Calculator methodology · Trust & method
Why Does the Makeham Term Matter More at Age 30 Than at Age 70 in a Life Expectancy Model?
The Makeham term is the constant background-mortality risk in the model — accidents and infections that do not rise with age. At 30 it is a large share of your total risk; by 70 the age-driven term dwarfs it.
Before the detail: if you want the whole engine first, read how a life expectancy calculator actually works, then come back for this one parameter.
What exactly is the Makeham term in the Gompertz-Makeham hazard function?
The Makeham term is the constant A in μ(x) = A + B·e^(cx). It captures age-independent background mortality — deaths from accidents, acute infections, and external causes — that occur at a roughly fixed rate regardless of how old you are.
The hazard function has two additive parts. The Makeham term A is a flat floor of risk that does not change with age x. The Gompertz term, B·e^(cx), is the age-driven part that climbs exponentially as you get older. Adding them gives your total instantaneous risk of dying. A represents extrinsic mortality — accidents, drowning, acute infections, environmental and occupational hazards — not the age-related degenerative disease the Gompertz term models.
The flat, age-independent background term was Makeham’s addition to Gompertz’s original exponential law, giving μ(x) = A + B·e^(cx); the constant A is why the combined model fits young-adult mortality far better than the pure exponential. Makeham, 1860; Gompertz, 1825
Because A and the Gompertz term carry different work, you cannot understand one without the other — see how the full Gompertz-Makeham hazard function is calculated. The real puzzle is why a constant can dominate at one age and vanish at another.
Why does the Makeham term represent a larger share of total hazard at age 30 than at age 70?
At age 30, B·e^(cx) is small, so the constant A forms a large fraction of total hazard μ(x). By age 70, B·e^(cx) has grown exponentially — roughly doubling every 8 years — so A’s proportional share falls to near-negligible levels, even though A itself never changed.
The key is that A stays fixed while the denominator beneath it expands. The share A ÷ μ(x) = A ÷ [A + B·e^(cx)] shrinks purely because B·e^(cx) grows, not because A moves. Between 30 and 70 the age-driven term compounds through roughly five 8-year doublings, so it swells by well over an order of magnitude while A holds its ground. The constant does not lose importance in absolute terms — it is simply outrun.
In WHO-calibrated high-income models the Makeham constant sits around A ≈ 0.0007 per year, while the Gompertz term doubles roughly every 8 years (slope c ≈ 0.087, i.e. ln 2 ÷ 8). Holding A fixed and letting the exponential run, the age-driven term grows from smaller than A at 30 to roughly 18× A by 70. Makeham constant and Gompertz doubling, WHO-calibrated life-table parameters
The same logic drives the other Gompertz parameter — see how the Gompertz slope parameter β differs by sex. But a constant is not just a number; it stands for specific kinds of death.
What types of death does the Makeham constant actually represent?
The Makeham term captures deaths from causes that do not increase with age: road accidents, drowning, acute infections, and occupational hazards. These risks are present at 20 and at 70 alike — the Gompertz term does not model them, so without A they would be missing from the young-adult hazard.
Actuarially, mortality splits into two families. Extrinsic (non-senescent) mortality — accidents, acute infections, environmental causes — is age-independent by definition and lives in the Makeham term A. Intrinsic (senescent) mortality — cardiovascular disease, cancer, organ failure — is exponentially age-dependent and lives in the Gompertz term. The combined model fits observed mortality well across roughly ages 25–85, and the Makeham term is most critical to fit accuracy in the young-adult band, ages 20–45, where extrinsic deaths dominate.
The Makeham term A holds extrinsic, non-senescent mortality (accidents, acute infection, external causes), while the Gompertz term holds senescent mortality (cardiovascular disease, cancer, organ failure) — a split that makes the model fit best across ages 25–85. Gompertz–Makeham cause structure, actuarial literature
This taxonomy is what makes the age-share shift matter for a real user’s number, not just the mathematics.
How does this age-dependent shift affect what the life expectancy calculator outputs?
When you enter age 30, the calculator’s hazard rate is materially shaped by A. At age 70 the Gompertz exponential dominates. This is why adjusting your current age produces nonlinear — not proportional — changes in the survival estimate: each decade adds exponentially more age-driven weight.
Because the age-driven term compounds, every decade of age adds exponentially more Gompertz weight, compressing remaining life expectancy faster than any straight-line model would predict. It also interacts with survivorship: a 50-year-old’s conditional life expectancy sits structurally above the naive life-expectancy-at-birth figure, because reaching 50 means the higher-risk members of the birth cohort have already dropped out. At age 30, A may account for roughly 40–60% of total μ(x); by age 70, its share typically falls below 5% in high-income country tables.
