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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Ansbach puts Europe's bravest politician under pressure

Angela Merkel must respond to a series of tragedies and criticisms of her refugee policy. 

Angela Merkel, the Chancellor of Germany, is supposed to be on holiday. Two separate attacks have put an end to that. The first, a mass shooting in Munich, was at first widely believed to be a terrorist attack, but later turned out to be the actions of a loner obsessed with US high school shootings. The second, where a man blew himself up in the town of Ansbach, caused less physical damage - three were seriously injured, but none killed. Nevertheless, this event may prove to affect even more people's lives. Because that man had come to Germany claiming to be a Syrian refugee. 

The attack came hours after a Syrian refugee murdered a pregnant Polish woman, a co-woker in a snack bar, in Reutlingen. All eyes will now be on Merkel who, more than any other European politician, is held responsible for Syrian refugees in Europe.

In 2015, when other European states were erecting barriers to keep out the million migrants and refugees marching north, Merkel kept Germany's borders open. The country has resettled 41,899 Syrians since 2013, according to the UNHCR, of which 20,067 came on humanitarian grounds and 21,832 through private sponsorship. That is twice as much as the UK has pledged to resettle by 2020. The actual number of Syrians in Germany is far higher - 90 per cent of the 102,400 Syrians applying for EU asylum in the first quarter of 2016 were registered there. 

Merkel is the bravest of Europe's politicians. Contrary to some assertions on the right, she did not invent the refugee crisis. Five years of brutal war in Syria did that. Merkel was simply the first of the continent's most prominent leaders to stop ignoring it. If Germany had not absorbed so many refugees, they would still be in central Europe and the Balkans, and we would be seeing even more pictures of starved children in informal camps than we do today. 

Equally, the problems facing Merkel now are not hers alone. These are the problems facing all of Europe's major states, whether or not they recognise them. 

Take the failed Syrian asylum seeker of Ansbach (his application was rejected but he could not be deported back to a warzone). In Germany, his application could at least be considered, and rejected. Europe as a whole has not invested in the processing centres required to determine who is a Syrian civilian, who might be a Syrian combatant and who is simply taking advantage of the black market in Syrian passports to masquerade as a refugee. 

Secondly, there is the subject of trauma. The Munich shooter appears to have had no links to Islamic State or Syria, but his act underlines the fact you do not need a grand political narrative to inflict hurt on others. Syrians who have experienced unspeakable violence either in their homeland or en route to Europe are left psychologically damaged. That is not to suggest they will turn to violence. But it is still safer to offer such people therapy than leave them to drift around Europe, unmonitored and unsupported, as other countries seem willing to do. 

Third, there is the question of lawlessness. Syrians have been blamed for everything from the Cologne attacks in January to creeping Islamist radicalisation. But apart from the fact that these reports can turn out to be overblown (two of the 58 men arrested over Cologne were Syrians), it is unclear what the alternative would be. Policies that force Syrians underground have already greatly empowered Europe's network of human traffickers and thugs.

So far, Merkel seems to be standing her ground. Her home affairs spokesman, Stephan Mayer, told the BBC that Germany had room to improve on its asylum policy, but stressed each attack was different. 

He said: "Horrible things take place in Syria. And it is the biggest humanitarian catastrophe, so it is completely wrong to blame Angela Merkel, or her refugee policies, for these incidents." Many will do, all the same.