Actuaries denote with \({}_tp_x\) the probability that a life alive aged exactly \(x\) years will survive a further \(t\) years or more.  The most basic result in survival analysis is the following relationship with the instantaneous mortality hazard, \(\mu_x\):

\[{}_tp_x = e^{-H_x(t)}\qquad(1)\]

where \(H_x(t)\) is the integrated hazard:

\[H_x(t) = \int_0^t\mu_{x+s}ds\qquad(2).\]

Impossibility has often featured in humourous fiction.  From Lewis Carroll's White Queen, who "believed as many as six impossible things before breakfast", to Douglas Adams' Restaurant at the End of the Universe, there is entertainment value in absurdity.

When asked what was most likely to blow a government off-course, Harold Macmillan allegedly replied "Events, dear boy, events!". Macmillan may not have actually uttered these words (Knowles, 2006, pages 33-34), but there's no denying that unexpected events can derail your plans.  I was recently faced with some unexpected events, albeit in a rather different context.

In Richards et al (2013) we described how actuaries can create mortality tables derived from a portfolio's own experience, rather than relying on tables published elsewhere. There are good reasons why actuaries need to be able to do this, and we came across a stark reminder of this while writing Richards & Macdonald (2024).

In a previous blog I looked at how continuous-time methods can provide real-time management information.  In that example we tracked the (almost daily) development of the mortality of two tranches of new annuities, as shown again in Figure 1.

Figure 1.  Cumulative hazard, \(\hat\Lambda(t)\), for new annuities written by French insurer.  Source: Richards and Macdonald (2024).

The sooner you know about a problem, the sooner you can do something about it.  I have written before about real-time updates to mortality estimates during shocks.  However, real-time methods also have application to everyday management questions.  Consider Figure 1(a), which shows a surge in new annuities in December 2014.  The volume of new annuities written in that month was large enough to shift the average age of the in-force annuit

Actuaries tasked with analysing a portfolio's mortality experience face a gap between what has happened in the outside world and the data they actually work with.  The various difference levels are depicted in Figure 1.

Figure 1.  The actuarial data onion.

2.8.7 Update

• A new resource area Training tab including datasets and exercises.
• Additional exclusion values for modelling specification.
• Additional liability and valuation outputs.
• License configurable SAML-based single-sign-on authentication.
• Platform, security and administrative support updates.
• Various fixes and handling improvements (documented in the release notes).