPP asks whether currency conversion equalizes the basket’s price at a particular date. Relative PPP asks whether the exchange rate changes in step with the inflation difference. An exchange rate can fail the absolute-PPP level comparison and still change at the rate relative PPP predicts. Conversely, a level may happen to match the absolute benchmark at one date even though the exchange rate does not track inflation over a later period.

Testing Absolute PPP with Basket Prices

To test absolute PPP, first identify a basket that is comparable across the two economies and determine its price in each currency. Divide the home price by the foreign price to obtain the benchmark in home currency per foreign currency. Then compare that benchmark with the observed exchange rate, using the same currency quote and date.

E^{\mathrm{PPP}}=\frac{P}{P^*}, \qquad \text{absolute-PPP gap}=E-E^{\mathrm{PPP}}

The percentage deviation of the observed rate from its benchmark can be expressed as \(E/E^{\mathrm{PPP}}-1\). A positive result means the observed quote is above the benchmark; a negative result means it is below. This comparison is meaningful only when the prices refer to sufficiently comparable baskets and the exchange rate is quoted in the stated direction.

Worked Example: Comparing Basket Prices with the Market Rate

A hypothetical standardized household basket costs 1,260 home-currency units in one country and 840 foreign-currency units in another. On the comparison date, the observed exchange rate is 1.80 home-currency units per foreign-currency unit. Calculate the absolute-PPP benchmark and assess the deviation.

Method. Calculate the basket-price ratio in the same quote as the observed exchange rate, then compare the two rates. The common mistake to avoid is reversing the price ratio and thereby producing a benchmark in foreign currency per home-currency unit.

The absolute-PPP benchmark is:

E^{\mathrm{PPP}}=\frac{1{,}260}{840}=1.50 \quad\text{home-currency units per foreign-currency unit.}

The observed rate is 1.80, so the absolute gap is \(1.80-1.50=0.30\) home-currency units per foreign-currency unit. Relative to the benchmark, the deviation is:

\frac{1.80}{1.50}-1=1.20-1=0.20=20\%.

Thus the observed exchange-rate quote is 20% above the absolute-PPP benchmark. As a check, converting the foreign basket at the observed rate costs \(1.80(840)=1{,}512\) home-currency units, compared with 1,260 for the home basket. The converted foreign-basket price is \(1{,}512/1{,}260=1.20\) times the home price, consistent with a 20% gap.

Price indexes require care in an absolute-PPP test. An index value such as 120 is generally a normalized measure, not a basket price expressed in currency units that can be directly divided by another country’s index. If two indexes use different base periods, their ratio does not by itself give the absolute exchange-rate benchmark. Absolute comparisons call for actual prices of comparable baskets, or a carefully constructed price-level measure that has a meaningful cross-country interpretation.

Testing Relative PPP with Price-Index Changes

Relative PPP is tested by comparing the exchange-rate change over a period with the inflation difference over exactly that period. If \(I_{H,0}\) and \(I_{H,1}\) are home price-index values at the start and end, the home gross inflation factor is \(I_{H,1}/I_{H,0}\). Define the foreign factor in the same way. This approach uses index changes, so the indexes’ arbitrary base values cancel.

1+\pi=\frac{I_{H,1}}{I_{H,0}}, \qquad 1+\pi^*=\frac{I_{F,1}}{I_{F,0}}, \qquad \left(\frac{E_1}{E_0}\right)^{\mathrm{PPP}} = \frac{1+\pi}{1+\pi^*}

The superscript \(\mathrm{PPP}\) in the last expression denotes the relative-PPP benchmark factor for the exchange-rate change. To produce an ending-rate prediction from a given starting rate, relative PPP applies that factor to the starting rate. Whether the starting rate is observed or is an absolute-PPP benchmark must be stated.

Worked Example: Testing a Two-Country Inflation Prediction

Over one year, home inflation is 4% and foreign inflation is 1%. The observed exchange rate rises from 1.50 to 1.59 home-currency units per foreign-currency unit. Assess whether the observed change matches relative PPP and calculate the relative-PPP ending-rate prediction based on the observed starting rate.

Method. Compare the observed exchange-rate ratio with the ratio of the two gross inflation rates. Then apply the relative-PPP assumption to the observed starting rate. The common mistake to avoid is treating that prediction as a result of absolute PPP.

