CIA Annual Meeting

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

2. Paper Presentation: Some Thoughts on Pension Plan Surplus by Claire Bilodeau, ASA, Ph.D. Associate Professor, Laval University on June 28, 2005

3. Presentation Outline • Private Pension Plans • Theoretical Approach • Application • Results • Conclusion • Further Research

4. Presentation Outline • Private Pension Plans • Theoretical Approach • Application • Results • Conclusion • Further Research

5. 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. 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. Presentation Outline • Private Pension Plans • Theoretical Approach • Application • Results • Conclusion • Further Research

8. Some key questions • What to give? • Whom to give it to?

9. Concerns of a pension plan • Solvency of the plan • Stability of the contributions Shared to some extent by all stakeholders (regulators, sponsor, participants)

10. 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. To apply criterion • Simulations are required • Resulting surplus discounted to 0 • Discounting at prevailing rate • Ordering of the surpluses

12. Random variables for simulations • Inflation • Rate of return (all components) • Wage increases Decrements and increments known Salary scale fixed

13. Some key questions • What to give? • Whom to give it to?

14. 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. Set of agents N • Active • Terminated, vested • Disabled • Retired • Sponsor (new entrants)

16. Characteristic function v • Criterion applied to each and every combination of agents

17. We want an allocation, a vector of shares xi. Sharing rules To each N and v, they associate an allocation.

18. Some desirable properties • Dummy axiom • Coalitional monotonicity

19. Dummy axiom for any i in N and all v in Γ(N)

20. Coalitional monotonicity and for any i in N and all v, w in Γ(N)

21. Sharing rules considered • Shapley value • Nucleolus

22. Shapley value It satisfies both properties.

23. 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. 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. Presentation Outline • Private Pension Plans • Theoretical Approach • Application • Results • Conclusion • Further Research

26. Model pension plan • Final earnings pension plan • 2% * service * last annual earnings • No indexation • No integration

27. Population • Stationary • Mature • 100 25-year-old entrants annually

28. Population – Initial Group Sizes

29. Benefits • Termination • Disability • Death • Retirement

30. Termination benefits • Refund of contributions, with interest, before vesting (2 years) • Deferred pension after vesting

31. Disability benefits • Pension deferred to age 60 • Service accrual during deferral

32. Death benefits • Refund of contributions, with interest, while active • Refund of actuarial value of deferred pension, during deferral • Nothing, after retirement

33. Retirement benefits • Normal at age 65 • Actuarial equivalents from 55 to 70 • Single annuity • No guaranteed period

34. Fixed valuation basis • Interest rate: 8% • Wage increase: 5%/year Funding method • Projected Unit Credit

35. Static investment policy • 50% stocks • 40% long-term bonds • 10% short-term bonds

36. 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. Start year • 1965 • Funded level of 150% • Not material

39. End year • 1986 • Funded level of 138% • Surplus of $19,211,884

40. Presentation Outline • Private Pension Plans • Theoretical Approach • Application • Results • Conclusion • Further Research

41. Criterion • Probability: 5% • Horizon: 3 years • Assets proportional to liabilities • Neutral starting values

42. Amount to distribute • $5,361,094 • 27.9% of the surplus • Surplus in excess of 27.5% of L • Surplus left ≈ 10 * NC

43. Treatment of sponsor (1st) • Separate treatment • Share in relation to contributions • 56.419% of surplus

44. Criticism of 1st treatment + Explicit account of all payments − Sensitive to split of normal cost

45. Impact of treatment on sharing rule • Worths modified • Shapley: sponsor in every subgroup • Nucleolus: sponsor in no subgroup

47. Observations • Odd results for nucleolus • Ordering according to coefficient of variation for Shapley value

48. Treatment of sponsor (2nd) • Excluded • No need for reduced game

50. Observations • Comparable results • Possible to give share to sponsor • Avoids modifying the worths