Hybrid Pension Schemes: Risk Allocation and Asset Liability Optimization

171 Pages·2005·1.54 MB·English

Hybrid Pension Schemes:
Risk Allocation and Asset Liability Optimization
DISSERTATION
oftheUniversityofSt. Gallen
GraduateSchoolofBusinessAdministration,
Economics,LawandSocialSciences(HSG)
toobtainthetitleofDoctorofEconomics
submittedby
RogerTitusBaumann
from
Aeschlen(Bern)
Approvedontheapplicationof
Prof. Dr. HeinzMu¨ller
and
Prof. Dr. HeinzZimmermann
DissertationNr. 3067
Difo-DruckGmbH,Bamberg2005
The University of St. Gallen, Graduate School of Business Administration, Economics,
LawandSocialSciences(HSG)herebyconsentstotheprintingofthepresentdissertation,
withoutherebyexpressinganyopinionontheviewshereinexpressed.
St. Gallen,June30,2005
ThePresident:
Prof. ErnstMohr,PhD
Acknowledgements
Alles,wasdieMenscheninBewegungsetzt,mussdurchihrenKopfhindurch;
aberwelcheGestaltesindiesemKopfannimmt,ha¨ngtsehrvondenUmsta¨ndenab.
–FriedrichEngels
InthesenseoftheabovequoteIwouldliketothankparticularlyProfessorHeinzMu¨ller
forhissupport. Ibecame interestedinthecombination ofoldagepensionschemes,risk
and portfolio optimization through our common projects. His flexibility to give me as
much time as I needed, and his willingness to discuss subject issues at any time, have
contributedtokeepmegroundedthroughouttheuncertainandbold ventureofwriting
athesis. IwouldalsoliketoexpressmysincerethankstoProfessorHeinzZimmermann
forhis assistance. His dedicationtothis area hasbeen trulyinspiring, and his economic
advicehasprovedinvaluable.
Equally, I would like to thank Stephan Mu¨ller for the numerous constructive discus-
sions on portfolio optimization. With his intuition and his analytical expertise he has
pointed out both possibilities as well as limits. Thanks also go to Thomas Pfiffner, Reto
Leibundgut, Evelyn Ribi and Alex Gulde for the proof reading of different parts of my
dissertation, and David Schiess for thoroughly checking the intricate appendix of chap-
ter five with the same generous and unselfish dedication for which I have always re-
spected and esteemed him on the football ground. Additional thanks go to Christian
JaagandBernhardTho¨nyforcriticalcommentsonmyideasaswellastherefreshingde-
batesabout”Godandtheworld”onairandduringthebreaks. Specialthanksalsogoto
ThomasHa¨ussler,whohasbeenashortheadaheadin languagesalready inhighschool
andwhohasgivenalastpolishtomyEnglish.
Furthermore,Iwouldalsoliketothankmymotherandmylatefather,whohaveknown
to motivate me by giving me self-confidence throughout the whole of my school days
without ever exerting pressure. Finally, I owe very much to Yvonne, who has been
particularly understanding during the final stage of my dissertation and who gives me
strength.
St. Gallen,July2005 RogerBaumann
iii

