Sunday, 3 April 2016

The most dangerous equation in the world

Each year Generali, one of the biggest insurers in the world, analyses the claims of its car insurance customers in the Netherlands. Results of that analysis can be found on their website. In their analysis, Generali relates the number of claims to where people live & drive, the age of the driver, the age of the car and the car brand. Some of these statistics provide insights that are to be expected. For example, you expect young drivers to have the highest claim rates, as their analysis confirms. Cars in less populated areas have the lowest claim rates, which seems plausible as well. There is however one finding that raised my eyebrows and that is that drivers of specific car brands have significant higher claim rates than others, suggesting that driving a car of a certain brand makes you either a better or a worse driver. This year Mazda drivers had the highest claim rates according to the Generali analysis, this was for the second year in a row. Drivers of a Citroen had the lowest claim rate, making them the safest drivers of 2015 according to Generali. So, is their truth in their finding and should you therefore avoid a Mazda driver or at least not buy a Mazda yourself? Generali’s statistics suggest you should, don’t they?

Putting it in perspective

Let’s take a closer look at the numbers. The claim rates themselves don’t tell much, but combining them with other data will. What will be interesting is to see how the distribution of car brands compares to the number of claims per brand. Unfortunately, but understandable, Generali only reports the relative difference in claims of a brand compared to the average claim rate. However, Generali claims that its findings apply to all drivers in the Netherlands, so it’s fair to assume that the distribution of car brands in their car insurance portfolio is similar to the overall distribution of car brands in the Netherlands. With data from the Netherlands Vehicle Authority (RDW) gathers, selecting only the brands reported by Generali, we find that 7,545,266 vehicles were registered in the Netherlands in 2015, with the following relative distribution over brands. Clearly Volkswagen, Opel and Peugeot are the biggest brands, while Skoda, Mitsubishi and Mazda are the smallest brands.

Does driving a Mazda make you the worst driver?

By plotting the population size per car brand against the relative claim performance per car brands an interesting pattern appears, the spread in claim performance is bigger when the population size decreases. So smaller car brands have a bigger spread in claim performance than bigger brands. This is the result of a not so well known statistical law, De Moivre’s Equation, which provides us with that standard deviation of the sampling distribution of the mean, σx=σ/ n. Howard Wainer named this equation the most dangerous equation in the world because too little people are aware of it and as a consequence made faulty decisions with serious impact. Look at the formula we see that the standard deviation of the mean is inversely proportional to the square root of the sample size. As a consequence car brands with a smaller number of cars in the Netherlands will have a larger variation in relative claim performance than bigger brands. To illustrate, a small brand with no claims will have the best claim performance in one year, while a small number of claims will make it the worst performing brand the next year. For the bigger brands this is not an issue. Note that the brand with the best claim performance last year was Skoda, this year Skoda was among the worst performers, De Moivre’s equation in action.


The most dangerous equation in the world


So, Generali’s claim that driving a Mazda makes you the worst driver in the road is much to strong. Whether you are a good or a bad driver depends on many things, but I doubt that it will be the brand of your car, and would require a much more detailed analysis. Hopefully Generali doesn’t take the brand of your car into account when calculating their premium levels. Chances are they either over or under price it when you choose to drive one of the rarer car brands. When you want to avoid that, better choose one of the larger car brands as it will be unlikely for them to end up being the worst performing category. Insurance claims are not the only subject affected by De Moivre’s equation. Wainer shows with some compelling examples what the consequences can be of being ignorant of the most dangerous equation of the world and why understanding variability is critical to avoid serious errors.
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