We have often written about how modelling the force of mortality, μ_x, is superior to using the rate of mortality, q_x.  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 μ_x 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. …

Trends in cause of death can be an instructive way of looking at past mortality, although we have previously seen that we have to be very careful that an apparent "trend" is not due to changes in recording.  Leaving aside the problems of shifting classification over time, what of the categories themselves?  Table 1 shows the top ten causes of death for males aged 70-74 in England and Wales in 2000.

Table 1. Top ten causes of death for males aged 70-74 in England & Wales in 2000.  Source: 20th Century Mortality.

There are two fundamentally different ways of thinking about how mortality evolves over time: (a) think of mortality as a time series (the approach of the Lee-Carter model and its generalizations in the Cairns-Blake-Dowd family); (b) think of mortality as a smooth surface (the approach of the 2D P-spline models of Currie, Durban and Eilers and the smooth versions of the Lee-Carter model).

In a time-series model the mortality rates of the population are presumed to follow an underlying process.  But how do we simulate observed rates in a smoothed model which has no such process?  In this post we describe how we can simulate future sample paths of mortality in this second family.  An illustration from a smoothed…

When projecting mortality rates it is common for people to ask what sort of changes in causes of death might be required to achieve a particular scenario.  Often one is asked to posit what causes of death have to be "eliminated", and the results can lead to the conclusion that a particular projection is unlikely and therefore too prudent.

However, this is sometimes due to forgetting that reducing mortality due to one cause of death can increase the rates of mortality due to another.  One example is prostate cancer, which was historically considered rare but by 2008 had become the fifth most common cause of death amongst males above age 85 in England and Wales (and the most common cancer cause of death, beating…

We have written before about how survival models make better use of available data.  Another way of viewing this is that survival models can make do with smaller data volumes than methods based on the rate of mortality, q_x.  But what do we mean by "data volumes"?  Should we measure this by claim events, by number of lives or by exposure time?  And how much is enough?

For survival models the most sensible measure is a combination of claim events and exposure time.  The number of lives is of secondary importance for survival models, since they naturally and easily span multi-year investigations.  For a survival model it is less important if 10,000 life-years of exposure is observed amongst 10,000 people…

Previously I wrote about how mortality rates by cause of death vary by deprivation index (and, by implication, socio-economic group). This substantially complicates any attempt to use cause-of-death data to make projections of mortality for annuity portfolios and defined-benefit pension schemes.

Another challenge lies in changes in classification over time. The most obvious change is that of the classification system itself: up until 2000 England & Wales used the ICD 9 system, whereas the ICD 10 system has been in use since 2001. Mapping from one to the other isn't always straightforward, not least because ICD 10 has many more causes of death than ICD 9.

One quick and obvious solution is to look at periods…

Last night an paper on the management of annuities was presented to the Faculty of Actuaries in Edinburgh.  The same paper will be presented again to the Institute of Actuaries on 22nd March in London.

I made some comments during the debate which can be found here.

Occasionally one comes across the expression "black box" with regards to a piece of software.  The user cannot see the inner workings of the software and may worry about what calculations are being performed.  It is worth taking a step back and looking at the dictionary definition of the term "black box":

Here we can see that the term "black box" is not pejorative when applied to computer system - indeed the majority of commercial software…

In an earlier blog I wrote about how stochastic volatility in run-off increases with age. This applies when you exactly know (or think you know) the current and future mortality rates.

Of course, in practice we are not certain about current or future rates.  What impact does this have?  A good way of exploring this is to use a stochastic projection model.  Table 1 shows the annuity factors at key ages using a best-estimate projection and a typical ICA or Solvency II stress test (50^th and 99.5^th percentiles, respectively).

Table 1. Annuity factors and relative capital increase for level annuity when moving from 50^th to 99.5^th percentile. ONS data for mortality of males in England & Wales, P-spline age-period…

Insurers and reinsurers throughout the EU are facing up to the implementation of Solvency II, a radical overhaul of regulatory standards for insurance business.  Recently we explored how much Solvency II demands stochastic models.  Another feature of Solvency II is the so-called "use test", which has been described as follows: