There are different reasons for a bounty including "to award to an existing answer". In some other case, the person is looking for updated answer.
When saw a bounty in the featured tab, The title shows this question has an open bounty worth
Instead if it shows the reason to put the bounty/ the notice, then it will be easy to decide whether we should open the question or not.
This will be helpful in all SE sites but since SO has large number of questions with bounty (At present, it is 399), it will be very much helpful in SO.
So, it will look like