Converting Percentage Changes Between Inverse FX Quotes

Summary

The document explains why percentage changes in two reciprocal exchange-rate quotes are not equal in magnitude. If one currency pair is quoted as the reciprocal of another, the second quote must be inverted at both the starting and ending observations. The percentage return is then computed from those two inverse levels, rather than by simply changing the sign of the first percentage return.

A numerical illustration starts with one quote and a higher ending value, calculates its percentage increase, and then computes the corresponding reciprocal quote and its percentage decrease. The example shows that the inverse quote's move is smaller in absolute percentage terms. The conversion can be written by applying the reciprocal transformation to the gross return, with the direction depending on which quote's percentage change is known. The figures illustrate the relationship for one move; quote direction and consistent start and end values are essential when applying it elsewhere.

Key ideas

• Reciprocal currency quotes have percentage changes that are not exact sign reversals.
• Invert both the initial and final exchange-rate levels to obtain the inverse quote's return.
• The gross return of one quote can be converted by taking its reciprocal and subtracting one.
• The size of the percentage move depends on the starting exchange-rate level.

Tags

# Exchange rate conversion


# Exchange rate conversion












If the EUR/USD exchangerate fell by -0,96%, how much has the USD/EUR exchange rate increased?

According to the below charts the number would be +0,97% (currently) but I cant figure out how these number are related.

EUR/USD

USD/EUR

Please state a formula to convert the percentage change of USD/EUR to the change of EUR/USD.

## Answer by Rime (score 2, accepted)

https://quant.stackexchange.com/a/20845

I just checked Google Finance and the `EUR/USD = 1.1190`.... for arguments sake lets say it goes up by `0.10` to `1.2190` the percentage change = `1.2190/1.1190-1 = +8.94%`

in terms of `USD/EUR` the beginning quote would be `1/1.1190 = 0.8937` but would be `1/1.2190 = 0.8203` after the `EUR/USD` went up by `0.10`. Therefore the change in terms of `USD/EUR = 0.8203/0.8937-1 = -8.21%`

@AlexC formula explains it best:

`1/(1-0.0821)-1= +8.94%` or `1/(1+0.0894)-1= -8.21%`

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.