# Geometric Progression Weighted Average ?

1. Dec 31, 2008

### musicgold

Hi,

I am trying to understand what Geometric Progression Weighted Average (GPWA) is in the context of the calculation of the US Dollar Index (USDX). I understand what a weighted average is but don’t understand what a GPWA is and when one should use it.

The following equation shows how the USDX is calculated using GPWA. In simple terms, the exchange rates of the US dollar with six major currencies are raised to a power value and the product of these terms is multiplied with an offset (50.14) to get the USDX. The power by which an exchange rate is raised represents the % of total US trade with that country. You can read more about the USDX here.
Note that my formula looks a bit different from the formula given in the article, but both formulae give the same answer. The term EUR / USD represents how many Euros can be purchased using one US dollar.

I am trying to understand the advantage of using trade weights as powers (i.e Ex. rate ^ weight) over simply using the weights as multiplying factors (i.e. Ex rate × weight). I know that using the weights as factors would result in a smaller number, but that is not a strong argument for using GPWA as one can always increase the magnitude of the offset to get a large value.

USDX = 50.14348112 × EUR / USD ^ 0.576 × JPY / USD ^ 0.136 × GBP / USD ^ 0.119 × CAD / USD ^ 0.091 × SEK / USD ^ 0.042 × CHF / USD ^0.036

I have calculated the USDX below using exchange rates as of Dec 30, 2008.

EUR / USD 0.7099
JPY / USD 90.36
GBP / USD 0.6924
CAN / USD 1.2185
SEK / USD 7.7565
CHF / USD 1.0575

USDX = 51.1435 × 0.821 × 1.845 × 0.957 × 1.018 × 1.090 × 1.002 = 82.44

Thanks,

MG.

Last edited: Dec 31, 2008
2. Dec 31, 2008

### NoMoreExams

3. Dec 31, 2008

### musicgold

NoMoreExams,

Thanks. No, I understand when to use an arithmetic mean ( when terms are added) and geometric mean (when terms are multiplied). I am not able to understand the use of 'weighted geometric mean'.

Last edited: Dec 31, 2008
4. Jan 2, 2009

Well, let's look at what the arithmetic and geometric means do.

Let's say you have a bunch of values $$x_1, \dotsc, x_n$$, and you want to find their arithmetic mean. You know, of course, that the arithmetic mean is $$\sum \frac{1}{n} x_i = \frac{x_1 + \dotsb + x_n}{n}$$.

Next, let's look at a weighted arithmetic mean. Then you have constants $$w_1, \dotsc, w_n$$ called weights, and to keep things simple, I'll say that $$\sum w_i = 1$$. Then the weighted arithmetic mean is $$\sum w_i x_i$$. If $$w_i = 1/n$$, then this is just the regular arithmetic mean.

We can use these same values and weights in a geometric mean. The weighted geometric mean of these values is defined as $$\prod x_i^{w_i}$$. If you have $$w_i = 1/n$$, then this is the usual geometric mean $$\prod x_i^{1/n} = \sqrt[n]{x_1 \dotsb x_n}$$.

If you want an arithmetic mean, then you must multiply the values by the weights. If you want a geometric mean, then you must use the weights as exponents. As for why a geometric mean is used, I'm sure there's a good reason, but I can't quite describe it.

5. Jan 3, 2009