Every reinsurance pricing basis rests on a mortality curve, and every mortality curve eventually has to survive a meeting. Someone — a pricing committee, a client actuary, occasionally a regulator — will point at age 42 and ask why the rate is what it is. "The model said so" has never once been an acceptable answer. This article is about how I build curves that answer that question: with actuarial laws where structure helps, GLM splines where the data insists, and diagnostics that show exactly where each one earns its place.

The context is real work. On term life and credit life reinsurance engagements for RGA, I built and calibrated mortality curves from experience studies, refined them with mortality improvement analysis from IIB and census data, and defended the resulting basis through sensitivity testing in AXIS. What follows is the method, not a theory of it.

Why raw experience is never enough

A seriatim experience study gives you crude rates: deaths over exposure, by age, duration, sex, smoker status, and whatever else your data supports. Those crude rates are the truth — and they are unusable as a basis. Cell counts thin out fast once you cross-classify. At age 61, non-smoker, duration 8, you might be dividing three deaths by a few hundred policy-years; the resulting rate can double or halve depending on one claim. Price off that and your rates zigzag with the noise. Graduation — fitting a smooth curve through the crude rates — is how you separate signal from sampling error.

The question is what kind of smoothness you impose, and that's where the two schools diverge.

Actuarial laws: structure you can explain

Law-based graduation fits a parametric form with a story behind it. Gompertz says the force of mortality grows exponentially with age; Makeham adds a constant term for age-independent causes. Two or three parameters describe the whole adult curve.

That parsimony buys three things that matter in pricing meetings:

  • — Extrapolation. Your experience study probably runs out of credible data around age 70 or 75. A fitted law extends into old ages on the strength of its functional form rather than a modeler's guess. That behavior at the tail is exactly where reserves and reinsurance recoveries are most sensitive.
  • — Stability. Refit next year's study and the parameters move a little; the curve doesn't reshape itself. Basis changes stay small and explainable.
  • — Auditability. Three parameters fit in a spreadsheet cell and a sentence. A reviewer can rebuild the entire curve from the documentation alone.

The cost is rigidity. Real portfolios have features the laws refuse to bend around — the accident hump in young male mortality, underwriting selection wearing off across early durations, a credit life book whose age mix reflects loan origination rather than population structure. Force a Gompertz–Makeham fit through those and the residuals will show systematic runs: the curve is smooth, and wrong in the same direction for ten ages at a stretch.

GLM splines: letting the data speak, within limits

The GLM formulation treats graduation as regression: deaths as the response, a log link, exposure as offset, and age entering through a spline basis rather than as a single slope. Penalized splines add a smoothness penalty so the fit follows genuine shape without chasing every noisy cell.

Three practical advantages drive my use of them:

First, shape. A spline picks up the accident hump and the select-period slope that a two-parameter law cannot express. Second, covariates. Sex, smoker status, band, and duration enter the same model with proper interaction structure, instead of being handled through side-table adjustment factors multiplied together and hoped about. Third, honest uncertainty. A GLM gives standard errors, so you know which region of the curve the data actually supports — which is precisely the input you need when deciding where judgment must take over.

The failure mode is the mirror image of the laws'. Splines extrapolate badly; beyond the last credible data point, an unconstrained spline will do something confident and absurd. And a spline fit is harder to hand to a reviewer: the "parameters" are dozens of basis coefficients with no individual meaning.

How I combine them in practice

The approach that has survived committee scrutiny is a hybrid with an explicit seam. Use the GLM spline where the data is dense — typically working ages, where the portfolio's exposure is concentrated and the shape features live. Use a law-based fit for the old-age tail, calibrated so the two segments meet with matching level and slope at a documented blend age. Below the blend age you can point to exposure counts justifying the flexible fit; above it, to the functional form carrying the extrapolation. Every choice has a stated reason, which is what "defensible" actually means.

Two disciplines keep the hybrid honest:

Actual-to-expected diagnostics, sliced hard. A/E ratios by age band, duration, calendar year, and size band, on both count and amount bases. A curve that looks fine in aggregate can be masking a 110% A/E on large policies offset by 90% on small ones — which is a pricing problem wearing a graduation costume. Residual runs, not just residual magnitudes, are what I read: systematic drift in one direction is the fingerprint of a functional form fighting the data.

Improvement handled separately. A curve fitted to a 2015–2023 study window describes the past. For long-duration reinsurance pricing, I analyzed mortality improvement from IIB and census data as its own exercise — trend fitted on population-scale data, then applied to the portfolio curve as explicit improvement vectors. Bundling improvement into the graduation muddies both estimates and makes the basis impossible to update cleanly when new trend data arrives.

Where machine learning fits — and where it doesn't

On the same engagements I used XGBoost, logistic, and linear models to assess risk relativities and flag high-risk lives. Those models are excellent at ranking: which segments carry more risk relative to the basis, which cells deserve underwriting attention. I treat their output as an overlay for segmentation and monitoring, not as a replacement for the curve. A gradient-boosted mortality surface may score well on held-out data, but it cannot be extrapolated with confidence, updated predictably, or explained at the level a pricing basis demands. The curve is the contract with the future; the ML layer is intelligence about the present.

If your basis is due for reconstruction — new experience data, a stale graduation, or a curve nobody can explain anymore — that reconstruction is work I do. Get in touch.