## Tables turned

Two years ago I asked the question whether we needed standard tables any more.  The question arose because most life offices and even many pension schemes have enough mortality-experience data to create their own portfolio-specific models.  Since these models can allow for more risk factors than just age and gender, and can allow for portfolio-specific variation by age, they are typically superior to standard tables.

However, I recently came across a rather rarified example where standard tables could still play a useful role.  The scenario in question involved a portfolio where retirals had only relatively recently been permitted into the fund.  The volume of data was respectable enough to create bespoke tables under normal circumstances, but the problem was that there was no data beyond age 75.  Although the statistical model fitted the data well enough, the extrapolated mortality rates far beyond age 75 were unacceptable.

In a sense this extrapolation by age is a kind of projection: instead of the usual projection of mortality rates for unknown future years, the model is projecting mortality rates for unknown ages.  Unlike projection by time, we have other contemporaneous mortality data, which we could use to judge the quality of the "projected" mortality rates at advanced ages.  This enabled us to decide that the model could not on its own extrapolate sensibly for this portfolio.

A rough and ready solution was to use the mortality rates in the standard table as the "baseline", and to measure the effect of risk factors as a simple upward or downward shift in rates.  While this approach is simplistic, it ensures that the pattern of mortality by age would be reasonable at advanced ages. This case is an exception, however, as it is usually better to use a parametric model instead of a standard table.  The limitation of a model based on a standard table is that risk factors are presumed constant across all ages.  Since most risk factors exhibit strong variation by age in their effect, this retricts the usefulness of models based on standard tables.

Assume we have a random variable, $$X$$, with expected value ... Read more