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.^[1][2]

The law is both a hazard decomposition and a parametric lifetime distribution. Once \(A\), \(B\), and \(C\) are specified, cumulative hazard, survival, death density, conditional survival, and mortality doubling time follow. The identity is the additive constant plus exponential hazard with age-indexed parameter meanings—not any S-shaped curve carrying the name Gompertz.

Structural Signature

A qualifying model contains these roles:

• Age origin and scale: a nonnegative age variable \(x\), with units declared so the rate parameters are interpretable.
• Instantaneous conditional risk: the force of mortality \(\mu(x)\), conditional on survival to age \(x\).
• Makeham component: a nonnegative age-independent hazard \(A\).
• Gompertz level: \(B\), setting the age-dependent component's baseline magnitude relative to the chosen age origin.
• Gompertz slope: \(C\), setting exponential acceleration with age and the doubling time \(\log 2/C\).
• Additive hazard composition: component hazards sum on the instantaneous-risk scale rather than on survival probabilities or death counts.
• Integrated survival law: cumulative hazard \(H(x)=\int_0^x\mu(u)\,du\) determines \(S(x)=e^{-H(x)}\).
• Adult-age validity window: fitting and interpretation are tied to ages where the constant-plus-exponential approximation is empirically defensible.

For the stated parameterization,

\[ H(x)=Ax+\frac{B}{C}\left(e^{Cx}-1\right), \qquad S(x)=\exp\!\left[-Ax-\frac{B}{C}\left(e^{Cx}-1\right)\right]. \]

The density of age at death is \(f(x)=\mu(x)S(x)\). Setting \(A=0\) recovers the two-parameter Gompertz law. Other sources write \(B c^x\) with \(c>1\); this is the same family under \(C=\log c\).

What It Is Not

The law is not the generic Gompertz growth curve used for tumor size, population saturation, or diffusion of innovations. Those applications usually model the level of a quantity with a sigmoidal function. Gompertz–Makeham models an instantaneous hazard whose age-dependent component grows exponentially.

It is not every increasing-hazard lifetime distribution. Weibull hazards grow as powers of age, logistic mortality models allow late-age deceleration, and gamma-frailty mixtures can flatten aggregate hazards through population heterogeneity. Nor is it an additive decomposition of observed probabilities of death over wide age intervals; hazards add instantaneously, while finite-interval probabilities derive through integration.

The Makeham term is not automatically a literal count of accidental deaths. It is a fitted age-independent hazard component. Cause-specific interpretation requires independent data and identifiability; a good aggregate fit alone does not prove distinct biological causes.

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.^[3]

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. Demographic studies therefore compare Gompertz–Makeham with logistic, frailty, and nonparametric alternatives rather than treating it as universal law.^[4][5]

Reliability analyses sometimes reuse the distribution for ageing technical systems, but transfer is literal only when the failure hazard genuinely supports a constant-plus-exponential form. The parameter names should then refer to failure rather than human causes of death.

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.

The most reliable recognition test is algebraic: plot or model \(\mu(x)-A\) on a logarithmic scale. Under the law, \(\log(\mu(x)-A)=\log B+Cx\) is linear over the chosen age range. If no nonnegative constant produces an approximately linear residual or if the slope changes systematically, the model's structural signature is unsupported.

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.

The compression also exposes identifiable interventions and uncertainties. A change mainly in \(A\) shifts hazards nearly uniformly; a change in \(B\) rescales the senescent component; a change in \(C\) alters the age gradient and doubling time. These interpretations are model-based and can be confounded by choice of age origin, population heterogeneity, or changing cause mixtures.

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 \(e^{C(x+d)}=2e^{Cx}\). The total hazard does not exactly double over \(d\) when \(A>0\); only the age-dependent component does. That distinction prevents a common inference error.

Changing the age origin from \(x\) to \(y=x-x_0\) leaves \(C\) unchanged but transforms the Gompertz level to \(B e^{Cx_0}\). Therefore cross-study comparison of \(B\) is meaningless unless age origins and time units match.

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. If maintenance resets effective age or hazards depend on load cycles rather than chronological time, a renewal or degradation model may be more appropriate.

