ASSAL XVII Annual Meeting The effects of Ageing on the Insurance Operations

1. ASSAL XVII Annual Meeting The effects of Ageing on the Insurance Operations Instituto de Seguros de Portugal 26/04/2006

2. Summary of the Presentation A. General considerations on Mortality Risk (2-4) B. Pricing and the Insurance Solvency Framework (5-9) C. Supervisory Upgrading of Regulatory Standards (10-12) D. Mortality Projections for Life Annuities (example) (13-31) E. Main Conclusions (32-33) 1

3. A. General considerations on Mortality Risk In order to better understand the effects of ageing on the insurance operations we should perhaps start by tying to understand the manifestation of those effects on the populations of whole countries as they are also valid for the insured subsets of national populations: 2

4. A. General considerations on Mortality Risk A.1.Some features apparently valid for every population: apparently stable and improving mortality trends observed over long periods of timeseem to correspond to long periods of political stability and socio-economical development that generate a continuously upgraded standard of living, easy and quick access to good quality healthcare and a generous and widely spread social and/or private retirement system. apparently unstable overtime mortality trends, with either abrupt decreases or improvements in life expectancyseem to correspond to periods of social, economical and political instability where some or all the above life improvement factors are suppressed to a larger or smaller extent. 3

5. A. General considerations on Mortality Risk A.2.Once identified a stable evolutionary mortality pattern, the larger the population, the smaller the volatility about the trend Therefore: Mortalityis subject tovolatilityand parameter uncertainty. Any disruption in the known life improvement factors of the populationsor their cohortsmay disrupt an apparently stable patternof continuously improving mortality trends. Where in presence of a factual or probabledisruptive influenceover the mortality pattern of a population,accrued prudential measures should applyto the projection and valuation of death or survival risk cash flows. 4

6. Pricing and the Insurance Solvency Framework Mortality Risk affects the Insurance Operations in multiple ways, to an extent that depends on the nature and type of mortality risk of each insurance operation. It clearly affects: thevaluation of the Liabilities (corresponding to the Technical Provisions)associated with the type of insurance operation; thevaluation of the Solvency Requirementsset above the level of the Liabilities; and, as the costs associated with holding the financial means necessary to cover the Global Solvency Requirements need to be financed through premium inflows,it also affects: the Pricing of each Contractcorresponding to a particular type of insurance operation; 5

7. Pricing and the Insurance Solvency Framework In order to better understand the links between the Technical Provisions, the Solvency Capital Requirements and the Insurance Premiums some concepts and common sense notions may help: Our basicReference Frameworkis that of theSolvency II Projectthat, in turn, seeks to be aligned with theAccounting Framework of the IASB for the Insurance Sector. B.1.Insurance Technical Provisions are deemed to represent the “Fair Value of Insurance Liabilities”, i.e. “the amount for which an asset could be exchanged or a liability settled (or transferred) between knowledgeable, willing parties in a arm’s length transaction.They may be interpreted as the sum of theBest Estimate(according to the foreseeable evolutionary trend, especially in the case of the Life Risk)and theMarket Value Marginof the Cost of Risk. 6

8. Pricing and the Insurance Solvency Framework B.2.Solvency Capital Requirements above the level of Technical Provisions are deemed to enable an insurance undertaking to absorb significant unforeseen losses beyond the value of Technical Provisions and meet all obligations (taking into account all significant quantifiable risks and including the effects of parameter uncertainty linked to the volatility of the observed data about the evolutionary trend, especially in the case of the Life Risk) over a specified time horizon to a defined confidence level. In doing so, the Solvency Capital Requirements should limit the risk that the level of available capital deteriorates to a level below the value of Technical Provisions at any time during the specified time horizon. B.3.Insurance Technical Provisions and the Cost of Solvency Capital Requirements The notions of Insurance Technical Provisions and Solvency 7

9. Pricing and the Insurance Solvency Framework Capital Requirements are interlinked to the extent that the Cost of holding the Solvency Requirements above Technical Provisions (the Cost-of-Capital) should also be a component of the latter. Therefore: Technical Provisionsmay be seen as the Market Cost of Risk, wheresome of the risks will behedgeable(a fully diversified assets’ portfoliowill existthat replicates the cash-flow structure and characteristics of the risky (liability) portfolio, its market value will correspond to the value of hedgeable risks)and some other risks will benon-hedgeable(in which case their value may be assessed as the sum of(i) the Expected present value of future liability cash-flows for non-hedgeable risks; and(ii) aMarket Value Margin for non-hedgeable risks, calculated as the present value of the cost of holding the future capital requirement for those risks) 8

10. Pricing and the Insurance Solvency Framework B.4.Pricing of Insurance Operations Once established the logical relationship between the concepts and notions mentioned under B.1., B.2. and B.3., the link with B.4. is easily apparent: In order to avoid Losses the Pricing of Insurance Products and Operations must equal or exceed the value of Technical Provisions at any time during the specified time horizon of the SCR. 9

