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In the topsy-turvy world of private equity math, what at first glance looks like a multibillion-dollar loss can in fact be a huge gain.

Such is the case with the Blackstone Group’s investment in the Hilton hotel group, which on Thursday returned to the public stock markets.

When Blackstone took Hilton private in 2007, paying a headline-grabbing price of $26 billion, it appeared to have been badly mistimed. One of the biggest deals during the last buyout boom, it came just months before financial markets seized up and travel — and hotel bookings — fell into a prolonged slump.

And when Hilton Worldwide Holdings’ stock began trading again on Thursday, the company had a market capitalization of just $20 billion. Doing the back-of-the-envelope math, that would suggest a loss in value of $6 billion.

But in fact, Blackstone has increased the value of its investment by nearly $10 billion through a combination of lucky timing, smart financial engineering and disciplined management.

“They paid a premium price at the peak of the market,” said Robert M. La Forgia, the chief financial officer of Hilton at the time of its sale, who now runs Apertor, a hospitality consultant firm. “But they were able to ride out the downturn, a significant real estate crisis and a financial crisis, and still come back and have a successful I.P.O.”

In buying Hilton, Blackstone contributed about $5.5 billion of cash to the deal, and borrowed about $20.5 billion from big banks.

It is an arrangement not unlike that of individuals who pay for homes with a down payment in cash coupled with a large mortgage.

“When they bought Hilton for $26 billion, it was like buying a very big house,” said Steven Kaplan, a professor at the University of Chicago Booth School of Business. “And they financed it like a house, taking on debt.”

Over the next couple of years, Blackstone used profits from Hilton to pay down that debt.

Often, private equity firms will take profits and borrow additional money to pay themselves special dividends, resulting in an early windfall that hedges their risk.

But by opting against dividends, and instead paying down debt, Blackstone was slowly but surely increasing the value of its equity in the company.

Then the financial crisis hit.

Hotel visits plunged, banks got nervous and Blackstone wrote down the value of its investment by more than half. This caused the value of the banks’ debt to plummet. In 2009, it looked as if the Hilton deal could be a disaster for the ages.

But in 2010, Blackstone approached its lenders and offered to restructure the deal. Led by its global head of real estate, Jonathan Gray, Blackstone offered to buy back some of the bank debt at a discount. Some lenders received just 35 cents on the dollar.

Other lenders converted their debt into preferred equity, receiving shares to sell in an eventual I.P.O. As part of the deal, Blackstone agreed to inject more capital into the business, bringing its total equity investment to around $6.5 billion.

“It was like refinancing your mortgage when interest rates were low,” Mr. Kaplan said. “They basically paid off their debt when it was very cheap to do so, because everybody was frightened and the price of their debt went very low.”

Since then, Blackstone has continued to pay off Hilton’s lenders with profits from the business. It has also cut costs from Hilton, expanded its international strategy and focused on the more profitable franchise model.

Through the I.P.O. on Thursday, Hilton raised about $2.4 billion. Some proceeds from the I.P.O. will go toward paying down the debt further, while some of it will go to the debt investors who converted their shares into preferred equity during the restructuring in 2010. Blackstone is not selling any of its shares.

Hilton today is larger and more profitable than it was when Blackstone bought it out, and the outlook for the hotel industry is good. The company is also in sound financial shape.

Between its regular debt servicing, the restructuring in 2010 and proceeds from the I.P.O. that will be used to pay lenders, Hilton will have about $12 billion in debt, down from $20.5 billion at the time of the buyout.

“They’ve accomplished a lot through leverage,” Mr. La Forgia said. “They almost lost the company, and might have without the debt restructuring.”

On Thursday, Hilton’s first day of trading, shares were up 7.5 percent to $21.50, giving the company a market capitalization of $21.2 billion. Adding the remaining $12 billion of debt gives it an enterprise value of about $33 billion. In other words, the overall value of the business actually increased by about 27 percent.

But by aggressively paying down its debt and renegotiating with the banks at an opportune time, Blackstone’s gains have been much more substantial.

With 76 percent of the equity, Blackstone’s stake in Hilton is worth $16.1 billion. That is a profit, on paper at least, of more than $9.5 billion.

That sounds like a lot of money, but on Wall Street, everything is relative.

Mr. Kaplan of the University of Chicago said that compared to an investment in the public markets, Blackstone’s investment in Hilton has been good but not great. Since the start of 2007, the Standard & Poor’s 500 stock index is up 25 percent. Blackstone more than doubled its money.

“This is a good deal if you’re measuring it relative to the public market,” Mr. Kaplan said. “But it’s not a home run.” Other alternative investments and asset classes have performed better over the last six years.

Even against Blackstone’s internal expectations, the Hilton deal, while an enormous winner, may not tick every box. Most private equity firms aim for an annual internal rate of return of about 18 to 20 percent. Spread over six years, the investment in Hilton looks to have yielded about 16 percent for Blackstone. “If you look at it against target returns, it’s not amazing,” Mr. Kaplan said.

But with its commanding stake in a newly public Hilton, Blackstone has nonetheless engineered one of the most successful deals in the firm’s history.

“In dollars, a $10 billion profit is a lot of money,” Mr. Kaplan said. “Even to them.”