A growing conviction that they could not prevail over dissident shareholders led chief executive Robert Mylod and the directors of Michigan National Corp. to make the agreement on Sunday to sell the company, sources close to the deal said.

The $1.56 billion, $110-a-share agreement with National Australia Bank Ltd. came as the dissidents had reportedly amassed enough votes to oust the board and call for the sale of the bank.

One source said that Mr. Mylod, who had repeatedly vowed to keep Michigan National independent, became convinced its largest shareholder was going to nominate an alternate slate of directors.

Mr. Mylod "couldn't be guaranteed of winning, despite all the things he had done in the past year," the source added, alluding to a cost-cutting plan the CEO had implemented.

Excoriated at the 1994 shareholders meeting and facing further pressure from that big shareholder, Mutual Shares, Michigan National directors "had a real concern that they weren't going to make it through the next annual meeting," added one former officer.

Although the directors of Michigan National were preferred independence, "they realized they needed to be realistic, given the bank's performance outlook and the pressure from Wall Street," the officer said.

The National Australia deal "provides a face-saving solution for the board and Mr. Mylod," this source added.

As part of the agreement, My. Mylod will resign when the transaction is completed - probably within six to eight months - but he will remain as chairman for an indeterminate period.

Douglas E. Ebert, Michigan National's president and chief operating officer, will become chief executive officer.

Mr. Mylod would not comment on his decision, but said from preliminary conversations, the company's major stockholders were happy with the deal.

"The price is fantastic," said Ray Garea of Heine Securities, parent of Mutual Shares and owner of 8.25% of total shares outstanding.

National Australia contacted Michigan National two years ago about a possible sale, but it was CS First Boston Corp. that brought the two together this time.

The investment bank advised Michigan National during its tender offer late last year, when the bank bought back 14% of its shares.

"The credit goes to First Boston," a source said.

Keefe, Bruyette & Woods Inc. advised Michigan National, and Morgan Stanley & Co. wrote the fairness opinion for National Australia.

Michigan National did not take other bids, a source said, but First Bank System was the main competition. Norwest Corp. was also rumored to be interested.

My. Mylod said he would entertain rival bids, but thought they were highly unlikely.

Because of a share repurchase program initiated by Mr. Mylod, a stock swap would have been impractical because of accounting rules that prevent repurchased stock from being used in a pooling.

While the cash agreement means investors will be taxed on their gains, it still represents a good deal for most of them, who bought in when the stock traded below $80.

The shares surged $12.875 Monday to close at $101.625.

Nonetheless, dissident shareholder Stanford C. "Bud" Stoddard, a former chairman, criticized the deal.

"Michigan National would be better served if it merged into a large, profitable, interstate banking company where you know what that bank can do for you," he said.

Mr. Stoddard also questioned the timing of the deal, asking whether Mr. Mylod had deluded shareholders when he offered them the tender offer, knowing the bank would be sold.

But Mr. Mylod said negotiations with NAB began in January, a month after the tender offer was completed.

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