Title: Apresentao do PowerPoint
1Tables de Mortalité
Instituto de Seguros de Portugal Le 10 mars 2008
2- Calculation of mathematical provisions
- Carried out on the basis of recognised actuarial
methods - The mortality table used in the calculation
should be chosen by the insurance undertaking
taking into account the nature of the liability
and the risk class of the product - No mortality table is prescribed
3- Calculation of mathematical provisions
- Longevity risk is mainly important in annuities
and in term assurance - With respect to term assurance companies are very
conservative in the choice of the mortality table
used to calculate premiums and mathematical
provisions (very high mortality rates compared to
observed rates) - In new life annuity contracts companies adequate
the choice of mortality tables to the effects of
mortality gains projected from recent experience
4- Calculation of mathematical provisions
- In old annuity contracts that were written on the
basis of old mortality tables, actuaries
regularly analyse the sufficiency of technical
basis and reassess the mathematical provisions
according to more recent mortality tables - The relative weight of life annuity mathematical
provisions represents about 2 of total
mathematical provisions from the life business
5Market Information to the Supervisor
Information on the Annual Mortality Recorded and
on the Annual Exposed-to-Risk (broken down by age
and sex) on the following types of Mortality Risk
- Endowments and Whole Life
- Universal Life types of policy
- Unit-linked and Index-linked types of policy
6Market Information to the Supervisor
Information on the Annual Mortality Recorded and
on the Annual Exposed-to-Risk (broken down by age
and sex) on the following types of Mortality
Risk (follow up)
- Pension Funds Annuitant Beneficiaries
- Number of Pension Fund Members
7- Supervisory process
- Responsible actuary report
- ISPs mortality studies
- Static and dynamic mortality tables
- Publication of papers and special studies
- ISP analysis of suitability of mortality tables
used
8- Supervisory process
- Responsible actuary report
- The responsible actuary should
- comment on the suitability of the mortality
tables used for the calculation of the
mathematical provision - produce a comparison between expected and actual
mortality rates - Whenever significant deviations exist, he should
measure the impact of using mortality tables that
are better adjusted to the experience and the
evolutionary perspectives of the mortality rates
9ISP Supervisory Process
- Feed-Back information from the Supervisor
- Under the Life Business Risk Assessment, ISP
conducts independent research and runs various
statistical methods (deterministic and
stochastic) to ascertain the Trend and Volatility
of the multiple variables and risk sources that
affect the Life Business
- Each year, ISP issues a Report on the Portuguese
Insurance and Pension Funds Market in which it
publishes Special Studies intended to feed-back
information onto the Insurance Undertakings and
their Responsible Actuaries on the above
mentioned risk sources, their possible modelling
techniques and the corresponding parameters.
10Mortality Projections for Life Annuities (example)
The force of mortality (?x) may be expressed as
the first derivative of the rate of mortality
(qx)
with
11If a mortality trend follows a Gompertz Law, then
If mortality were static, then the complete
expectation of Life would be
, or, in summary
with
where
Is the Euler constant
12Mortality Projections for Life Annuities (example)
Let us suppose now, that for every age the force
of mortality tends to dim out as time goes by, in
such a way that an individual which t years
before had age x and was subject to a force of
mortality ?x , is now aged xt and is subject to
a force of mortality lower than ?xt (from t
years ago). The new force of mortality will now
be
Where translates the annual averaged
relative decrease in the force of mortality for
every age
If we further admit another assumption, that the
size relation between the forces of mortality in
successively higher ages is approximately
constant over time, i.e.
