'Frustrated', meet 'disappointed': Cigna spurns Anthem's $54B bid

Cigna believes a merger with Anthem would create a health insurance powerhouse, but the Connecticut-based insurer is still rejecting a $53.8 billion takeover bid from its larger rival, saying the offer is not in the best interest of shareholders.

"We are deeply disappointed with your recent actions," the company wrote in a letter signed by Cigna CEO David Cordani and Chairman Isaiah Harris, rebuking Anthem's board for taking their discussions public over the weekend.

"We have been engaged in good faith discussions with Anthem to determine whether a potential strategic combination is in the best interests of Cigna's shareholders," the letter added.

On Saturday, Anthem confirmed the two had been in serious talks since late May, even as it declared it was "frustrated" with Cigna's demands. The company revealed it had raised its offer to $184 a share, after Cigna rejected three prior offers starting on June 3rd. The offer represents more than a 35 percent premium to Cigna's May 28th close, when the firms began discussions in earnest.

Anthem CEO Joseph Swedish pointed the finger at Cordani and Cigna's board for insisting on retaining control of the combined firm—demands he was actually willing to meet. Swedish said Cordani would be named Chief Operating Officer of the combined firm, and Cignas would get 6 seats on the 14-member board.

"Despite the compelling aspects of these proposals, you continued to insist that Mr. Cordani be immediately appointed CEO of the combined company, or that we reach an acceptable position for his specific roles and responsibilities as well as the timing for a transition to CEO," Swedish wrote in a letter released Saturday.

In addition, Cigna demanded "that there be an almost equal Board split of directors in the combined company and that certain changes related to senior leadership changes," he added, a request that was apparently unacceptable to Anthem.

"I certainly have heard this from shareholders of both Anthem and Cigna, that most obvious successor to Swedish in an Anthem/Cigna combination is Cordani," said Leerink analyst Ana Gupte. "He has to just trust himself that would get that role," she said.

Read More'Frustrated' Anthem makes new offer to Cigna as Aetna moves in on Humana

In their counter letter, Cigna executives say their concerns go to Anthem's management.

"You are…facing a number of major issues, including Anthem's lack of a growth strategy, complications relating to your membership in the Blue Cross Blue Shield Association (BCBSA) and the related antitrust actions, and other significant challenges, such as the massive data breach you experienced in February," Cordani and Harris wrote.

"In fact, these fundamental issues, and your inability to address them in the context of a strategic combination, caused your management team, at your direction, to terminate our prior discussions earlier this year," they added.

With both sides hostile, Leerink's Gupte says it will now come down to shareholder pressure. Cigna stockholders may be hard pressed to turn down $184 per share on the table.

"This is a very attractive multiple. I think the shareholders of Cigna are likely very pleased," Gupte said.

With that in mind, "I'm not sure Anthem will go up much more from here," the analyst added. "Cigna could still try to make a bid for Humana, but I think there will certainly be some pressure from both Cigna and Anthem shareholders for the deal to get consummated."

Meanwhile, Humana is reportedly in talks to be acquired by Aetna, according to a report in The Wall Street Journal. Neither side has confirmed discussions, but the publication reports negotiations are ongoing.