Pricing, risk and solvency requirements [Elektronische Ressource] : an analysis of investment guarantees embedded in individual pension products ; a regime switching approach / Wojtek Piaskowski

Pricing, Risk and Solvency Requirements:
An Analysis of Investment Guarantees
Embedded in Individual Pension
Products - A Regime Switching Approach
Inauguraldissertation zur Erlangung des akademischen Grades
eines Doktors der Wirtschaftswissenschaften
der Universita¨t Mannheim
Wojtek Piaskowski
vorgelegt im Herbstsemester 2009Dekan: Prof. Dr. Hans H. Bauer
Referent: Prof. Dr. Peter Albrecht
Korreferent: Prof. Dr. Hans-Jochen Bartels
Tag der mundlic¨ hen Prufung:¨ 30. September 2009
iiContents
List of Figures ix
List of Tables xvii
Acknowledgments xxiii
1 Introduction 1
I Stochastic Model 11
2 Stochastic models in finance 13
2.1 Stylizedfactsaboutfinancialtimeseries ... ......... 14
2.1.1 Asymmetryandleptokurtosis ..... 14
2.1.2 Conditionalheteroskedasticity ..... ......... 16
2.1.3 Leverageeffect .. ........... 17
2.1.4 Non-continuoustrading......... ......... 17
2.1.5 Meanreversion .. 17
2.2 Mathematicalpreliminaries .......... ......... 18
2.3 Markovchains....... ........... 21
2.3.1 DefinitionoftheMarkovchain..... ......... 21
2.3.2 Transition probabilities......... 26
iii2.3.3 Stoppingtime ... ........... ......... 27
2.3.4 StationarityofMarkovchains ..... 31
2.4 Continuousstochasticmodelsinfinance ... ......... 34
2.4.1 Diffusionmodels . ........... 34
2.4.2 Discretizationofdiffusionmodels ... ......... 43
2.5 Literatureoverview(MSmodels) ....... 44
2.6 Finitemixturedistributions .......... ......... 49
2.6.1 Definitionoffinitemixturedistribution 49
2.6.2 Momentsofmixed-normalprocess... ......... 51
2.6.3 Examplesofmixturedistributions... 53
2.7 Markovswitchingmodel . ........... ......... 57
2.7.1 DefinitionofMarkovswitchingmodel. 57
2.7.2 SpecialcasesofMarkovswitchingmodels........ 60
2.7.2.1 Brownian motion with Markov switching . . . 60
2.7.2.2 VasiˇcekmodelwithMarkovswitching .... 62
2.8 EstimationoftheMarkovswitchingmodel .. ......... 63
2.8.1 Log-likelihoodfunction ......... 63
2.8.2 Inferenceabouttheunobservablestatevariable .... 64
2.8.2.1 Filter of Hamilton (1989) . . ......... 64
2.8.2.2 Smoother of Kim (1994) . . . 65
2.8.3 EMAlgorithm... ........... ......... 67
2.9 EmpiricalresultsoftheMS .......... 70
2.9.1 DAX-REXPportfolios ......... ......... 70
2.9.2 Descriptionoftheestimation ..... 71
2.9.3 Estimationresults. ........... ......... 72
3 Testing Markov switching models 81
3.1 Introduction ........ ........... ......... 81
iv3.2 Informationcriteriontests ........... ......... 82
3.3 ProblemswithtestingofMarkovswitchingmodels....... 89
3.4 Simpletests ........ ......... 92
3.4.1 Waldtest ..... ........... 92
3.4.2 Likelihoodratiotest .......... ......... 97
3.5 Testsbasedonscores ...100
3.5.1 Scores ....... ........... .........100
3.5.2 ScoretestofNewey-Tauchen-White..102
3.5.3 Lagrange multiplier test of Hamilton (1996) ......108
3.6 Testsbasedonsimulationoftheteststatistic .........113
3.6.1 Likelihood ratio test of Hansen (1992, 1996) ......114
3.6.2 Likelihood ratio test of Garcia (1998) . .........120
3.7 Conclusion......... ...........129
II Investment Guarantees Embedded in Individual
Pension Products 133
4 Pricing of investment guarantees 135
4.1 Financialmarket ..... ........... .........136
4.1.1 Moneymarketaccountandriskystock136
4.1.2 Contingentclaim . .........137
4.1.3 Self-financingtradingstrategy .....138
4.2 Optionpricing....... ........... .........141
4.2.1 Equivalentmartingalemeasure ....141
4.2.2 Optionpricingformula ......... .........144
4.3 Existenceofthesolution(Arbitrage) .....146
4.4 Uniqueness of the solution (Completeness of the market) . . . 148
v4.5 Esscher risk-neutral probability measure . . . .........149
4.6 Optionpricing....... ...........154
4.6.1 Black-Scholesmarket .......... .........154
4.6.2 MarkovSwitchingmarket .......157
4.6.2.1 Bollen-Hardyoptionpricingformula .....158
4.6.2.2 Webboptionpricingformula .........159
4.7 Quantitativeresults.... ...........164
4.7.1 Outlineofthestudy .......... .........164
4.7.1.1 Designoftheguarantee ...164
4.7.1.2 Investmentstrategies..... .........165
4.7.1.3 Designofthestudy......168
4.7.2 Guaranteecost .. ........... .........171
4.7.2.1 Impact of the initial state on the guarantee
cost ... .........172
4.7.2.2 Impact of the stochastic model on the guar-
anteecost ........... .........176
4.7.2.3 Impact of the guarantee level on the guaran-
teecost . .........179
4.7.2.4 Impact of the investment strategy on the guar-
anteecost ........... .........181