The relative-PPP benchmark factor is:

\frac{1+\pi}{1+\pi^*} = \frac{1.04}{1.01} = 1.02970297\ldots

Applying this factor to the observed starting rate gives the predicted ending rate:

E_1^{\mathrm{pred}} = 1.50(1.02970297\ldots) = 1.54455445\ldots

The predicted rate is 1.5446 home-currency units per foreign-currency unit, rounded to four decimal places. The actual exchange-rate ratio is \(1.59/1.50=1.06\), a 6% increase, rather than the benchmark increase of about 2.9703%. The actual ending rate exceeds the relative-PPP prediction by \(1.59-1.54455445\ldots=0.04544555\ldots\), or about 2.9423% of the prediction. Therefore the observed change does not match the relative-PPP benchmark over this period.

The calculation does not establish an absolute-PPP benchmark: no comparable basket prices were supplied. It applies the separate relative-PPP hypothesis to the observed starting rate. All rate figures are in home-currency units per foreign-currency unit; the predicted ending rate is rounded only at the final step.

Why the Benchmark-Rate Derivation Needs Care

The distinction between a PPP benchmark and an observed rate follows directly from the definitions. Suppose \(E_t^{\mathrm{PPP}}=P_t/P_t^*\) denotes the absolute-PPP benchmark at date \(t\). Let \(1+\pi=P_1/P_0\) and \(1+\pi^*=P_1^*/P_0^*\). Taking the ratio of the absolute-PPP benchmarks gives:

\frac{E_1^{\mathrm{PPP}}}{E_0^{\mathrm{PPP}}} = \frac{P_1/P_1^*}{P_0/P_0^*} = \frac{P_1/P_0}{P_1^*/P_0^*} = \frac{1+\pi}{1+\pi^*}

This derivation establishes a relationship between the two absolute-PPP benchmark rates. It does not, on its own, establish that the ratio of observed rates \(E_1/E_0\) equals the inflation factor. That equality is the additional relative-PPP hypothesis. Consequently, using an observed starting rate in a relative-PPP prediction is an application of that hypothesis, not a conclusion obtained merely by dividing price levels.

If the observed starting rate equals its absolute-PPP benchmark, \(E_0=E_0^{\mathrm{PPP}}\), and relative PPP also holds over the period, then the predicted observed ending rate equals the ending absolute-PPP benchmark. Without that initial equality, a relative-PPP prediction based on the observed starting rate need not equal the ending absolute-PPP benchmark.

Theorem: Inflation determines the change in the absolute-PPP benchmark through \(E_1^{\mathrm{PPP}}/E_0^{\mathrm{PPP}}=(1+\pi)/(1+\pi^*)\). Predicting the change in the observed exchange rate by the same factor requires the separate relative-PPP assumption \(E_1/E_0=(1+\pi)/(1+\pi^*)\).

A Practical Comparison with Index Data

With observed price indexes, use the index ratios rather than subtracting index values. For instance, a home index moving from 100 to 115 indicates 15% inflation, while a foreign index moving from 100 to 108 indicates 8% inflation. The relative-PPP benchmark factor is \(1.15/1.08\), not \(1+(15\%-8\%)\). The latter is only a rough approximation, because relative PPP uses a ratio of gross inflation factors.

Worked Example: Using Price Indexes to Assess Relative PPP

A home price index rises from 100 to 115 while a foreign price index rises from 100 to 108. The observed exchange rate at the start is 1.20 home-currency units per foreign-currency unit, and the observed ending rate is 1.25. Calculate the relative-PPP prediction and compare the observed change.

Method. Obtain each inflation factor from its index ratio, calculate the relative-PPP factor, and apply it to the observed initial rate. The common mistake to avoid is using the difference between the two inflation rates as if it were the exact factor.

The home and foreign gross inflation factors are \(115/100=1.15\) and \(108/100=1.08\). Thus:

\left(\frac{E_1}{E_0}\right)^{\mathrm{PPP}} = \frac{1.15}{1.08} = 1.06481481\ldots

The predicted ending rate is \(1.20(1.06481481\ldots)=1.27777778\ldots\), or 1.2778 to four decimal places. The observed exchange-rate ratio is \(1.25/1.20=1.04166667\ldots\), so the observed rate rose by about 4.1667%, compared with the relative-PPP benchmark increase of about 6.4815%. The observed ending rate is \(1.25-1.27777778\ldots=-0.02777778\ldots\) below the prediction, about 2.1739% below it.

