This “SDR deflator” is calculated as a weighted average of the GDP deflators of Japan, the United Kingdom, the United States, and the euro area, the currencies currently used in the SDR. Weights are the amount of each country’s currency in one SDR unit; note that both the composition of the SDR and the relative exchange rates for each currency change over time. The SDR deflator is first calculated in SDR terms, and then converted to U.S. dollars using an SDR to dollar conversion factor that is calculated using the Atlas method. Note that weights vary over time with the composition of the SDR and relative exchange rates.
Details of the calculation follow. For any economy i for year t:
Let be the quantity of local currency in the SDR
Let be the exchange rate between local currency and the SDR
Let be GDP in local currency at current prices
Let be GDP in local currency at constant prices relative to a base year a
Let be the local currency GDP deflator measured relative to a base year a (currently, the year 2000 is used in the calculations)
The weight of each currency in the SDR (expressed in terms of SDRs) is given by
Deflators for each economy in SDR terms are defined by
Which can be rewritten
Where is the implicit deflator in local currency terms, defined as . Thus deflators for each economy in SDR terms are calculated by multiplying by the implicit GDP deflator by the ratio of the exchange rate (local currency to SDR) in year t to the exchange rate in base year a.
The weights from equation (1) and the deflators from equation (2) are then used to calculate a weighted average of the annual inflation rates derived from the deflators for each economy; a Fisher index (the geometric mean of a Laspeyres and Paasche index) is used:
The SDR deflator (in SDR terms) is constructed from the Fisher index:
The SDR deflator (in U.S. dollar terms) is then calculated as
Where the conversion factor from SDR to U.S. dollars is calculated using the Atlas method, i.e.,
- The formula used to calculate the SDR deflator in dollar terms is missing the base year exchange rate, which would rescale it relative to the base year value of the SDR. However, because the deflator is only used to calculate the inflation rate (t-2 to t and from t-1 to t) in the Atlas exchange rate formula and, similarly, is used to update the income thresholds by the annual rate of inflation, the omission of the base year exchange rate, a constant, has no effect on either.
- The country composition of the euro area GDP deflator has changed over time, the deflator used in the calculation reflects the membership in the euro area at each time period after 1998.
- Prior to 1999, the SDR was comprised of the German mark, French franc, British pound, U.S. dollar, and Japanese yen. From 1999 onward, the euro was included in the SDR and the mark and franc were dropped.