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:

Solvency II is a major overhaul of the reserving rules for insurers throughout the European Union.  An important consideration for annuity writers is how it will relate to longevity trend risk.  For example, consider the following text about "Statistical quality standards" as they refer to insurers' internal models:

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…

When investigating risk in an annuity portfolio, a key task is to simulate the future lifetime for each annuitant.  Survival models make this particularly easy, as covered in an earlier posting on simulating lifetimes.

One of the first things which strikes practitioners is that volatility in run-off valuations increases with the average age of a portfolio.  The reason for this is that the variation in future lifetime gets larger relative to the average future lifetime.  One way of looking at this is to use the coefficient of variation, which is simply the standard error of the future lifetime divided by the mean:

coefficient of variation = standard error / mean

The coefficient of variation is normalised in that…

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 an earlier post we discussed how a survival model was directly equivalent to assuming future lifetime was a random variable.  One consequence of this is that survival models make it quick and simple to simulate a policyholder's future lifetime for the purposes of ICAs and Solvency II.

The survival curve is the proportion of lives surviving to each age, i.e. _tp_x in actuarial parlance.  Below is a sample survival curve in red for a life aged x, showing how to read off the probability of survival to age x+t:

For simulation purposes we simply reverse this procedure: we generate a pseudo-random number uniformly distributed over the interval (0, 1), place it on the vertical axis and look up the age at death x+t.

A huge…

 Every good statistician knows that correlation does not imply causation.  Just because two things appear linked does not mean they are. However, with historical data we often don't have the luxury of carrying out controlled, scientific experiments to see if A really does cause B.

One particularly interesting example is the possible link between influenza and coronary heart-disease (CHD) mortality. The following chart is derived from data from the Massachusetts Department of Public Health Registry of Vital Records and Statistics, and is inspired by a similar chart by Azambuja and Levins (2007):

While we know that correlation does not imply causation, there does seem to be a rising incidence of CHD along…