### The Curse of Cause of Death Models

#### (Mar 22, 2018)

Stephen's earlier blog explained the origin of the very useful result relating the life-table survival probability $${}_tp_x$$ and the hazard rate $$\mu_{x+t}$$, namely:

${}_tp_x = \exp \left( - \int_0^t \mu_{x+s} \, ds \right). \qquad (1)$

To complete the picture, we add the assumption that the future lifetime of a person now aged $$x$$ is a random variable, denoted by $$T_x$$, and the connection with expression (1) which is:

${}_tp_x = \Pr[ T_x > t ]. \qquad (2)$

The 'package' of random lifetime $$T_x$$, hazard rate $$\mu_{x+t}$$, survival function $${}_tp_x$$, and expression (1) tying everything together, sums up the mathematics of a survival model.  For the statistics, we have observations…

### A/E in A&E

#### (Apr 25, 2010)

We have often written about how modelling the force of mortality, μx, is superior to using the rate of mortality, qx.  This is all very well when you are building a formal model, but what about when you just want to quickly compare rates?  As it happens, the μx approach is quicker and more reliable, especially for portfolios with competing risks.

Consider a portfolio of term assurances where the policyholder can either lapse the policy or die.  For simplicity we will assume that each policyholder has only one policy, although in practice this is not the case and deduplication is required.  Suppose you want to compare the mortality rates between two portfolios which have very different lapse rates. …