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

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

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

Lost in translation

(Dec 30, 2009)

Actuaries have a long-standing habit of using different terminology to statisticians.  This page lists some common terms used by actuaries in mortality work and their "translation" for a non-actuarial audience.  The terms and notation are those used by actuaries in the UK, but in every country I have visited the local actuaries have used similar notation.

Table 1. Common actuarial terms and their definition for statisticians.

Actuarial term  Actuarial notation
Statistical description
central exposed to risk Ecx The time exposed to risk of dying at age x.
curve of deaths tpxmuxplust Probability density function for the future lifetime of an individual currently alive and aged exactly x.
force of mortality mux

In…

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Tags: central exposed-to-risk, curve of deaths, force of mortality, initial exposed-to-risk, mortality law, mortality rate, survival rate, waiting time, survival models

Run-off volatility

(Dec 5, 2009)

When investigating risk in an annuity portfolio, a key task is to simulate the future lifetime for each annuitant.  Survival models make this particularly easy, as covered in an earlier posting on simulating lifetimes.

One of the first things which strikes practitioners is that volatility in run-off valuations increases with the average age of a portfolio.  The reason for this is that the variation in future lifetime gets larger relative to the average future lifetime.  One way of looking at this is to use the coefficient of variation, which is simply the standard error of the future lifetime divided by the mean:

coefficient of variation = standard error / mean

The coefficient of variation is normalised in that…

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Tags: simulation, curve of deaths, coefficient of variation, ICA, Solvency II

Features of the survival curve

(Sep 10, 2008)

 The survival curve is simply the proportion of lives surviving to each age.  Below is an example for males at initial age 60 in the United Kingdom, using the Interim Life Table from the Government Actuary's Department:

Survival curve for males in United Kingdom between 2004 and 2006

The survival curve starts at 1 (or 100%) as everyone is alive at outset, and decreases monotonically towards zero (or 0%) as people die. The survival curve is better known to actuaries as tpx, the probability of a life aged x surviving to age x+t.  An oft-unappreciated feature of the survival curve is that the area underneath it is simply the life expectancy.

Instead of plotting the survival curve, exactly the same data can be used plot the distribution of age at death:

Distribution of age at death for males in United Kingdom between 2004 and 2006

The graph above is known to actuaries…

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Tags: survival analysis, survival curve, curve of deaths

Mortality transformation

(Sep 1, 2008)

A tool often used by demographers is the distribution of age at death in a population.  This is known to actuaries as the curve of deaths, and the past 170 years have seen a rather remarkable transformation in this curve.  In the mid-19th century mortality was characterised by a very high rate of mortality in the early years of life, as shown in the chart below using the third English Life Table for males:

Distribution of age at death according to English Life Table 3

The dark band in the chart above shows the shortest adult age range in which half of all deaths occur.  From the late 20th century onwards mortality looks very different, as shown below for English Life Table 15 for males:

Distribution of age at death according to English Life Table 15

As before, the shortest age range covering half of all deaths is marked in a darker colour.  There…

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Tags: mortality transformation, curve of deaths, mortality compression

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