On Integrated Chance Constraint in ALM for Pension Funds

29 Pages·2015·0.96 MB·English

Introduction
Intoductiontostochasticprogramming
ApplicationtoALMforpensionfunds
Conclusion
References
On Integrated Chance Constraint in ALM for Pension Funds
Oslo, Norway
Youssouf Toukourou
(joint work with F. Dufresne)
[email protected]
DepartmentofActuarialScience–UniversityofLausanne,
Extranef134,CH-1015UnilDorigny
Colloquium of the International Actuarial Association
June 7-10, 2015
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM
Introduction
Intoductiontostochasticprogramming
ApplicationtoALMforpensionfunds
Conclusion
References
1 Introduction
2 Intoduction to stochastic programming
3 Application to ALM for pension funds
4 Conclusion
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM
Introduction
Intoductiontostochasticprogramming
ApplicationtoALMforpensionfunds
Conclusion
References
1 Introduction
2 Intoductiontostochasticprogramming
3 ApplicationtoALMforpensionfunds
4 Conclusion
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM
Introduction
Intoductiontostochasticprogramming
ApplicationtoALMforpensionfunds
Conclusion
References
Asset Liability Management (ALM) for pension fund
The pension fund ALM problem is an optimization problem.
Stochastic programming is a good tool for pension fund ALMs
Ziemba and al. [1986], Carino and al. [1994], Consigli and Dempster [1998].
Constraint: Guarantee sufficient solvency over the studying period
Expected amount of shortage: One-year risk Integrated Chance Constraint
(OICC) (Vlerk and al. [2003])
Novelty: The risk parameter is scale free, time dependent. OICC is exended to
multiperiod ICC (MICC) as well.
Objective: Analyse and measure the impact of ICCs by the means of an example.
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM
Introduction
InAtpopdluiccattioionnttoosAtoLcMhasftoircppernoCsgoironanmclfumusniinodngs SRtisokchcaosntsictrlaininetasrprogram(SLP)
References
1 Introduction
2 Intoductiontostochasticprogramming
Stochasticlinearprogram(SLP)
Riskconstraints
3 ApplicationtoALMforpensionfunds
4 Conclusion
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM
Introduction
InAtpopdluiccattioionnttoosAtoLcMhasftoircppernoCsgoironanmclfumusniinodngs SRtisokchcaosntsictrlaininetasrprogram(SLP)
References
Stochastic linear program: Structure
Linear program (LP) Stochastic LP (SLP)
min C(cid:62)X min C(cid:62)X
X X
s.t.BX (cid:83)d s.t.B(ω)X (cid:83)d(ω), ω⊂Ω
X ∈X ⊂Rm X ∈X ⊂Rm
Algo: Simplex Algo: L-shaped (two-stage)
Interior point methods Bender’s decomposition (multi-stage)
e.g.: One period asset allocation.
Solution methods may significantly differ:
LP is easily handleable in most cases
In some cases, one may determine deterministic equivalent of the SLP
SLP algorithms often take advantage of the problem structure
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM
Introduction
InAtpopdluiccattioionnttoosAtoLcMhasftoircppernoCsgoironanmclfumusniinodngs SRtisokchcaosntsictrlaininetasrprogram(SLP)
References
Classical SLP with recourse
The SLP with recourse is stated as
Z =min C(cid:62)X +Eω[Q(X,ω)]
X
s. t. BX (cid:83) d
X ∈X ⊂Rd
where
Q(X,ω):=min f(ω)yω
yω
s. t. Eωyω =eω−AωX
yω ∈Yω
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM
Introduction
InAtpopdluiccattioionnttoosAtoLcMhasftoircppernoCsgoironanmclfumusniinodngs SRtisokchcaosntsictrlaininetasrprogram(SLP)
References
Two-stage SLP
Deterministic equivalent of the two-stage SLP
min C(cid:62)X +Eω[fωyω]
X,y
s. t.BX (cid:83) d
AωX +Eωyω =eω
X ∈X; yω ∈Yω;ω∈Ω
S scenarios with probability ps,s ∈S ={1,··· ,S}
S
min C(cid:62)X +(cid:88)psfsys
X,y
s=1
s. t.BX (cid:83) d
AsX +Esys =es; s ∈S
X ∈X; ys ∈Y
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM
Introduction
InAtpopdluiccattioionnttoosAtoLcMhasftoircppernoCsgoironanmclfumusniinodngs SRtisokchcaosntsictrlaininetasrprogram(SLP)
References
Scenarios and decisions: Kouwenberg [2001]
1
2
Discretetimeperiodofoneyear.
Anodeisapossibleoutcomeofthestochastic
eventatagiventimet,denoted(t,s)in
scenarios.
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Eachpathofnodesfromt=0tot=3is
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calledascenario.
33 Decisionsaremadeatthebeginningofeach
34 period(node)withfullknowledgeofthepast.
ScenariotreesaretheworkhorseofMulti-stage
SP.
Optimality is defined in terms of current
39 costs plus expected future costs.
40 Only time 0 variables’ values are of interest.
0 event 1 event 2 event 3 Non-anticipativity...
Figure:Ascenariotreewith40scenariosand66nodes.
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM
Introduction
InAtpopdluiccattioionnttoosAtoLcMhasftoircppernoCsgoironanmclfumusniinodngs SRtisokchcaosntsictrlaininetasrprogram(SLP)
References
Multi-period SLP: Dempster and al. [1998]
Whenobjectivesandconstraintsarelinearandrandomnesscanbeapproachedbyaset
of S scenarios, multiperiod linear programming is defined as:
(cid:40) (cid:40) (cid:40) (cid:41) (cid:41)(cid:41)
mX0inC0(cid:62)X0+E0 mX1inC1(cid:62),sX1,s+E1 mX2inC2(cid:62),sX2,s+···+ET−1,s mXTinCT(cid:62),sXT,s ···
s.t. B0X0 =b0
D1s01X0 + B1s0X1 =b1
D2s01X0 + D2s12X1 + B2s1X2 =b2
(1)
... ...
DTs01X0 + DTs12X1 + DTs13X2 DTsT−1XT−1 + BTsT−1XT =bT
lt≤Xt≤ut;t=1,···,T
X∈X
with s is the history of the process up to time t and over the time horizon [1,...,T].
t
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Y.A.F.Toukourou&F.Dufresne IntegratedChanceConstraintsinALM