I don't want to discourage you from doing this, but since you're being allowed to do this because the wild animals are killing the domesticated animals that you should go after them with something you're almost 100% gauranteed of killing them with?

I'd hate to see a conversation along these lines;

"Well Bob, did you get anything today? We've had another two chickens killed this morning."
"No Ma'am, I wasn't able to, I (Missed/Wounded/Etc) one though."
"Why's that Bob?"
"Well you see, shooting a bow many things can go wrong. I'm sure this time it was because of a (Branch/Twig/Blade of Grass/Breeze) that got in my way."

If they're killing the animals you need to get them taken care of in the quickest most efficient way possible in my eyes, not the way that's going to bring you the most enjoyment.