Currency exchange

Exchange rate

Notice which currency is on the left: it is the one worth $1$. If $1\text{EUR}=1.25\text{CHF}$, the euro is the reference currency and $1.25$ is what one euro costs measured in Swiss francs.

With that information, everything reduces to a proportion. To go from A to B, the amount in A is multiplied by $r$; to go back from B to A, the amount in B is divided by $r$:

$$\text{amount in B}=r\cdot (\text{amount in A})\text{amount in A}=\frac{\text{amount in B}}{r}$$

A common-sense check is useful: if the currency you are getting is "worth less" (you receive more units than you hand over), the number must increase; if it is worth more, it must decrease.

Buying and selling

A bank or bureau de change does not pay the same for a currency as it charges for it, and this is how it makes its profit. That is why two rates appear on its boards:

• Sell: the bureau sells you foreign currency (you need it for travelling). You receive few units of the foreign currency for each unit of yours.
• Buy: the bureau buys from you the foreign currency you have left over. You have to hand over more units of the foreign currency for each unit of yours.

What matters is reading the table from the bureau's point of view and choosing the column that matches what you do. When buy and sell rates are given in the same units, the sell figure is the one that gives you less foreign money for each unit of your currency, and the buy figure is the one that requires you to hand back more.

Commissions

As well as the exchange rate, many bureaux charge a commission on the transaction. There are two forms:

• Percentage commission of $p\%$: that fraction of the amount is deducted. What remains is $\left(1-\frac{p}{100}\right)$ times the amount. With a $3\%$ commission, $97\%$ remains, that is, a factor of $0.97$.
• Fixed commission: a fixed amount is subtracted per transaction, regardless of how much you exchange.

If the question is asked the other way round (how much you must hand over to receive a given amount), the steps are undone in reverse order: first divide or multiply by the exchange rate, then undo the commission. With a fixed commission, write an equation and solve it. For example, with rate $r$ and fixed commission $f$ we have $s=r(b-f)$, where $b$ is the amount handed over and $s$ the amount received.

Exchanging and exchanging back

If you exchange money and, without spending it, change it back into your original currency, you do not get the same amount back. Each transaction carries a commission, and the buy and sell rates do not match. The difference is a loss, which is measured as follows:

$$\text{loss}=\text{initial amount}-\text{final amount},\text{percentage loss}=\frac{\text{loss}}{\text{initial amount}}\cdot 100\%$$

Even if the exchange rate were identical in both directions, the commissions alone would be enough to lose money. With a $2\%$ percentage commission each time, for example, the total factor is $0.98\cdot 0.98=0.9604$: you keep $96.04\%$, not $96\%$.