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

v1 · 2026-08-30 · History

Domain-specific #

1944

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.

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

Current abstraction Gompertz–Makeham Law of Mortality Domain-specific

Parents (1) — more general patterns this builds on

• 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 (5) — routes to 3 parentless roots

• Gompertz–Makeham Law of Mortality → Probability Distribution → Random Variable → Function (Mapping)

Show alternative paths (4)

Neighborhood in Abstraction Space

Gompertz–Makeham Law of Mortality sits in a sparse region of the domain-specific corpus (95^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Machine-Learning Learning Curve — 0.78
• Feature scaling — 0.77
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• Robust Regression — 0.76
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Computed from structural-signature embeddings · 2026-09-08