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.