200 years of Gompertz

Today is the 200th anniversary of Benjamin Gompertz's reading of his famous paper before the Royal Society of London.  Generations of actuaries and demographers are familiar with his law of mortality:

\[\mu_x = e^{\alpha+\beta x}\qquad(1),\]

where \(\mu_x\) is the mortality hazard at age \(x\) and \(\alpha\) and \(\beta\) are real numbers free to vary on the real line. Equation (1) is the modern expression of Gompertz's law, which is preferred because it obviates the requirement to impose constraints when estimating parameters. 

[Side note: Gompertz's original paper used \(\mu_x = aq^x\), where \(a\) and \(q\) are positive real numbers. The \(q\) here has absolutely nothing to do with the \(q_x\) notation commonly used for mortality rates, which is another reason to prefer equation (1).]

As equation (1) suggests, Gompertz's 1825 law was expressed in continuous time, as is evident from his reference to "infinitely smaIl intervals of time" (page 518). He referred to his measure as the "intensity of [...] mortality" (page 518), a term still occasionally used today as a synonym for the mortality hazard.

Gompertz' paper addressed many practical actuarial problems of his day, including:

1. Interpolating mortality rates at any age.  At a time when mortality data were typically published in ten-year intervals of age, there was a pressing commercial need for actuaries to calculate mortality rates at any age.  Gompertz's model could be fitted to any mortality data, and mortality rates thus derived for any intervening age.  Indeed, although it wasn't regarded as significant at the time, Gompertz's model also allowed extrapolation of mortality rates to ages beyond the data.

2. A closed-form expression for the survival probability between any two ages.  In a pre-computer age, this reduced the amount of manual calculation required (and the scope for error).

3. A closed-form expression for the joint survival probability.  Calculating survival probabilities dependent on joint lives is a lot easier when you just add hazard rates!

It is interesting for the modern reader to note that Gompertz tabulated survival probabilities, not annual mortality rates.  With his modelling of the mortality hazard and his focus on survival probabilities, Gompertz was perhaps the earliest advocate of survival models for actuarial work.