The Fallacy of Individual Event Probability – A Real Life Example

Identifying the proportion of a population willing to do X with the probability that any individual in that population will do X is a fallacy – the Fallacy of Individual Event Probability. What follows is a real example of it in action.

A very large bank of my acquaintence has a gadget that calculates the value of a given loan L of amount £A and term T months for a APR of r, call it V(r; L). It also has a gadget to calculate the probability that a customer in a credit segment S will accept an APR of r for the loan L, call it P(r). Maximise the product – the expected value - of the two by changing r. The APR that maximises the expected value is what the customer gets.

The most important thing the bank needs to know is what the default rate or bad debt will be from lending to a customer. Ideally, every customer should pay back the original loan and the interest, but in practice some don't. Credit analysts spend a lot of time trying to work out parameters that will tell them who doesn't and to what extent. In particular, they worked out that people in the segment S will lose x% of the original loan on average. That number is used to calculate the price r_max. Let the probability of purchase at r_max be p_max. At the lower-quality end of the risk scale, the maximising APR is surprisingly high and the corresponding probability of purchase is astonishingly low – around twenty per cent and sometimes even lower.

Take a look at the buying psychology. Each of the customers in the segment has a price beyond which they won't go, either because they can't afford the repayments, because they think they can better elsewhere, because the sheer folly of the entire enterprise hits them when they hear the cost or a dozen other reasons. Call that APR the shut-off point r_shut. Let the proportion of customers in this segment willing to accept a price of r or higher be D(r). When you set the APR via the maximisation, you are only getting those customers for whom r_shut >= r_max.

Here's why the bank has a problem. They are treating P(r) as the probability that any customer in the segment will buy at the rate r: their calculations assume that every customer in the segment has a probability of p_max of accepting the r_max. This makes the use of the loss rate of x% in the calculations correct. But it doesn't work like that. Why not? Because it's a logical impossibility: an individual customer either accepts the price or not, there's no probability. Uncertainty, yes, probability, no. What their gadget actually tells them is D(r) – the proportion of people willing to put up with a rate of r or greater. But this means they need the loss rate for that sub-segment, not for the whole segment, when they calculate the value of the loan. They have gone from a proportion of a population to a probability over the whole population and been mislead about the nature of the model they need in so doing.

Like many fallacies, a suitably benign world will not punish anyone too harshly for commiting it. To get round it in this case, the bank either has to calculate the appropriate loss rate, or has to show that within the segment S, for each subsegment defined by a value of r, the loss rate varies only randomly and with a small variance at that.