Inverse of an average vs the average of an inverse: what is the difference?
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INVERSE OF AN AVERAGE vs THE AVERAGE OF INVERSES: Why are they not the same?

Every number has an “inverse” which is calculated by dividing 1 by that number. For example, the inverse of 5 is 1/5 (or 0.2)

Consider a set of numbers and their inverses:

Number                                           Inverse

A                                                       1/A

B                                                       1/B

C                                                       1/C

D                                                      1/D

E                                                       1/E

F                                                       1/F

 

  1. Calculate the average of the set (A+B+C+D+E+F)/6 = G, and then calculate the inverse of that average (1/G).
  1. Calculate the average of all the inverses:   (1/A + 1/B + 1/C + 1/D + 1/E + 1/F)/6

 

You would expect the 2 results to be the same, but this is not true!

 

Here’s a very simple example which shows the difference in the methods and the answers using a set of two numbers (2 and 4):

 

The average of 2 and 4 is 3. The “Inverse of the Average” is therefore  (or 0.333)

The inverse of 2 is ½. The inverse of 4 is ¼. The “Average of the Inverses” is (½ + ¼)/2 which is equal to  (or 0.375)

 

Thus, we can see that the Inverse of the Average does not equal the Average of the Inverses.

 

How is this relevant to FX Rates?

Many companies use Period Average FX rates for Revaluation/Translation. The averages are usually calculated at the end of each accounting period, using daily FX rates for that period.

 

How are Period Averages calculated?

Let’s look at an example, using USD and EUR for the month of March:

You have USD>EUR daily rates for all 31 days in the month. To calculate the Period Average it is clear to add up all 31 daily rates and then divide by 31.

But what about EUR>USD? There are two commonly used methods:

  • For each USD>EUR daily rate take the inverse to get the EUR>USD rate. Then calculate the average of the 31 EUR>USD rates, i.e. use the Average of the Inverses
  • Take the USD>EUR average for the period and calculate the inverse, i.e. use the Inverse of the Average.

Both the above methods are equally valid but will give different results.

How can I get more information on this subject?

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