Trade-weighted US dollar index

The index is computed as the geometric mean of the bilateral exchange rates of the included currencies. The weight assigned to the value of each currency in the calculation is based on trade data, and is updated annually (the value of the index itself is updated much more frequently than the weightings).[2] The index value at time ${\displaystyle t}$ is given by the formula:[2]

${\displaystyle I_{t}=I_{t-1}\times \prod _{j=1}^{N(t)}\left({\frac {e_{j,t}}{e_{j,t-1}}}\right)^{w_{j,t}}}$.

where

• ${\displaystyle I_{t}}$ and ${\displaystyle I_{t-1}}$ are the values of the index at times ${\displaystyle t}$ and ${\displaystyle t-1}$
• ${\displaystyle N(t)}$ is the number of currencies in the index at time ${\displaystyle t}$
• ${\displaystyle e_{j,t}}$ and ${\displaystyle e_{j,t-1}}$ are the amount of currency ${\displaystyle j}$ required to purchase one U.S. Dollar at times ${\displaystyle t}$ and ${\displaystyle t-1}$[4]
• ${\displaystyle w_{j,t}}$ is the weight of currency ${\displaystyle j}$ at time ${\displaystyle t}$
• and ${\displaystyle \sum _{j=1}^{N(t)}w_{j,t}=1}$

The real exchange rate is a more informative measure of the dollar's worth since it accounts for countries whose currencies experience differing rates of inflation from that of the United States. This is compensated for by adjusting the exchange rates in the formula using the consumer price index of the respective countries. In this more general case the index value is given by:[2]

${\displaystyle I_{t}=I_{t-1}\times \prod _{j=1}^{N(t)}\left({\frac {e_{j,t}\cdot {\frac {p_{t}}{p_{j,t}}}}{e_{j,t-1}\cdot {\frac {p_{t-1}}{p_{j,t-1}}}}}\right)^{w_{j,t}}}$.

where

• ${\displaystyle p_{t}}$ and ${\displaystyle p_{t-1}}$ are the values of the US consumer price index at times ${\displaystyle t}$ and ${\displaystyle t-1}$
• and ${\displaystyle p_{j,t}}$ and ${\displaystyle p_{j,t-1}}$ are the values of the country ${\displaystyle j}$'s consumer price index at times ${\displaystyle t}$ and ${\displaystyle t-1}$