Modern Problems in Insurance Mathematics

388 Pages·2014·6.514 MB·English

by  Dmitrii Silvestrov, Anders Martin-Löf (eds.)

EAA Series
Dmitrii Silvestrov
Anders Martin-Löf Editors
Modern
Problems in
Insurance
Mathematics
EAA Series
Editors-in-Chief
Hansjoerg Albrecher University of Lausanne, Lausanne, Switzerland
Ulrich Orbanz University Salzburg, Salzburg, Austria
Editors
Michael Koller ETH Zurich, Zurich, Switzerland
Ermanno Pitacco Università di Trieste, Trieste, Italy
Christian Hipp Universität Karlsruhe, Karlsruhe, Germany
Antoon Pelsser Maastricht University, Maastricht, The Netherlands
Alexander J. McNeil Heriot-Watt University, Edinburgh, UK
EAAseriesissuccessoroftheEAALectureNotesandsupportedbytheEuropean
Actuarial Academy (EAA GmbH), founded on the 29 August, 2005 in Cologne
(Germany) by the Actuarial Associations of Austria, Germany, the Netherlands
and Switzerland. EAA offers actuarial education including examination, perma-
nent education for certified actuaries and consulting on actuarial education.
actuarial-academy.com
For furthertitlespublished inthis series, please goto
http://www.springer.com/series/7879
Dmitrii Silvestrov Anders Martin-Löf
•
Editors
Modern Problems
in Insurance Mathematics
123
Editors
Dmitrii Silvestrov Anders Martin-Löf
Department of Mathematics Department of Mathematics
Stockholm University Stockholm University
Stockholm Stockholm
Sweden Sweden
ISSN 1869-6929 ISSN 1869-6937 (electronic)
EAASeries
ISBN 978-3-319-06652-3 ISBN 978-3-319-06653-0 (eBook)
DOI 10.1007/978-3-319-06653-0
Springer ChamHeidelberg New YorkDordrecht London
LibraryofCongressControlNumber:2014939957
Mathematics Subject Classification: 91B30, 01A70, 60F10, 60G44, 60J10, 60J20, 60J27, 60K05,
60K15,11K45,62J05,62M05,91G10,91G20,68M11,68U35
(cid:2)SpringerInternationalPublishingSwitzerland2014
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor
informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodology now known or hereafter developed. Exempted from this legal reservation are brief
excerpts in connection with reviews or scholarly analysis or material supplied specifically for the
purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe
work. Duplication of this publication or parts thereof is permitted only under the provisions of
theCopyright Law of the Publisher’s location, in its current version, and permission for use must
always be obtained from Springer. Permissions for use may be obtained through RightsLink at the
CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt
fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
This book is a compilation of 21 of the papers presented at the International
Cramér Symposium on Insurance Mathematics (ICSIM) held at Stockholm Uni-
versity on 11–14 June, 2013.
Eachchapterisdevotedtoasinglepaper,andthechaptersaregroupedintofive
parts, each part representing one of the main topics of the symposium:
Part I—International Cramér Symposium on Insurance Mathematics.
Part II—Modern Risk Theory and Its Applications.
Part III—Stochastic Modelling of Insurance Business.
Part IV—New Mathematical Problems in Life and Non-Life Insurance.
Part V—Related Topics in Applied and Financial Mathematics.
Part I begins with the short chapter by Silvestrov and Martin-Löf (Chap. 1),
whichpresentsasummaryofthesymposium.ThechapterbyMartin-Löf(Chap.2)
is devoted to the work of Harald Cramér in the area of insurance mathematics.
Djehiche and Sandström’s chapter (Chap. 3) presents historical notes on the
Scandinavian Actuarial Journal.
PartIIbeginswiththechapterbySchmidli(Chap.4),whichpresentsnewexplicit
formulas, based on Gerber–Shiu functions, for the value of the discounted capital
injections in a classical compound Poisson risk model. D. Silvestrov’s chapter
(Chap.5)presentsasurveyofresultsonimprovedasymptoticsforruinprobabilities
in the form of exponential asymptotic expansions, necessary and sufficient condi-
tionsandexplicitratesofconvergenceintheclassicalCramér–Lundberg,stableand
diffusionapproximations.Twochapters,byNiandPetersson,continuethislineof
research.ThechapterbyNi(Chap.6)presentsasymptoticexponentialexpansions
for ruin probabilities for the Cramér–Lundberg risk model with non-polynomial
type perturbations. Petersson’s chapter (Chap. 7) presents asymptotic exponential
expansions for ruin probabilities for the discrete time analogue of the perturbed
Cramér–Lundbergriskmodel.Thispartalsoincludestwochaptersdevotedtothe
studyofriskmodelswithheavy-taileddistributions.ThechapterbyKonstantinides