As a check on the factor, the home inflation rate exceeds foreign inflation, so the benchmark exchange rate in home currency per foreign currency should rise. The predicted rate does rise from 1.20 to 1.2778. The observed rate also rises, but by less than the benchmark predicts.

Interpreting Deviations and Their Causes

A PPP gap is a comparison with a benchmark, not automatically evidence of an error in the exchange-rate data. Even careful comparisons can show deviations because the assumptions behind PPP are demanding. A basket may include goods that cannot be traded internationally, costs of transporting goods may be substantial, and market frictions may prevent prices from adjusting quickly or equally across locations.

Transport costs and trade barriers. Shipping, insurance, tariffs, and other border costs create a wedge between the local prices of traded goods. A price difference smaller than these costs may not make cross-border trade worthwhile, so arbitrage need not eliminate it.
Nontraded goods and services. Housing, many local services, and other goods that are difficult to move across borders can have different prices across countries. If they form a substantial part of a consumer price index, the index may move differently from the prices of internationally traded goods.
Market frictions and pricing practices. Different taxes, regulations, competition, contracts, and costs of changing prices can slow or limit price adjustment. Firms may also set different prices in different markets.
Basket and measurement differences. National indexes may use different goods, weights, quality adjustments, or collection methods. Their inflation rates can therefore differ even when they are intended to measure broad price changes.

For these reasons, the result of a test should identify the comparison, time period, and data used. A single absolute-PPP comparison is a statement about an observed level relative to a chosen benchmark basket. A single relative-PPP comparison is a statement about exchange-rate and inflation changes over a specified interval. Neither comparison alone establishes that the relationship holds generally or will hold in the future.

Common Exam Traps

Mixing up the currency quote.
If \(E\) is home currency per foreign currency, the absolute benchmark is \(P/P^*\). Reversing that ratio changes the quote and makes the comparison invalid unless the observed rate is also inverted.

Confusing a benchmark-rate movement with an observed-rate movement.
The ratio of absolute-PPP benchmark rates follows from price levels. The same relationship for observed rates is a relative-PPP assumption, not an algebraic consequence of absolute PPP.

Using inflation-rate subtraction as an exact calculation.
Use \((1+\pi)/(1+\pi^*)\). The difference \(\pi-\pi^*\) is a useful small-rate approximation, but it is not the exact relative-PPP factor.

Comparing index levels as if they were comparable basket prices.
Index changes can measure inflation over time. Absolute PPP requires price levels with a meaningful common basket and currency interpretation.

Key takeaway: For absolute PPP, compare a properly measured observed exchange rate with the ratio of comparable basket prices. For relative PPP, compare the observed exchange-rate ratio with the ratio of gross inflation factors. Keep benchmark-rate calculations separate from assumptions about observed rates.

Check Your Understanding

For each problem, use home currency per foreign-currency unit and keep the stated comparison period consistent.

A comparable basket costs 960 home-currency units and 600 foreign-currency units. The observed exchange rate is 1.70. Find the absolute-PPP benchmark and the percentage by which the observed rate is above or below it. (Answer: benchmark \(=1.60\); observed rate is 6.25% above it.)
The observed exchange rate is 0.80 at the start of a year. Home inflation is 2% and foreign inflation is 5%. Under relative PPP, what ending rate is predicted from the observed starting rate? (Answer: \(0.80(1.02/1.05)=0.7771\), to four decimal places.)
A home price index rises from 108 to 117, and a foreign price index rises from 104 to 110. If the observed starting exchange rate is 2.00, calculate the relative-PPP predicted ending rate. (Answer: \(2.00[(117/108)/(110/104)]=2.0485\), to four decimal places.)
The observed exchange rate rises from 1.25 to 1.30. Home inflation is 6% and foreign inflation is 2%. Calculate the relative-PPP predicted ending rate based on the observed starting rate and state whether the observed ending rate is above or below it. (Answer: predicted rate \(=1.25(1.06/1.02)=1.2990\), to four decimal places; observed 1.30 is above the prediction.)
Two countries have identical observed inflation rates over a year, but their observed exchange rate changes. Does that observation establish a failure of absolute PPP? What does it say about relative PPP under the stated rates? (Answer: It does not by itself establish a failure of absolute PPP, which is a level comparison requiring comparable basket prices. With equal inflation rates, relative PPP predicts no exchange-rate change; a nonzero observed change does not match that period’s relative-PPP prediction.)

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