Contents
ListofFigures viii
Notation ix
1 Introduction 1
1.1 Premises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 PensionSchemes,FinancialRiskandYieldGuarantee . . . . . . . . . . . . 6
1.3 LiteratureOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 ContentsandResultsofThesis . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 TheModelBasics 19
2.1 AssumptionsandDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 HybridPensionScheme . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 FinancialMarket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.3 Wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.4 Liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.5 FundingRatio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.6 StoppingTime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.7 UtilityFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.8 TheExpectedUtilityHypothesis . . . . . . . . . . . . . . . . . . . . 25
2.1.9 ParetoOptimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 TheHamilton-Jacobi-Bellmann equation . . . . . . . . . . . . . . . . . . . . 26
2.2.1 GeneralHJBEquation . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 OptimalTerminalValue . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.3 Time-HomogeneousSetting . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.4 Long-TermGrowthRate . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.5 AUsefulTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.A.1 ProofofTheorem2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.A.2 ProofofTheorem2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . 36
v
vi Contents
3 AssetLiabilityOptimizationwithExogenousLiability 39
3.1 Merton’sProblemRevisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 OptimalConsumptionandBequestMotive . . . . . . . . . . . . . . 39
3.1.2 TerminalWealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 OptimalFundingRatio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 GeneralModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 SpecialCase: NoNetContributions . . . . . . . . . . . . . . . . . . 46
3.3 TheImpactoftheYieldGuarantee . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1 UnderfundedPension . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 OverfundedPension . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.A.1 ProofofProposition3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.A.2 DistributionFunctionofτ . . . . . . . . . . . . . . . . . . . . . . . 57
4 MinimumYieldGuaranteeandSurplus 61
4.1 LowMinimumYieldGuaranteeandBoundedFundingRatio . . . . . . . 62
4.1.1 ModelSetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 HighMinimumYieldGuaranteeandSurplus . . . . . . . . . . . . . . . . . 69
4.2.1 GeneralModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.2 SpecialCase: CIES-Utility . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.3 SpecialCase: CIES-UtilityandConstraintonFundingRatio . . . . 75
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5 IndividualRisk-SharingModels 83
5.1 TheBasicsoftheSettings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 TheReferenceYieldPlan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.1 GeneralModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.2 SpecialCase: NoNetContributionsandCRRAUtility . . . . . . . 92
5.2.3 ImpactoftheReferenceYield . . . . . . . . . . . . . . . . . . . . . . 95
5.3 FreePensionSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3.1 GeneralModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3.2 SpecialCase: NoNetContributionsandCRRAUtility . . . . . . . 106
Contents vii
5.3.3 ImpactoftheQuasi-Guarantee . . . . . . . . . . . . . . . . . . . . . 109
5.4 ContractModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4.1 GeneralModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4.2 SpecialCase: NoNetContributionsandCRRAUtility . . . . . . . 114
5.4.3 ImpactoftheQuasi-Guarantee . . . . . . . . . . . . . . . . . . . . . 124
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.A.1 ProofofProposition5.1 . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.A.2 ProofofCorollary5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.A.3 ProofofProposition5.4 . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.A.4 ProofofProposition5.5 . . . . . . . . . . . . . . . . . . . . . . . . . 148
Bibliography 151
List of Figures
3.1 Upperboundaryrˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
0,T
3.2 Upperboundaryrˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
0,p
3.3 Distributionfuctionofτ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1 TrajectoriesofW, L (or X )and F. . . . . . . . . . . . . . . . . . . . . . . . . . 64
t t t t
4.2 DevelopmentofX and F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
t t
4.3 Relationofrˇ andr¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
0 0
5.1 Referenceyieldplanoryieldguaranteesetting? Theemployee’sperspective. 94
5.2 Referenceyieldplanoryieldguaranteesetting? Theplansponsor’sperspective. 97
5.3 Referenceyieldsetting: Contracts. . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Referenceyieldplanwithcontractoryieldguaranteesetting? Theemployee’s
perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5 Referenceyieldplanwithcontractoryieldguaranteesetting? Theplanspon-
sor’sperspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6 Optimafortheplansponsorandtheemployeeforconstantportfolio. . . . . 107
5.7 Freepensionchoice: Thechoice. . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.8 Freepensionchoice vs. autonomousinvestmentand yield guaranteesetting:
Boundariesforr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
0
5.9 Contractsetting: Offers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.10 Contractsetting: Contract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.11 Contractsetting: Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.12 Contractsetting: Solutions3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.13 Contractsetting: ParetoboundariesR . . . . . . . . . . . . . . . . . . . . 122
e,Pareto
5.14 Example for parameters where the contract setting is worse than the yield
guaranteesetting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.15 Contractsetting: Paretooptimality. . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.16 Contractsetting: Possibler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
0∗
viii
Notation
Uppercase Letters
B point,seeproposition5.1,
C netcontributionprocess
D domain
D diagonalmatrixwithelementsa
a
E expectationoperator
F fundingratioprocess
I indirectutilityfunctionoftheemployee
J indirectutilityfunctionoftheplansponsor, J-function
L liability process
M indexfor”Mertonsolution”;martingaleprocess
P probabilityoperator
P probabilitymeasure
Q radicandofthesolutionofacubicequation
R riskaversionoftheCRRAutility
R employee’sriskaversion
e
R plansponsor’sriskaversion
p
S priceprocess
T timehorizon
U utilityfunction
W wealthprocess
X accruedretirementassetsprocess
Z Wienerprocess
ix
x Notation
Lowercase Letters
a,b,c,d,g,h,k,q realsupportparameters
b vectorprocess
c individualcontributionprocess
i
f (.),g(.),h(.),m(.),y(.) real-valuedsupportfunctions
i,j integerindexes
k,m,n dimensions
l liability suplementprocess
p probability
r returnoftherisklessasset
yieldguarantee,minimumyield,
r
0
referenceyield,quasi-guarantee
s servicecostsprocess
t pointoftime
timeindependentpart
u(.)
oftheutilityfunction
w process
x arguments
x argument;proportionofthegrowthoptimalportfolio
z argument
Uppercase Greek Letters
Γ,Θ,Ξ staticobjectivefunctions
Π portfoliovalueprocess
Σ
variance-covariance matrix
Φ distributionfunction
Ψ constraintprocess
Ω setofstates