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Different assets have different motivations and components for price determination. For example, an equity stock price might be considered to be include the value of underlying company, its potential for future growth, and its value as a store of wealth (kind of like a Yap stone).

The price of a currency would seem to be largely determined by the balance of trade with the country or countries involved. For example, if there is a net interest in buying the products of a country, then its currency will be in demand and the price will decrease, but if the citizens of the country are net buyers of foreign goods, then the price will tend to decrease. Thus, it is self balancing, because as long as the net buying of foreign goods continues, the value of the currency will decline reducing the amount they can buy, until it becomes so low that the country is forced to become a net seller, and the value of the currency begins to increase again.

Is there an econometric method for valuing currencies that takes such factors into account?

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Yes, the earlier answer about currency value gives some links, including a link to an IMF publication that presents models (methods) with some notes on how to calibrate them and further links to literature. There they describe among others also the macroeconomic balance framework that takes into account the internal and external balance at the same time.

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