If the EUR/USD exchangerate fell by -0,96%, how much has the USD/EUR exchange rate increased?
According to the below charts the number would be +0,97% (currently) but I cant figure out how these number are related.
Please state a formula to convert the percentage change of USD/EUR to the change of EUR/USD.
On
Hint
Let the original $€\to\$$ exchange rate be $r$. Then the original $\$\to€$ exchange rate was $\frac1r$. Now the new $€\to\$$ exchange rate is
$$r'= r \cdot (1-0.0096)$$
The task is now to write $\frac1{r'}$ in terms of $\frac1r\cdot (1+a)$ where $a$ is the rate increase you are looking for.
In order to derive the formula, it is best to assume the exchange rate has some value, in fact that it has a "before" value and an "after" value, and to give names to those values.
Let's say that initially (at time $t_0$), $\unicode{0x20AC}1 = \$E_0$, that is, we can exchange $E_0$ dollars per euro, so $E_0$ is the EUR/USD rate of exchange in the notation of this page.
Then initially $\$1 = \unicode{0x20AC}\frac1{E_0}$, that is, the USD/EUR rate is $\frac1{E_0}$.
Suppose that at a later time $t_1$ the EUR/USD rate has changed to $E_1$. Then that USD/EUR rate at that time is $\frac1{E_1}$.
The percentage change in the EUR/USD rate during this time period is given by the formula
\begin{align} \%\text{ change in EUR/USD} &= \frac{E_1 - E_0}{E_0} \times 100\% \\ &= \left(\frac{E_1}{E_0} - 1\right) \times 100\%. \end{align}
The percentage change in the USD/EUR rate during this same time period is
\begin{align} \%\text{ change in USD/EUR} &= \frac{\frac{1}{E_1} - \frac{1}{E_0}}{\frac{1}{E_0}} \times 100\% = \left(\frac{\left(\frac{1}{E_1}\right)} {\left(\frac{1}{E_0}\right)} - 1\right) \times 100\% \\ &= \left(\frac{E_0}{E_1} - 1\right) \times 100\%. \end{align}
That is,
\begin{align} 1 + \frac{\%\text{ change in EUR/USD}}{100\%} &= \frac{E_1}{E_0} \\ 1 + \frac{\%\text{ change in USD/EUR}}{100\%} &= \frac{E_0}{E_1} = \frac{1}{\left(\frac{E_1}{E_0}\right)}.\\ \end{align}
Therefore $$ 1 + \frac{\%\text{ change in USD/EUR}}{100\%} = \frac{1}{1 + \frac{\%\text{ change in EUR/USD}}{100\%}}. $$
As you can see, all the $E_0$ and $E_1$ symbols have disappeared from this formula, leaving only the known change in one exchange rate and the desired-to-be-known change in the other exchange rate as variables. It is not necessary to know the actual exchange rates at any time in order to apply this formula.
If you want a formula that has just the percentage change in the USD/EUR exchange rate on the left-hand side, you can subtract $1$ from both sides of the equation above and then multiply both sides by $100\%$, but I find the equation more intuitive as it is.