13_AS_7_lec_a

# The death data and the census data dont match let

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Unformatted text preview: aths, and give the range of ages of the lives at the beginning of the rate interval. Determine the value of f , stating any assumptions you make. Actuarial Statistics – Module 7: Exposed to risk Examples Calendar year rate intervals Solution I 1 2 Everybody changes age label on the same day-1 Jan. This is a calendar year rate interval. The actual age of a person with age label x on Jan 1, must between x (birthday is on 1 Jan) and x + 1 (birthday is on 31 Dec). The death data and the census data don’t match. Let time 0 = 1 Jan 2006 time 1 = 1 Jan 2007 27/35 Actuarial Statistics – Module 7: Exposed to risk Examples Calendar year rate intervals Solution II Px (t )=no. of lives at time t aged x last birthday at the previous or coincident 1 Jan. Then 1 c Ex = Px (t )dt 0 = = 1 (P (1 Jan 2006) + Px (Just before 1 Jan 2007)) 2x 1 1 (Px (0) + Px +1 (1 Jan 2007)) = (Px (0) + Px +1 (1)) 2 2 Note from deﬁnition: Px (0)=no. of lives at time 0 (1 Jan 2006) aged x last birthday at the previous or coincident 1 Jan. Px (0)=no. of lives on 1 Jan 2006+0 aged x next birthday. 28/35 Actuarial Statistics – Module 7: Exposed to risk Examples Calendar year rate intervals Solution III Noting that age x last birthday is same as age x + 1 next birthday yields Px (0) = Px +1 (0) The same logic implies Px +1 (1) = Px +2 (1) So ﬁnally 3 29/35 1 c Ex = (Px +1 (0) + Px +2 (1)) 2 From 1. we know the range of ages at the start of the rate interval is (x , x + 1). Hence, the average age at the start of d 1 the rate interval is x + 2 and µx = Exc estimates ˆ x µ(x + 1 )+ 1 = µx +1 . This means f = 1. 2 2 We assumed that the birthdays are uniformly spread across the calendar. Actuarial Statistics – Module 7: Exposed to risk Examples Policy year rate intervals 1 Introduction Central vs Initial Exposed to Risk Complete data Incomplete data 2 Census approximations Introduction “Calendar Year” rate interval “Policy Year” rate interval 3 Examples Deﬁnition of x Trapezium approximation Estimation Principle of correspondence Calendar year rate intervals Policy year rate intervals 30/35 Actuarial Statistics – Module 7: Exposed to risk Examples Policy year rate intervals Example Consider the age deﬁnition “age next birthday at the policy anniversary preceding death”. Determine 1 the minimum/maximum possible actual ages at the start of the rate interval 2 the average actual age at the start of the rate interval. The deﬁnition of age gives rise to a policy year rate interval starting on the policy anniversary: 1 2 30/35 At the start of the rate interval, actual ages range from x − 1 (those lives with x th birthdays occur in a year’s time, just before the next policy anniversary) to x (those lives with x th birthdays occur immediately after the policy anniversary) Assuming birthdays are distributed uniformly over policy years, average actual age at start of the rate interval is x − 1 . 2 Actuarial Statistics – Module 7: Exposed to risk Examples Policy year rate intervals Example of exam question An...
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## This document was uploaded on 04/03/2014.

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