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

Less is More: when weakness is a strength

(Jun 1, 2018)

A mathematical model that obtains extensive and useful results from the fewest and weakest assumptions possible is a compelling example of the art.  A survival model is a case in point.  The only material assumption we make is the existence of a hazard rate, \(\mu_{x+t}\), a function of age \(x+t\) such that the probability of death in a short time \(dt\) after age \(x+t\), denoted by \({}_{dt}q_{x+t}\), is:

\[{}_{dt}q_{x+t} = \mu_{x+t}dt + o(dt)\qquad (1)\]

(see Stephen's earlier blog on this topic).  It would be hard to think of a weaker mathematical description of mortality as an age-related process.  But from it much follows:

  • If we observe a life age \(x_i\) for a time \(t_i\), and define \(d_i = 1\) if the…

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Tags: survival models, Poisson distribution

(GDP)Renewing our mail-list

(May 25, 2018)

A short and simple administrative announcement ...

In common with many other organisations, we are celebrating the arrival of the EU General Data Protection Regulation (GDPR) by renewing our mailing list. We only use our mailing list for relatively infrequent communication about our blogs, research and software. We don't sell or pass on anyone's contact details.

In order to keep things simple, we are going to start from a clean slate. So, even if you had previously joined our mailing list, in this post-GDPR world, we're going to ask for you to reconfirm your desire to hear from us. If you don't do this, you won't receive mailshots from us again (but obviously can still find out what we are up to by visiting us here).

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Tags: GDPR, data protection

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