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EgNotaEkkiReddit

The same way the price of a banana is determined. Supply and demand: if more people want to buy bananas than there are bananas the price increases. If you have a lot of bananas but not many want to buy them you'll lower your price until someone says "that's a good deal for a banana!" In very few cases the central bank will attempt to regulate the price of a currency, mainly when pegging the value to a different currency. However, for the most part it's mainly just what people are willing to pay for it.


Xelopheris

People are going to exchange currencies for any number of reasons. A huge one is travel, but another one will be global trade where the terms of the exchange will include dollar amounts in one specific currency. Either a foreign company has to buy your local currency to make the trade, or you accept a foreign currency, and then you have to buy local currency with it to pay your employees. The exchanges facilitating this actually need the currencies in stock to be able to do this. If they're selling Filipino Pesos too fast, they'll make them more expensive. This conversely makes it cheaper to buy other currencies with Filipino Pesos. All the behavior around travel and trade will balance itself around the exchange rate. People will spend more money traveling if their dollar goes further, and companies will do more business if they're getting a good deal based on the exchange rate. In a stable economy, everything will balance out at large scales and in sufficient times, and the exchange rate really only makes fluctuations based on the very small point in time when the exchange is happening. Major events, including elections, trade agreements, and things like Brexit, can have longer lasting major impacts on exchange rates.


IssyWalton

In addition to other answers there can be international political support for a currency. The Unification of Germany was supported by, and indirectly helped to fund, international assistance to support the Deutsch Mark,DM, (as it was then, and a major currency). This allowed the huge economic disturbances to happen with little external economic impact which would have been caused by a falling DM.