One of the trickiest questions facing bailed-out banks that still have to repay federal aid may have just gotten easier.

The Treasury Department last week released its most comprehensive report on how it has unloaded warrants in more than four dozen lenders that have repaid the Troubled Asset Relief Program.

The data gives banks more insight into how much their warrants are worth and shows that investors have a healthy demand for these somewhat-novel securities at auction, experts said.

Failure to agree on price was a big reason that 13 of the 50 banks that repaid the government as of June 30 decided to let the Treasury auction their warrants rather than repurchase them.

The 28-page "Warrant Disposition Report" is the Treasury's second six-month update on this topic. It goes deeper than the first one in January and gives the more than 200 banks that still hold federal aid new grist for negotiations with Uncle Sam, experts say. After repaying the government, banks have about two weeks to decide whether to buy back the warrants or let them go to market.

KeyCorp, Huntington Bancshares Inc., M&T Bank Corp. and a handful of other midsize banks still have their federal aid.

"The more you know how these auctions have gone, there is going to be less of a gap between what you can negotiate and what you can get at auction," said Linus Wilson, an assistant professor of finance at the University of Louisiana at Lafayette who has studied warrant sales. "The banks that are negotiating to buy back their warrants will look at both of these reports very closely."

The warrants give the holder the right to buy a company's common shares at a set price for up to 10 years from their issue date. Most expire in 2018 or 2019.

The Treasury took them as a reward for helping banks through the crisis but came under pressure to sell them late last year. Figuring out exactly how much the warrants were worth was tough. These types of securities had never really traded en masse before. A lot of banks' share prices are lower than the so-called "strike price" of the warrants, complicating valuations. The heads of some companies like JPMorgan Chase & Co. said the Treasury was asking for too much money. They opted to let the government sell its warrants to outside investors.

The report showed that investor interest has tended to be highest in auctions that involved smaller companies with the fewest warrants up for grabs, like TCF Financial Corp. and First Financial Bancorp.

TCF's auction was 18.3 times oversubscribed, meaning that bids exceeded the number of warrants offered for sale by more than 18 times. First Financial's auction was 10.3 times oversubscribed. The ratio at JPMorgan Chase's auction was 6.5 times and at Wells Fargo & Co.'s, 3.9.

The report also showed that only one of the 13 banks that went the auction route — Wells Fargo — had repurchased its own securities. Though it was public knowledge that Wells had won back some of its warrants at auction in May, it was not clear whether other companies had done so.

Experts say there are several reasons why most banks did not buy any of their warrants.

For one thing, they had made it clear that they were not desperate to repurchase them when they let them be sold off, so they probably did not bid aggressively.

Also, the timing of Wells Fargo's auction gave it a unique chance to repurchase its own shares. A large offering that included warrants from Zions Bancorp. just before the Wells Fargo auction on May 20 probably sapped demand, analysts said. Also, the markets were unusually soft that day, which may have prompted Wells to repurchase about 70 million of the 110 million of warrants up for grabs to ensure that the auction did not fail for lack of demand.

"Had Wells Fargo not participated in the auction, it would have gone quite poorly," said Gary Townsend, the chief executive officer of Hill-Townsend Capital LLC. "I think they were in there supporting the auction."

The report also said that seven of the 13 never bid to repurchase their warrants in the run-up to their auctions. The seven are: Bank of America Corp., Washington Federal Inc., Signature Bank, Texas Capital Bancshares, PNC Financial Services Group Inc., Comerica Inc. and First Financial Bancorp.

Wilson, the Louisiana professor, speculated that some banks were wary of bidding on their own warrants for fear of sending a mixed signal with a lowball bid.

"They either don't want to repurchase them or they are reluctant to make a bid because they know their bid is going to be disclosed," Wilson said. "In one sense shareholders want you to get your best deal. In another sense, your shareholders don't want you to say your warrants aren't worth very much."

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