So he conducted a two-part study to try to put a price on those intangibles.

First he calculated the cost of switching from iOS to a competing platform — Google’s (GOOG) Android, say.

Then he conducted a survey of 1,000 iPhone and iPad customers to find out how much of a discount a competing product would have to offer before those customers would consider switching.

He reported his findings in a note to clients Friday. The key takeaways:

• iOS users are a loyal bunch. 94% are likely or highly likely to stick with Apple for their next smartphone or tablet. More than one in five said there isn’t a discount high enough to persuade them to switch.
• It costs a lot to switch. The explicit switching cost (to replace music, apps, etc.) comes to an average of $122 to $301 per device. The implicit switching costs (the “pain in the neck factor,” losing Apple’s better-than-average customer service, etc.) make the platform even stickier.
• Installed base matters. Shope estimates that by Saturday (when Apple’s third fiscal quarter ends), the installed base of iOS units will exceed 281 million, up 72% from the same quarter last year. Using a blended average selling price reduced by Apple’s gross margin, platform erosion etc., he calculates that the lifetime value to Apple of each iOS customer is $1,053.
• Apple’s greatest value is its customer base. Multiplying 281 million by $1,053, Shope gets a value for Apple’s installed base of roughly $295 billion. That’s about 55% of Apple’s current market capitalization. And that, he adds, “is before taking into consideration Apple’s $110 billion cash balance, the rapid growth of the installed base, the value of the Macintosh and traditional iPod businesses, and the related content, services and peripherals streams.”

There are risks, Shope writes, that the value of the platform will decrease significantly in the future, among them failure to innovate, regulatory scrutiny, market saturation, platform competition and a deterioration of switching costs. But he adds that these risks “are more than captured in Apple’s currently discounted valuation.”

Apple remains Goldman Sachs’ (GS) top pick in the sector with a 12-month target price of $850. It closed Thursday at $569.05.