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Gompertz–Makeham Law of Mortality

An adult-lifetime model decomposes instantaneous mortality into an age-independent Makeham hazard plus a Gompertz hazard that rises exponentially with age.

Version
v2 · 2026-09-19 · History
Domain-specific #
1944
Domain group
Social Sciences
Origin domain
Economics & Finance
Aliases
Makeham law of mortality, Gompertz-Makeham distribution

Core Idea

The Gompertz–Makeham law of mortality models the force of mortality at age \(x\) as the sum of a constant background hazard and an exponentially increasing age-related hazard:

\[ \mu(x)=A+B e^{Cx},\qquad A\geq0,\ B>0,\ C>0. \]

The constant \(A\) is the Makeham term, conventionally interpreted as an age-independent background contribution. The Gompertz component \(B e^{Cx}\) captures the regular adult-age increase in mortality risk. Benjamin Gompertz introduced the exponential-age law in 1825; William Makeham added a constant component in nineteenth-century actuarial work.

How would you explain it like I'm…

Steady Plus Growing Risk

People who study how long people live use a simple recipe for the chance of dying at each age. One part is a small risk that stays the same at every age, like the chance of an accident. The other part starts tiny and keeps getting bigger and bigger as a grown-up gets older, multiplying again and again. Add the two parts and you get the Gompertz–Makeham Law of Mortality.

Steady Risk Plus Growing Risk

The Gompertz–Makeham Law of Mortality is a formula that describes how the risk of dying changes with age. It adds together two pieces. The first is a background risk that's the same no matter how old you are. The second grows exponentially with age, which means it multiplies by the same amount over each equal stretch of years, so for adults it doubles again and again. Once you know three numbers that set these pieces, you can figure out things like how many people are likely to survive to each age.

Constant-Plus-Exponential Mortality Law

The Gompertz–Makeham law models the force of mortality, roughly the instantaneous risk of death at age x, as μ(x) = A + B·e^(Cx). The constant A, the Makeham term, is usually read as a background risk that doesn't depend on age. The term B·e^(Cx), from Gompertz, grows exponentially with age, capturing the steady rise of adult mortality; with C > 0 this age-related part doubles over a fixed number of years. Benjamin Gompertz introduced the exponential part in 1825, and William Makeham later added the constant. Once A, B and C are known, you can calculate survival probabilities and the distribution of ages at death. The defining feature is constant plus exponential, not just any curve with Gompertz's name on it.

 

The Gompertz–Makeham Law of Mortality specifies the force of mortality (hazard rate) at age x as μ(x) = A + B e^(Cx), with A ≥ 0, B > 0, C > 0. The Makeham constant A is conventionally read as an age-independent background hazard, while the Gompertz component B e^(Cx) models the regular exponential rise in adult mortality, with C setting the rate of increase and hence a mortality doubling time of ln(2)/C for that component. Gompertz introduced the exponential-age law in 1825, and Makeham added the constant in nineteenth-century actuarial work. The law is both a decomposition of the hazard and a full parametric lifetime distribution: integrating μ gives the cumulative hazard, from which survival S(x) = exp(−∫μ), the death density, and conditional survival all follow. This makes it a workhorse in actuarial science and demography. Its identity is the additive constant-plus-exponential hazard with these age-indexed parameter meanings; it should not be confused with the generic S-shaped Gompertz growth curve.

Scope of Application

The model is used in actuarial science for life contingencies, annuity and pension valuation, mortality tables, and longevity-risk work; in demography and gerontology for adult mortality schedules; and in survival analysis as a named parametric lifetime distribution. Modern actuarial texts derive survival and conditional-lifetime quantities directly from its force of mortality.

Its strongest empirical role is a parsimonious approximation over adult and many older ages. It is not designed to capture elevated infant and childhood mortality. At very high ages, sparse data, age misreporting, cohort selection, and heterogeneity complicate whether mortality continues exponentially, decelerates, or approaches a plateau.

Clarity

The model makes three quantities that are often conflated explicit. The hazard \(\mu(x)\) is an instantaneous conditional rate, not the probability of death at exact age \(x\). The survival function \(S(x)\) is the probability of reaching beyond age \(x\). The finite-interval death probability conditional on survival to \(x\) is \(1-S(x+t)/S(x)\), not simply \(t\mu(x)\) except as a small-interval approximation.

Manages Complexity

Three parameters compress an entire adult survival schedule. \(A\) sets background risk, \(B\) fixes the starting level of age-related risk, and \(C\) fixes its rate of acceleration. Integration yields survival without separately parameterizing every age-specific death rate. This supports comparison across cohorts, pricing calculations, scenario stress, and extrapolation—provided the validity interval is declared.

Abstract Reasoning

Conditional survival for an individual already alive at age \(x\) is

\[ {}_tp_x=\frac{S(x+t)}{S(x)} =\exp\!\left[-At-\frac{B}{C}e^{Cx}\left(e^{Ct}-1\right)\right]. \]

This equation shows that the constant component contributes \(At\) regardless of attained age, while the exponential component is multiplied by \(e^{Cx}\). The Gompertz component doubles every \(d=\log 2/C\) units because \(e^{C(x+d)}=2e^{Cx}\).

Knowledge Transfer

Within mortality modeling, the role inventory transfers across national life tables, insured populations, cohorts, sexes, and causes when each analysis uses age, force of mortality, constant background, exponential senescent component, and integrated survival. The same computational machinery produces life expectancy, survival probabilities, and actuarial present values.

Transfer to biological or technical reliability preserves the mathematics but changes the domain accent. Age becomes operating time, death becomes failure, and the Makeham term becomes age-independent baseline failure.

Relationships to Other Abstractions

Local relationship map for Gompertz–Makeham Law of MortalityParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Gompertz–MakehamLaw of MortalityDOMAINDomain-specific abstraction: Probability Distribution — is a kind ofProbabilityDistributionDOMAIN

Current abstraction Gompertz–Makeham Law of Mortality Domain-specific

Parents (1) — outgoing typed relations

  • Gompertz–Makeham Law of Mortality is a kind of Probability Distribution Domain-specific

    Gompertz–Makeham is a strict specialization of Probability Distribution: the hazard uniquely determines a named lifetime law and its density, survival, and quantiles.

Hierarchy paths (3) — routes to 2 parentless roots

Neighborhood in Abstraction Space

Gompertz–Makeham Law of Mortality sits in a sparse region of the domain-specific corpus (96th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25