2310-Bilodeau - Logo CIA Meetings
Download
Report
Transcript 2310-Bilodeau - Logo CIA Meetings
CIA Annual Meeting
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Paper Presentation:
Some Thoughts
on Pension Plan Surplus
by Claire Bilodeau, ASA, Ph.D.
Associate Professor, Laval University
on June 28, 2005
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Presentation Outline
•
•
•
•
•
•
Private Pension Plans
Theoretical Approach
Application
Results
Conclusion
Further Research
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Presentation Outline
•
•
•
•
•
Private Pension Plans
Theoretical Approach
Application
Results
Conclusion
Further Research
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
•
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
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
•
Reality does not follow assumptions
•
If no control…
•
•
A/L → 0 or A/L → ∞
Need for control
•
•
Existing control for deficits
How about surpluses?
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Presentation Outline
•
•
•
•
•
Private Pension Plans
Theoretical Approach
Application
Results
Conclusion
Further Research
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Some key questions
•
•
What to give?
Whom to give it to?
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Concerns of a pension plan
•
•
Solvency of the plan
Stability of the contributions
Shared to some extent by all stakeholders
(regulators, sponsor, participants)
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
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.
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
To apply criterion
•
•
•
•
Simulations are required
Resulting surplus discounted to 0
Discounting at prevailing rate
Ordering of the surpluses
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Random variables for simulations
• Inflation
• Rate of return (all components)
• Wage increases
Decrements and increments known
Salary scale fixed
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Some key questions
•
•
What to give?
Whom to give it to?
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
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
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Set of agents N
•
•
•
•
•
Active
Terminated, vested
Disabled
Retired
Sponsor (new entrants)
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Characteristic function v
•
Criterion applied
to each and every
combination of agents
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
We want an allocation,
a vector of shares xi.
Sharing rules
To each N and v,
they associate an allocation.
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Some desirable properties
•
•
Dummy axiom
Coalitional monotonicity
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Dummy axiom
v( S {i}) v( S ), S N \ {i}
i (v) 0
for any i in N and all v in Γ(N)
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Coalitional monotonicity
v(S ) w(S ), S such that i S
and
v(S ) w(S ), S such that i S
i (v) i ( w)
for any i in N and all v, w in Γ(N)
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Sharing rules considered
•
•
Shapley value
Nucleolus
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Shapley value
Y
i
( n s 1)!s!
n!
S:iS
[v( S {i} v( s)]
It satisfies both properties.
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
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 xi zi , i k
but xk 1 z k 1
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Nucleolus
Excess vector e( x; S )
x v( S )
iS
i
N 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.
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Presentation Outline
• Private Pension Plans
• Theoretical Approach
Application
• Results
• Conclusion
• Further Research
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Model pension plan
•
•
•
•
Final earnings pension plan
2% * service * last annual earnings
No indexation
No integration
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Population
•
•
•
Stationary
Mature
100 25-year-old entrants annually
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Population – Initial Group Sizes
Subgroup
Sponsor
Size
Percent
1
–
1359
28.04%
Disabled, vested lives
88
1.82%
Disabled, retired lives
105
2.23%
Retired lives
275
5.67%
Terminated, vested lives
1995
41.16%
Terminated, retired lives
1022
21.08%
Active lives
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Benefits
•
•
•
•
Termination
Disability
Death
Retirement
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Termination benefits
•
•
Refund of contributions, with
interest, before vesting (2 years)
Deferred pension after vesting
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Disability benefits
•
•
Pension deferred to age 60
Service accrual during deferral
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Death benefits
•
•
•
Refund of contributions, with
interest, while active
Refund of actuarial value of
deferred pension, during deferral
Nothing, after retirement
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Retirement benefits
•
•
•
•
Normal at age 65
Actuarial equivalents from 55 to 70
Single annuity
No guaranteed period
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Fixed valuation basis
•
•
Interest rate: 8%
Wage increase: 5%/year
Funding method
•
Projected Unit Credit
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Static investment policy
•
•
•
50% stocks
40% long-term bonds
10% short-term bonds
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
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)
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Start year
•
•
•
1965
Funded level of 150%
Not material
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
80,000
70,000
Assets
Liabilities
Dollars (000's)
60,000
50,000
40,000
30,000
20,000
10,000
0
1965
1970
1975
Year
1980
1985
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
End year
•
•
•
1986
Funded level of 138%
Surplus of $19,211,884
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Presentation Outline
• Private Pension Plans
• Theoretical Approach
• Application
Results
• Conclusion
• Further Research
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Criterion
•
•
•
•
Probability: 5%
Horizon: 3 years
Assets proportional to liabilities
Neutral starting values
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Amount to distribute
•
•
•
•
$5,361,094
27.9% of the surplus
Surplus in excess of 27.5% of L
Surplus left ≈ 10 * NC
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Treatment of sponsor (1st)
•
•
•
Separate treatment
Share in relation to contributions
56.419% of surplus
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Criticism of 1st treatment
+ Explicit account of all payments
− Sensitive to split of normal cost
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Impact of treatment on sharing rule
•
•
•
Worths modified
Shapley: sponsor in every subgroup
Nucleolus: sponsor in no subgroup
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Benefit increase in %
Shares with Sponsor Separate
10.00%
8.69%
8.00%
6.33%
6.00%
4.00%
6.85%
4.99%
4.26%
3.62%
3.16%
2.00%
0.00%
0.00%
Shapley value
Nucleolus
Sharing rule
Active lives
Disabled lives
Retired lives
Terminated lives
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Observations
•
•
Odd results for nucleolus
Ordering according to coefficient
of variation for Shapley value
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Treatment of sponsor (2nd)
•
•
Excluded
No need for reduced game
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Benefit increase in %
Shares with Sponsor Excluded
20.00%
16.15%
13.87%
12.09%
15.00%
10.00%
15.81%
13.18%
11.77%
7.36%
6.94%
5.00%
0.00%
Shapley value
Nucleolus
Sharing rule
Active lives
Disabled lives
Retired lives
Terminated lives
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Observations
•
•
•
Comparable results
Possible to give share to sponsor
Avoids modifying the worths
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Choice of treatment of the sponsor
•
•
•
•
Separate undesirable
Avoid reducing the games
Fix share in some other way
No impact on others’ relative
shares
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Choice of sharing rule
•
•
•
Properties Shapley value
Simplicity Shapley value
Robustness Shapley value
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Presentation Outline
• Private Pension Plans
• Theoretical Approach
• Application
• Results
Conclusion
• Further Research
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
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.
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Considerations for setting criterion
•
•
•
•
Solvency
Stability of contributions
Horizon
Level of prudence
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Simulations
•
•
Identify key variables
Model to generate plausible
futures
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Sharing of the surplus
•
•
•
•
•
Cooperative game theory
Worths of each subset
No game reduction
Shapley value preserves ordering
Sponsor’s share independently
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Limitations
•
•
•
Initial allocation drives results
Exponential number of worths
Sponsor’s share undefined
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
Presentation Outline
• Private Pension Plans
• Theoretical Approach
• Application
• Results
• Conclusion
Further Research
LOOKING BACK…focused on the future
CIA Annual Meeting
Session 2310
What this has not done
•
•
•
Repeated application
Impact on different groups
Impact over time
LOOKING BACK…focused on the future