Credit Arjen Toet CC BY

Strange excesses: on insurance and probabilities

How insurance can sometimes be a gamble in more than one way

Are you a gambler? Let’s not count lottery draws, or even the occasional punt on a horse or the result of a football match. If we exclude these, chances are you wouldn’t consider yourself the gambling kind. And yet…

…we are all gamblers. When we take out insurance, we gamble. We bet against the insurer that our house will burn down, that we will die prematurely, or that we will crash our car. If our house never catches fire, we live to be 90, or we are very safe drivers, we lose: all our premiums are ‘lost’. The insurance company takes the opposite bet: it loses if it needs to pay out for the insured event if that amount is larger than the premiums we have paid.

And when we don’t take out insurance — say travel cover — we gamble too. We save ourselves the premium, but there is a risk that we fall over and break our arm, catch malaria or whatever. Should that happen, we have to bear all the costs.

Yet few among us have any idea of the actual risk that our house will burn down, that we’ll die before the kids have flown the nest, or that we will collide with another vehicle. We may not know the risk, but it is vital for insurance companies to do so: to stay in business, insurers need to be very, very good at estimating the risks involved. If they overestimate them, they’ll charge premiums that are higher than they need to be, and end up more expensive than their competitors who are better at judging it. If they underestimate the risks, they will quickly go bankrupt.

Still, we come across quirky phenomena in the insurance market. If you’ve every used a comparison site to see what it would cost to insure your car, you will have noticed that, as you scroll down from the best quotes at the top, the premiums go up to twice, three times even six times or more the cheapest premium. As these quotes are based on exactly the same information, you would expect insurers to provide if not identical, then at least similar quotes — unless they classify the various risk factors very differently.

The premium takes into account many different aspects, including your age, your occupation, where you live, the car you drive (how old it is, what type, what it’s worth) and so on. Converting these factors into risk is more black magic (with a sprinkle of statistics) than hard science, but it is what makes the distinction between profit and loss for an insurer.

Yet we are not entirely stuck with the insurer’s assessment of the risk we pose. We can convince the insurer that we are a better risk than they think, and get to pay a lower premium. That is what the voluntary deductible excess is for — the part of any claim that we will pay ourselves before the insurance pays out.

Apart from this minor collision with a submarine, I am a safe driver, really. (image: Marinmuseum Karlskrona CC BY)

A few weeks ago, I needed to renew my own vehicle insurance, and I wanted to see what the effect was of the voluntary excess on the premium. Is there an optimum excess amount? I opened a spreadsheet, and ended up spending an instructive couple of hours.

I reasoned that, if I am a very safe driver, a high excess will hardly bother me, as I am unlikely to have to dip in my piggy bank for it. This is, in effect, a wager between the insurer and me. Their stake is the discount to the premium, and they bet that I will make a claim. If that happens and the claim is larger than the excess amount, they win that amount (in that they cut the pay-out by that sum). For me it’s the opposite: if I don’t make a claim, I pocket the discount, but if I do, I lose the excess.

Is opting for a voluntary excess a good deal? That depends: a £500 excess that gets you a reduction of just £1 would seem to be a terrible deal. But we can actually work out what would make a deductible excess an attractive proposition. Imagine a driver whose risk profile predicts that, on average, he will have on average an accident once every 10 years (hence, a 10% chance they will make a claim this year — this is the UK average), with an average claim size (in the UK) of £3,000. With no excess, the breakeven point for the premium would then be £300 per year (let us ignore the insurer’s costs and profit margin to keep things simple). With an excess of £100, the average pay-out would drop to £2,900 — so the breakeven premium would be reduced to £290 per year.

Such a deal would be neutral: what you gain in lower premiums, you would lose in the excess when you make a claim. To really gain through a voluntary excess, you’d need to look for a deal where you get a larger cut in the premium. For my chosen insurer, I found that an excess of £150 reduced the premium by nearly £24. If we apply this to the UK averages, then the pay-out would now be £2,850 (£3,000 — £150), and the annual premium would be £276 (£300 — £24). The original risk of 10% (£300/£3,000), has now dropped to £276/£2,850 or 9.68%. The lower premium signifies that agreeing to a £150 excess has improved my risk profile.

And that does indeed make me better off: my gross saving is £24 per year, of which I must use £15 as a provision to cover he excess. Over 10 years (the average period in which I make one claim — 10%, remember?) I will have accumulated the £150 I need to pay myself, but also an additional £90. Hurrah!

But wait, we’re not finished yet: what if we increase the voluntary excess? Raising it to £250 cuts the premium by a further £11. My risk profile improves a little (to 9.64%: £265/£2750) and I now pocket £10 per year (I need to hold back £25 to cover the excess over 10 years). How about a £350 excess? Now things are getting strange: the premium is now down to £258, which means my gross saving is £42. But I also need to set aside £35 to cover the excess, so my net saving is down to £7. And it gets worse: raising the excess to £450 cuts the premium by just over £1. With a premium of £257, my risk level is 10.08%, higher than I started with! And that is reflected in the fact that I no longer make a net saving: I pay £43 less than originally, but I need to hold back £45 for the excess, so I am losing out to the tune of £2 per year.

What do we learn from this? First, a voluntary excess can indeed be a strong signal that I am (or at least believe to be) a lower risk than standard — and that I am willing to put my money where my mouth is. If I am happy to pick up the first slice of the claim (and not make frivolous claims for minor dings) the insurer is happy to reduce my premium.

More importantly, we should not take anything for granted. Insurers can exploit the fact that we cannot intuitively distinguish a good deal from a bad deal — we may be taken in by a reduction in the premium that does not properly reflect the reduction in the risk. In this case, there was clearly an optimum at the £250 level. I would have lost out had I gone for the highest excess.

And most important of all… thank goodness for the reward of intellectual and moral satisfaction. I realized a net saving of £10 on my premium, and I could have ended up £2 worse off. That result hardly justified playing around with a comparison website and a spreadsheet for a couple of hours. But the knowledge that I saw through the insurer’s shenanigans? Priceless!

Originally published at on January 11, 2019.

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Accidental behavioural economist in search of wisdom. Uses insights from (behavioural) economics in organization development. On Twitter as @koenfucius