Why do you think Bilbo shudders when he hands Bard the stone? Use text evidence.

Bilbo shudders because he doesn't want to let the stone go.... possibly, also because he knows what is in store when Thorin hears what he has done.

"This is the Arkenstone of Thrain," said Bilbo, "the Heart of the Mountain; and it is also the heart of Thorin. He values it above a river of gold. I give it to you. It will aid you in your bargaining." Then Bilbo, not without a shudder, not without a glance of longing, handed the marvellous stone to Bard, and he held it in his hand, as though dazed.