Benjamin Gompertz's Mortality Model

CHAPTER THREE: METHODOLOGY.
3.1.1 Mortality Modeling.

Life assurance is a product designed for financial managed as far as uncertainty an individual will survive is concerned. Therefore, X is used to represent the lifetime random variable and its associated model for mortality are the basic building block in actuarial mathematics.

3.1.2 Survival Distributions.

X is a non-negative random variable representing an individual lifetime in a cohort or population.
Let f(x) represents the probability density function of X and let the cumulative distribution be = (X x) = Be the probability of a person surviving to age x. be the force of mortality which is given by

the number of people die each day during any particular year of age. , and are the discrete counterparts of continuous functions , and

3.1.3 Mathematical Mortality Model Abraham De Moivre in 1725, suggested that probability of survival from birth until age x could be expressed as a linear function of age. In terms of hazard rate the model can be expressed as Where is the highest attainable age.

However the most successful and influential mortality law belongs to Benjamin Gompertz which he came up with in 1825. In terms of hazard rates is given by Gompertz law is an exponential function and it is a reasonable assumption for middle ages and older ages.

In 1860 William Makeham noted Gompertz equation failed to capture the behavior of mortality at higher ages and added a constant term, A in order to correct for deficiency. In terms of hazard rates is given by The constant term is sometimes interpreted as an allowance for accidental deaths, not depending on age.
The rationale behind the laws is based on an observation made by Benjamin Gompertz in the early 1800s.

-log
Taking logs and changing signs again gives