Frailty models

(May 1, 2017)

A population consists of individuals, each with their own genetics, lifestyle, and yes, their very own force of mortality.  National mortality data, such as held by the Office for National Statistics (ONS), are observed only at the population level and the variation in the force of mortality across individuals of the same age is forever hidden.  The purpose of this blog is to show how we can attempt to model this hidden heterogeneity.

Let us suppose that Gompertz really was right and that the mortality of each individual follows their own Gompertz law from ages 40 to 90, say.  Notice that this is very different from saying that the mortality of the entire population (of individuals) follows the Gompertz law.  Let…

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Tags: frailty

Changing patterns of mortality

(Apr 28, 2017)

In an earlier post we introduced the idea of the so-called curve of deaths, which is simply the distribution of age at death.  This is intimately bound up with survival models and the idea of future lifetime as a random variable.

The development of the distribution of deaths by age can reveal a lot about a population.  Animation 1 shows the development of the distribution of deaths for males in England and Wales since 1961.  It shows the very welcome fall in infant mortality on the left, but it also shows the steady rightward drift in the distribution at older ages.

Animation 1. Male deaths by age in England and Wales since 1961.  Click on the chart to restart the animation.

ONS deaths by age for males in England and Wales since 1961

Animation 1 shows how the modal age at death drifts…

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Tags: 1919, curve of deaths

On the (funding) level

(Apr 18, 2017)

When I read Allan Martin's earlier blog on how pension-scheme reserves routinely fail to include expenses, I was so surprised I had to ask him if it was really true.  As a former life-insurance actuary, any reserve which didn't include an allowance for expenses simply wasn't a complete assessment of the liability in my view.  My amazement was reawakened recently when I received a statement about my preserved pension from a former employer.  An excerpt from the statement is shown below:

Table 1. Excerpt from summary funding statement.

  Actuarial valuation
as at
31 December 2013
Assets £2,904m
Estimated amount required
to provide benefits
Surplus/Deficit £330m
Funding level 113%


On the face of it,…

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Tags: pension-scheme funding, expenses, market consistency

Special Delivery

(Apr 4, 2017)

Drug molecules, without special intervention, don't apply only where we want them to. Indeed, late last year this fact landed pharmaceutical giant Reckitt Benckiser in trouble with the Australian regulator. Their "specific pain" range, despite bold claims on the packaging, was deemed to not target back pain, migraine or indeed any other specific pain. As a result, the premium pricing was ruled misleading and earned the company a AU$6 million fine. Pharmaceutical marketing departments will likely be unhappy that a regulator has decided the standard approach of deploying medication via simple blood circulation should no longer be represented as targeted.

Of course, this issue isn't simply about misleading…

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Tags: mortality, cancer, targeting

The Alias Problem

(Mar 18, 2017)

A problem that can crop up during mortality modelling is that of aliasing, specifically extrinsic aliasing.  The situation can be illustrated by an example of the sort of data available for a pension scheme.  Members accruing benefits have a job classification (say), which can be used to model mortality differentials.  Let's assume the classification is M (manual) and O (office), and that this coding is available for all member pensioners.  Of course, pension schemes also contain benefits paid to surviving spouses, but unless they also worked for the same employer they won't have a job classification.  We therefore have a job-classification factor with three levels: M, O and S (for spouses).

Another common…

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Tags: extrinsic aliasing

Top of the tree

(Feb 20, 2017)

What do civil servants and monkeys have in common (ignoring a purportedly greater than average interest in bananas)? This question isn't an invitation to heap scorn on the political establishment. After all, in these interesting times such an invitation seems hardly necessary. I'm referring, instead to the Whitehall studies, examining health and mortality outcomes for members of the British Civil Service. The original study began in 1968 and ran for a decade, while Whitehall II began in 1985. The findings from these prospective cohort studies continue to influence policy both nationally and internationally into the present day.

The central findings from Whitehall, however, still carry an element of…

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Tags: mortality, social status, primates, monkeys

Division of labour

(Jan 10, 2017)

At this time of year insurers have commenced their annual valuation of liabilities, part of which involves setting a mortality basis.  When doing so it is common for actuaries to separate the basis into two components: (i) the current, or period, mortality rates and (ii) the projection of the future path of mortality rates (usually mortality improvements).  This sub-division is carried over into the regular Solvency II assessment of capital requirements, where there is always a minimum of two sub-risks for longevity:

  1. Mis-estimation risk, i.e. the uncertainty over the current level of mortality.
  2. Trend risk, i.e. the uncertainty over the future direction of improvements.

In practice a Solvency II assessment…

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Tags: Valuation, Solvency II, mis-estimation risk, trend risk

Habit (re)forming

(Jan 3, 2017)

Behavioural risk factors such as smoking and excessive alcohol consumption are significant drivers of mortality and morbidity. In 2014, the UK NHS estimated smoking to be responsible for 78,000 deaths, or 17% of all deaths recorded that year. By comparison, ONS statistics that year attributed around 8,700 deaths directly to alcohol, which on the same basis would therefore equate to around 1.9% of deaths. Whilst these estimates put the relative mortality impacts in context, they take no account of wider social consequences, which are clearly another important area of debate.

The growth in use of e-cigarettes might be considered a large-scale public-health experiment. Despite fears to the contrary, a recent…

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Tags: mortality, smoking, alcohol

Season's Greetings to all our readers!

(Dec 19, 2016)

\[y = \frac{\log_e\left(\frac{x}{m}-sa\right)}{r^2}\]

\[\Rightarrow yr^2 = \log_e\left(\frac{x}{m}-sa\right)\]

\[\Rightarrow e^{yr^2} = \frac{x}{m}-sa\]

\[\Rightarrow me^{yr^2} = x-msa\]

\[\Rightarrow me^{rry} = x-mas\]

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Signal or noise?

(Nov 25, 2016)

Each year since 2009 the CMI in the UK has released a spreadsheet tool for actuaries to use for mortality projections.  I have written about this tool a number of times, including how one might go about setting the long-term rate.  The CMI now wants to change how the spreadsheet is calibrated and has proposed the following model in CMI (2016a):

\[\log m_{x,y} = \alpha_x + \beta_x(y-\bar y) + \kappa_y + \gamma_{y-x}\qquad (1)\]

which the CMI calls the APCI model.  \(m_x\) is the central rate of mortality at age \(x\) in year \(y\) and \(\alpha_x\), \(\beta_x\), \(\kappa_y\) and \(\gamma_{y-x}\) are vectors of parameters to be estimated.  \(\bar y\) is the average year, which is used to centre the time index around…

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Tags: CMI, APCI, APC, Lee-Carter, Age-Period, smoothing

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