Coalition ministers were so keen to ensure the group became long-term backers of the company some were asked in advance what price they would pay for shares.
But the spending watchdog yesterday revealed 12 of them sold some or all of their shares within weeks as their price soared.
Standard Life last night confirmed it had sold 99% of the shares it bought within months.
As anger grew over what Labour described as a "fire sale", there were calls for Business Secretary Vince Cable, who oversaw the privatisation, to resign.
Brian Scott of the Unite union said: "The Government must name the 16 priority investors they did a deal with. The UK taxpayer needs to know why they were selected as priority investors.
"It is unbelievable that any government thought it could reach a gentleman's agreement with rapacious investors who were out to make a fast buck."
At a press conference, David Cameron said taxpayers benefited in three ways from the sale , including "the capital receipt … the benefit that this is now a profit-making company paying taxes into the Exchequer"
He added: "This is a successful company doing well, doing well for Britain."
Mr Cable also refused to back down over for the sale price, telling MPs: "The last thing I intend to do is apologise".
Given previous fears that ministers would fail to find buyers for the Royal Mail, he said: "I would have thought that caution in this context had a lot to commend it."
Labour's shadow trade and investment minister Ian Murray said: "Vince Cable said he wanted the Royal Mail shares in the hands of long-term investors but most of those investors have now sold out, making considerable profits, and who would blame them given the knock-down price that they were initially sold at? The spending watchdog has said the taxpayer has lost out to the tune of £750m."
Mr Cable's shadow, Labour's Chuka Umunna, called the privatisation a "first-class disaster" and said people the Business Secretary had referred to as "spivs and gamblers" are laughing all the way to the bank.
The National Audit Office (NAO) report found 12 priority investors sold all or some of their holdings within the first few weeks of trading.
The report said Mr Cable's cautious approach led to shares priced at a level "substantially below" the initial trading price.
On the first day of trading last year, Royal Mail's shares closed at 455p, 38% higher than their sale price, boosting their value by £750. The 16 priority investors were allocated shares worth £728 million.
Labour MP Adrian Bailey, who chairs the business select committee, said the report underlined allegations that the Government sold Royal Mail "at any cost", and he said Mr Cable should consider his position.
The SNP spokesman for postal services, Mike Weir, said: "It is completely indefensible as part of Westminster's rush to sell of Royal Mail that taxpayers have been short-changed."