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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Why the Liberal Democrats by-election surge is not all it seems

The Lib Dems chalked up impressive results in Stoke and Copeland. But just how much of a fight back is it?

By the now conventional post-Brexit logic, Stoke and Copeland ought to have been uniquely inhospitable for the Lib Dems. 

The party lost its deposit in both seats in 2015, and has no representation on either council. So too were the referendum odds stacked against it: in Stoke, the so-called Brexit capital of Britain, 70 per cent of voters backed Leave last June, as did 62 per cent in Copeland. And, as Stephen has written before, the Lib Dems’ mini-revival has so far been most pronounced in affluent, Conservative-leaning areas which swung for remain. 

So what explains the modest – but impressive – surges in their vote share in yesterday’s contests? In Stoke, where they finished fifth in 2015, the party won 9.8 per cent of the vote, up 5.7 percentage points. They also more than doubled their vote share in Copeland, where they beat Ukip for third with 7.3 per cent share of the vote.

The Brexit explanation is a tempting and not entirely invalid one. Each seat’s not insignificant pro-EU minority was more or less ignored by most of the national media, for whom the existence of remainers in what we’re now obliged to call “left-behind Britain” is often a nuance too far. With the Prime Minister Theresa May pushing for a hard Brexit and Labour leader Jeremy Corbyn waving it through, Lib Dem leader Tim Farron has made the pro-EU narrative his own. As was the case for Charles Kennedy in the Iraq War years, this confers upon the Lib Dems a status and platform they were denied as the junior partners in coalition. 

While their stance on Europe is slowly but surely helping the Lib Dems rebuild their pre-2015 demographic core - students, graduates and middle-class professionals employed in the public sector – last night’s results, particularly in Stoke, also give them reason for mild disappointment. 

In Stoke, campaign staffers privately predicted they might manage to beat Ukip for second or third place. The party ran a full campaign for the first time in several years, and canvassing returns suggested significant numbers of Labour voters, mainly public sector workers disenchanted with Corbyn’s stance on Europe, were set to vote Lib Dem. Nor were they intimidated by the Brexit factor: recent council by-elections in Sunderland and Rotheram, which both voted decisively to leave, saw the Lib Dems win seats for the first time on massive swings. 

So it could well be argued that their candidate, local cardiologist Zulfiqar Ali, ought to have done better. Staffordshire University’s campus, which Tim Farron visited as part of a voter registration drive, falls within the seat’s boundaries. Ali, unlike his Labour competitor Gareth Snell and Ukip leader Paul Nuttall, didn’t have his campaign derailed or disrupted by negative media attention. Unlike the Tory candidate Jack Brereton, he had the benefit of being older than 25. And, like 15 per cent of the electorate, he is of Kashmiri origin.  

In public and in private, Lib Dems say the fact that Stoke was a two-horse race between Labour and Ukip ultimately worked to their disadvantage. The prospect of Nuttall as their MP may well have been enough to convince a good number of the Labour waverers mentioned earlier to back Snell. 

With his party hovering at around 10 per cent in national polls, last night’s results give Farron cause for optimism – especially after their near-wipeout in 2015. But it’s easy to forget the bigger picture in all of this. The party have chalked up a string of impressive parliamentary by-election results – second in Witney, a spectacular win in Richmond Park, third in Sleaford and Copeland, and a strong fourth in Stoke. 

However, most of these results represent a reversion to, or indeed an underperformance compared to, the party’s pre-2015 norm. With the notable exception of Richmond’s Sarah Olney, who only joined the Lib Dems after the last general election, these candidates haven’t - or the Lib Dem vote - come from nowhere. Zulfiqar Ali previously sat on the council in Stoke and had fought the seat before, and Witney’s Liz Leffman and Sleaford’s Ross Pepper are both popular local councillors. And for all the excited commentary about Richmond, it was, of course, held by the Lib Dems for 13 years before Zac Goldsmith won it for the Tories in 2010. 

The EU referendum may have given the Lib Dems a new lease of life, but, as their #LibDemFightback trope suggests, they’re best understood as a revanchist, and not insurgent, force. Much has been said about Brexit realigning our politics, but, for now at least, the party’s new normal is looking quite a lot like the old one.