Makeham's invaluable constant

In 1860 William Makeham published a famous paper. In it he extended Gompertz's mortality law to include a constant term:

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

where \(\mu_x\) is the mortality hazard and \(\epsilon\), \(\alpha\) and \(\beta\) are parameters free to vary on the real line.  Equation (1) is the modern expression of Makeham's law, which avoids having to impose constraints on the parameters during estimation.  Makeham expressed his law differently to equation (1), but his 1860 paper is interesting because the parameters are estimated from twenty-year survival probabilities (pages 302-304).  Having estimated the three parameters, Makeham could then deduce estimates of mortality rates at any intervening age (page 306).  In 1866 Makeham published a further paper with an explicit formula for his mortality law and a closed-form expression for the survival function (both on page 315).

[Side note: Richards (2012, Tables 1 & 2) lists a number of functional forms for the mortality hazard, \(\mu_x\), and the corresponding formulae for the integrated hazard, \(H_x(t)\).  From this one can quickly obtain a closed-form expression for the \(t\)-year survival probability from age \(x\), \({}_tp_x\), as \({}_tp_x=e^{-H_x(t)}\).]

In 1867 Makeham presented a more detailed paper on the same model to the Institute of Actuaries. Like Gompertz before him, Makeham knew the advantages of working in continuous time over discrete time.  He began his 1867 paper by calling the hazard "indispensable" to work in mortality.  Makeham coined the term "force of mortality", which is still used by Anglophone actuaries. (Gompertz called this same quantity the "intensity of mortality", while modern statisticians use the term "mortality hazard".) Makeham's argument was that an interval-based mortality rate is an imperfect measure of mortality when mortality is high, as it doesn't account for the changing (reducing) number of lives exposed to risk during the interval. Makeham's solution to this problem was to shrink the interval to zero ("diminish \(\Delta x\) without limit") and work in continuous time.  Like Gompertz, Makeham used the continuous-time mortality hazard and survival probabilities, i.e. survival models.

Makeham's law in equation (1) adds a constant term to Gompertz's law, and Makeham proved the value of this in his 1867 paper using five different example data sets.  However, the importance of the 1867 paper went much further than merely adding a parameter to Gompertz's model:

1. Makeham showed how competing decrements can be handled in continuous time (pages 329-330).

2. Makeham further showed that continuous-time methods work when there is immigration (new entrants) and emigration (withdrawals) (pages 332-333).

[Side note: Richards & Macdonald (2025) give a detailed treatment of the practical benefits to the modern actuary of working in continuous time.]

At the time Makeham wrote his papers there were advocates for using raw data, rather than fitting mortality models. To such critics Makeham said "the very worst course that could possibly be adopted is to pin our faith upon the crude results of observation", and that actuaries would "hoodwink ourselves into the belief that in so doing we are following the path indicated by experience". Makeham was therefore an early advocate of using models to smooth out random variation.