Billionaire investor Carl Icahn (pictured) has called on Apple to increase its share buyback programme, claiming the world's largest company is "dramatically undervalued".

In a letter to Tim Cook, Apple's chief executive, Mr Icahn said the iPhone maker should embark on a tender offer for its shares that would "meaningfully accelerate and increase the magnitude of share repurchases".

The activist investor insisted he would not tender any of his Apple shares if the company followed his recommendations. He said his valuation analysis of Apple suggested its shares should trade at $203 (€160) a share, nearly double their current price.

“The market misunderstands and dramatically undervalues Apple,” he said in the letter, which forecast that Apple’s earnings per share would grow 44 per cent for 2015 – above consensus forecasts.

Mr Icahn’s investment case for Apple rests on an ambitious set of forecasts for its products, including the prediction that it will release an UltraHD television set within two years.

“We think television represents a large opportunity for Apple, one that reaches far beyond ‘the hobby’ that Apple TV currently represents,” he wrote.

“Televisions are a centrepiece to the modern living room and thereby a promising gateway into the home for Apple’s growing ecosystem,” Mr Icahn said, including the caveat that Apple “has not announced plans for a TV set and may never do so”.

Mr Cook said earlier this year that he no longer believed television was a hobby for the company, after selling $1 billion worth of its set-top boxes last year.

Mr Icahn estimates Apple could sell 12 million TV sets costing an average of $1,500 in 2016. The veteran investor also believes Apple’s Watch can sell 20 million units in its first year on sale, rising to 72.5 million by 2017. – Copyright The Financial Times Ltd 2014