Longevity trend risk is different from most other risks an insurer faces because the risk lies in the long-term trajectory taken by mortality rates. This trend unfolds over many years as an accumulation of small changes, so a natural approach is to calculate reserves using a long-term stress projection from a stochastic model, as shown in Figure 1.

Figure 1. Central projection from a Lee-Carter model, together with the lower 99.5th confidence level derived from multiplying the projection standard error by Z=-2.58 (this being the lower 99.5th point of the N(0,1) distribution).

The stressed-trend approach behind Figure 1 can be used for a variety of models - Table 1 shows the best-estimate and 99.5th percentile…

If you'll forgive the nautical metaphor, forecasting longevity over the past few decades has proven to be anything but plain sailing. Those plotting a course with unshakable certainty have usually ended up storm-tossed and floating in a barrel. Since the future is unknowable, any methodology applied to forecasting must have uncertainty at its heart, which is why we advocate using stochastic projection models (and more than one at that).

Stochastic projection models are tricky beasts, however, with many methods of forecasting (ARIMA time series, drift models, penalty projections, bivariate projections, and others), each with different characteristics. One concern that arises from this sophistication…

In an earlier post we looked at the implications for savers of the historically low interest rates in the UK.  Low interest rates are a policy response to the unusual economic conditions in which the developed world currently finds itself.  Besides being bad for savers, low interest rates increase the value of liabilities for pension schemes and can thus aggravate pension deficits.

However, these record low interest rates only apply in the short term, with a steep rise to a more normal longer-term rate.  This is shown in Figure 1, which plots the redemption yields for UK government gilts on 16th December 2011.  The points appear to follow a smooth underlying pattern, known as the yield curve.

Figure 1. Redemption…

The CMI recently asked for an overview note on survival models.  Since this subject is of wider actuarial interest, we wanted to make this publically available. An electronic copy can be downloaded from the link on the right.

Mortality statistics occasionally make the news, usually with some eye-catching statements.  Here is a recent example from the BBC:

This statement is indeed eye-catching.  It is also incorrect.  Figure 1 shows the mortality rates for assured lives of life companies in the United Kingdom. If we take the "fifth and sixth decades of life" to mean the age range 40-59, then Figure 1 shows that mortality rates are at their lowest for people in their third…

When an insurer or reinsurer takes on a new insurance risk, there are two things of special interest: the best estimate of the risk and the tail risk.  The best estimate is about the current expectation of claim levels or costs, while tail risk is about how bad things could get if the company were unlucky.  The UK's ICA regime and the pending EU Solvency II regime are both concerned with tail risk at the 99.5th percentile.

When investigating tail risk, we are talking about the right-hand tail of a loss distribution, i.e. the part which contains the rare-but-costly scenarios.  It is therefore critically important that the model used does a good job of estimating the size and shape of this tail.  Since the loss is a random…

In just over one year's time, insurers throughout the European Union will be prohibited from using a person's gender to price insurance risks.  This court ruling applies to individual contracts, but not (apparently) to group arrangements.

For insurers it is time to review their pricing bases and systems in good time for December 2012.  Gender has been used as a risk factor in insurance for a very long time: the oldest copy of the Journal of the Institute of Actuaries on my bookshelf shows that actuaries were separately calculating mortality tables by gender at least as far back as 1871.  As a result, it is quite possible that pricing and quotation systems have some subtle oddities, and these will need to be identified…

Late last year I drew up a table of actuarial terms and their translation for statisticians.  I had thought that it was a uniquely actuarial trait to use different names compared to other disciplines.  It turns out that statisticians are almost as guilty.  Table 1 shows some common statistical terms in mortality modelling and their description for non-statisticians.

Table 1. Some statistical terms and their definition for mathematicians and engineers.

Statistical term	Notation	Description
hazard function	varies	The instantaneous failure rate.
observed information matrix		The curvature of the log-likelihood function, i.e. the negative of the matrix of second partial derivatives.  This is the same…

In an earlier post we looked at how to create a proxy for ill-health early retirements based on age at commencement.  This is an example of dealing with missing data - we infer a useful proxy to replace the lost or missing health status at retirement.

Another common problem occurs during data or system migrations, where historical experience data is often not carried across to a new administration system.  Migrations happen when a life office consolidates multiple systems into one, or when a pension scheme changes administrator.  System migrations aren't easy, and migrating past historical data is usually one of the last tasks on the priority list.  As a result, data migration is unfortunately one of the first…

Graduation is the process whereby smooth mortality rates are created from crude mortality rates.  Smoothness is an important part of graduation, but another is the extrapolation of mortality rates to ages at which data may be unreliable or even non-existent.  An example would be pension-fund work where reliable mortality-experience data might be available up to age 100, but where actuaries required mortality rates up to age 120 (say) for the purposes of calculating pension reserves.  An illustration of this dual smoothing and extrapolation is given in Figure 1.

Figure 1. log(mortality) by age for males in United States of America.  Crude rates shown for ages 20-100, together with graduated rates outside…