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

Some points for integration

(Mar 2, 2016)

The survivor function from age \(x\) to age \(x+t\), denoted \({}_tp_x\) by actuaries, is a useful tool in mortality work.  As mentioned in one of our earliest blogs, a basic feature is that the expected time lived is the area under the survival curve, i.e. the integral of \({}_tp_x\).  This is easy to express in visual terms, but it often requires numerical integration if there is no closed-form expression for the integral of the survival curve.  In this article we look at some of the options available to actuaries who need to integrate numerically.

A general result is that the survivor function has the following form:

\[{}_tp_x = e^{-H_x(t)}\]

where \(H_x(t)\) is the integrated hazard function:

\[H_x(t) = \int_0^t…

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Tags: life expectancy, survival curve, numerical integration, adaptive quadrature, Trapezoidal Rule, Simpson's Rule, Simpsons's 3/8 Rule

Forward thinking

(Nov 10, 2010)

A forward contract is an agreement between two parties to buy or sell an asset at a specified price at a date in the future. It is typically a private arrangement used by one or both parties to manage their risk, or where one party wishes to speculate.

A new innovation is the idea of a survivor forward, or S-forward, which is based on the concept of the survival curve. The two parties will agree on what the neutral or best-estimate survival probability will be to a certain age, and the actual survival probability will determine who pays whom and how much. If the two parties do not agree on a neutral survival probability, one may pay the other a premium. Since the future survival curve involves considerable uncertainty,…

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Tags: survivor forward, S-forward, survival curve

Simulation and survival

(Dec 6, 2009)

In an earlier post we discussed how a survival model was directly equivalent to assuming future lifetime was a random variable.  One consequence of this is that survival models make it quick and simple to simulate a policyholder's future lifetime for the purposes of ICAs and Solvency II.

The survival curve is the proportion of lives surviving to each age, i.e. tpx in actuarial parlance.  Below is a sample survival curve in red for a life aged x, showing how to read off the probability of survival to age x+t:

Simulating future lifetime from the survival curve

For simulation purposes we simply reverse this procedure: we generate a pseudo-random number uniformly distributed over the interval (0, 1), place it on the vertical axis and look up the age at death x+t.

A huge…

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Tags: survival curve, ICA, Solvency II, integrated hazard function

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

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