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: Jeremy Corbyn turns "the nasty party" back on Theresa May

The Labour leader exploited Conservative splits over disability benefits.

It didn't take long for Theresa May to herald the Conservatives' Copeland by-election victory at PMQs (and one couldn't blame her). But Jeremy Corbyn swiftly brought her down to earth. The Labour leader denounced the government for "sneaking out" its decision to overrule a court judgement calling for Personal Independence Payments (PIPs) to be extended to those with severe mental health problems.

Rather than merely expressing his own outrage, Corbyn drew on that of others. He smartly quoted Tory backbencher Heidi Allen, one of the tax credit rebels, who has called on May to "think agan" and "honour" the court's rulings. The Prime Minister protested that the government was merely returning PIPs to their "original intention" and was already spending more than ever on those with mental health conditions. But Corbyn had more ammunition, denouncing Conservative policy chair George Freeman for his suggestion that those "taking pills" for anxiety aren't "really disabled". After May branded Labour "the nasty party" in her conference speech, Corbyn suggested that the Tories were once again worthy of her epithet.

May emphasised that Freeman had apologised and, as so often, warned that the "extra support" promised by Labour would be impossible without the "strong economy" guaranteed by the Conservatives. "The one thing we know about Labour is that they would bankrupt Britain," she declared. Unlike on previous occasions, Corbyn had a ready riposte, reminding the Tories that they had increased the national debt by more than every previous Labour government.

But May saved her jibe of choice for the end, recalling shadow cabinet minister Cat Smith's assertion that the Copeland result was an "incredible achivement" for her party. "I think that word actually sums up the Right Honourable Gentleman's leadership. In-cred-ible," May concluded, with a rather surreal Thatcher-esque flourish.

Yet many economists and EU experts say the same of her Brexit plan. Having repeatedly hailed the UK's "strong economy" (which has so far proved resilient), May had better hope that single market withdrawal does not wreck it. But on Brexit, as on disability benefits, it is Conservative rebels, not Corbyn, who will determine her fate.

George Eaton is political editor of the New Statesman.