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Unformatted text preview: investigation into mortality covered the period 1 Jan 2006 to 1
Jan 2007, and the following data were recorded for each x and y :
dx =no of deaths aged x next birthday at the previous or
coincident policy anniversary
Py (t )=no of lives on 1 Jan in year 2006 + t with age label y ,
where y is the age last birthday at entry plus curtate duration
1 2 31/35 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 .
Determine the value of f , stating any assumptions you make. Actuarial Statistics – Module 7: Exposed to risk
Examples
Policy year rate intervals Solution I 1 The death data and the census data don’t match. Deﬁne the
consistent census function:
Px (t ) =no. of policyholders at time t aged x next birthday at
the previous or coincident policy anniversary. and let
time 0 = 1 Jan 2006
time 1 = 1 Jan 2007 Assume that Px (t ) varies linearly between time 0 and time 1.
Then
1
P (0) + Px (1)
c
Px (t )dt = x
Ex =
2
0
32/35 Actuarial Statistics – Module 7: Exposed to risk
Examples
Policy year rate intervals Solution II
and
Px (0) = no. of policyholders at time 0 (1 Jan 2006)
aged x next birthday
at the previous or coincident policy anniversary
= no. of policyholders at time 0 (1 Jan 2006)
aged x1 last birthday
at the previous or coincident policy anniversary
Since the curtate duration is the whole no. of years that have
elapsed since the policy was purchased, we have
age last birthday at entry plus curtate duration
= age last birthday at the previous
33/35 or coincident policy anniversary Actuarial Statistics – Module 7: Exposed to risk
Examples
Policy year rate intervals Solution III So Px (0) = no. of policyholders at time 0 (1 Jan 2006) aged
x last birthday at the previous or coincident policy anniversary.
Therefore,
Px (0) = Px −1 (0)
and similarly
Px (1) = Px −1 (1)
so that
c
Ex = 34/35 Px −1 (0) + Px −1 (1)
2 Actuarial Statistics – Module 7: Exposed to risk
Examples
Policy year rate intervals Solution IV
2 Since the age label x is determined by reference to the
“previous or coincident policy anniversary”, we have a policy
year rate interval.
The actual age of a person aged x at the start of the rate
interval ranges from x − 1 (birthday is at the end of the rate
interval) to x (birthday is at the start of the rate interval).
Hence the average at the start of the rate interval is
1
1
2 (x − 1 + x ) = x − 2 and the average in the middle of the rate
interval is x .
Here we assume that birthdays are uniformly distributed
between policy anniversaries. As a consequence µ =
ˆ dx
c
Ex estimates µx , and
f =0 35/35...
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This document was uploaded on 04/03/2014.
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