The Valuation of Hybrid Pension Plan ARCH06V40N1 pres3 · August 2005 @ 40th ARC 3 Introduction and...
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August 2005 @ 40th ARC 1 The Valuation of a Hybrid Pension Plan Kai Chen Supervisor: Mary Hardy Acknowledge the UW Institute for Quantitative Finance and Insurance for their support.
Transcript of The Valuation of Hybrid Pension Plan ARCH06V40N1 pres3 · August 2005 @ 40th ARC 3 Introduction and...
August 2005 @ 40th ARC
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The Valuation of a Hybrid Pension Plan
Kai ChenSupervisor: Mary Hardy
Acknowledge the UW Institute for Quantitative Finance and Insurance for their support.
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Outline
Introduction and BackgroundModels and AssumptionsNumerical Techniques and ResultsEquilibrium Pricing ModelComments and Future Work
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Introduction and Background
Two basic kinds of pension plans:Defined Benefit(DB) and Defined Contribution(DC)
In Italy and Australia, the hybrid pension plan is used by some pension fund managers.
During the year 2002, The State of Florida’s 600,000 public employees were given the unprecedented ability to convert their DB pension plan into a DC pension plan.
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Introduction and Background
“Greater of” benefit is composed by DB and DC. It guarantees minimum pension benefits.
The payoff is same as the exchange option’s payoff.
McGill University offers the hybrid pension plan with minimum DB guarantee.
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Description of Benefits(DB)
The DB benefits depend on the current salary at exit time t:
:The actuarial value at time t:The accrual rate:The salary at time t:The years of service:The annuity factor
ttTt EanSTtxD −××××α= )12(65),,(
),,( TtxDα
tSn
)(1265a
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Description of Benefits(DC)
The DC benefits depend on the monthly contribution:
Where:The contribution rate:The time interval
tt ShchfTtxAThtxA ⋅⋅++⋅=+ )1(),,(),,(
ch
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Assumptions
Two random variables:
Let , we get
f
s
dZtfdttfdfdZtsdttsds),(),(
),(),(σ+μ=σ+μ=
dtdZdZ sffs ρ=
))1()(,(),( 2fsfssf dZdZtfdttfdf ρ−+ρσ+μ=
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Simulation Process
Use the risk-adjusted stochastic difference equations:
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Numerical Results
First use Sherris’(1995) scenarios and compare with Sherris’ (1995) results.
Generate 10,000 paths to simulate the rate of salary growth and the crediting rate. The contribution rate is 15%.
Consider the difference between two accounts
)0),,,(),,(( TtxATtxDMax −
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6 Scenarios
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Cost of Max of DB and DC Accounts
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Cost of Guarantee
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Remarks
Need more information to choose the scenarioParameters look outdatedThe market is incomplete. DC and DB accounts are not traded.Mixture of P-Measure and Q-MeasureConsider the equilibrium model
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Equilibrium Pricing Model
The equilibrium approach plays a very important role in modern finance. It can be used in the incomplete market.
Many major pricing models can be derived by the equilibrium approach, such as CAPM, B-S option pricing formula and CIR model.
The equilibrium approach considers the utility function and agent’s consumptions.
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Equilibrium Model
Use power utility function:
The price of the option:
0,1,/)( ≠β<ββ= βxxu
)]('[])('[
0SuEPayoffSuE
P ptt
p ⋅=
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Updated Parameters
Based on last 50 years Canadian data fromReport on Canadian Economic Statistics 1924-2003
We use updated parameters
.15.0,05.0,0
,1.0,07.0,033.0,055.0
===ρ
=σ=μ=σ=μ
crsf
ff
ss
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Equilibrium Model
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The distribution of the mean cost under Nature’s Measure
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Comments
The cost under Nature’s measure can be expressed as the equilibrium price with appropriate parameter selection.
The highest cost of guarantee occurs for employees with entry age 40.
The cost under Nature’s measure should be lower than the equilibrium price.
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Future Work
The relationship between Nature’s measure, equilibrium approach and Q-measure.
Risk management on this hybrid pension with minimum DB guarantee.
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Thank You!
