# Information Matrix

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### Don't fear the integral!

Actuaries denote with $${}_tp_x$$ the probability that a life alive aged exactly $$x$$ years will survive a further $$t$$ years or more.  The most basic result in survival analysis is the following relationship with the instantaneous mortality hazard, $$\mu_x$$:

${}_tp_x = e^{-H_x(t)}\qquad(1)$

where $$H_x(t)$$ is the integrated hazard:

$H_x(t) = \int_0^t\mu_{x+s}ds\qquad(2).$

Written by: Stephen Richards

### Doing our homework

In Richards et al (2013) we described how actuaries can create mortality tables derived from a portfolio's own experience, rather than relying on tables published elsewhere.  There are good reasons why actuaries need to be able to do this, and we came across a stark reminder of this while writing Richards & Macdonald (2024).

Written by: Stephen Richards

### Valuing liabilities with survival models

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, $$v^t$$.

Written by: Stephen Richards

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.
Written by: Stephen Richards

### Some points for integration

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.

Written by: Stephen Richards

### Forward thinking

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.
Written by: Stephen Richards

### Simulation and survival

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.
Written by: Stephen Richards

### Fifteen-year (h)itch

Effective risk modelling is about grouping people with shared characteristics which affect this risk.  In mortality analysis by far the most important risk factor is age, so it is not a good idea to mix the young and old if it can be avoided.
Written by: Stephen Richards