In a (much) earlier blog, Angus introduced the product-integral representation of the survival function:
\[{}_tp_x = \prod_0^t(1-\mu_{x+s}ds),\qquad(1)\]
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A common approach to teaching students about mortality is to view survival as a Bernoulli trial over one year. This view proposes that, if a life alive now is aged \(x\), whether the life dies in the coming year is a Bernoulli trial with the probability of death equal to \(q_x\). With enough observations, one can estimate \(\hat q_x\), which is the basis of the life tables historically used by actuaries.
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On 29th August 2024 Professor Angus S. Macdonald and Dr. Stephen J. Richards will give a joint seminar on contemporary mortality modelling, in principle and in practice. Attendance is free and pre-registration is not required.
The seminar will take place at 11:45hrs in Room T.01 in the Colin Maclaurin building at Heriot-Watt's Riccarton campus. See also the campus map for location and parking.
The Institute and Faculty of Actuaries (IFoA) is hosting a meeting on contemporary mortality modelling on 11th March 2025. Booking details can be found on the IFoA website.
A PDF of the two papers can be downloaded here.
Longevitas is pleased to announce that the Government Actuary’s Department has licensed the Longevitas survival-modelling software. Further details can be found in the press release.
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:
\[H_x(t) = \int_0^t\mu_{x+s}ds\qquad(2).\]
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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.
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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.
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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 (2025).
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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).
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