Mme Calment's Other Secret?

(Dec 9, 2018)

Favourite stories can, in the process of retelling, turn into legends. But might it eventually become difficult to distinguish between legend and myth? Indeed, are we longevity watchers about to lose a favourite story? Consider what follows, dear readers, and decide for yourselves...

Claims to extreme lifespan require independent verification for good reason. If practitioners are to make sense of the course of human longevity, the data gathered on centenarians and supercentenarians must be of high quality. Take for example, Mbah Gotho, who purportedly died aged 146 years old in 2016. Since Indonesia didn't officially record births until 1900 there existed something of a verification deficit. That deficit…

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Tags: longevity, supercentenarian

The Long Shadow of the Life Table

(Oct 27, 2018)

For centuries, the life table has been at the centre of actuarial work.  It sets out the gradual extinction of a hypothetical population, often a birth cohort.  At age \(x\), \(l_x\) individuals are still alive, and \(d_x\) of them die during the next year, so in a year's time \(l_{x+1} = l_x - d_x\) are still alive, and so on.  The following table is an extract from a life table, describing 100,000 persons in a birth cohort, of whom the last dies age 114.

Age \(x\) \(l_x\) \(d_x\)
0 100,000 362
1 99,638 236
2 99,402 225
3 99,177 229
4 98,948 234
\(\ldots\) \(\ldots\) \(\ldots\)
\(\ldots\) \(\ldots\) \(\ldots\)
111 18 10
112 8 5
113 3 2
114 1 1
115 0  

The life table is simple, intuitive, and requires hardly any mathematics to…

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Tags: life table

Hedging or betting?

(Sep 27, 2018)

Last week I presented at Longevity 14 in Amsterdam.  A recurring topic at this conference series is index-based approaches to managing longevity risk.  Indeed, this topic crops up so reliably, one could call it a hardy perennial.

For a long time insurers and pension schemes were sceptical of derivatives-based solutions to managing longevity risk.  Part of this scepticism was due to basis risk - why enter into a contract based on population mortality when a portfolio has very specific mortality characteristics?  In particular, most portfolios tend to have a concentration of risk in a relatively small subset of lives.  Another reason for scepticism was price - it was often cheaper to reinsure the entire risk…

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Tags: basis risk, concentration risk, model risk

'D' is for deficiency

(Aug 28, 2018)

The United Kingdom has long had persistent regional disparities in mortality, and thus in life expectancy.  A large part of this is due to socio-economic mix, as shown in a much earlier blog.  However, as Gavin showed in a comparison of three UK cities, socio-economic variation cannot wholly explain Glasgow's excess mortality.  There are many possible contributory factors, but in this blog we focus on one: sunshine.

An obvious difference between Scotland and the rest of the United Kingdom is that it gets less sunshine.  Figure 1 shows the monthly average hours of sunshine between 1981 and 2010:

Figure 1. Average monthly hours of sunshine across the U.K., 1981-2010. Source: Met Office.

And even when Scotland…

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Tags: Scotland, sunshine, vitamin D

Smooth Models Meet Lumpy Data

(Aug 15, 2018)

Most of the survival models used by actuaries are smooth or piecewise smooth; think of a Gompertz model for the hazard rate, or constant hazard rates at individual ages.  When we need a cumulative quantity, we use an integral, as in the cumulative hazard function, \(\Lambda_x(t)\):

\[ \Lambda_x(t) = \int_0^t \mu_{x+s} \, ds. \qquad (1) \]

Mortality data, on the other hand, are nearly always lumpy.  A finite number of people, \(d_{x+t_i}\) say, die at a discrete time \(t_i\), one of a set of observed times of death \(t_1, t_2, \ldots, t_r\).  Then when we need a cumulative quantity, we use a sum.  We saw in a previous blog that if \(l_{x+t_i^-}\) was the number of persons being observed just before time \(t_i\), then…

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Tags: Nelson-Aalen

Valuing liabilities with survival models

(Aug 2, 2018)

Regular readers of this blog will know that we are strong advocates of the benefits of modelling mortality in continuous time via survival models.  What is less widely appreciated is that a great many financial liabilities can be valued with just two curves, each entirely determined by the force of mortality, \(\mu_{x+t}\), and a discount function, \(v^t\).

The first of these useful curves is the discounted survival function, \({}_tp_xv^t\), where \({}_tp_x\) is the probability of survival from age \(x\) to age \(x+t\).  If you know the force of mortality, then you know the survival probability from the following fundamental relationship:

\[{}_tp_x=\exp\left(-\int_0^t\mu_{x+s}ds\right).\]

For…

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Tags: survival curve, curve of deaths

More than one kind of information

(Jul 19, 2018)

This collection of blogs is called Information Matrix, and it is named after an important quantity in statistics.  If we are fitting a parametric model of the hazard rate, with log-likelihood:

\[ \ell( \alpha_1, \ldots, \alpha_n ) \]

as a function of \(n\) parameters \(\alpha_1, \ldots, \alpha_n\), then the information matrix is the matrix of second-order partial derivatives of \(\ell\). That is, the matrix \({\cal I}\) with \(ij\)th component:

\[ {\cal I}_{ij} = \frac{\partial^2 \ell}{\partial \alpha_i \partial \alpha_j}. \]

It is important because \(-{\cal I}^{-1}\) evaluated at the fitted maximum \((\hat{\alpha}_1, \ldots, \hat{\alpha}_n)\) approximates the variance-covariance matrix of…

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Tags: information, indicator process

Testing the tests

(Jul 1, 2018)

Examining residuals is a key aspect of testing a model's fit.  In two previous blogs I first introduced two competing definitions of a residual for a grouped count, while later I showed how deviance residuals were superior to the older-style Pearson residuals.  If a model is correct, then the deviance residuals by age should look like random N(0,1) variables.  In particular, they should be independent with no obvious pattern linking the residual at one age with the next, i.e. there should be no autocorrelation.

In this article we will look at three alternative test statistics for lag-1 autocorrelation, i.e. correlation with the neighbouring value.  Each test statistic is based on the Pearson correlation…

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Tags: deviance residuals, autocorrelation, Fisher transform

Socio-economic differentials: convergence and divergence

(Jun 18, 2018)

Many western countries, including the UK, have recently experienced a slowdown in mortality improvements.  This might lead to the conclusion that the age of increasing life expectancies is over.  But is that the case for everyone?  Or are there some groups in the UK who are still experiencing mortality improvements?  The short answer is that mortality rates are still falling for the least deprived half of the population in England, while mortality improvements since 2011 have been virtually zero for the most deprived third.  This has important consequences for reserving for pensions and annuities, so let us explore in a bit more detail.  The findings in this blog are based on some early results of research…

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Tags: mortality convergence, mortality improvements, concentration risk, basis risk

Getting animated about longevity

(Jun 7, 2018)

We'll be the first to admit that what we have here doesn't exactly provide Pixar levels of entertainment.  However, with the release of v2.7.9 users of the Projections Toolkit can now generate animations of fitted past mortality curves and their extrapolation into the future.  Such animations can help analysts understand the behaviour of a forecast, as well as being a particularly useful way of communicating with non-specialists.  Below is a selection of animations from a smoothed Lee-Carter model fitted to the data for males in England & Wales between ages 50 and 104.

Figure 1 shows the logarithm of the force of mortality in the data region (1971-2015) and the forecast region.  It shows how mortality is…

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Tags: survival curve, curve of deaths, mortality compression

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