Unformatted text preview: t the policy
anniversary preceding death”
at the start of the rate interval (policy year) ages range from x
exact to x + 1 exact
assume a uniform distribution of birthdays with respect to
policy anniversaries
1
q estimates qx + 2 and µ estimates µx +1 14/35 Actuarial Statistics – Module 7: Exposed to risk
Census approximations
“Policy Year” rate interval Distribution of policy anniversaries over the year
Assumptions about distribution of policy anniversaries need to be
treated with care. For example:
some policies tend to be taken out in large numbers just
before the end of the tax year
there might a tendency to take out policies just before dates
upon which premium would rise
under group schemes, where insurance cover for employees is
provided by the employer, the policy anniversaries might all be
the same (in which case we have both a policy year rate
interval and a calendar year rate interval) 15/35 Actuarial Statistics – Module 7: Exposed to risk
Examples
Deﬁnition of x 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
16/35 Actuarial Statistics – Module 7: Exposed to risk
Examples
Deﬁnition of x Example: Age last birthday A mortality investigation covers the period 1 Jan 2001 to 31 Dec
2003. In this investigation, the age label used is “age last
birthday”.
Give the range of dates for which the lives in the following table
c
contribute to Ex for all relevant x ’s in the investigation period.
Assume that the day of entry counts in the exposed to risk but the
day of exit does not.
Date of birth Date of Entry Date of exit Reason for exit
A
25.04.69
07.08.99
30.10.02
Death
B
01.07.69
12.09.02
 16/35 Actuarial Statistics – Module 7: Exposed to risk
Examples
Deﬁnition of x A: The life joins the investigation at age 31 last birthday. It’s
contribution to
c
E31 : 01.01.01 to 24.04.01
c
E32 : 25.04.01 to 24.04.02
c
E33 : 25.04.02 to 29.10.02
B: The second life joins the investigation at age 33 last birthday.
It’s contribution to
c
E33 12.09.02 to 30.06.03
c 01.07.03 to 31.12.03
E34 17/35 Actuarial Statistics – Module 7: Exposed to risk
Examples
Deﬁnition of x Example: Age next birthday
Now suppose that for the same mortality investigation covers the
period 1 Jan 2001 to 31 Dec 2003, the age label used is “age next
birthday”.
c
Give the range of dates for which life A contribute to Ex at each
age where it makes a contribution.
Date of birth Date of Entry Date of exit Reason for exit
25.04.69
07.08.99
30.10.02
Death
The life joins the investigation at age 32 next birthday. It’s
contribution to
c
E32 : 01.01.01 to 24.04.01
c
E33 : 25.04.01 to 24.04.02
c
E34 : 25.04.02 to 29.10.02
18/35 Actuarial Statistics – Module 7: Exposed to risk
Examples
Trapezium approximation 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...
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