Tag Archives: accuracy

From Premium to Contribution: Recovering Solidarity

This post was originally written and published in French, De la prime à la cotisation: retrouver la solidarité

Well, let’s start with something everyone can observe. Insurance has a bad image, a bad press. It is often suspected of being a cold bureaucracy, a paperwork industry, a partner that looks for loopholes precisely when you need it. We talk about premiums the way we talk about a price, and we end up judging insurance the way we judge a purchase. Did I “get my money’s worth” this year. Did I “lose” money if I had no claim. Was I a good customer if I kept quiet. With questions like these, the very idea of solidarity quickly feels out of place.

And yet, if we set aside the forms and the marketing campaigns, insurance is first and foremost a social technology. It makes a simple reality livable. Some events are rare, hit hard, and cannot be financed individually without tipping into ruin. Insurance says the following. We do not know when, we do not know who, we only know that one day someone will have an accident, fall ill, see their home damaged, or cause harm to someone else. And because we cannot decide in advance who that person will be, we choose to be many to carry the associated financial burden. We pool a small share of our resources, and we agree on rules so that, when the day comes, the burden is bearable. For everyone. It is not only a service, it is a common.

I had already tried to look at the issue from a concrete angle, the angle of claims settlement, and the way it can disenchant a tool that is, in principle, virtuous, in an earlier post When insurance falls apart, the silent crisis of claims settlement. And there are other posts too, revolving around pricing, perception, and what we truly expect from an insurance contract, such as The value of life, The paradoxes of segmentation and discrimination in insurance, Cheaper personalized insurance premiums thanks to AI, or Insurance, a zero sum game. But what was missing was a broader lens, more sociological, and probably more political too. And as often, going back to classic works in the social sciences helps clarify what is at stake.

Continue reading From Premium to Contribution: Recovering Solidarity

De la prime à la cotisation: retrouver la solidarité

Bon, commençons par un constat que tout le monde peut faire. L’assurance a mauvaise image, mauvaise presse. On la soupçonne d’être une bureaucratie froide, une industrie de paperasse, un partenaire qui cherche l’échappatoire au moment où l’on en a besoin. On parle de primes comme on parlerait d’un prix, et on finit par juger l’assurance comme on juge un achat. Ai-je “rentabilisé” mon contrat cette année ? Ai-je “perdu” de l’argent si je n’ai pas eu de sinistre. Ai-je été un bon client si je suis resté discret ? Avec ce genre de questions, l’idée même de solidarité paraît vite hors sujet.

Pourtant, si l’on met de côté les formulaires et les campagnes marketing, l’assurance est d’abord une technologie sociale. Elle sert à rendre vivable une réalité simple. Certains événements arrivent rarement, frappent fort, et ne sont pas finançables individuellement sans basculer dans la ruine. L’assurance dit ceci. On ne sait pas quand, on ne sait pas qui, on sait seulement qu’un jour quelqu’un aura un accident, tombera malade, verra sa maison endommagée, ou causera un tort à autrui. Et comme on ne peut pas décider à l’avance qui sera cette personne, on décide d’être plusieurs à porter le poids financier associé. On met en commun une petite part de nos moyens, et on s’accorde sur des règles pour que, le jour venu, la charge soit supportable. Pour toutes et tous. Ce n’est pas seulement un service, c’est un commun.

J’avais déjà essayé de regarder le problème par un angle concret, l’angle de l’indemnisation, et de la manière dont elle peut désenchanter un outil a priori vertueux, dans mon ancien billet Quand l’assurance se défait, la crise silencieuse de l’indemnisation. Et puis il y a d’autres billets, qui tournent autour de la tarification, de la perception, et de ce que l’on attend vraiment d’un contrat d’assurance, comme La valeur de la vie, Les paradoxes de la segmentation et de la discrimination en assurance, Des primes d’assurance personnalisées moins chères grâce à l’IA ?, ou encore L’assurance, un jeu à somme nulle?. Mais il manquait une grille de lecture plus large, plus sociologique, plus politique probablement. Et comme souvent, se replonger dans des classiques des sciences sociales aide à voir ce qui est en jeu.
Continue reading De la prime à la cotisation: retrouver la solidarité

“Life insurers can predict when you’ll die with about 98% accuracy”

The other day, I was reading a rather interesting article, “How GLP-1s Are Breaking Life Insurance” (a title reminiscent of the dozens of articles over the past 35 years that have predicted the death of insurance), with this short paragraph, which echoed an article I had just published (in French), on segmentation in insurance.

Life insurers can predict when you’ll die with about 98% accuracy. This ruthless precision comes from from decades and decades of mortality data they use to figure out how much to charge you every year, so that the money they earn (from you and by investing your premiums) will easily cover what they’ll need to pay out later. Of course, not everyone gets the same deal. Underwriting is the dark art that allows an insurer to figure out if you’re a good bet or a risky one (etc)

Even though I have mastered this “obscure art” (at least, I have published several manuals on the subject), I understand, in a way, that many people may be confused… But what troubled me most was the first sentence (which I will repeat here in bold so we can spend a little more time on it—I will also remove the reference, which doesn’t say much about this mysterious figure)

Life insurers can predict when you’ll die with about 98% accuracy. 

Coincidentally, I am also revisiting a presentation I gave a year ago at the Cerisy conference, which dealt with almost the same topic, or rather statistical certainty when aggregating data, as opposed to individual epistemic uncertainty.

In any case, I thought this might be an opportunity to rewrite a blog post specifically on this question: what does it mean “98% accuracy”?

