Tables de Mortalité

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 • ISP’s 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 theLife 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 Table Mortality Table 1.000.000 1,000 900.000 0,900 800.000 q x 0,800 700.000 ( ) q To determine the value of tg In graph 0,700 One should take into account the fact that m 600.000 The cathets of the triangles should be taken x 0,600 500.000 In their correct scale l(x) = Nº individuals alive at age (x) 0,500 400.000 Prob. of 1 individual of age x) 300.000 0,400 Dying over 1 year = q(x) 200.000 0,300 ( ) 100.000 m = q tg 0,200 x ( ) m = q tg 0 x 0,100 ( ) 0 5 m = q tg 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 x 0,000 age (x) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 age (x) • 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. × m = m × k t e + x t x æ ö m æ ö m m 1 ç ÷ ç ÷ = × + × = × = + + x t k t k ln x t x t and also e k t ln hence , then ç ÷ ç ÷ m m m t è ø è ø x x x m æ ö ç ÷ x m f x ¥ m k è ø e o k o x 1 - × z ò = = × × × e e e dz k x x k k Z 1 n m æ ö - ç ÷ x ¥ ¥ m m k æ ö è ø x 1 - × z ò × × = - g - - ç ÷ x å e dz ln k × k n n ! è ø Z = 1 n 1 g = 0 , 5772157 ... 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 tyears before had age x and was subject to a force of mortalityx , is now aged x+t and is subject to a force of mortality lower than x+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 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 Tables 1,000 [ ] + q T t [ ] x q T 0,900 x [ ] m + T t 0,800 x [ ] m T x 0,700 × m = m × k t e 0,600 + x t x dying in the course of 1 year = q(x) 0,500 Prob. of 1 individual aged (x) 0,400 m = m × - × t r t e x x 0,300 0,200 0,100 0,000 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 age (x) • Mortality Projections for Life Annuities(example) The practical application of the theoretical concepts involving the variablesk and rmay 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 variablesk 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 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:

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.

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. qx 2004 2001 1998 Idades (x) • Mortality Projections for Life Annuities(example) Ages (x)

30. Statistical Quality Tests for Mortality Projections

31. Mortality Projections: Variance Error Correction