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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 day1 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
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