4.7.2.5 Impactoftimeontheguaranteecost .....184
4.7.2.6 Impact of the contract term on the guarantee
cost ... ........... .........189
4.7.2.7 Impact of the contribution payment scheme
ontheguaranteecost .... .........190
4.7.2.8 Interrelation between guarantee cost and ex-
pectedprofit ......... .........195
vi4.8 Conclusion......... ........... .........200
5 Risk analysis and solvency requirements 205
5.1 Riskdefinition....... ........... .........205
5.2 Riskmeasure207
5.2.1 Dispersionriskmeasures ........ .........207
5.2.2 Quantileriskmeasures .........208
5.2.3 Shortfallriskmeasures .........209
5.3 Designofthestudy .... ...........213
5.4 Riskanalysis ....... .........215
5.4.1 Impactoftheinitialstateontheguaranteerisk ....215
5.4.2 Impact of the stochastic model on the guarantee risk . 220
5.4.3 Impact of the guarantee level on the guarantee risk . . 222
5.4.4 Impact of the investment strategy on the guarantee risk 225
5.4.5 Impactoftimeontheguaranteerisk . .........230
5.4.6 Impact of the contract term on the guarantee risk . . . 238
5.4.7 Impact of the contribution payment scheme on the
guaranteerisk ... ........... .........240
5.5 Solvencyrequirements . .249
5.6 Interrelation between solvency requirements and expected profit253
5.7 Conclusion......... ........... .........256
6 Main results and further research 261
6.1 Mainresults ........ ........... .........261
6.2 Furtherresearch......266
A Proofs for Chapter 3 269
A.1 AssumptionsfortheHansentest........ .........269
A.2 ProofofLemma2..... ...........271
viiB MS estimation (1.1975-12.2004) – parameters 285
C States of Markov switching models (1.1975-12.2004) 299
D Conditional moments (1.1975-12.2004) – graphs 313
E Histogram of the log-returns (1.1975-12.2004) 353
F Information Criterion Tests (1.1975-12.2004) 357
G Wald and likelihood ratio test (1.1975-12.2004) 371
H Newey-Tauchen-White test (1.1975-12.2004) 379
I Lagrange multiplier test (1.1975-12.2004) 385
J Garcia test (1.1975-12.2004) 389
K Impact of the guarantee level on the guarantee risk 403
L Impact of time and the contract term on the guarantee short-
fall risk (life-cycle strategies) 411
M Impact of the contribution payment scheme on the guarantee
shortfall risk (life-cycle strategies) 421
Bibliography 433
viiiList of Figures
1.1 Monthly DAX30 and its log-returns: 1975-2004 ........ 6
2.1 Normal distribution vs. monthly log-returns of DAX30 (1975-
2004) . ........... ........... ......... 15
2.2 Examplesofmixednormals .......... 54
2.3 Filtered and smoothed probabilities: REXP(0%)-DAX30(100%)
-MS(1-2) ......... ........... ......... 67
3.1 Distribution of the Garcia’s SupC statistic for DAX30.....127
4.1 Impact of the initial state on the normalized guarantee cost . . 173
4.2 Impact of the stochastic model on the normalized guarantee
cost(buy-and-holdstrategies) ......... .........177
4.3 Impact of the guarantee level on the normalized guarantee cost 180
4.4 Sensitivity of the normalized guarantee cost to changes in the
guaranteelevel ...... ........... .........181
4.5 Impact of the investment strategy on the normalized guarantee
cost . ........... .........182
4.6 Impact of time and the contract term on the normalized guar-
anteecost(buy-and-holdstrategies) ...... .........185
4.7 Impact of time and the contract term on the guarantee cost
(life-cyclestrategies) ... ........... .........186
ix4.8 Impact of the contribution payment scheme on the normalized
guaranteecost(buy-and-holdstrategies) ... .........191
4.9 Impact of the contribution payment scheme on the normalized
guaranteecost(life-cyclestrategies) ...... .........193
5.1 Impact of the initial state on risk measures (periodic contri-
bution)........... ........... .........216
5.2 Impact of the initial state on risk measures (single contribution)218
5.3 Impactofthestochasticmodelonriskmeasures ........221
5.4 Impactoftheguaranteelevelonriskmeasures .........223
5.5 Sensitivity of risk measures to changes in the guarantee level
(purestockportfolio) ... ........... .........224
5.6 Impact of the investment strategy on the guarantee risk (buy-
and-holdstrategies).... .........226
5.7 Impact of the investment strategy on the guarantee risk (all
strategies) ......... ........... .........228
5.8 Impact of time and the contract term on the guarantee short-
fallrisk(buy-and-holdstrategies) ....... .........232
5.9 Impact of time and the contract term on the guarantee short-
fallrisk(moderatestrategy) .......... .........234
5.10 Impact of time and the contract term on the guarantee short-
fall risk (aggressive strategy).......... .........236