Apresentao do PowerPoint

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Tables de Mortalité
Instituto de Seguros de Portugal Le 10 mars 2008
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• 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

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• 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

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• 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

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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

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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

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• 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

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• 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

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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.

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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
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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

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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
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Mortality Projections for Life Annuities (example)
The prior equation also implies that
where
hence, finally
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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
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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)
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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

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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

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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.

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Mortality Projections for Life Annuities (example)
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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
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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
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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).

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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.

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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

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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

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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

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• 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

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