When 98 % Sounds Like Clairvoyance… but Isn’t

Obviously, this sentense

Life insurers can predict when you’ll die with about 98 % accuracy

sounds spooky, almost like they know your personal death date within a one-in-fifty margin. In reality, it’s a statement about cohort‐level forecasting, not individual fate.

As mentioned in the next sentence of the article,

This ruthless precision comes from from decades and decades of mortality data

and indeed for centuries, insurers build mortality tables from massive historical data, slicing people into homogeneous bands (age, sex, health metrics). If you group, say, 50 000 sixty-year-old non-smoking males, the table might assign each a 1 % chance of dying next year. Actuarial models then predict (with high accuracy, or reliability) the number of deaths in that group, but not which individuals. Actually, it’s not AI, nor magic… it’s elementary statistics, or to be a little more precise, the law of large numbers, and the central limit theorem.

Each year, they have to make prediction about the number of people that might die in the portfolio. Mathematically, they deal with a binomial random variable D \sim \mathcal{B}in(n, p), where n is the number of people alive, and p the probability to die next year (estimated from past historical data). The number of deaths D is a random variable, with expectation  \mathbb{E}[D]=np and variance np(1 - p) . The statement “98 % accuracy” simply means that in “98 % of portfolio‐year observations”, the observe* deaths fall within ±2\% of the expected deaths, or mathematically \mathbb{P}\Bigl(\bigl|D - np\bigr|\le 0.02\,np\Bigr)\approx0.98 It doesn’t mean \mathbb{P}(\text{“you” will die exactly in June 2027})=0.98.(which is where the initial sentence is clearly misleading)

“98% Accuracy” ?

Let’s make this concrete. We can simulate portfolios with death probability p=1\% and we want to find the cohort size n so that the coverage within \pm 2 \% hits roughly 98 %. By the Central Limit Theorem (i.e. a normal approximation on the number of deaths): \mathbb{P}\bigl(|D - np|\le \delta\,Np\bigr)\approx 2\,\Phi\!\bigl(\displaystyle\frac{\delta\,np}{\sqrt{np(1-p)}}\bigr)-1 so solving for n with \delta=0.02 gives:

> p <- 0.01 # death probability
> delta<- 0.02 # ±2% tolerance
> z <- qnorm(0.99)
>
> # required cohort size
> N_req <- (z^2 * (1 - p)) / (p * delta^2)
> cat("Required N for 98% accuracy ≈", round(N_req), "\n")
Required N for 98% accuracy ≈ 1339444

or

> simulate_accuracy <- function(N, p, trials = 10000, tol = 0.02) {
+ counts <- rbinom(trials, size = N, prob = p)
+ expected <- N * p
+ rel_err <- abs(counts - expected) / expected
+ mean(rel_err <= tol)
+ }
> N1 <- 1340000
> p <- 0.01
> acc1 <- simulate_accuracy(N1, p)
> cat(sprintf(
+ "With N = %d and p = %.2f, accuracy ≈ %.2f%% of years within ±2%%\n",
+ N1, p, acc1 * 100
+ ))
With N = 1340000 and p = 0.01, accuracy ≈ 97.94% of years within ±2%

I.e., with about n=1.4 million lives in your cohort, you would naturally have 98% accuracy. Each increase in n shrinks the relative dispersion \sqrt{(1-p)/(np)}, so coverage goes up.

Uncertainty and Ambiguity

Beyond the pure “coin‐flip” randomness (aleatory uncertainty) of each life’s Bernoulli trial, there’s also “epistemic uncertainty” in our death probability p, since we don’t observe the true p directly but must estimate it from historical life tables, or ML models. If our model’s p is itself uncertain, say p\pm1\% (a very small calibration error) or, in worse cases, p\pm50\%, then our expected death count np could be systematically off by that margin before we even toss a coin. With poor predictive power, i.e. p\pm50\% (say uniformy distributed on that interval), it is still possible to achieve “98% accuracy”. Or more precisely what should be n so that \mathbb{P}(∣D−np∣\leq 0.02np)\geq0.98 with 95\% chance?

Specifically here, to guarantee that 95% chance,  if we suppose that p is uniformly distributed on [0.5\%,1.5\%] observe that the 5-th percentile of that uniform is p=0.55\%, and then\mathbb{P}\bigl(\lvert D - np\rvert \le 0.02\,np\bigr)\approx2\,\Phi\!\Bigl(\frac{0.02\,n\,p}{\sqrt{n\,p\,(1-p)}}\Bigr)-1\;\ge\;0.98
Thus, the two‐tailed 98 % requirement corresponds to \frac{0.02\,n\,p}{\sqrt{n\,p\,(1-p)}}\;\ge\;z_{0.99}\approx2.326so solving for n gives n\;\ge\;\frac{z_{0.99}^2\,(1-p_{0.05})}{p_{0.05}\,(0.02)^2}\;=\;\frac{2.326^2\times(1-0.0055)}{0.0055\times0.0004}\;\approx\;2.45\times10^6Hence, with about n=2.45 million lives in your cohort so that, for at least 95 \% of the possible p values in [0.5 %, 1.5 %], the year‐to‐year death counts fall within \pm2 \% of the mean in at least 98 % of years.

Group Accuracy vs. Individual Uncertainty

So, overall, eachieving “98% accuracy” is not that difficult… but clearly, it is a misleading statement, that conflate group-level confidence with individual-level certainty.  “98 % accurate” has to do with statistics, or actuarial science, but it is not mystical foresight for you specifically.