What two 100-year-old companies - IBM and Wells Fargo - have learned about staying relevant

If Wells Fargo hadn't gone out mobile app five years ago, it would be out of business, said its CEO and Chairman John Stumpf.

"Today half our customers are mobile and, not only are they only on mobile, that's their predominant use. If we weren't on mobile, we would be out of business. And that's just in five years," Stumpf said.

On stage at the Fortune Global Forum, the CEO of a 164-year-old banking company shared the spotlight with Ginni Rometty, CEO of the 104-year-old tech company IBM.

Both agreed that most important thing for their companies is to constantly be reinventing yourself. Otherwise, Stumpf joked, we would still be seeing the Wells Fargo horse-drawn carriages on the highways.

"The companies that are most enduring and have the best opportunity in the future are those that can reinvent in a way that's managed chaos," Stumpf said.

Rometty's IBM is showcase for a company that needs to change to stay relevant. She's not apologetic for more than $8 billion in divestitures during her tenure as CEO.

"Reinvention is not about protecting your past," Rometty said. "We did hardware for 60 years. Don't protect your past, and don't define yourself as a product."

Instead, CEOs at the crux of the market shift as everything goes digital must make big bets as "stewards for the long term," Rometty said.

It's not enough to be digital now because all companies will be digital. The question for today's CEOs is to answer what's next. IBM is focused on cloud, big data, and mobility, but cognitive is the future, Rometty said. Her big bet is clearly on its intelligence platform Watson and all of its applications.

Neither Rometty nor Stumpf are looking at the competition within their categories as guidance. "You are defined by your clients, you are not defined by your competition," Rometty said.

For Wells Fargo, a lot of innovation and reinvention is by focusing entirely on the customer and their needs. A bank can no longer be only a physical location, and the company would have died five years ago, Stumpf said, if it hadn't gone mobile.

"The people who influence us the most are outside," Stumpf said, listing companies like Amazon and Google. "They take complexity and make it simple. That has a huge influence. If we can't tell a customer exactly what is in their account after a 25 year relationship with them, but they can ask Google and find out something in 3 nanoseconds, how would we look?"