Let's build some intuition for how this works.

Suppose the US Dollar and Euro interest rates are identical. Plugging identical rates into the formula makes the fraction equal to one, meaning the forward rate would be exactly the same as the spots rate. In other words, if there's no interest rate differential between the two currencies, there's no difference between spot and forward prices. However, when the two interest rates differ, the spot and forward rates will diverge. The greater the interest rates differential, the larger the gap between the spot and forward prices. For instance, if US Dollar interest rates increase from 5.395% to 6.395%, the forward FX rates would rise accordingly.

The formula also helps us understand the direction of the adjustments. A higher interest rate for the quoted currency leads to its depreciation on a forward basis, while a lower interest rate for the quoted currency leads to its appreciation on a forward basis. To see this in action, let's revisit our earlier example. In that case, US Dollars had a higher interest rate than Euros, and a Euro Dollar spot rate of 1.095 moved to a forward rate of 1.10391 over six months. This reflects US Dollar depreciation and Euro appreciation on a forward basis. Perfectly aligning with what the formula predicts.

But what's the intuition behind this relationship? It boils down to a fundamental concept. Theoretically, there should be no profit on an FX hedged carry trade.

A carry trade involves borrowing in a low yielding currency, converting it into a high yielding currency, and investing at the higher interest rates on the surface. This seems like a risk-free way to earn a return, but the strategy carries exchange rate risk.

At the end of the investment period, you'll need to convert the high yielding currency back to the low yielding one to repay the loan. If the low yielding currency has strengthened in the meantime, the exchange rate loss could wipe out your interest rate gain or worse, leave you with a loss.

To eliminate this risk, you could use an FX forward to lock in the exchange rate today. However, by doing so, you also eliminate the risk from the carry trade. And as we know in finance, no risk means no return.

In an efficient market this means that an FX hedged carry trade shouldn't generate a profit. For this to hold, the forward rates must adjust to offset the interest rate differential. This means the lower yielding currency will appreciate on a forward basis and the higher yielding currency will depreciate.

This leads to another important insight. The forward rates is driven not only by the spot rates, but also by the interest rates of the two currencies. Consequently, entering into an FX forward exposes you to market risks related to all three factors, the spot FX rates, as well as the interest rates in the two currencies involved.