Faculty of Business and Economics
Centre for Actuarial Studies
ACTL30003 Contingencies
Note 1 – Overview and Select Life Tables
David Pitt
Lecture 1 - Week 1
Contingencies
Note 1
Lecture 1, Week 1
1 / 12
Overview
An Overview
Heavily relying on the knowledge of probability theory and financial
mathematics
Associated with life insurance and annuities, but not general
insurance or health insurance
Actuarial techniques for calculating premiums and policy values for
life insurance products and for annuities
Contingencies
Note 1
Lecture 1, Week 1
2 / 12
Overview
Insurance benefits
Whole life insurance: pays benefit on death
Term insurance: pays benefit on death within a specified period
Pure endowment insurance: pays lump sum benefit on survival for a
certain number of years
Endowment insurance: pays benefit on survival to a specified term or
on prior death within the term
Deferred insurance: pays benefit on death occurring after a deferred
period
Contingencies
Note 1
Lecture 1, Week 1
3 / 12
Overview
Annuity benefits
Whole life annuity: pays regular payment while annuitant is alive
Term annuity: same as whole life annuity except that payments stop
at a certain point
Deferred annuity: pays regular amount once annuitant reaches certain
age
Guaranteed annuity: pays benefits for a fixed number of years, and on
expiry of the guaranteed term, for the remainder of the life of the
annuitant
Contingencies
Note 1
Lecture 1, Week 1
4 / 12
Select and Ultimate tables
Selection
Ultimate life table
for a standard/general population, which is a mix of healthy and sick
Insured lives have di↵erent mortality experience to the general
population.
Insurers will not insure lives who are very ill ) select lives are healthier
Some impaired lives may be insured at an additional rate ) some
select lives are less healthy
The di↵erence between select lives and the standard population is
called the selection e↵ect.
The underwriting selection e↵ect wears o↵ over a time period, called
the select period. Mortality then reverts to the population average
(ultimate mortality).
Contingencies
Note 1
Lecture 1, Week 1
5 / 12
Select and Ultimate tables
Select table notation
[x]: age at selection
[x] + t: current age is x + t, select at age x
t p[x]+d :
the probability that a life currently aged x + d who was
select at age x survives to age x + d + t
t q[x]+d :
the probability that a life currently aged x + d who was
select at age x dies before age x + d + t
µ[x]+d : the force of mortality at age x + d for an individual who was
select at age x
D: Select period
Contingencies
Note 1
Lecture 1, Week 1
6 / 12
Select and Ultimate tables
Relations used in select life tables
t p[x]+d
t q[x]+d
t+u p[x]+d
t p[x]+d
l[x]+d+t
l[x]+d
l[x]+d l[x]+d+t
=
l[x]+d
= t p[x]+d · u p[x]+d+t
=
= e
R d+t
d
µ[x]+u du
Note that [x] + d 6= [x + d]
When d
Contingencies
D, we can drop ‘[ ]’ ) l[x]+d = lx+d and µ[x]+d = µx+d
Note 1
Lecture 1, Week 1
7 / 12
Lecture 1, Week 1
8 / 12
Select and Ultimate tables
A Screen-shot of the H2005 Mortality Table
Contingencies
Note 1
Select and Ultimate tables
Example 1
Suppose Jane was a newly select life on 1/1/2010 and that she turned 60
on 1/1/2011. Assuming H2005 select mortality, calculate the probability
on 1/1/2011 that Jane will be alive on 1/1/2015.
Solution:
Contingencies
Note 1
Lecture 1, Week 1
9 / 12
Select and Ultimate tables
Example 2
You are given the ultimate mortality table and the select mortality rates as
shown below:
Age (x)
60
61
62
63
64
lx
798,148
782,354
765,374
747,157
727,660
qx
0.019788
0.021704
0.023801
0.026095
0.028604
q[x]
0.009894
0.010852
0.011901
0.013048
0.014302
q[x 1]+1
0.014841
0.016278
0.017 851
0.019571
0.021453
Complete the following select life table:
[x]
60
61
62
Contingencies
l[x]
l[x]+1
Note 1
lx+2
Lecture 1, Week 1
10 / 12
Select and Ultimate tables
Solution:
[x]
60
61
62
Contingencies
l[x]
l[x]+1
Note 1
lx+2
Lecture 1, Week 1
11 / 12
Select and Ultimate tables
Reading
AMLCR (either edition) Sections 3.7 to 3.9. (Sections 3.4 to 3.6 are useful
background reading.)
Expectations
Explain why insured lives and annuitants are considered as select
lives, and what the di↵erence is between these two groups.
Calculate probabilities of survival/death for select lives given a select
survival model or a select mortality table.
Construct a select mortality table from an ultimate table given select
mortality rates.
Contingencies
Note 1
Lecture 1, Week 1
12 / 12