[Sapient]

They may lose money on the first asteroid, but once they have the first one, everything should become exponentially cheaper as very little will need to be launched into LEO, and once they have a water rich rock, they can manufacture all the fuel they need, virtually for free. Once they have brought enough metals into LEO, along with some sort of manufacturing plant, they can build more miners, and they will only have to launch the complex parts into LEO, building the rest in orbit.

At some point, mining these asteroids will, for all intents and purposes, be free.

Edit: Another alternative is that these bored billionaires want to actually change the world just for fun, or to see if they can.