In the debate about how fast mortality will improve in the future, sometimes it is useful to remind ourselves how far we have come.  For example, while reading a paper from one of the leading actuarial researchers on mortality, Robert Beard, I came across the following statement:

Beard doesn't give a source for this, but using English Life Table No. 12 (which draws on data from 1960-62), the probability of a male life surviving to age 90 was around 3%. …

With the exception of dressmaking, bias is generally undesirable.  This is particularly the case when projecting future mortality rates for reserving for pension liabilities.  More precisely, a bias towards over-stating mortality rates would be particularly bad because it would lead to under-reserving.

One method for forecasting future mortality is to project changes by cause of death.  I have written previously about the numerous technical challenges facing the cause-of-death approach.  However, there is also a fundamental academic objection to this approach: it is biased.  This is covered by a number of researchers, two of whom we quote below:

One of the most written-about models for stochastic mortality projections is that from Lee & Carter (1992).  As Iain described in an earlier post, the genius of the Lee-Carter model lies in reducing a two-dimensional forecasting problem (age and time) to a simpler one-dimensional problem (time only).

A little-appreciated fact is that there are two ways of approaching the time-series projection of future mortality rates.  A simple method is to treat the future mortality index as a simple random walk with drift.  This makes the strong simplifying assumption that the mortality trend changes at a constant rate (apart from the random noise).  Figure 1 shows an example projection for males in England &…

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…

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…

In my previous post I illustrated the effects of over-dispersion in population data.  Of course, an actuary could quite properly ask: why use ONS data?  The CMI data set on assured lives might be felt to be a better guide to the mortality of pensioners, although Stephen has raised a question mark over this assumption in the past.

Figure 1 illustrates what happens with the CMI data set. The over-dispersion parameter is much smaller at 1.82, so the Poisson model gives a reasonable forecast.  Note that the over-dispersion in the CMI data comes from a different source, namely the presence of duplicates causing extra variability in death counts.  However, the same approach to over-dispersion works regardless of the…

Actuaries need to project mortality rates into the far future for calculating present values of pension and annuity liabilities.  In an earlier post Stephen wrote about the advantages of stochastic projection methods.  One method we might try is the two-dimensional P-spline method with the simple assumption that the number of deaths at age i in year j follows a Poisson distribution (Brouhns, et al, 2002).  Figure 1 shows observed and fitted log mortalities for the cross-section of the mortality surface for age 70 with this method.

Figure 1.  Observed log(mortality) rates with fitted P-spline for underlying average.  ONS data for males in England & Wales.

At first sight, all seems well - the fit seems perfectly…

Stochastic projections of future mortality are increasingly used not just to set future best-estimates, but also to inform on stress tests such as for ICAs in the UK.  By the time the Solvency II regime comes into force, I expect most major insurers across the EU will be using stochastic models for mortality projections (if they are not already doing so).

Nevertheless, there are still some people who harbour doubts about stochastic models.  One way to address such skepticism is to look at how such models would have performed in predicting the recent past.  This is known as a back-test.  The idea is that we take a long data series on mortality (here the ONS data for males in England & Wales from 1961-2007) and carry…

In a recent post on basis risk in mortality projections, I floated the idea of forecasting with limited data and even suggested that it would be possible to use the method to produce a family of consistent forecasts for different classes of business.  The present post describes an example of how this idea works in practice.

Forecasting with limited data depends on the simple idea of using actual portfolio data to adjust a separate forecast made with some very much larger reference data set.  We use the following example:

1. Reference data: CMI assured lives data between years 1950-2005, covering ages 40-89.  We assume the data have been graduated and a forecast has been made to 2048, say.  The output from this process…

With many stochastic models of mortality, projections of future mortality rates are done using a time series.  In a landmark paper, Currie, Durban and Eilers (2004) introduced the idea of using P-splines as an alternative means of generating a forecast.  P-splines formed the basis of a projection tool the CMI made freely available in 2005.

An oft-heard criticism of P-spline projections is that they can be volatile from year to year when adding new data.  This criticism was repeated in a recent paper by the CMI's projections working party: