13_AS_7_lec_a

# Of lives under observation aged x 1 last birthday at

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Unformatted text preview: are deﬁned via age last birthday Adjustments are needed: Census data to use should be Px ,t : no. of lives under observation aged x nearest birthday at time t (i.e. no. of lives 1 under observation with the x th birthday in (t − 2 , t + 1 ).) 2 Px ,t is the no. of lives under observation aged x last birthday at time t (i.e. no. of lives under observation with the x th birthday falling in (t − 1, t )) Px −1,t is no. of lives under observation aged x − 1 last birthday at time t (i.e. no. of lives under observation with the x th birthday falling in (t , t + 1)) Assume that birthdays are uniform over a (calender) year. Hence 23/35 1 1 Px ,t ≈ Px −1,t + Px ,t 2 2 Actuarial Statistics – Module 7: Exposed to risk Examples Calendar 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 24/35 Actuarial Statistics – Module 7: Exposed to risk Examples Calendar year rate intervals Example Suppose the actual birthday is on the 1st of August 1950, and death occurs in 1990. We want to compare age deﬁnition: age last birthday (life year rate interval), with age deﬁnition: age on the birthday in the same calendar year (calendar year rate interval), at three diﬀerent dates: Date Recorded age x Actual age Recorded age (age on the birthday in the same calendar year) age last birthday 5 1 Jan 40 39 12 39 11 1 Jul 40 39 12 39 4 1 Dec 40 40 12 40 24/35 Actuarial Statistics – Module 7: Exposed to risk Examples Calendar year rate intervals Example For the age deﬁnition “age nearest birthday on 1 May 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 calendar rate interval starts on 1 May: 1 2 25/35 At the start of the rate interval (1 May), individuals with age label x have actual ages ranging from x − 1 (those lives with 2 x th birthday in just less than 6 months’ time) to x + 1 (those 2 lives who just had their x th birthday less than 6 months ago) Assuming that birthdays are uniformly distributed over the calendar year, the average actual age at the start of the rate interval is x . Actuarial Statistics – Module 7: Exposed to risk Examples Calendar year rate intervals Example of exam question A mortality investigation covered the period 1 Jan 2006 to 1 Jan 2007, and the following data were recorded for each age x : dx =no. of deaths aged x last birthday at the previous or coincident 1 Jan. Px (t )=no. of lives on 1 Jan in year 2006 + t aged x next birthday 1 2 Obtain an expression for the central exposed to risk in terms of the available census data that may be used to estimate the force of mortality µx +f , stating your assumptions. 3 26/35 Describe the rate interval implied by the classiﬁcation of de...
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