Shares in Regis Corp. partly recovered from a steep drop Thursday after the company made a rare public rebuttal of a financial analyst.

Regis’ shares fell as much as 19 percent in the first 40 minutes of trading as investors reacted to an estimate by a Piper Jaffray analyst that the company faced tens of millions of dollars in new costs to comply with the federal government’s extension of overtime pay.

The new rule, finalized on Wednesday and taking effect in December, doubles to $47,476 the maximum annual income a salaried worker can earn and still be automatically eligible for overtime pay. The rule will affect many businesses, particularly nonprofit firms, retailers, hotels and restaurants that have many management workers with salaries below the new threshold.

In a report issued before the market opened, Piper Jaffray analyst Stephanie Wissink estimated that Regis, an Edina firm that operates and franchises hair salons across the country, would encounter $81 million in annual incremental costs because of the rule. As a result of the expected new cost, she downgraded her rating on the company’s stock.

Regis itself had not provided an estimate of the cost for complying with the rule. In an interview, Wissink said she and her colleagues put together estimates for about 125 publicly traded companies that will be affected by the rule.

“We assumed Regis would fall in line with the broader group of consumer companies that we looked at,” Wissink said. “What they shared with me later in a follow-up conversation is that is not the case.”

In addition to contacting Wissink, Regis executives quickly drafted a six-sentence statement and issued it shortly after 9 a.m., a time when its stock had lost nearly one-fifth of its value. It said most Regis salon managers are paid by the hour rather than receiving a salary, meaning they wouldn’t be affected by the new rule.

It also provided a hard figure. “These new rules could increase our costs by up to $5 million per year,” the company said, adding it is looking at ways to reduce that.

Wissink issued a new report that reversed the downgrade. Regis shares bounced upward and finished the day 9.1 percent lower at $12.36.

“The stock did recalibrate and I think it settled in a position that was reflective of their comments on the initial impact,” Wissink said.

She called it an “intense day” and added, “We can appreciate where we need to take responsibility.”

Representatives at Regis did not respond to requests for comment. The company operates salons under the Supercuts, SmartStyle, MasterCuts and Regis brands, among others