The Tyranny of the Straight Line

There is a ritual in the delegated-authority market as dependable as renewal season itself. A carrier, having studied the business plan, admired the pricing model and signed the binder, watches the first months of a new young-driver telematics scheme arrive down the API: the bordereau is dead, the data streams continuously and the loss ratio now updates itself on a dashboard like a patient’s heart monitor. The number is ugly. Somewhere a chart refreshes, an eyebrow ascends, and a capacity line that took eighteen months to court begins composing its exit. Modernity has not cured the panic; it has merely put it on a faster feed. What once took a reserving committee a quarter to misread can now be misread before lunch by anybody.

I write as a former owner of a telematics MGA in the UK, so I declare the stake: I have sat in that meeting more times than a cardiologist would advise, on the side of the table that does not get to leave. And here is the thing: the number on the chart was almost always correct — the conclusion drawn from it almost never was.

The case for the ruler

Let us first do the 365ths method the courtesy of stating its case at its strongest, because it is not a stupid convention and the people who apply it are not stupid people.

The method earns premium in equal daily instalments across the policy year: one three-hundred-and-sixty-fifth per day, claims incurred measured against premium earned. It is simple. It is auditable. It is consistent across books, borders and decades, which is why reporting frameworks are built on it. And for the overwhelming bulk of motor business it is a perfectly fair proxy for exposure, because a forty-year-old accountant’s July is about as dangerous as her March. Where hazard is level across the policy term, the straight line and the truth are close enough to shake hands. The 365ths is a good map of flat country.

The difficulty begins when someone drives the map into the mountains.

Jordan, in June

Consider Jordan, seventeen, who passes his test in the second week of June and is on cover by the third. Jordan is not a villain in this essay; he is its audience, and statistically he is also its patient. The epidemiology on drivers like Jordan is about as settled as anything in road-safety research gets, and it says one thing with tedious unanimity: his risk is not level. It is front-loaded, and steeply.

In the American SHRP2 naturalistic driving study, drivers aged 16 to 19 crashed at roughly 30 per million miles against 5.3 for adults aged 35 to 54 (Seacrist et al., 2016). More to the point, the excess is not spread evenly across the novice’s year. New Jersey licensing data show crash rates for novice drivers at their highest in the very first month of independent driving, declining substantially across the first year (Curry et al., 2015); Masten and Foss (2010) found the same immediate post-licensure peak in North Carolina. A review of eleven studies concluded that inexperience and youth each contribute, and that the improvement with the first months of solo driving is rapid (McCartt et al., 2009). These are American datasets, so carry the figures with that label attached; but the shape of the curve, the steep early decay, is the consistent finding across jurisdictions, and it is the shape, not the American numerator, that concerns us here.

Jordan’s hazard, in other words, decays like a fresh radioactive isotope. His most dangerous month is his first. His safest is his twelfth. The 365ths method prices his June and his following May as identical twins. They are not even cousins.

To the published literature I can add a humbler grade of evidence, but a longer run of it. For more than a decade, my own emerging claims experience reproduced this curve with the punctuality of a tide table: cohort after cohort of newly licensed drivers, crashing early, settling down, month after month, year after year. That is testimony, not a peer-reviewed dataset, and I present it as testimony. But when a decade of monthly claims runs keeps drawing the same shape the epidemiologists drew first, one is entitled to stop calling it a hypothesis.

The arithmetic of the growing book

On a single policy this hardly matters. Claims arrive early, premium earns evenly, the first quarter looks dire, the final quarter looks angelic, and the policy year in aggregate lands exactly where it was priced. The straight line tells the truth eventually; it merely lies about the journey.

But a new scheme is not a single policy. A new scheme is a portfolio in permanent adolescence, because every month of growth restocks the book with drivers in their most dangerous weeks. Take a stylised example, offered as illustration rather than measurement: suppose, consistent in direction with the New Jersey decay data, that a novice’s hazard in his first quarter on cover runs at twice his own annual average, and his final quarter at half of it. In a static, seasoned book these effects wash out. In a book that is doubling every few months, first-quarter Jordans dominate the exposure, and the incurred-to-365ths-earned loss ratio prints materially above the ultimate that every individual policy is on course to deliver.

And that is my whole complaint: the measured loss ratio can run hot while every single policy performs precisely to plan. The metric is not detecting bad underwriting. It is detecting growth, and growth is the one thing the carrier itself demanded in the business plan it approved. The MGA is then invited to explain, month after month, why the book looks worse than the book is, to an audience holding a ruler against a curve and concluding the curve is broken.

Nor is this illusion a peculiarity of teenage motorists. I have watched the same conjuring trick performed on Japanese three-disease critical illness plans, where small benefits and light acceptance criteria invite the least healthy to arrive first and claim early, and on funeral insurance, where the hazard is likewise shaped by duration rather than spread evenly along it. Different products, different continents, identical arithmetic: wherever risk varies across the policy term and the book is growing, the time-earned loss ratio measures the growth and calls it the underwriting. Again, this is one practitioner’s observation across a few markets, offered at that grade; the mechanism, however, is not anecdote. It is division.

The problem is simply this: the ruler is straight and the risk is not.

The cost of the panic

One could forgive all this as a harmless accounting foible were the consequences not so perverse. Capacity that withdraws from a young-driver telematics book in month nine is abandoning the scheme at the exact point the cohort mathematics turns friendly, and abandoning the one corner of UK motor with a plausible public-health dividend attached. Industry analysis by LexisNexis Risk Solutions (2018) reported casualties among drivers aged 17 to 19 down 35 per cent since 2011, against 16 per cent for drivers generally, over a period in which roughly four in five newly insured drivers of that age took telematics policies. Those are market figures rather than a controlled trial, and should be carried at that grade; but a Dutch field experiment found pay-as-you-drive incentives produced measurable reductions in speeding among young drivers (Bolderdijk et al., 2011), which points the same way. The boxes and the feedback loops do, on the available evidence, what the brochures claim.

Is it too much to ask that an industry founded on the mathematics of contingent probability distinguish a level hazard from a decaying one? Apparently so. The actuarial profession that can price a hurricane in the Gulf declines, in these meetings, to price the difference between Jordan’s first month and his eleventh.

For the brokers and MGA principals presently having this argument with a twitching carrier, four lines to write on the back of the slip:

• Agree the measurement basis before binding: year-one performance judged on cohort development, by month since inception, alongside the standard 365ths line, never instead of judgement.
• Put the decay curve in the underwriting submission, citations attached, and have the carrier’s Head of Pricing initial the page.
• Report loss ratios by policy-month cohort with every monthly report, unprompted, so the hot early print arrives in their dashboard pre-explained rather than post-panicked.
• Set out in writing what “on plan” looks like at months three, six and twelve. A carrier that has signed the curve cannot later claim to be ambushed by it.

Good luck to all of you who take on this challenge. Take comfort in this: you are not arguing with the data. You are arguing with a ruler.