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The mortality table used in the calculation should be chosen by the insurance ... John H. Pollard 'Improving Mortality: A Rule of Thumb and Regulatory Tool' ... – PowerPoint PPT presentation

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Title: Apresentao do PowerPoint


1
Tables 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

5
Market 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
  • Death Risk
  • Term Assurances
  • Survival Risk
  • Pure Endowments
  • Endowments and Whole Life
  • Universal Life types of policy
  • Unit-linked and Index-linked types of policy

6
Market 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)
  • Annuitants Risk
  • Annuities
  • 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

9
ISP 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.

10
Mortality Projections for Life Annuities (example)
The force of mortality (?x) may be expressed as
the first derivative of the rate of mortality
(qx)
with
11
If 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

12
Mortality 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
13
Mortality Projections for Life Annuities (example)
The prior equation also implies that
where
hence, finally
14
Mortality 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
15
Mortality 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)
16
Mortality 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

17
Mortality 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

18
Mortality 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.

19
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20
Mortality Projections for Life Annuities (example)
21
Mortality 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
22
6. 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
23
6. 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).

24
Mortality 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.

25
Mortality 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

26
Mortality 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

27
Mortality 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

29
Mortality Projections for Life Annuities (example)
Ages (x)
30
Statistical Quality Tests for Mortality
Projections
31
Mortality Projections Variance Error Correction
32
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