Calculate Currency Exchange Rates

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When trying to calculate currency exchange rates, there is more than a simple ratio to take into account.

Basic maths says that if EUR/USD ratio is, lets say 1.15, and you want to know how many EUR you will get from 100 USD, then you just should divide 100/1.15 and obtain 86.96€. In the opposite, to get 100 EUR converted to USD you need to multiply 100 * 1.15, which equals to 115$. An easy way to remember this is to multiply across left-to-right and divide across right-to-left.

But from foxer currency calculator’s perspective, things are slightly different and more complex.

Widening the maths

At TUDICOR we consider every spot exchange rates publication as a universe in itself. It contains the major currencies known by the institution and provides the exact values against the base currency at a certain date. The Base currency is the currency managed by the Central Bank of the country. It is the key and the pillar to all possible conversions available in the context and can’t be defined by itself, but for comparison to the provided values. No other currency values can be inferred out of this report.

But there is one important exception to this: pegged currencies.

Oppositely to free floating currencies, and as explained previously in this article, pegged currencies do have a fixed ratio against some other country currency. The Dirham of the United Arab Emirates (AED), for example, is linked at 3.6725 per USD, acting in this case, the USD as a bridge for the calculation.

And this is important because it means that, in a general way, if any bridge currency is present in a spot exchange rates report, all those currencies that are linked to it are convertible too.

For instance, the Swiss National Bank publication contains just 4 currencies: USD, EUR, GBP & JPY. But the USD have 15 country currencies pegged to it, the GBP 2 and the Euro 22 ( including the historical ones ), so, at the end, the currencies available for the SNB publication are more than 40.

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An overall approach

Now that we have an extended perspective, we can realize that currency conversion effectively is “a simple mathematical operation from unit X to unit Y”. But the variety of roles between floating, pegged & bridge currencies and their relative positions are far from simple, because every different structure implies more or less parameters in different relationships. This is why we call this conversion’s geometry.

The image below shows the calculations to be done for a floating to floating currency pair.
float to float currency calculation diagram

And this other one the calculations to be done for a pegged to pegged currency pair.
pegged to pegged currency calculation diagram

As said before, currency conversion can be not so simple when all possible cases are observed.

Casuistry table by currency roles

Considering all mentioned currencies roles against a common base currency, we can summarize the following possibilities for a currency conversion operation. For the table below we have taken USD & EUR as floating currencies and AED ( Emirates Dirham ) & XOF ( CFA Franc ) as pegged ones. The Base currency is irrelevant. Please, note that combinations that can lead to equality have been omitted.

INCOMING CURRENCYPEGGED TOOUTGOING CURRENCYPEGGED TOCONVERSION TYPEFORMULA
USDBASEFloatToBaseX / IR
BASEUSDBaseToFloatX * IR
USDEURFloatToFloat( X / IR ) * OR
AEDUSDEURPeggedToFloat( X / IR ) * OR
USDXOFEURFloatToPegged( X / IR ) * OR
AEDUSDUSDPeggedToBridgeX / ( IR / IPR )
USDAEDUSDBridgeToPeggedX * ( OR / OPR )
AEDUSDXOFEURPeggedToPegged( X / IR * IPR ) * ( OR * OPR )
(*) IR, OR, IPR & OPR Stand for Incoming Ratio, Outgoing Ratio, Incoming Peg Ratio & Outgoing Peg Ratio

For the given formulas to be right, your data provider should bring you consistent Base pairs in the form of 1 Base Currency = N Ending currency. It may seem obvious, but not all Central Banks deliver data in this way, and this is the reason why in foxer currency calculator we introduced the UNIQUE logic parameter in version 9.8.5.

Testing

If your system is running foxer, you can just select and copy the table below by pressing CTRL+C. This Cross Table contains the most common variations pointed out in this article and should give you a very similar results to the table at right. ( On the popup, click on the f icon to get values only ).

Please note there is no Bank selector in the table, so you can also test by selecting a different bank. Once again, you can expect subtle differences because of the provided exchange rates, but results should be roughly similar.

AED EUR GBP USD XOF
AED 100 100 100 100 100
EUR 100 100 100 100 100
GBP 100 100 100 100 100
USD 100 100 100 100 100
XOF 100 100 100 100 100
AED EUR GBP USD XOF
AED 100 425,05 500.67 367.25 0.65
EUR 23.53 100 117.79 86.4 0.15
GBP 19.97 84.9 100 73.35 0.13
USD 27.23 115.74 136.33 100 0.18
XOF 15,432.30 65,595.70 77,265.18 56,675.11 100

Final notes

Because of optimization algorithms, foxer currency calculator does not exactly follow this schema, but it is very similar.

Remember you can always check exchange rates, peg ratios and how calculations are done by opening foxer‘s Conversion Report.

Here you can see the line for USD to EUR from Bank Of England / 2021.10.13:

0020 = "100" => ( 100.0000 / GBP/USD=1.36330 ) * GBP/EUR=1.17790 => "86.40€ ( $100 ) "

And here the one for AED to XOF:

0031 = "100" => ( 100.0000 / GBP/AED=5.00672(USD/AED=3.6725) ) * GBP/XOF=772.65175(EUR/XOF=655.957) => "15,432.30F. ( 100Dh ) "

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Foreign exchange rates

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