Would Newcastle have to pay back £4bn if its Wonga sponsorship was a loan?

Interest is tricky.

When Wonga announced their intention to sponsor Newcastle United FC, it generated no small amount of opprobrium. Despite the company's best attempts to generate a positive image for itself, it is still largely seen as a payday loan company, preying on the poor for a quick buck. So it was no surprise that smart a demonstration of that fact very quickly made the rounds:

Anything with over 2,000 retweets is going to get fact-checked a lot, and debates soon broke out over whether the number was accurate. Is it?

Wonga's "representative APR" is 4214 per cent. When you take out a loan with it, it decides at the beginning of the period what your interest is, and charges it to you on the total amount of capital borrowed over that period. In other words, it doesn't compound the interest - which makes sense, because it would be hard to compound anything over a loan as short month. As a result, if you were charged an annual interest rate of 4214 per cent, then at the end of a four year period you would have to pay back: 

£24,000,000 + £24,000,000 x 42.14 x 4 = £4,069,440,000.00                                 

(That's the capital, plus four years interest.) A shade over £4bn. So James Dixon is correct.

Except that the 4214 per cent APR is already compounded. As Wonga explains, industry regulations require it to present interest at an annual rate even if it doesn't make annual loans. To do this, it is required to take the amount of interest you would pay on its longest loan, a month-long one, and act as though you rolled it over, taking out larger and larger loans to pay off the interest as you go along. If we compounded Newcastle's loan similarly, then after four years it would owe:

£24,000,000 x (1+42.14)^4 = £83,125,028,034,051.84                                 

That is £83 quadrillion. It's over one hundred times world GDP, and in the ballpark for the total value of everything on earth.

But Wonga would maintain that using that interest rate is unfair. Although they are required to present their representative APR in that manner, they have never, and would never, charge it to a customer. The annual rate of interest which they actually charge is "just" 360 per cent, and the rest is made up of the compounding which they are forced to assume. If Newcastle's loan was taken out at that rate, it would have to pay back:

£24,000,000 + £24,000,000 × 3.6 x 4 = £369,600,000.00                                 

£370m is still quite a lot to pay for £24m, but it's nowhere near billions. And in actual fact, Newcastle wouldn't even pay that much. It's not a person, it's a business, and Wonga have - controversially - launched a division exclusively for lending to businesses. The largest and longest loan it offers is £15,000 for a year, which costs £19,350 to pay back, implying an APR of 29 per cent. If Newcastle borrowed £24m for four years at that rate, then if the interest compounded, it would equal:

£24,000,000 × (1+0.29)^4 = £66,461,491.44                                 

And if it was charged in one lump sum, it would equal:

£24,000,000 + £24,000,000 × 0.29 x 4 = £51,840,000.00                                 

The root of the problem is that Wonga isn't actually in the business of making multi-year, multi-million-pound loans. The assumptions we make in trying to squeeze their business model into a shape that lets us make that comparison are important, because they're the difference between paying back £52m and £83qdrn.

Front page of Wonga.com

Alex Hern is a technology reporter for the Guardian. He was formerly staff writer at the New Statesman. You should follow Alex on Twitter.

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PMQs review: Theresa May shows how her confidence has grown

After her Brexit speech, the PM declared of Jeremy Corbyn: "I've got a plan - he doesn't have a clue". 

The woman derided as “Theresa Maybe” believes she has neutralised that charge. Following her Brexit speech, Theresa May cut a far more confident figure at today's PMQs. Jeremy Corbyn inevitably devoted all six of his questions to Europe but failed to land a definitive blow.

He began by denouncing May for “sidelining parliament” at the very moment the UK was supposedly reclaiming sovereignty (though he yesterday praised her for guaranteeing MPs would get a vote). “It’s not so much the Iron Lady as the irony lady,” he quipped. But May, who has sometimes faltered against Corbyn, had a ready retort. The Labour leader, she noted, had denounced the government for planning to leave the single market while simultaneously seeking “access” to it. Yet “access”, she went on, was precisely what Corbyn had demanded (seemingly having confused it with full membership). "I've got a plan - he doesn't have a clue,” she declared.

When Corbyn recalled May’s economic warnings during the referendum (“Does she now disagree with herself?”), the PM was able to reply: “I said if we voted to leave the EU the sky would not fall in and look at what has happened to our economic situation since we voted to leave the EU”.

Corbyn’s subsequent question on whether May would pay for single market access was less wounding than it might have been because she has consistently refused to rule out budget contributions (though yesterday emphasised that the days of “vast” payments were over).

When the Labour leader ended by rightly hailing the contribution immigrants made to public services (“The real pressure on public services comes from a government that slashed billions”), May took full opportunity of the chance to have the last word, launching a full-frontal attack on his leadership and a defence of hers. “There is indeed a difference - when I look at the issue of Brexit or any other issues like the NHS or social care, I consider the issue, I set out my plan and I stick to it. It's called leadership, he should try it some time.”

For May, life will soon get harder. Once Article 50 is triggered, it is the EU 27, not the UK, that will take back control (the withdrawal agreement must be approved by at least 72 per cent of member states). With MPs now guaranteed a vote on the final outcome, parliament will also reassert itself. But for now, May can reflect with satisfaction on her strengthened position.

George Eaton is political editor of the New Statesman.