That is the strangest down-vote I've ever received on a textbook answer straight from Finance 101.
Simple example:
company A has 100 shares outstanding and only 1 asset: $100
The value of each share is $1.
It then decides a share buy-back is a good strategy, and buys back 50 shares for $50.
It now has 50 shares outstanding and $50 of assets.
The value of each share is still $1.
The buy-back has no effect on share price.
In any case -- to answer your question -- in your example, it would not have any effect on share price as it seems the company is just a holding company for $100. However, in the case of Apple they have other assets besides the cash (brand, intellectual property) and those assets produce more cash -- so all the cash the other assets generate in the future gets distributed to a smaller number of shareholders.
But you answered as a fact that a share buyback increases share price, and sorry but I don't agree with that point - as demonstrated, for an asset holding company it is not the case.
And for a company with cash generation capabilities there is still a subtle point about whether the company can create more value with the cash than the shareholders; if the company can create more value than shareholders, then removing cash from the company via a share buyback should reduce share price. If it cannot, then removing cash from the company should increase the share price.
Since Apple is simply hoarding the cash, we've already ruled this out -- Apple themselves admits they do not have any way to deploy the cash that would produce more returns than just stashing it in a 0.07% interest bearing account. The question then just comes down to how they disburse the cash (keep vs buyback vs dividends)
In my view you really should qualify your earlier statement that "Buy-backs are more tax efficient because they increase stock price" as it is really not a generally true statement. If it was, I would be in the business of buying companies, then making them use any free cash to buy back some shares from me (my remaining shares then somehow go up in value), then selling the remainder back to the market. Free money, it would be great!
The 80s were well known for corporate mavens buying undervalued companies, splitting them up, and selling the assets individually. The practice resulted in a net profit.
But that's bit different from saying that a stock buyback will always increase share price.