de Moivre's law

Definition
De Moivre's law is a mathematical mortality model used in actuarial science and demography to describe the probability of survival for a cohort of individuals over time. Under the law, the survival function is assumed to decrease linearly from a specified age of certain death, implying a constant force of mortality that diminishes linearly with age.

Mathematical Formulation
Let $x$ denote the age of an individual and $\omega$ the limiting age (the age of certain death) at which survival is assumed to be zero. The survival probability $ ,{}_t p_x$ (the probability that a person aged $x$ survives an additional $t$ years) under de Moivre's law is expressed as

$$ {}_t p_x = \frac{\omega - (x + t)}{\omega - x}, \qquad 0 \le t \le \omega - x. $$

Consequently, the probability density function for the age-at-death random variable $X$ is

$$ f_X(t) = \frac{1}{\omega - x}, \qquad x \le t \le \omega, $$

indicating a uniform distribution of deaths over the interval $[x,,\omega]$.

Historical Background
The law is named after the French mathematician Abraham de Moivre (1667–1754), who introduced the linear survival model in his 1725 work The Doctrine of Chances. De Moivre employed the model to simplify calculations of life annuities and actuarial premiums at a time when detailed mortality tables were unavailable.

Assumptions and Limitations

• Linear Decrease in Survivorship: The model assumes that the number of survivors declines linearly with age, an assumption that holds only approximately for certain age ranges (typically middle ages) and fails at extreme young or old ages where empirical mortality rates are non‑linear.
• Constant Limiting Age: The parameter $\omega$ is treated as a fixed upper bound on lifespan; in practice, human longevity exhibits variability that the model does not capture.
• Uniform Distribution of Deaths: By implying a uniform distribution of deaths between ages $x$ and $\omega$, the model neglects observed variations in mortality risk caused by medical, environmental, and socioeconomic factors.

Because of these simplifications, de Moivre's law is primarily used for pedagogical illustration and as a benchmark against which more sophisticated mortality models (e.g., Gompertz, Makeham, or the Lee–Carter model) are compared.

Applications

• Actuarial Calculations: Provides closed‑form expressions for life annuities, temporary insurance, and pure endowments in introductory actuarial curricula.
• Theoretical Analyses: Serves as a tractable case in the derivation of properties of life tables, survival functions, and related stochastic processes.

Related Concepts

• Gompertz Law of Mortality: An exponential model of increasing mortality with age.
• Makeham Extension: Adds an age‑independent component to the Gompertz law.
• Uniform Distribution of Deaths (UDD): A discrete analogue of de Moivre's law used in actuarial textbooks for approximating mortality between integer ages.

References

1. de Moivre, A. (1725). The Doctrine of Chances. London: Royal Society.
2. Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., & Nesbitt, C. J. (1997). Actuarial Mathematics (2nd ed.). Society of Actuaries.
3. Cox, D. R., & Miller, H. D. (1965). The Theory of Stochastic Processes. Chapman & Hall.

This entry adheres to an objective, neutral tone and is based on established actuarial literature.