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					    Tables de Mortalité




Instituto de Seguros de Portugal
        Le 10 mars 2008
                                   1
       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


                                                           2
       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
                                                               3
        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




                                                                   4
         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
                                                             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: (follow up)

     •Annuitants Risk
       Annuities
       Pension Funds Annuitant Beneficiaries
        Number of Pension Fund Members



                                                             6
                    Supervisory process

• Responsible actuary report

• ISP’s mortality studies

   • Static and dynamic mortality tables

   • Publication of papers and special studies

• ISP analysis of suitability of mortality tables used



                                                         7
                   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


                                                                8
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.
                                                                    9
                                            Mortality Projections for Life Annuities (example)
                   The force of mortality (m x) may be expressed as the first derivative of the rate of mortality (qx):

                                                                                                                                with




                                                          Mortality Table                                                                                Mortality Table

                                         1.000.000
                                                                                                                               1,000
                                          900.000
                                                                                                                               0,900


                                                                                             Prob. of 1 individual of age x)
                                                                                                                                                                                                                                               qx
l(x) = Nº individuals alive at age (x)




                                          800.000
                                                                                                                               0,800
                                          700.000


                                                                                             Dying over 1 year = q(x)
                                          600.000
                                                                                                                               0,700
                                                                                                                                                              To determine the value of tg ( )In graph
                                                                                                                                                                                              q                                      mx
                                                                                                                               0,600                          One should take into account the fact that
                                          500.000                                                                                                              The cathets of the triangles should be taken
                                                                                                                               0,500                                      In their correct scale
                                          400.000

                                          300.000
                                                                                                                               0,400
                                          200.000                                                                              0,300
                                                                                                                                                                                                                                    m x = tg ( )
                                                                                                                                                                                                                                             q
                                          100.000                                                                              0,200
                                                                                                                                                                                                                                    m x = tg ( )
                                                                                                                                                                                                                                             q
                                                 0                                                                             0,100                                                                                            m x = tg ( )
                                                                                                                                                                                                                                         q
                                                          25




                                                                      50



                                                                      65

                                                                                  75



                                                                                  90
                                                     10
                                                          15
                                                          20

                                                                 30
                                                                 35
                                                                      40
                                                                      45

                                                                      55
                                                                      60

                                                                                  70

                                                                                  80
                                                                                  85

                                                                                        95
                                                      0
                                                      5




                                                                                       100
                                                                                       105




                                                                                                                               0,000
                                                                                                                                       0
                                                                                                                                           5
                                                                                                                                               10


                                                                                                                                                         20


                                                                                                                                                                   30


                                                                                                                                                                             40


                                                                                                                                                                                       50


                                                                                                                                                                                                 60
                                                                                                                                                    15


                                                                                                                                                              25


                                                                                                                                                                        35


                                                                                                                                                                                  45


                                                                                                                                                                                            55


                                                                                                                                                                                                      65
                                                                                                                                                                                                           70
                                                                                                                                                                                                                75
                                                                                                                                                                                                                     80
                                                                                                                                                                                                                          85
                                                                                                                                                                                                                               90
                                                                                                                                                                                                                                    95
                                                                                                                                                                                                                                         100
                                                                                                                                                                                                                                               105
                                                                        age (x)

                                                                                                                                                                                  age (x)
                                                                                                                                                                                                                                                     10
If a mortality trend follows a Gompertz Law, then
                                                              m x + t = m x  e k t
          m x +t                           m                                            1  m x +t   
  hence          = e k t , then k  t = ln x+t 
                                            m                         and also k =        ln
                                                                                                      
                                                                                                       
          mx                                x                                           t  mx       

If mortality were static, then the complete expectation of Life would be
                                                                                             m 
                     mx
                                              mx                                           f x
          o      e   k                  - z                                           o
                                                                                              k 
                          
                               1
          ex =                     e          k     dz ,   or, in summary            ex =            with
                     k     1Z                                                                 k

                                                                   m x n
                                                                   -
                                  mx
                                                       m       k 
                                          dz = -g - ln x  -        
                           -z
                  Z e
                     1             k

                 1                                      k  n = 1 n  n!
                          whereg         = 0,5772157...               Is the Euler constant
                                                                                                              11
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 mx , is now aged x+t and is subject
 to a force of mortality lower than mx+t (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
                                                                                                                             12
 John H. Pollard –“Improving Mortality: A Rule of Thumb and Regulatory Tool” – Journal of Actuarial Practice Vol. 10, 2002
Mortality Projections for Life Annuities (example)
 The prior equation also implies that:




  where                         hence, finally




                                                     13
               Projections for Life Annuities (example)
Mortality application of the theoretical concepts involving the variables k
 The practical
  and r may be illustrated in the graph bellow:
                                                                                  Mortality Tables


                                                                                                                                               qx [ + t]
                                              1,000
       dying in the course of 1 year = q(x)




                                                                                                                                                   T
                                              0,900                                                                                            qx[ ]
                                                                                                                                                   T
                                                                                                                                    m x [T + t]
       Prob. of 1 individual aged (x)




                                              0,800
                                                                                                                                    m x[ ]
                                                                                                                                         T
                                              0,700

                                              0,600                                                             m x +t = m x  e k  t
                                              0,500

                                                                                                                m tx = m x  e - rt
                                              0,400

                                              0,300

                                              0,200

                                              0,100

                                              0,000




                                                                                                                                                        100
                                                                                                                                                              105
                                                              10
                                                                   15




                                                                                       35
                                                                                            40




                                                                                                                60
                                                                                                                     65
                                                                                                                          70




                                                                                                                                              90
                                                                                                                                                   95
                                                                        20
                                                                             25
                                                                                  30



                                                                                                 45
                                                                                                      50
                                                                                                           55




                                                                                                                               75
                                                                                                                                    80
                                                                                                                                         85
                                                      0
                                                          5




                                                                                                                                                                    110
                                                                                                      age (x)                                                             14
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)




                                                                                            15
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 k=0.05 for
   the age band from 20 to 50 years and with k=0.09 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; r=0.05 for the
   age band from 20 to 50 years and r=0 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:


                                                                                           16
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”
                                                                             17
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.
                                                                                  18
19
Mortality Projections for Life Annuities (example)




                                                     20
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




                                                                                         21
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
                                                                             22
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).

                                                                              23
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.
                                                                               24
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:
                                                          25
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




                                                                          26
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




                                                                               27
                    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

                                                              28
Mortality Projections for Life Annuities (example)

   qx




               2004

        2001



1998


                        Ages (x)
                      Idades(x)
                                                     29
Statistical Quality Tests for Mortality Projections




                                                      30
Mortality Projections: Variance Error Correction




                                                   31
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