Activist and billionaire investor Carl Icahn has developed a very public history with Apple (AAPL -0.35%) this year. It began with some tweets in August that he had initiated a stake in the company. He expressed that Apple was undervalued and recommended that management boost the share repurchase program. Now he's back again, following up on his request by filing a precatory proposal asking shareholders to recommend a larger buyback.

Again, Icahn did not fail to inform his Twitter followers:

The precatory proposal is a request for a nonbinding vote, even if the majority is in favor. His request does not specify an exact amount for the repurchase.

But does a larger buyback at these levels make sense?

Even at $565, Icahn persists
Apple is trading considerably higher than when Icahn first began to stir up ruckus. In August, when he tweeted he had a large position in Apple and that he had spoken to Apple CEO Tim Cook about a larger buyback, Apple shares were trading below $500. As of this writing, shares are trading at about $565 -- about 23% higher than when Icahn tweeted about his Apple position in August. His request for a buyback even at $565 shows just how undervalued Icahn really believes Apple is.

That said, buying back shares at $565 isn't going to get the same results as it would when Apple was trading below $500. And $565 isn't what Icahn had in mind originally; in an open letter to Cook that Icahn made public in October, Icahn proposed a buyback at $525 per share. Notably, however, he did tell CNBC after he shared the letter that the buyback would still be compelling at the $550 to $575 level.

Apple's attractive fundamentals
At just 14 times earnings, it's tough for anyone to call this cash cow expensive. From almost every angle, Apple stock looks attractive. But how does Icahn view the stock?

For Icahn, it basically comes down to the company's cash on hand ($148.6 billion) and its ability to generate even more cash. He likes to view Apple after backing off net cash. Today, Apple trades at about 11 times earnings using this method. Furthermore, Icahn believes that "Apple will grow earnings per share at a rate well in excess of the S&P 500 for the foreseeable future." And boosting Apple's already-massive share repurchase program of $60 billion more would help Apple boost EPS even more aggressively. It makes sense.

But it's important to keep in mind that Carl Icahn is not generally a tech investor. So his advice should always be taken with a grain of salt. He's not the only person on the Street suggesting investors buy shares of Apple, though. More than half of the analysts polled by Thomson Reuters believe that Apple is a buy, and only three of 53 analysts believe Apple is rated an underperform or sell, based on recommendations from last month. Even more, 18 analysts held strong buy ratings on the stock.

Icahn buyback or not, his actions do highlight the fundamental attractiveness of the company's stock.

What do you think? Is Apple still undervalued at $565? Even more, is it undervalued enough for Apple to boost its share repurchase program?