1. CIA Annual Meeting LOOKING BACK…focused on the future

2. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Paper Presentation: Some Thoughts on Pension Plan Surplus by Claire Bilodeau, ASA, Ph.D. Associate Professor, Laval University on June 28, 2005 Paper Presentation: Some Thoughts on Pension Plan Surplus by Claire Bilodeau, ASA, Ph.D. Associate Professor, Laval University on June 28, 2005

3. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Presentation Outline Private Pension Plans Theoretical Approach Application Results Conclusion Further Research Presentation Outline Private Pension Plans Theoretical Approach Application Results Conclusion Further Research

4. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Presentation Outline  Private Pension Plans Theoretical Approach Application Results Conclusion Further Research Presentation Outline  Private Pension Plans Theoretical Approach Application Results Conclusion Further Research

5. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Pre-funding Contributions paid now To fund future benefits Assumptions vs. reality Both the same  A/L = 1 always Not the same  A/L varies over time Pre-funding Contributions paid now To fund future benefits Assumptions vs. reality Both the same  A/L = 1 always Not the same  A/L varies over time

6. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Reality does not follow assumptions If no control… A/L → 0 or A/L → ∞ Need for control Existing control for deficits How about surpluses? Reality does not follow assumptions If no control… A/L → 0 or A/L → ∞ Need for control Existing control for deficits How about surpluses?

7. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Presentation Outline Private Pension Plans  Theoretical Approach Application Results Conclusion Further Research Presentation Outline Private Pension Plans  Theoretical Approach Application Results Conclusion Further Research

8. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Some key questions What to give? Whom to give it to? Some key questions What to give? Whom to give it to?

9. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Concerns of a pension plan Solvency of the plan Stability of the contributions Shared to some extent by all stakeholders (regulators, sponsor, participants) Concerns of a pension plan Solvency of the plan Stability of the contributions Shared to some extent by all stakeholders (regulators, sponsor, participants)

10. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Suggested criterion The amount must be such that, with a given level of probability, the funded level will be above 100% at the next actuarial valuation. Suggested criterion The amount must be such that, with a given level of probability, the funded level will be above 100% at the next actuarial valuation.

11. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future To apply criterion Simulations are required Resulting surplus discounted to 0 Discounting at prevailing rate Ordering of the surpluses To apply criterion Simulations are required Resulting surplus discounted to 0 Discounting at prevailing rate Ordering of the surpluses

12. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Random variables for simulations Inflation Rate of return (all components) Wage increases Decrements and increments known Salary scale fixed Random variables for simulations Inflation Rate of return (all components) Wage increases Decrements and increments known Salary scale fixed

13. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Some key questions What to give? Whom to give it to? Some key questions What to give? Whom to give it to?

14. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Cooperative game theory Splits cost or profit among players Depends on contract enforceability Elements needed to apply theory Set of agents N Characteristic function v Cooperative game theory Splits cost or profit among players Depends on contract enforceability Elements needed to apply theory Set of agents N Characteristic function v

15. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Set of agents N Active Terminated, vested Disabled Retired Sponsor (new entrants) Set of agents N Active Terminated, vested Disabled Retired Sponsor (new entrants)

16. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Characteristic function v Criterion applied to each and every combination of agents Characteristic function v Criterion applied to each and every combination of agents

17. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future We want an allocation, a vector of shares x i. Sharing rules To each N and v, they associate an allocation. We want an allocation, a vector of shares x i. Sharing rules To each N and v, they associate an allocation.

18. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Some desirable properties Dummy axiom Coalitional monotonicity Some desirable properties Dummy axiom Coalitional monotonicity

19. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Dummy axiom for any i in N and all v in Γ(N) Dummy axiom for any i in N and all v in Γ(N)

20. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Coalitional monotonicity and for any i in N and all v, w in Γ(N) Coalitional monotonicity and for any i in N and all v, w in Γ(N)

21. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Sharing rules considered Shapley value Nucleolus Sharing rules considered Shapley value Nucleolus

22. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Shapley value It satisfies both properties. Shapley value It satisfies both properties.

23. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Leximin ordering Two vectors, x and z, both of size n. Coordinates in increasing order. x is leximin preferred to z if there exists an integer k < n such that but Leximin ordering Two vectors, x and z, both of size n. Coordinates in increasing order. x is leximin preferred to z if there exists an integer k < n such that but

24. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Nucleolus Excess vector is the allocation such that its excess is leximin preferred to that of any other allocation It satisfies the dummy axiom, but not coalitional monotonicity. Nucleolus Excess vector is the allocation such that its excess is leximin preferred to that of any other allocation It satisfies the dummy axiom, but not coalitional monotonicity.

25. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Presentation Outline Private Pension Plans Theoretical Approach  Application Results Conclusion Further Research Presentation Outline Private Pension Plans Theoretical Approach  Application Results Conclusion Further Research

26. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Model pension plan Final earnings pension plan 2% * service * last annual earnings No indexation No integration Model pension plan Final earnings pension plan 2% * service * last annual earnings No indexation No integration

27. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Population Stationary Mature 100 25-year-old entrants annually Population Stationary Mature 100 25-year-old entrants annually

28. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Population – Initial Group Sizes SubgroupSizePercent Sponsor1– Active lives135928.04% Disabled, vested lives881.82% Disabled, retired lives1052.23% Retired lives2755.67% Terminated, vested lives199541.16% Terminated, retired lives102221.08%

29. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Benefits Termination Disability Death Retirement Benefits Termination Disability Death Retirement

30. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Termination benefits Refund of contributions, with interest, before vesting (2 years) Deferred pension after vesting Termination benefits Refund of contributions, with interest, before vesting (2 years) Deferred pension after vesting

31. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Disability benefits Pension deferred to age 60 Service accrual during deferral Disability benefits Pension deferred to age 60 Service accrual during deferral

32. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Death benefits Refund of contributions, with interest, while active Refund of actuarial value of deferred pension, during deferral Nothing, after retirement Death benefits Refund of contributions, with interest, while active Refund of actuarial value of deferred pension, during deferral Nothing, after retirement

33. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Retirement benefits Normal at age 65 Actuarial equivalents from 55 to 70 Single annuity No guaranteed period Retirement benefits Normal at age 65 Actuarial equivalents from 55 to 70 Single annuity No guaranteed period

34. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Fixed valuation basis Interest rate: 8% Wage increase: 5%/year Funding method Projected Unit Credit Fixed valuation basis Interest rate: 8% Wage increase: 5%/year Funding method Projected Unit Credit

35. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Static investment policy 50% stocks 40% long-term bonds 10% short-term bonds Static investment policy 50% stocks 40% long-term bonds 10% short-term bonds

36. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Inflation, rates of return and wage increase Past Canadian statistics (CIA Report) Future (10,000 scenarios) Wilkie model (economic data) Sharp’s model (wage increases) Inflation, rates of return and wage increase Past Canadian statistics (CIA Report) Future (10,000 scenarios) Wilkie model (economic data) Sharp’s model (wage increases)

37. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Start year 1965 Funded level of 150% Not material Start year 1965 Funded level of 150% Not material

38. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future

39. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future End year 1986 Funded level of 138% Surplus of $19,211,884 End year 1986 Funded level of 138% Surplus of $19,211,884

40. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Presentation Outline Private Pension Plans Theoretical Approach Application  Results Conclusion Further Research Presentation Outline Private Pension Plans Theoretical Approach Application  Results Conclusion Further Research

41. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Criterion Probability: 5% Horizon: 3 years Assets proportional to liabilities Neutral starting values Criterion Probability: 5% Horizon: 3 years Assets proportional to liabilities Neutral starting values

42. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Amount to distribute $5,361,094 27.9% of the surplus Surplus in excess of 27.5% of L Surplus left ≈ 10 * NC Amount to distribute $5,361,094 27.9% of the surplus Surplus in excess of 27.5% of L Surplus left ≈ 10 * NC

43. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Treatment of sponsor (1 st ) Separate treatment Share in relation to contributions 56.419% of surplus Treatment of sponsor (1 st ) Separate treatment Share in relation to contributions 56.419% of surplus

44. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Criticism of 1 st treatment + Explicit account of all payments − Sensitive to split of normal cost Criticism of 1 st treatment + Explicit account of all payments − Sensitive to split of normal cost

45. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Impact of treatment on sharing rule Worths modified Shapley: sponsor in every subgroup Nucleolus: sponsor in no subgroup Impact of treatment on sharing rule Worths modified Shapley: sponsor in every subgroup Nucleolus: sponsor in no subgroup

46. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future

47. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Observations Odd results for nucleolus Ordering according to coefficient of variation for Shapley value Observations Odd results for nucleolus Ordering according to coefficient of variation for Shapley value

48. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Treatment of sponsor (2 nd ) Excluded No need for reduced game Treatment of sponsor (2 nd ) Excluded No need for reduced game

49. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future

50. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Observations Comparable results Possible to give share to sponsor Avoids modifying the worths Observations Comparable results Possible to give share to sponsor Avoids modifying the worths

51. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Choice of treatment of the sponsor Separate undesirable Avoid reducing the games Fix share in some other way No impact on others’ relative shares Choice of treatment of the sponsor Separate undesirable Avoid reducing the games Fix share in some other way No impact on others’ relative shares

52. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Choice of sharing rule Properties  Shapley value Simplicity  Shapley value Robustness  Shapley value Choice of sharing rule Properties  Shapley value Simplicity  Shapley value Robustness  Shapley value

53. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Presentation Outline Private Pension Plans Theoretical Approach Application Results  Conclusion Further Research Presentation Outline Private Pension Plans Theoretical Approach Application Results  Conclusion Further Research

54. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Suggested criterion The amount must be such that, with a given level of probability, the funded level will be above 100% at the next actuarial valuation. Suggested criterion The amount must be such that, with a given level of probability, the funded level will be above 100% at the next actuarial valuation.

55. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Considerations for setting criterion Solvency Stability of contributions Horizon Level of prudence Considerations for setting criterion Solvency Stability of contributions Horizon Level of prudence

56. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Simulations Identify key variables Model to generate plausible futures Simulations Identify key variables Model to generate plausible futures

57. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Sharing of the surplus Cooperative game theory Worths of each subset No game reduction Shapley value preserves ordering Sponsor’s share independently Sharing of the surplus Cooperative game theory Worths of each subset No game reduction Shapley value preserves ordering Sponsor’s share independently

58. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Limitations Initial allocation drives results Exponential number of worths Sponsor’s share undefined Limitations Initial allocation drives results Exponential number of worths Sponsor’s share undefined

59. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future Presentation Outline Private Pension Plans Theoretical Approach Application Results Conclusion  Further Research Presentation Outline Private Pension Plans Theoretical Approach Application Results Conclusion  Further Research

60. CIA Annual Meeting Session 2310 LOOKING BACK…focused on the future What this has not done Repeated application Impact on different groups Impact over time What this has not done Repeated application Impact on different groups Impact over time