Interpreting Currency Pairs as Value Ratios and Conversion Factors

# Answer by Rodolfo Oviedo (score 3, accepted)


# Why does the EUR/USD exchange rate is in fact USD/EUR from a mathematical point of view? Why finance does not use the mathematical notation?












Why does the EUR/USD exchange rate is in fact USD/EUR from a mathematical point of view?

By not using the mathematical notation it's a nightmare to compare exchange rates, so why is everyone still doing it?

Edit: I think I wasn't clear enough, what I would like to understand is why or how the dominant notation came to be the conventional way, and who initiated it. If people agree it's confusing, then why not change it?

## Answer by Rodolfo Oviedo (score 3, accepted)

https://quant.stackexchange.com/a/78121

(1) Suppose `1 EUR = 1.08 USD`. Here, EUR and USD are units of measure in the value dimension.

(2) Dividing by USD, `EUR/USD = 1.08`, which is the notation used by traders. Intuitively, 1 EUR is worth more than 1 USD by a factor of 1.08. In other words, the value of 1 EUR is 1.08 times the value of 1 USD.

Example of a computation using this notation. Let the exchange rate of the euro in terms of the dollar be `EUR/USD = 1.08` and let the exchange rate of the dollar in terms of Mexican pesos be `USD/MXN = 17.22`. Multiplying the corresponding sides of the previous two equations, we get the currency rate of the euro in terms of the Mexican Peso: (EUR/USD)(USD/MXN) = 1.08 × 17.22, or `EUR/MXN = 18.5976`.

(3) Dividing (1) by EUR, `1 = 1.08 USD/EUR`.

The right hand of the last equation is called a conversion factor by physicists. Because any conversion factor equals 1, it can multiply any quantity measuring value without altering its value. For example,

```
1000 EUR = 1000 EUR × 1 = 1000 EUR × 1.08 USD/EUR = 1080 USD ,
```

where the second equality follows from (3).

Conclusion (following the second comment by the OP below): Most of us are accustomed to interpreting EUR/USD as units of measurement accompanying a number, as in (3). So, if someone says that EUR/USD is 1.08, the first instinct of someone who is not a currency trader is to interpret that statement as 1.08 EUR/USD, which is wrong because the correct ratio of units that accompanies 1.08 is USD/EUR, hence 1.08 USD/EUR. However, if we read that EUR/USD is 1.08, we should instead interpret it as EUR/USD = 1.08, or the division of the value of 1 EUR by that of 1 USD.

Note: The argument above is the same used for physical quantities. The following is an example in length dimension. Using the same numbering as above,

(1) Let `1 m = 100 cm`

(2) Dividing by cm, `m/cm = 100`. A meter is 100 times the length of a centimeter.

(3) Dividing (1) by 100 cm, `1 m / (100 cm) = 1`, or `0.01 m/cm = 1`

Reexpressing in meters a quantity expressed in centimeters:

```
76 cm = 76 cm × 1 = 76 cm × 0.01 m/cm = 0.76 m ,
```

where the second equality follows from (3).

## Answer by Attack68 (score 2)

https://quant.stackexchange.com/a/40664

I can give an answer the second part of your edit. In my experience terminology, once established and active with a critical mass tends to propagate and then remain. It doesn't really change much. Particularly in trading, terminology has evolved around the central points of liquidity, and becomes embedded in many systems; computerised, vocal and written contracts. Often it forms based on the fastest way to express something when agreeing a trade. In the cross-currency swap market for example I speculate that the sole reason that the voice broker market quotes "euro-dollar" (in that order of currencies) as opposed to "dollar-yen", when they are technically the same similar product, is simply the nature that it rolls off a western tongue faster and easier than the converse. And in the dollar-yen case it avoids having to have notionals like "1.1 trillion yen please", cause you default to USD.

As a concrete example, some terminology that appeared in the EUR IRS interdealer market 2y ago is the name of the price "10Y gadget", which is the yield differential between 10Y bond futures and 10Y swaps. I found this an awful name and lobbied to change it to "10Y spread", which is common in GBP and USD. But when you find there is no real (simultaneous) will to change it doesn't get much traction and as long as everyone uses it and understands it you just have to go with it.

Perhaps there is no legitimate explanation for the appearance of the '/', and after battling the initial confusion no one cares anymore.

## Answer by nbbo2 (score 2)

https://quant.stackexchange.com/a/64262

You ask how the notation EUR/USD = 1.2 originated. To put it more bluntly how could anyone have thought up to write the exchange rate in such an illogical fashion, which seems backwards.

You can think of it not as an exchange rate between currencies but as a comparison of the "strength" of one currency to the other. In the old days 'strength' might have been based on a comparison of the amount of gold backing one unit of each currency (so eur/usd = 1.2 says that a euro has 20% more ounces of gold in it than a dollar). Or, when gold ceased to be used, 'strength' might be based on a basket of goods which a unit of currency can purchase (Gustav Cassel's Purchasing Power theory). So EUR/USD = 1.2 says with one EUR you can buy 1.2 times the consumer goods basket than you can buy with 1 USD. Or 1.2 times as many barrels of oil, if you prefer.

Then it makes sense. And dm63 made a similar point here in a comment. But I am not sure if this will help or confuse things further.