Examples

1. Pure Gompertz case. Set \(A=0\). Then \(\mu(x)=Be^{Cx}\), and the entire hazard doubles every \(\log2/C\) age units.
2. Background-dominated early adult ages. When \(Be^{Cx}\ll A\), total mortality is approximately constant. As age increases, the exponential component eventually dominates and the log hazard becomes approximately linear with slope \(C\).
3. Conditional survival. An insurer evaluating survival from age \(x\) to \(x+t\) uses \({}_tp_x\), not \(S(t)\), because the Gompertz term depends on attained age.
4. Infant-mortality counterexample. A mortality schedule that falls sharply after birth and later rises cannot be fitted across all ages by a monotonically increasing constant-plus-exponential hazard without systematic error.
5. Late-age boundary. If cleaned data show persistent hazard deceleration, a logistic or frailty-extended model can fit a feature the classical law cannot express.^[5]

Structural Tensions

• Parsimony vs. age-specific realism. Three parameters make calculation and comparison easy but cannot represent every childhood, cohort, or late-life feature. Diagnostic: inspect residuals by age and declare the fitted interval before extrapolating.
• Component interpretability vs. identifiability. Constant and exponential terms invite causal labels, but aggregate data may not uniquely identify causes. Diagnostic: require cause-specific or external evidence before interpreting \(A\) as accidents or \(Be^{Cx}\) as one biological mechanism.
• Extrapolation vs. oldest-old uncertainty. Exponential continuation gives definite forecasts where data are sparse. Diagnostic: compare Gompertz–Makeham forecasts with logistic, frailty, and nonparametric alternatives under age-quality checks.
• Hazard doubling vs. total-risk doubling. \(C\) fixes doubling only for the Gompertz component when \(A>0\). Diagnostic: subtract the estimated Makeham term before asserting a doubling interval.
• Autonomy vs. Probability Distribution closure. Probability Distribution supplies generic mass, density, and survival machinery but not this constant-plus-exponential hazard or parameter roles. Diagnostic: require the exact hazard decomposition and adult-age interpretation before using the named node.

Structural–Framed Character

The law is structurally explicit: age, two component hazards, additive combination, integration, and survival consequences are stable across applications. It is framed by actuarial and demographic meanings—force of mortality, attained age, background risk, senescent acceleration, and adult validity. Its formula transfers more broadly than its human-mortality interpretation.

Structural Core vs. Domain Accent

The structural core is a survival distribution generated by an additive constant and exponential hazard. The domain accent interprets time as age, the event as death, the constant as Makeham background mortality, and the exponential as Gompertz adult-age mortality. Removing these roles yields generic parametric survival analysis, already covered by Probability Distribution and hazard modeling.

The stable formula-plus-role package justifies a domain-specific node because actuarial and demographic practice repeatedly uses its parameters, nesting, conditional survival, and validity boundaries as one named model.

Instantiates / Related Primes

Gompertz–Makeham is a strict specialization of Probability Distribution: the hazard uniquely determines a named lifetime law and its density, survival, and quantiles. It relates to Temporal Decay and Degradation only at a broad aging-pattern level and to Renewal Process only when lifetimes become interarrival distributions in a separate recurrent-event model. Neither is a tighter parent.

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
• Regression — 0.76
• Robust Regression — 0.76
• Focused Information Criterion — 0.76

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With

• Gompertz distribution: the \(A=0\) nested case.
• Gompertz curve: a sigmoidal level curve, not necessarily a mortality hazard.
• Weibull distribution: typically has power-law rather than exponential-age hazard.
• Logistic mortality model: permits late-age hazard deceleration or plateau behavior.
• Gamma-Gompertz model: introduces unobserved heterogeneity/frailty and changes the aggregate hazard.
• Life table: a tabular empirical or projected mortality schedule, which may or may not use this law.

References

[1] Benjamin Gompertz, “On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies,” Philosophical Transactions of the Royal Society of London 115 (1825): 513–583, DOI 10.1098/rstl.1825.0026. registry ↩

[2] William Matthew Makeham, “On the Law of Mortality,” Journal of the Institute of Actuaries 13, no. 6 (1867): 325–358, https://www.cambridge.org/core/journals/journal-of-the-institute-of-actuaries/article/abs/on-the-law-of-mortality/646923122A0951B92642564F0210F424, DOI 10.1017/S2046166600003238. registry ↩

[3] David C. M. Dickson, Mary R. Hardy, and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks, 3^rd ed., Cambridge University Press, 2020, chapters on survival models and Gompertz–Makeham laws. registry ↩

[4] Leonid A. Gavrilov and Natalia S. Gavrilova, “Historical Evolution of Old-Age Mortality and New Approaches to Mortality Forecasting,” North American Actuarial Journal 15, no. 4 (2011): 433–458, DOI 10.1080/10920277.2011.10597629. registry ↩

[5] Shiro Horiuchi and John R. Wilmoth, “Deceleration in the Age Pattern of Mortality at Older Ages,” Demography 35 (1998): 391–412, DOI 10.2307/3004009. registry ↩a ↩b