Addressing Conservative party members at the party's spring conference in London today, he said that he had learnt lessons over the past week.
"It has not been a great week. I know that I should have handled this better, I could have handled this better. I know there are lessons to learn and I will learn them," he said at the start of his address to Tory activists.
In reference to partial statements from his Downing Street office, he added: "Don't blame Number 10 Downing Street or nameless advisers, blame me. I was obviously very angry about what people were saying about my dad. I loved my dad, I miss him every day. He was a wonderful father and I'm very proud of everything he did."
"But I mustn't let that cloud the picture. The facts are these: I bought shares in a unit trust, shares that are like any other sorts of shares and I paid taxes on them in exactly the same way. I sold those shares. In fact, I sold all the shares that I owned, on becoming prime minister," he said.
He announced that he will publish his tax returns for the past few years in an attempt to settle the row over his taxes.
He said: "Later on I will be publishing the information that goes into my tax return, not just for this year but the years gone past because I want to be completely open and transparent about these things."
"I will be the first prime minister, the first leader of a major political party, to do that and I think it is the right thing to do," he added.
The Panama Papers revelations have sparked political reaction around the world, including India, where high-profile figures have been implicated for tax avoidance.
Cameron had been under increasing pressure as he refused to give details of his family's money held offshore in a Caribbean tax haven.
His late father Ian Cameron's name is allegedly named in the over 11 million leaked documents from the Panama-based law firm Mossack Fonseca that were shared by the International Consortium of Investigative Journalists (ICIJ) with 107 media organisations around the world.
According to the Consortium, Ian Cameron used Mossack Fonseca's services to shield profits from his investment fund, Blairmore Holdings Inc, with a series of expensive and complicated arrangements.
Ian Cameron, who died in 2010, was a director of Blairmore, an investment fund run from the Bahamas but named after the family's ancestral home in Aberdeenshire.
On Thursday night, in a television interview, Cameron admitted he had profited from his father's off-shore fund.
"I did own stocks and shares in the past, quite naturally because my father was a stockbroker. I sold them all in 2010, because if I was going to become Prime Minister I didn't want anyone to say you have other agendas, vested interests. (My wife) Samantha and I had a joint account. We owned 5,000 units in Blairmore Investment Trust, which we sold in January 2010. That was worth something like 30,000 pounds," he told 'ITV News'.
The Opposition Labour party had accused him of "hypocrisy" asLabour's deputy leader Tom Watson said Cameron had in the past called people who invested in similar schemes "morally wrong".
Another Labour MP, John Mann, yesterday called on the Prime Minister to resign and referred him to Parliament's standards watchdog.
He claimed that Cameron should have registered his shareholding when he was in Opposition because it could have influenced his conduct.
Downing Street said the PM had followed the rules.
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