Calendar year population aged 55 last birthday on 1

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Unformatted text preview: Examples Definition of x Trapezium approximation Estimation Principle of correspondence Calendar year rate intervals Policy year rate intervals 19/35 Actuarial Statistics – Module 7: Exposed to risk Examples Trapezium approximation Given the following data, we want to estimate the central exposed c to risk aged 55 last birthday E55 for the 3-year period 2001-2004. Calendar year Population aged 55 last birthday on 1 January 2001 46,233 2002 42,399 2003 42,618 2004 42,020 Using the census approximation: [P55 (01.01.2001) + P55 (01.01.2002)] 2 [P55 (01.01.2002) + P55 (01.01.2003)] + 2 [P55 (01.01.2003) + P55 (01.01.2004)] + 2 = 129, 143.5 c E55 ≈ 19/35 Actuarial Statistics – Module 7: Exposed to risk Examples Estimation 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 20/35 Actuarial Statistics – Module 7: Exposed to risk Examples Estimation Example: q vs µ You have details of the number of deaths aged 40 nearest birthday in a recent investigation. The actual age at the start and at the end of the life year rate 1 interval for age label 40 is 39 2 and 40 1 , respectively. 2 q estimates q39 1 ˆ 2 µ estimates µ40 ˆ 20/35 Actuarial Statistics – Module 7: Exposed to risk Examples Estimation Example An investigation into mortality covered the period 1 Jan 2006 to 1 Jan 2007, and the following data were recorded for each x : dx =no of deaths aged x last birthday Px (t )=no of lives on 1 Jan in year 2006 + t aged x last birthday 1 2 21/35 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. Determine the value of f , stating any assumptions you make. Actuarial Statistics – Module 7: Exposed to risk Examples Estimation 1 The death data and the census data match. Assuming that Px (t ) varies linearly over calendar year 2006, the central exposed to risk for age label x is 1 c Ex = Px (t )dt = 0 2 Since the age label x is defined to be the age last birthday, the actual age at the start and the end of the life year rate interval are x and x + 1, respectively. d So µx = Exc estimates µx + 1 , ie ˆ x 2 f= 22/35 Px (0) + Px (1) 2 1 2 Actuarial Statistics – Module 7: Exposed to risk Examples Principle of correspondence 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 23/35 Actuarial Statistics – Module 7: Exposed to risk Examples Principle of correspondence Assume deaths are defined as age nearest birthday, and census data...
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