11. C. Supervisory Upgrading of Regulatory Standards One of the present concerns of ISP is that, once the new IFRS Standards and the EU Insurance Directive are adopted, no Life Business Technical Provisions fall below the Fair Value of Insurance Liabilities. • As the IASB’s concept of Fair Value of Insurance Liabilities has not yet produced a stable definition, ISP is presently taking the corresponding SOLVENCY II Working Concept as a reference, which, for the moment, means: Working Concept of Fair Value of Insurance Liabilities The Financially Discounted Value of the Liabilities’ Cash-Flows corresponding to a 75% confidence level according to the Term Structure of the Yield Curve corresponding to AA rated corporate bonds. 10

12. C. Supervisory Upgrading of Regulatory Standards • Projecting the cash-flows of Insurance Liabilities at a 75% confidence level requires that the specific characteristics of each insurance contract are taken into account, namely if the contract is “With Profits” or “Without Profits”, if it possesses Lapsation and Surrender Options, a Guaranteed Interest Rate, etc. • One of the least complicated exercises of Liability Cash-Flow Projection at a 75% confidence level corresponds to the case of a “Without Profits”, Immediate Life Assurance Annuity contract, where only the value of the Periodic Payoff, the Projected Mortality Rate (by age and sex) and the Term Structure of Interest Rates are involved. 11

13. C. Supervisory Upgrading of Regulatory Standards • As this issue is specially important to Pension Schemes, we propose to briefly illustrate how that may be done in respect of the most important element in the valuation  the Mortality Projection 12

14. D. Mortality Projections for Life Annuities (example) 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 l(x) = Nº individuals alive at age (x) The cathets of the triangles should be taken x 0,600 500.000 Prob. of 1 individual of age x) In their correct scale 0,500 400.000 Dying over 1 year = q(x) 300.000 0,400 200.000 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) The force of mortality (x) may be expressed as the first derivative of the rate of mortality (qx ): with 13

15. D. Mortality Projections for Life Annuities (example) × 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 14

16. D. 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 15 John H. Pollard –“Improving Mortality: A Rule of Thumb and Regulatory Tool” – Journal of Actuarial Practice Vol. 10, 2002

17. D. Mortality Projections for Life Annuities (example) The prior equation also implies that: where hence, finally 16

18. D. Mortality Projections for Life Annuities (example) 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 over the period of 1 year = q(x) 0,500 Prob. of 1individual 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) The practical application of the theoretical concepts involving the variablesk and rmay be illustrated in the graph bellow: 17

19. D. 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) 18

20. D. 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: 19

21. D. 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” 20

22. D. 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. 21

23. 22

24. D. Mortality Projections for Life Annuities (example) 23

25. D. Mortality Projections for Life Annuities (example) As may be seen in the graph below, between the young ages and age 50 there are multiple decrement 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 24

26. D. 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 25

27. D. 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). 26

28. D. 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 ()=75% which implies that   0.67285 • 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. 27

29. D. 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: 28

30. D. Mortality Projections for Life Annuities (example) • As explained before, knowing the Stable Population Structure of our cohort of annuitants enables us to estimate the parameter uncertainty of the trend to a chosen degree of approximation or (in other words) to a chosen confidence level, using the formula: • Strictly speaking, if we assume a normal distribution for the deviations about the trend,the confidence level of an annuity for age x is somewhat less thanthe value ofan annuity for age x calculated from the qx set to the same confidence level. This is because, for successive ages, the deviations about the trend flow both above and below the trend and the annuity reflects the value of a multi-year survival probability. However, calculating the value of the annuity from the the qx set to the required confidence level allows us to estimate the parameter uncertainty of the trend, which is a necessary step for assessing the Solvency Capital Requirement. 29

31. D. 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 30

32. D. 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 31

33. E. Main Conclusions E.1.In modern Risk-based Solvency Systems (like the ongoing Solvency II Project) the dynamic aspects of mortality affect both: thePricing of the Insurance contracts, theValuation of Technical Provisionsand theSolvency Capital Requirementsabove the Technical Provisions. E.2.In the near future, all aspects of Insurance Valuation and Management will be influenced by the analysis of dynamic evolutionary trends and the volatility about the trend(especially in the Life Assurance Business)based on stochastic modelling techniques. 32

34. E. Main Conclusions E.3.As the size of volatility of risks about their dynamic trends depends on the size of the Portfolio of Risks , as does size of parameter uncertainty: the size and the extent of risk diversification of the Portfolio of Risks will influence the amount of Solvency Capital Requirements (and hence the Cost-of-Capital and the Technical Provisions), becoming elements of critical importance to preserve the capacity of insurance undertakings to remain competitive in their markets. taking a pro-active attitude today in order to create solutions and avoid those future problems will very likely increase the probability of survival for many insurance undertakings. 33