Questions like this are hard to answer because they are so vague. What does 'play modern titles' mean? You mean will the game boot up and run, even if it's at 20fps? Or does 'play' to you means a certain minimum framerate? Also, it's not just about your videocard, your CPU and potentially the amount of system ram you have will also determine how a game 'plays'.
To answer your question in a preliminary way, an Nvidia GT 730 DDR5 is the cheapest new gaming videocard you can get. It's not a good card really, but you CAN 'play' newer games on it at low resolution and settings, although I'd recommend lower than 720p for that card.
A used card would be much better. an AMD HD 7770 or 7790 would be a better deal than a GT 730 in that price range.