To understand why all this adds up to trouble demands an appreciation of the key financial concept behind the explosion in motor financing. “Residual value” is the complex calculation used by manufacturers’ financial services divisions and other lenders when they create personal contract plans — the leasing agreements used in most car sales today, under which the customer effectively pays for the cost of the expected loss in value of their vehicle over the life of the contract

It estimates what the car being sold will be worth at the end of the agreement, with the manufacturer offering a guaranteed resale price, subject to terms and conditions, when the personal contract plan expires. If a company wants to encourage sales, it will use an optimistic number, or higher estimated value. A more cautious carmaker might opt for a gloomier prediction, making the contract more expensive but reducing the risk of a loss when the car is resold if the customer decides not to buy it outright.

FCE Bank, Ford’s lending business, applies a relatively conservative 5 per cent to 8 per cent discount. However, this is less to do with its fears about a market fall and more about generating “equity for the customer at the end of the contract”, giving the buyer more incentive to take out a new PCP for a brand new car when the agreement ends.

Not everybody is so conservative. Last month analysts at Sanford Bernstein examined leasing deals offered by Blue Chilli Cars, a specialist motor finance broker. Using the terms on offer for a range of popular cars, including the Volkswagen Golf and Mercedes-Benz C Class, they calculated the implied residual values being offered by some lenders and compared them with the used car price provided by Parkers, a British second-hand car valuation company.

In the samples they examined, the analysts found a difference of £1,777 to £4,794 between the guaranteed price offered by the lender which, to be clear, is not Blue Chilli itself, and the likely resale value of the cars at the end of the contracts.

Multiply this by hundreds of thousands of cars and several lenders and you have the prospect of big losses down the line.

The chief executive of a British bank believes that the car finance industry is playing a dangerous game, having used personal contract plans to boost customer demand for their products in the years after the financial crisis.

“These guys have effectively got themselves trapped into a renewal cycle, with all the PCPs written between 2011 and 2013 falling due,” he said. “The market has since normalised and that means that the same monthly payment that once got you a top-of-the-range car doesn’t today and that doesn’t really appeal to the consumer. However, if you are the lender and the manufacturer, you have more of the value chain to play around with.

“Now [car manufacturers] have profits from lending, which they didn’t have previously, they can make a combined price to incentivise the customer to renew on a like-for-like basis.”

Not all auto industry managers are delighted with this development and the Sanford Bernstein analysts quoted one unnamed German executive describing how “the finance guys constantly ask us to support higher residual values, as it makes their offers more competitive. But since we are underwriting these values, it is often a difficult conversation.”

Sanford Bernstein estimates that the residual value exposures built up by Europe’s leading car manufacturers could leave them with heavy losses if the market changes. They calculate that the cost of writedowns on diesel cars alone resulting from new regulation could reach €1.7 billion. Rises in interest rates would be a blow, too, with a one percentage point increase knocking pre-tax earnings from VW’s financial services unit by 30 per cent.

While this may be painful for the carmakers, it would be far from the carnage wrought on consumers by the sub-prime crisis. Adrian Dally, head of motor finance at the Financing and Leasing Association, rejects claims that the auto lending industry is heading for a fall comparable with the American housing market in the run-up to the financial crisis of 2008.

“No, there is no bubble here,” he said. “The dynamics are very different. Houses go up and down in value. With vehicles, the value of those assets only go down but in a very predictable way. It’s faster for some brands than others, but lending in relation to cars is much safer than sub-prime mortgages. Also, the sub-prime mortgages were packaged and sold on and on and on. Motor finance is very well underwritten.”

That kind of optimism is not universal. “Like everything in the debt-driven world, the manufacturers have brought forward future consumption and inflated demand,” the money manager at the US bank said. “At some point, this is going to unwind and they will be on the wrong end of the trade.”

The problem could get worse as carmakers drive sales of electric cars and other alternatively fuelled vehicles. Unlike with established technology, where depreciation can be predicted with accuracy, the residual value calculations on the new generation of cars will be more difficult because of the pace at which engineering changes and the desire by manufacturers to grab a large market share. Financing terms, therefore, might become even more aggressive.

Analysts at Moody’s, the credit ratings agency, have also aired their fears, saying in a report last month: “Carmakers’ new auto technologies combined with attractive financing terms and risk retention will prove a risky strategy to stay on top of the market, but [it] is one to which they have few strategic alternatives,” Falk Frey, senior vice-president at Moody’s, wrote.

With Über, Google, Apple, Tesla and others making big investments in driverless technology and electric cars, motor finance is providing a glimpse into a future where most cars are rented, not bought. Coping with the revolution will be the biggest test yet for the industry.