The Trade Weighted US dollar Index, also known as the broad index, is a measure of the value of the US Dollar relative to other world currencies. It is similar to the US Dollar Index in that its numerical value is determined as a weighted average of the price of various currencies relative to the dollar, however it differs in which currencies are used and how their relative values are weighted.

## History

The trade weighted dollar index was introduced in 1998 for two primary reasons. The first being the introduction of the

euro, which eliminated several of the currencies in the standard dollar index; the second being to keep pace with new developments in US trade.

## Included currencies

In the standard US Dollar Index, a significant weight is given to the euro. In order to more accurately reflect the strength of the dollar relative to other world currencies, the

Federal Reserve created the Trade Weighted US Dollar Index, which includes a bigger collection of currencies than the US Dollar Index. The regions included are:

- Europe (euro countries)
- Canada
- Japan
- Mexico
- China
- United Kingdom
- Taiwan
- Korea
- Singapore

- Hong Kong
- Malaysia
- Brazil
- Switzerland
- Thailand
- Philippines
- Australia
- Indonesia
- India

- Israel
- Saudi Arabia
- Russia
- Sweden
- Argentina
- Venezuela
- Chile
- Colombia

## Mathematical formulation

### Based on nominal exchange rates

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). More formally the index value at time

$t$ is given by the formula:

- $I\_t\; =\; I\_\{t-1\}\; times\; prod\_\{j\; =\; 1\}^\{N(t)\}\; left(frac\{e\_\{j,t\}\}\{e\_\{j,t-1\}\}\; right)^\{w\_\{j,t\}\}$.

where

- $I\_t$ and $I\_\{t-1\}$ are the values of the index at times $t$ and $t-1$
- $N(t)$ is the number of currencies in the index at time $t$
- $e\_\{j,t\}$ and $e\_\{j,t-1\}$ are the exchange rates of currency $j$ at times $t$ and $t-1$
- $w\_\{j,t\}$ is the weight of currency $j$ at time $t$
- and $sum\_\{j=1\}^\{N(t)\}\; w\_\{j,t\}\; =\; 1$

### Based on real exchange rates

In order to account for countries whose currencies experience differing rates of inflation from that of the

United States the real exchange rate is a more informative measure of the dollar's worth. 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:

- $I\_t\; =\; I\_\{t-1\}\; times\; prod\_\{j\; =\; 1\}^\{N(t)\}\; left(frac\{e\_\{j,t\}\; cdot\; frac\{p\_t\}\{p\_\{j,t\}\}\}\{e\_\{j,t-1\}cdot\; frac\{p\_\{t-1\}\}\{p\_\{j,t-1\}\}\}\; right)^\{w\_\{j,t\}\}$.

where

- $p\_t$ and $p\_\{t-1\}$ are the values of the US consumer price index at times $t$ and $t-1$
- and $p\_\{j,t\}$ and $p\_\{j,t-1\}$ are the values of the country $j$'s consumer price index at times $t$ and $t-1$

## References