Studiometry saves exchange rates to items when you create the items. This means that, when you create a work item, a snapshot of the exchange rate will be stored with that item. The same goes for invoices and for payments.
Now when you create an invoice and send it to your client, there is a delay between you sending the invoice and the client sending the payment. During this delay, exchange rates can fluctuate, but the amount the client sends into you will not.
The invoice will always show as "paid" when the client has sent in the same payment amount as the invoiced amount, because they have fulfilled the invoice. The project balance, however, will let you see if you have made or lost money as a whole because of fluctuations in exchange rates between invoicing and payment.