China has four of the five most valuable financial technology start-ups in the world, according to CB Insights, with Ant Financial leading the way at $60 billion. And investments in financial technology rose 64 percent in China last year, while they were falling 29 percent in the United States, according to CB Insights.

In Africa, the financial industry has been shaken up by the rise of new forms of mobile phone-based payment systems like M-Pesa, in Kenya, which was started by the local mobile phone operator rather than a bank, and has risen to become the dominant form of payment in the country.

The obvious reason that financial start-ups have not achieved the same level of growth in the United States is that most Americans already have access to a relatively functional set of financial products, unlike in Africa and China.

Some start-ups looked at becoming banks, so they could offer deposit insurance — among other things — but they generally found that getting a bank charter required more time and money than is available to even the most successful start-ups. This left most start-ups reliant on banks to hold and move any money they collected from customers.

“Operating in America, you needed to get a gatekeeper to approve your access to the networks,” said Josh Reich, who sold his banking start-up, Simple, to the Spanish banking giant BBVA. “If we wanted to touch their money, we needed to be part of the banking system.”

Online lenders were particularly ambitious in trying to take on the banks from outside the system. Lending Club, which makes personal loans, and OnDeck, which focuses on small business loans, initially grew swiftly and went public, but both companies have run up against the limits of how fast a lending business can grow without being a bank.

The online lenders have struggled with the high cost of acquiring new customers through marketing, which is not as necessary in other industries that have been disrupted by technology.