Actuaries have a long-standing habit of using different terminology to statisticians.  This page lists some common terms used by actuaries in mortality work and their "translation" for a non-actuarial audience.  The terms and notation are those used by actuaries in the UK, but in every country I have visited the local actuaries have used similar notation.

Table 1. Common actuarial terms and their definition for statisticians.

Actuarial term	Actuarial notation	Statistical description
central exposed to risk		The time exposed to risk of dying at age x.
curve of deaths		Probability density function for the future lifetime of an individual currently alive and aged exactly x.
force of mortality		In…

When investigating risk in an annuity portfolio, a key task is to simulate the future lifetime for each annuitant.  Survival models make this particularly easy, as covered in an earlier posting on simulating lifetimes.

One of the first things which strikes practitioners is that volatility in run-off valuations increases with the average age of a portfolio.  The reason for this is that the variation in future lifetime gets larger relative to the average future lifetime.  One way of looking at this is to use the coefficient of variation, which is simply the standard error of the future lifetime divided by the mean:

coefficient of variation = standard error / mean

The coefficient of variation is normalised in that…

 The survival curve is simply the proportion of lives surviving to each age.  Below is an example for males at initial age 60 in the United Kingdom, using the Interim Life Table from the Government Actuary's Department:

The survival curve starts at 1 (or 100%) as everyone is alive at outset, and decreases monotonically towards zero (or 0%) as people die. The survival curve is better known to actuaries as _tp_x, the probability of a life aged x surviving to age x+t.  An oft-unappreciated feature of the survival curve is that the area underneath it is simply the life expectancy.

Instead of plotting the survival curve, exactly the same data can be used plot the distribution of age at death:

The graph above is known to actuaries…

A tool often used by demographers is the distribution of age at death in a population.  This is known to actuaries as the curve of deaths, and the past 170 years have seen a rather remarkable transformation in this curve.  In the mid-19th century mortality was characterised by a very high rate of mortality in the early years of life, as shown in the chart below using the third English Life Table for males:

The dark band in the chart above shows the shortest adult age range in which half of all deaths occur.  From the late 20th century onwards mortality looks very different, as shown below for English Life Table 15 for males:

As before, the shortest age range covering half of all deaths is marked in a darker colour.  There…