I doubt a company could take itself private. Who would own it? Would it own itself?
MaysonL's explanation in the grandfather comment to this one is right. You do have to take into account that the 85% that will now be the new 100% will have less capital on hand, though.
If the stock market price is efficient, the new apple should only have a market cap of 85% of the old one. If it's more (or less), the stockmarket is not really rational..