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^tů

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

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