After years of speculation, Apple finally made its play for the payments market last month, unveiling a compelling mix of NFC technology, tokenisation and biometrics designed to make paying at the till with a phone simple, quick and secure.

In an open letter to Apple CEO Tim Cook, Icahn, who owns 53 million shares in the company, says he expects the new payments feature to have "limited financial impact" in 2015 because retailers need to catch up and roll out NFC infrastructure.

However, things will then pick up quickly as Apple takes a significant slice of the US market's pie. Says the letter: "We estimate that, based upon Apple Pay's rumoured fee of 15 bps of all spend on credit and debit cards (US card spend was $4.2 trillion in 2012) and merchant deployment of NFC reaching 80%+ in 2017, Apple in the U.S. could generate revenues (also equivalent to gross margins, as the variable costs are de minimis) of $2.5 billion in FY 2017 if it reaches 30% market share of all spend on US credit and debit cards."

While Icahn's figures for NFC rollout and Apple's share of card spend appear wildly optimistic - the 30% market share quote is particularly overly-ambitious - he is even more bullish for the firm's longer term prospects because it dominates the premium market, meaning that its customers spend more than their peers, leaving it "unusually well positioned to succeed with Apple Pay where others could not".

More generally, Icahn - who recently got his wish when eBay agreed to spin off PayPal - is urging Apple to use some of its $133 billion cash pile to buy back more shares. The investor is pledging to keep hold of his own shares, saying that the company is dramatically undervalued and should be trading at $203, around double its current position.