Gompertz vs Makeham: Laws of Mortality

Understanding how our mortality plays out with age is a crucial yet daunting task. Two prominent laws, Gompertz's Law and Makeham's Law, offer valuable insights into this delicate dance between life and death. While both hold significant weight, they differ in their approach and applicability.

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Gompertz's Law

Gompertz's Law, proposed by Benjamin Gompertz in 1825, models the exponential increase in mortality rates with age. It is expressed as λ(t) = λ_0⋅e^(α⋅t), where λ(t) is the mortality rate at age t, λ_0​ is the baseline mortality rate, α is a constant determining the rate of increase with age, and e is the base of the natural logarithm.

Age-Dependent Exponential Increase:

Gompertz's Law captures the idea that the risk of death accelerates exponentially as individuals age. This law is particularly applicable to adult and elderly populations.

Use in Actuarial Science:

Gompertz's Law is widely used in actuarial science for modeling mortality rates in life insurance. It helps insurers understand the age-specific risk of mortality and is valuable in determining appropriate premium rates.

Long-Term Predictions:

Gompertz's Law is often employed for long-term predictions of mortality rates and is suitable for describing mortality patterns in relatively older populations.

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Makeham's Law

Makeham's Law, introduced by William Makeham in the mid-19th century, extends Gompertz's Law by adding a constant term. The model is expressed as λ(t)= A+B⋅e^(α⋅t), where A is the constant term, B is a coefficient, and the other terms have the same meanings as in Gompertz's Law.

Addition of Constant Term:

Makeham's Law acknowledges that there is a baseline risk of mortality that is independent of age. The additional constant term (A) accounts for factors such as external causes of death or genetic predispositions that contribute to mortality regardless of age.

Applicability Across Ages:

Makeham's Law is often considered more suitable for describing mortality patterns across a broader range of ages, including both younger and older populations. The constant term allows for the incorporation of factors influencing mortality at all ages.

Actuarial Applications:

Makeham's Law, like Gompertz's Law, is used in actuarial science. It provides a more flexible model that can be applied to a wider age range, making it useful for comprehensive mortality analysis in life insurance and pension planning.

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Comparison:

Gompertz vs Makeham:

• Gompertz's Law is simpler and effective for describing mortality in older age groups.

• Makeham's Law is an extension that adds a constant term, making it more versatile for modeling mortality across various age ranges.

• Both laws are used in actuarial science, but the choice between them depends on the age distribution of the population under consideration and the specific factors influencing mortality.

In summary, while Gompertz's Law focuses on the exponential increase in mortality with age, Makeham's Law extends this concept by incorporating a constant term to account for additional mortality risks, providing a more comprehensive model for mortality analysis across different age groups.