Victims of the payment ­protection insurance mis-selling scandal could have been under-compensated by £1 billion, an investigation has found.

Customers who may possibly have been affected had PPI on credit cards issued by Lloyds Banking Group, Barclays, MBNA and Capital One, it was reported.

All these banks claim to make every effort to pay the correct amount of compensation.

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The estimated shortfall ­uncovered by the BBC arose because some fees and charges that were triggered by mis-sold PPI policies have been left out of compensation calculations, which also involve adding interest, thereby dramatically reducing the amounts some customers would have otherwise received.

It commissioned independent consultant Cliff D'Arcy, who spent 12 years as a banker working in the PPI industry, to estimate the overall size of the shortfall and how much banks would have to pay to make it good.

Mr D'Arcy said: "I'm confident that the figure will be somewhere in the region of a billion pounds of extra compensation."

His calculation was based on factors such as typical penalty charges on credit cards, how many credit card users had PPI, how many of those paid charges and what interest they had to pay.

The Financial Ombudsman Service (FOS), which resolves disputes between consumers and financial firms, said fees and charges triggered by mis-sold PPl premiums should be paid back.

Principal Ombudsman Caroline Wayman said: "If a fee is the result of the mis-sold PPI it should be given back, and if it's not included that would be a mistake."

Ms Wayman recommended that credit card customers who suspect their fees and charges were left out of their compensation calculations go back to their banks to ask what happened.

Which? said the Financial Conduct Authority (FCA) should investigate if any bank is found to have made systematic mistakes.

Lloyds Banking Group said: "When a customer lets us know that they may have incurred other costs because of their PPI policy, we will investigate and make an appropriate refund."

Barclays acknowledged that it has previously operated a month-by-month assessment of whether PPI premiums had triggered additional fees which, it said, met the requirements of the regulations and was independently assured. Barclays said it is now introducing a new, "enhanced" method.

MBNA said: "We are confident that our redress is correct, we have considered our methodology carefully and in detail. Our confidence is reinforced through external independent reviews."

Capital One said: "We aim to pay redress that puts the customer back in the position they would have been in if they had not had PPI."