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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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