Lecture 6 - 1.8 A r-Year Select and Ultimate Life Table In...

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1.8 A r -Year Select and Ultimate Life Table In an aggregate life table, the force of mortality of ( x ) , which is μ x ( t ) = μ x + t = - d dt l x + t l x + t , depends only on attained ages and hence the actuarial functions such as l x + t , d x + t , and q x + t depend only on attained ages x + t as well. For example, μ 30 (20) = μ 30+20 = μ 35+15 = μ 35 (15) = μ 50 . However, if we have additional information about a life at age x (for example, the life bought an insurance policy at age x or the life is disabled at age x ), then the force of mortality μ x ( t ) = μ x + t should depend on both the age x and the duration t and thus the actuarial functions such as l x + t , d x + t , and q x + t should also depend on both x and t . 1
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For example, if we consider three lives all aged 40 and assume that life 1 was disabled at 30, life 2 bought a life insurance at 35, and life 3 is unemployed at age 40, then the probability that life 1 will die next year should depend on the age 30 and the duration of 10 years (we may denote this probability by q 30+10
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