His decision to stand down contrasts with the decision of Joe Palmer, the PIA's chairman, to stay at the helm after his former company, Legal & General, was fined pounds 180,000 in February for a series of regulatory failures.
The Securities and Investments Board, the industry's senior regulator, criticised Barclays Life over a series of rule breaches between September 1991 and June last year.
The subsidiary failed in its training of the 1,500-strong sales force and its supervision of the way they sold insurance products.
Mr Bignall, aged 53, has been chairman of Barclays Life for the past three years, having joined Barclays after leaving school at 18.
For the past 18 months he has been a member of the PIA board, helping to persuade banks and building societies to join the new regulatory body.
He said yesterday he accepted the SIB's rebuke, which related largely to his period as chairman at Barclays Life. 'We are not disputing that it is justified,' he said. 'When I arrived here it was evident that we needed to bring things up to a higher standard.
'It is now clear that, despite the reforms we brought to bear, I was not aggressive enough and did not institute changes in a number of areas. In the light of these criticisms I do not think it appropriate that I should stay on the board.
'The public should have confidence in the PIA and its board should be unblemished.'
He declined to comment on whether his resignation should be followed by that of Mr Palmer, saying: 'Everyone has to reach their own conclusion, but my personal view is there for all to see.'
A PIA spokesman said Mr Bignall's resignation was to be regretted. It was a personal decision.
Since the SIB's investigation last July, Barclays Life has almost doubled the number of its compliance staff while sales staff have now been reduced to about 1,100.
The company, which previously paid staff by commission only, is now moving to a salaried basis.
They will be paid an annual salary of between pounds 14,000 and pounds 17,000, plus bonuses.
A telephone hotline has been set up on 0800 494949 for customers with queries.
The SIB's rebuke is the heaviest penalty it can impose. Insurance companies have been able to choose whether to be regulated by the SIB or by Lautro, which has the power to impose unlimited fines.
Barclays Life has formally applied to join the newly-formed PIA, which has taken Lautro's place and also has the power to fine its members.
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