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Policy year rate intervals
10/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
“Calendar Year” rate interval Calendar Year Rate Intervals
A calendar year rate interval is a rate interval that starts on a ﬁxed
date in the calendar year.
Everyone’s age label changes on that ﬁxed date.
Example: age on the birthday in the same calendar year.
Investigation over whole year 2008.
Life A was born in August 1950 and died on 1 Jan 2008.
The death time 1 Jan 2008 and the 58th birthday is in year
2008, so the age label for the death is 58. Life A is included in
d58 .
Life B was born on 1 Jan 1980 and life C was born 31 Dec
1980.
For any time t in 2008, both lives B and C contribute to
P (28, t ) if they are alive at time t . This is because t here is in
the same calendar year as their 28th birthday.
10/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
“Calendar Year” rate interval q vs µ At the start of the rate interval (1 Jan) for age label x , actual ages
range from x − 1 exact (those lives with x th birthdays on 31
December this year) to x exact (those lives with x th birthdays on 1
Jan this year).
Hence, if we assume that birthdays are uniformly distributed over
the calendar year, the average age at the start of rate interval (1
1
Jan), is x − 2 . This means that in this context:
q estimate qx − 1
ˆ 2 µ estimate µ(x − 1 )+ 1
ˆ
2 11/35 2 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
“Policy Year” rate interval 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
12/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
“Policy Year” rate interval Policy year rate interval I Policy duration  length of time since policy purchased
Policy anniversary  integer policy duration (date policy taken
out)
For example, if an individual bought an endowment policy on
1 August 2006, the third policy anniversary would be 1
August 2009.
A policy year rate interval: a rate interval determined by the
policy anniversary.
Age label changes on a day that is ﬁxed relative to this policy
anniversary. 12/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
“Policy Year” rate interval Policy year rate interval II
The change in age label need not occur on the policy
anniversary.
For example, a policy was bought on 1 August 2006. Age
deﬁnition is age last birthday at the nearest policy anniversary.
Age label changes when crossing midway between
anniversaries.
Example for deaths classiﬁed by policy year:
dx =no. of deaths, age x last birthday at the policy
anniversary preceding (or coincident with) the date of death
over the investigation period
The classiﬁcation of deaths by policy year might arise if all we
know is
age last birthday (or next birthday, etc) at time the policy was
purchased
the policy duration at death
13/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
“Policy Year” rate interval q vs µ Recall q and µ estimate qy and µy + 1 , respectively, where y is
2
the average age at the start of the rate interval
For example, if deaths are “age x last birthday a...
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