Obesity is a public-health concern throughout the developed world, since it is linked to a variety of chronic conditions such as diabetes.  Obesity is also of interest to insurers, since it is linked to excess mortality from a wide range of causes, including heart disease.  The most commonly used measure of obesity is the body-mass index (BMI), which is calculated from a person's height and weight. The British Heart Foundation has an online BMI calculator which is free to use.

Now, the United Kingdom doesn't often take a leading role in the EU, but sadly obesity is one area where it rather stands out, as shown in Table 1.

Table 1. Prevalence of obesity among adults in selected large EU countries (Source: Health Interview…

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…

The rationale of any business is to make a profit.  This is usually achieved by selling things for more than they cost to make or supply.  The annuities business is no different: an insurer in the UK will typically charge around 5% more than it expects the annuity to cost.  The twist is that it takes many years to find out how much profit is actually made (or if any profit was made at all).

Of course, mortality is a random process as no-one can say for sure when an annuitant will die.  This means that the profit on an annuity is also a random variable.  Figure 1 shows the probability distribution of the profit on a level annuity payable annually in advance to a male aged 60.

Figure 1. Probability distribution of discounted profit…

One of the most important means of checking a model's fit is to look at the residuals, i.e. the standardised differences between the actual data observed and what the model predicts.  One common definition, known as the Pearson residual, is as follows:

where r is the residual, D is the observed number of deaths and E is the expected number of deaths. This definition is quick and easy to apply, and works well where there are relatively large numbers of observed and expected deaths.  If the underlying model used to generate the expected values in E is correct, the residuals should have an approximate N(0, 1) distribution.  The sum of the r² values can be compared with the appropriate point of a χ² (chi-squared)…

We've created a short graphical summary of the application of postcode-driven lifestyle within actuarial mortality models. It shows how the incorporation of postcode geodemographics as a proxy for socio-economic group can overcome the limitations involved in pricing by benefit size alone.

Feel free to download the postcode pricing information sheet.  It may be used for any purpose you like, as long as it is kept whole and unmodified and the copyright message is unaltered.

Examining past trends in cause of death can be very instructive.  However, in some quarters it has become popular to try to extrapolate trends in causes of death to create a forecast of future mortality rates.  This has a superficial appeal: using a more-detailed breakdown of mortality data feels like it should result in a better-quality forecast.

However, extrapolating trends is tricky because some important causes of death are driven by the same underlying factors.  For example, smoking increases mortality due to heart disease and numerous cancers.  This means that trends by cause of death are not independent, and forecasting correlated time series is problematic.  Furthermore, some causes of death…

In statistical terminology, a factor is a categorisation which contains two or more mutually exclusive values called levels.  These levels may have a natural order, in which case the variable is said to be an ordinal factor.  An example might be year of birth: 1931 must lie between 1930 and 1932.  Another example would be benefit size band: the 9th decile of sums assured must lie between the 8th and 10th deciles.

In contrast to ordinal factors, a categorical factor is a variable where the levels do not have an obvious order.  An example of a categorical factor would be gender: all you can say about females is that they are categorically different from males, but whether you list males before females or vice versa is…

Mortality rates increase exponentially with age.  This can make comparisons difficult, as shown in Figure 1 below, which shows the period mortality rates for males in England and Wales at ten-year intervals.  It is clear that mortality has fallen at older ages, but it is much less clear what is happening at younger ages.

Figure 1. Crude mortality hazard rates for males in England and Wales

The vertical scale of any figure is determined by the extremes, which in mortality work means the rates at the youngest and oldest ages. The problem with Figure 1 is that the rates at age 90 are up to 300 times the rates at age 40, which means that even large relative changes at younger ages are invisible.  The usual solution to this…

We have written previously about the importance of the independence assumption when modelling mortality for annuities and pensions. In a recent presentation to the Royal Statistical Society I showed the audience how life insurers deduplicate their annuity data and how they use postcodes to identify socio-economic status.

When I pointed out the strong link between income, status and multiple policies, a member of the audience asked about the impact of failing to deduplicate. This is an interesting question, since getting mortality assumptions correct for annuity pricing is particularly important due to the great sensitivity of profitability on reserve levels.

We therefore fitted a simple Perks model…

In recent years insurers have looked to making better use of the data they already have. The appeal is simple: if you have already collected the data, then it is like leaving money on the table if it is not being exploited to the full. Worse, if your competitors make better use of their data, you can be selected against and lose money.

The biggest change has been in insurers' attitude towards the use of postcodes. Postcodes have to be collected and maintained anyway as part of normal business, so any extra value which can be squeezed out of them is a low-cost bonus.  As we will see, this can sometimes even be a zero-cost bonus.

Every UK residential postcode can be assigned a geodemographic type to describe the sort of people…