Information Matrix

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:

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

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).

Regular readers of this blog will know that we are strong advocates of the benefits of modelling mortality in continuous time via survival models. What is less widely appreciated is that a great many financial liabilities can be valued with just two curves, each entirely determined by the force of mortality, \(\mu_{x+t}\), and a discount function, \(v^t\).

The survivor function from age \(x\) to age \(x+t\), denoted \({}_tp_x\) by actuaries, is a useful tool in mortality work. As mentioned in one of our earliest blogs, a basic feature is that the expected time lived is the area under the survival curve, i.e. the integral of \({}_tp_x\). This is easy to express in visual terms, but it often requires numerical integration if there is no closed-form expression for the integral of the survival curve.