Working the model at typical high-income parameters, the Makeham constant supplies on the order of 40–60% of total hazard at age 30 but under 5% by age 70 — the arithmetic consequence of a fixed A sitting beneath an exponential that has grown ~18-fold. Model arithmetic from calibrated A ≈ 0.0007 and 8-year Gompertz doubling
That nonlinearity is exactly why conditional life expectancy rises as you age. The size of A, though, is not the same everywhere.
Does the Makeham term vary by country or sex cohort?
Yes. Countries with higher accident rates, conflict exposure, or infectious-disease burden carry a larger calibrated A value. Males typically show a higher A than females in the same country, reflecting greater extrinsic — accidental and external-cause — mortality risk.
A is not a universal constant; it is fitted per cohort. Populations with elevated accident, homicide, or infectious-disease mortality produce a higher A, while high-income, low-external-risk countries produce a lower one. Within a country, male cohorts typically carry a higher A than female cohorts, consistent with men’s greater accidental and external-cause mortality. The calibration is drawn from country-sex period life tables published by the WHO Global Health Observatory and the UN World Population Prospects 2024 revision.
The country-sex Makeham and Gompertz parameters are calibrated to period life tables from the WHO Global Health Observatory and the UN World Population Prospects 2024 revision, so A reflects each cohort’s real extrinsic-mortality burden rather than a single global figure. WHO Global Health Observatory; UN World Population Prospects, 2024 revision
See the downstream effect in how country and sex cohort selection changes your estimate. Given all this, what breaks if a model simply leaves A out?
Why does omitting the Makeham term produce biased life expectancy estimates for younger users?
Without A, a pure Gompertz model understates total hazard at ages 20–45, because it ignores extrinsic mortality that is a large share of young-adult deaths. That missing risk inflates predicted survival, overstating a young user’s life expectancy — the error is largest exactly where A matters most.
A pure Gompertz curve has no floor: as age falls, its hazard shrinks toward zero, which is wrong, because accidents and infections still kill young adults. Including A corrects this, improving model fit across ages 20–45 where accidental deaths make up a disproportionate share of total mortality. There is a historical echo here too: much of the twentieth-century mortality decline came from cutting the Makeham component — better sanitation, infection control, safer roads and workplaces — rather than slowing the Gompertz ageing term.
Dropping A biases a mortality model upward for young adults by omitting extrinsic deaths; including it is what lets the Gompertz–Makeham form fit the 20–45 band, where much of the historical mortality decline came from reducing background — not senescent — mortality. Gompertz–Makeham model fit and mortality-decline attribution, actuarial literature
So the Makeham term earns its place by keeping young-adult estimates honest, and the calculator’s whole point is honesty about the number. Now that you can see how one constant governs your risk at 30 and fades by 70, do the useful thing with it: enter your real age and watch the estimate move nonlinearly, not on a straight line — the steepening is the Gompertz term overtaking Makeham in real time. Read the output as a survivorship-adjusted range rather than a fixed date, choose your true country and sex so A is fitted to your cohort, and treat a young-adult number as one that already accounts for accidents and infections, not just ageing. When you are ready, get your survivorship-adjusted life expectancy estimate.
Frequently asked questions about the Makeham term
What is the Makeham constant in a mortality model?
The Makeham constant is the age-independent term A in the hazard function μ(x) = A + B·e^(cx). It represents background mortality — accidents, acute infections, and external causes — that occurs at a roughly fixed rate at every adult age, separate from age-related degenerative disease.
Why does the Gompertz term dominate at older ages?
Because the Gompertz term grows exponentially, roughly doubling every 8 years, while the Makeham constant stays fixed. By age 70 the exponential term is on the order of 18× the Makeham constant, so it supplies almost all of total hazard and A’s share falls below 5%.
What does the hazard rate mean in a life expectancy calculator?
The hazard rate is your instantaneous risk of dying at a given age, given you have survived to it. The calculator builds it from A + B·e^(cx), then accumulates it across ages to produce your survival curve and percentile life-expectancy range.
How does the Gompertz-Makeham model change with age?
The constant Makeham term stays flat while the Gompertz term climbs exponentially. So young-adult hazard is dominated by background risk (A), and older-age hazard is dominated by the age-driven term — which is why raising your age input changes the estimate nonlinearly.
What causes background mortality in actuarial models?
Background mortality comes from extrinsic, non-senescent causes: road accidents, drowning, acute infections, occupational and environmental hazards. These are largely age-independent, so actuaries model them with the constant Makeham term rather than the age-driven Gompertz term.
Is the Makeham term important for people over 70?
Its proportional share is small over 70 — typically under 5% of total hazard — because the Gompertz term dominates. It still exists in absolute terms, but its influence on the estimate is minor compared with its large role for users in their twenties and thirties.
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