Price Stability for Health Insurance, a Tale
Once Upon a Time … the Inverted Cycle of Production
Insurance can be a delight for stats nerds and any practitioner of uncertainty.
Its charm comes from the “inverted cycle of production.” Traditional businesses incur costs before selling a product. Consequently, they know the cost at the time of sale and can set the price with a good idea of the induced margin.
In insurance, we sell coverages: engagements of reimbursement if certain events occur. We only learn the cost after selling the coverages and waiting for a time to get an idea of the frequency of the risk. Setting a price that allows us to meet a certain margin is thus a fine art.
In health insurance, we update our prices every year. When we sign a new customer, we estimate the cost of covering their population. Then, we start reimbursing insured members and collecting premiums: we monitor both quantities.
At Alan, we sell insurance at cost and try to match exactly the amount of premiums collected with the cost of health claims. Every year, we review all our offers for all customers and adjust our prices to keep this balance.
We want to make this process:
- Fair: your price is aligned with how much you consume
- Transparent: it’s easy to follow and understand how your price will evolve
- Stable: your price should not be subject to significant year-over-year variations
Easy, right?
Not so much because of our nemesis: volatility
The Villain: Volatility
Consider your own healthcare consumption: in some years, you may face a difficult health condition and need to visit doctors or even go to the hospital. You could also choose to undergo eye surgery to eliminate the need for glasses, or address dental issues that have been causing pain for some time. On the other hand, there may be years when you are generally healthy and require very little healthcare.
👉 Everyone’s health story is unique, largely unpredictable, and volatile.
In years with expensive health events, your insurer will reimburse much more than you contributed. The insurer could balance premiums and claims by simply setting the price to match what you consumed.
This would likely not be fair as the principle of insurance is to materialize solidarity between those who are “lucky” with their health and those “unlucky”. Also, this approach would not yield stable prices.
The Hero: Central Limit Theorem
Hence, the core idea of insurance: mutualization. Assuming that your health risk and my health risk are independent, we can use my money during my “lucky” years to fund your reimbursement during your “unlucky” years, and vice versa. The group of people on which we apply this principle is called a mutualization segment.
In an ideal world, we would like to come up with a price for each mutualization segment that reflects exactly their intrinsic expected health consumption, given a set of known characteristics (age, gender, location etc.): the best theoretical price.
However, we never directly observe this theoretical price due to volatility. Here comes the stats to the rescue 💡: under a few reasonable hypotheses, the average health consumption within the mutualization group will be normally distributed around the best theoretical price!
There’s more good news: we know how quickly the empirical average converges towards the theoretical expectation. This convergence rate is the square root of the sample size we consider.
For example, if we only look at the health consumption of one person, we can’t confidently predict their future consumption. But if we group that person with 100 others, the average consumption over all of them becomes a much more stable estimator. The larger the group, the more stable the estimator. For a very large group of people, the average consumption in a given year is a reliable basis for predicting their expected consumption in the following year.
The Magic Sword: Hypothesis Testing
Okay, but how does this help us in practice? Should we simply make mutualization segments large enough to iron out the noise and gain sufficient confidence in pricing?
In France, employees are usually mutualized at their company level, meaning they share health risks with their coworkers. This results in mutualization segments of various sizes, sometimes with 10,000 people for larger companies or just a few hundred for smaller ones. We don't get to choose the mutualization segment size.
So here is how hypothesis testing helped us solve our business issue of fair and stable pricing.
For each year:
- We know what is your current price
- We know the size of your company
- We know the average consumption within your company
- We know that this consumption is normally distributed around your true optimal price
👉 we can test the following hypothesis: “With 80% confidence, can we reject that your consumption is centered on your current price?”
If we can reject, then we will update your price. If not, we assume the price is correct and we keep it unchanged.
Implicitly, this approach allows for two layers of mutualization:
- The first layer is between employees of a company.
- The second layer is between companies, for the residual noise between their price and consumption, as long as this noise is not statistically significant.
This rule is transparent, meaning each customer can see where they stand on the interval, and it creates stability with an 80% chance of success.
However, is it truly fair? If a company consumes slightly less than it pays each year, is it fair to maintain this status quo year after year?
Likely not. We still need one final trick.
The Final Boss: Time
Here is the last big idea: instead of just mutualizing employees with their coworkers today, why not mutualize them with their past selves as well?
By examining companies not only as they exist today, but also as they existed in the past, we can combine past profits with current losses, increase our sample size, and gain confidence in our estimates. As a result, we can reduce the size of our confidence intervals!
In practice, if a company with 50 or fewer employees consumes 20% more than their price in the first year, we will not change the price. This is because we cannot conclude whether the increase is due to underpricing or volatility. However, after 2 or 3 years, if the trend is consistent, we will reprice it as soon as the confidence interval becomes smaller than 20%.
We now have a methodology:
- Fair: everyone pays for what they consume over time.
- Transparent: the rule is simple and public, so it’s easy to understand.
- Stable: most companies are not repriced every year.
…and voilà!
Epilogue: balancing the total
Wait… But from the insurer’s point of view, the total consumption will only match the total price if the companies are evenly distributed around a 100% ratio.
Indeed! We need to ensure that the cloud of points (companies) is centered. Do we have a methodology for that?
If you're interested in the answer, please drop us a line at jobs@alan.eu and come meet us 😉
