The fact that many current health-care plans do not offer all the benefits required under Obamacare means that many premiums are likely to jump dramatically, Aetna CEO Mark Bertolini told CNBC. Bertolini, appearing on Tuesday's "Closing Bell," said that most Americans with current plans are below the 60 percent essential Affordable Care Act benefit requirement, and individual plan participants will have to pay a minimum of a 20 percent increase to upgrade, he said.

Mark Bertolini, CEO of Aetna. Adam Jeffery | CNBC

Citing the current economy and reduced Medicare rates, Bertolini said the health-care company must find a way to bridge the gap between what it will cost seniors to pay out of pocket and what Aetna will be able to cover.

"Aetna alone will pass through to its customers over $1 billion in taxes and fees associated with the Affordable Care Act that need to go into the pricing," Bertolini said.

(Read more: Aetna CEO: Obamacare sites are flawed)