Geometric Progression Weighted Average ?

[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 http://www.akmos.com/forex/usdx/".
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.

 

[NoMoreExams]

Are you basically asking why would we use geometric mean instead of arithmetic mean? If so googling "using geometric mean" produces a few links of interest such as http://pages.stern.nyu.edu/~wsilber/Geometric Average Versus Arithmetic Average.pdf

 

[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'.

 

[adriank]

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.

 

[musicgold]

adriank,

Thanks. Now it is clear to me.

As for why GM has been chosen over AM, I am still trying to understand that. The following is an interesting link I came across while searching for material.
Also, I found the following quote an important one and I am trying to apply it in the USDX case.

The big idea behind means is this. You have a bunch of different numbers. You want to replace each of these different numbers by the same number, in such a way that the net effect (the result of combining the numbers) is unchanged.

http://www.cse.unsw.edu.au/~teachadmin/info/harmonic3.html"