and
then
hence
John H. Pollard Improving Mortality A Rule of
Thumb and Regulatory Tool Journal of Actuarial
Practice Vol. 10, 2002
13Mortality Projections for Life Annuities (example)
The prior equation also implies that
where
hence, finally
14Mortality Projections for Life Annuities (example)
The practical application of the theoretical
concepts involving the variables k and r may be
illustrated in the graph bellow
15Mortality Projections for Life Annuities (example)
In order to increase the goodness of fit of the
mortality data by using the theoretical Gompertz
Law model involving the variables k and r, it is
sometimes best to assume that r has different
values for different age ranges (we may, for
example, use r1 for the younger ages and r2
for the older ages)
16Mortality Projections for Life Annuities (example)
As may be seen, the previous graph illustrates
several features related to the Portuguese
mortality of male insured lives of the
survival-risk-type of life assurance contracts
(basically, endowment, pure endowment and savings
type of policies) for the period between 2000 and
2002
- The mortality trend for the period 2000-2002
(centred in 2001) is adequately fitted to the
observed mortality data and has been projected
from the Gompertz adjusted mortality trend
corresponding to the period between 1995 and
1999, with k0.05 for the age band from 20 to 50
years and with k0.09 for the age band from 51 to
100 years. The parameter r, which translates the
annual averaged relative decrease in the force of
mortality for every age assumes two possible
values r0.05 for the age band from 20 to 50
years and r0 for the age band from 51 to 100
years - Some minor adjustments to the formulae had to be
introduced, for example, the formula for the
force of mortality for the age band from 51 to
100 years is best based on the force of mortality
at age 36, multiplied by a
scaling factor - than if it were directly based on the force of
mortality at age 51
17Mortality Projections for Life Annuities (example)
- Further to that, some upper and lower boundaries
have also been added to the graph. Those
boundaries have been calculated according to
given confidence levels in respect of the
mortality volatility (in this case
and - ) calculated with the
normal approximation to the binomial
distribution, with mean
and volatility
- The upper boundary may, therefore, be calculated
as
- And the lower boundary may be calculated as
- Those approximations to the normal distribution
are quite acceptable, except at the older ages,
where sometimes there are too few lives in
, the Exposed-to-risk
18Mortality Projections for Life Annuities (example)
As for the rest, the process is relatively
straightforward
- From the Exposed-to-Risk ( )at each
individual age, and from the observed mortality (
) we calculate both the Central Rate of
Mortality ( ) and the Initial Gross
Mortality Rate ( ) and assess the Adjusted
Force of Mortality - ( ) using spline graduation
- We then calculate the parameters for the Gompertz
model that produce - in a way that replicates as close as possible
the
- The details of the process are, perhaps, best
illustrated in the table presented in the next
page
- This process has been tested for male, as well as
for female lives, so far with very encouraging
results, but we should not forget that we are
only comparing data whose mid-point in time is
distant only some 4 or 5 years from each other
and that we need to find a more suitable solution
for the upper and lower boundaries at the very
old ages.
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20Mortality Projections for Life Annuities (example)
21Mortality Projections for Life Annuities (example)
As may be seen in the graph below, between the
young ages and age 50 there are multiple
decremental causes beyond mortality among the
universe of beneficiaries and annuitants of
Pension Funds. That impairs mortality conclusions
for the initial rates, which have to be derived
from the mortality of the population of the
survival-risk-type of Life Assurance
226. Mortality Projections for Life Annuities
(example)
- In general, the mortality rates derived for
annuitants have to be based on the mortality
experience of Pension funds Beneficiaries and
Annuitants from age 50 onwards but, between age
20 and age 49 they must be extrapolated from the
stable trends of relative mortality forces
between the Pension Funds Population and that of
the survival-risk-type of Life Assurance.
Annuitants (Males)
Ages 20?40
Ages 41?49
Ages 50 ? ?
Where T is the Year of Projection and 2006 is the
Reference Base Year
236. Mortality Projections for Life Annuities
(example)
Annuitants (Females)
Ages 20?34
Ages 35?44
Ages 45 ? ?
Where T is the Year of Projection and 2006 is the
Reference Base Year
- The above formulae roughly imply (for both males
and females) a Mortality Gain (in life
expectancy) of 1 year in each 10 or 12 years of
elapsed time, for every age (from age 50 onwards).
24Mortality Projections for Life Annuities (example)
Annuitants
- As was mentioned before, for assessing the
mortality rates at the desired confidence level
we may use the following formulae
In our case ?(?)99,5 which implies that ? ?
2,575835
- Now, to use the above formulae we need to know
two things ? The dynamic mortality trend for
every age at onset, and ? the numeric population
structure.
25Mortality Projections for Life Annuities (example)
- In order to calculate the trend for the dynamic
mortality experience of annuitants we need to use
the earlier mentioned formulae and construct a
Mortality Matrix
26Mortality Projections for Life Annuities (example)
- In order to calculate a Stable Population
Structure we need to smoothen the averaged
proportionate structures from several years
experience
27Mortality Projections for Life Annuities (example)
- We are now able to project the dynamic mortality
experience for different ages at onset and for
different confidence levels
28- Supervisory process
- ISP analysis of suitability of mortality tables
used - ISP receives annually information regarding the
mortality tables used in the calculation of the
mathematical provisions -
- This information is compared with the overall
mortality experience of the market and with
mortality projections -
- ISP makes recommendations to actuaries and
insurance companies to reassess the calculation
of mathematical provisions with more recent
tables whenever necessary
29Mortality Projections for Life Annuities (example)
Ages (x)
30Statistical Quality Tests for Mortality
Projections
31Mortality Projections Variance Error Correction
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