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So I am having trouble understanding the UIP equilibrium condition. Namely, according to UIP, when the home country has a higher interest rate than the foreign country, its currency will depreciate over time. This is highly counter-intuitive to me, since I would expect higher interest rates to increase demand for the home currency, thereby raising the currency's price, leading to an appreciation of the home currency. What is the mechanism that would lead to the home country's currency depreciating? What is the logic behind this?

There were similar questions on this forum, but I did not find an answer that could explain it well to me.

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The mechanism is positive expected value trading. You take a loan in the currency with the lower interest rate (adjusted for expected exchange rate change) convert it to the currency with the higher interest rate and invest it. When you get the payout, you convert it back to the original currency. The expected value of this sum is larger than what you owe, including interest, thus you (or large risk neutral traders) are incentivized to do this.

The trades above increase present day demand for the high interest rate currency and increase future demand for the low interest rate currency, moving both the exchange rate and the expected exchange closer towards what is needed for parity.

An example with USD($) and EUR(€):

Suppose $$ 1 + i_{$} < E_{€/$} \cdot (1 + i_{€}) \cdot E_{$/€}^e $$ Then traders are incentivized to take dollar loans, convert this to euros, collect the euro interest, and convert it back to dollars. Due to the trades there will be more present day demand for euros, making euros more expensive in dollar terms. Thus the €/\$ exchange rate will decrease. (You get fewer euros for a dollar.) In the future, when the traders reconvert euros to dollars repay their dollar loans, the trades create more demand for dollars, making them more expensive in euro terms, thus the expected \$/€ exchange rate will also decrease.

Thus (in this form) both exchange rates on the right hand side decreased, bringing the inequality closer to equality. One could make the argument that the demand resulting from the trades has a similar effect on the interest rates.

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