and Kountzakis (Chap. 8) introduces a new expected shortfall-like risk measure.
Rassoul’s chapter (Chap. 9) presents empirical estimates for ruin probabilities in
riskmodelswithheavy-taileddistributions.
v
vi Preface
Part III begins with the chapter by Aas, Neef, Raabe and Vårli (Chap. 10),
devoted to a simulation-based Asset Liability Management (ALM) model for
computingthemarketvalueoftheliabilitiesforalifeinsurancecompany,whichis
one of the key aspect of the Solvency II regulatory framework. This part also
includesthechapterbyGünther,Tvete,Aas,Hagen,KvifteandBorgan(Chap.11),
devotedtomodellinginsuranceclaimswithaPoissonrandomeffectsmodelanda
statistical analysis of the prediction performance of this model. The chapter by
D’Amico, Gismondi, Janssen and Manca (Chap. 12) proposes effective stochastic
modelling and computational methods based on alternating renewal processes in
disabilityinsurance claims studies.
Part IV begins with the chapter by Ekheden and Hössjer (Chap. 13) on sto-
chastic modelling of mortality, based on new advanced methods of variance
decomposition. This part also includes the chapter by Yu. Kartashov, Golomoziy
and N. Kartashov (Chap. 14), where new advanced results on stability of Markov
chains are applied to an analysis of the impact of stress factors on the price of
widow’s pensions. Tzougas and Frangos’ chapter (Chap. 15) presents a new
method for the design of an optimal Bonus-Malus system using the Sichel dis-
tribution for modelling of the claim frequencies. The chapter by Mahmoudvand
andAziznasiri(Chap.16)isalsodevotedtothe studyofBonus-Malussystemsin
open and closed portfolios of insurance policies.
Part V begins with the chapter by De Gregorio and Macci (Chap. 17), devoted
to the study of large deviations for a damped telegraph process, which may yield
largedeviationestimatesforlevelcrossingprobabilities,suchasruinprobabilities
forsomeinsurancemodels.MalmbergandHössjer’schapter(Chap.18)presentsa
model of probabilistic choice when the set of options is infinite. The chapter by
Engström and S. Silvestrov (Chap. 19) is devoted to a study of PageRank algo-
rithms,whichareusedtoranknodesinnetworksandenableeffectivesearchtools
forinformationdatabases.Thishasapplicationsinmanydifferentareas,including
insurance.ThispartalsoincludesthechapterbyOgutu,Lundengård,S.Silvestrov
and Weke (Chap. 20), where Vandermonde type matrix analysis is applied to
problemsofhighordermomentfittingforlatticetreeandjump-diffusionmodelsof
priceprocesses.Finally,MelnikovandSmirnov’schapter(Chap.21)isdevotedto
thestudyofoptionpricingandCVaR-optimalpartialhedgingintheframeworkof
the two-state telegraph market model.
Allchaptershavebeen reviewed,andwearegratefultothereviewersfortheir
work.
Thebookcomprisesselectedrefereedcontributionsfromseverallargeresearch
communitiesinmoderninsurancemathematicsanditsapplications.Wehopethat
thebookwillbeausefulsourceofinspirationforabroadspectrumofresearchers,
research students and experts from the insurance business. In this way, the book
willcontributetothedevelopmentofresearchandacademy–industryco-operation
in the area of insurance mathematics and its applications.
Stockholm, December 2013 Dmitrii Silvestrov
Anders Martin-Löf
Contents
Part I International Cramér Symposium on Insurance Mathematics
1 International Cramér Symposium on Insurance
Mathematics (ICSIM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Dmitrii Silvestrov and Anders Martin-Löf
1.1 ICSIM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Sponsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Committees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Harald Cramér and Insurance Mathematics . . . . . . . . . . . . . . . . 7
Anders Martin-Löf
2.1 The Early Years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Career and Theoretical Development . . . . . . . . . . . . . . . . . . 8
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 100 Years of the Scandinavian Actuarial Journal . . . . . . . . . . . . 15
Boualem Djehiche and Arne Sandström
3.1 Foundation of the Journal . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 The Scandinavian Actuarial Journal . . . . . . . . . . . . . . . . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Part II Modern Risk Theory and Its Applications
4 A Note on Gerber–Shiu Functions with an Application . . . . . . . . 21
Hanspeter Schmidli
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Gerber–Shiu Functions in the Non-Discounted Case a¼0 . . . 22
4.3 Change of Measure and the Discounted Case. . . . . . . . . . . . . 26
4.4 Discounted Capital Injections. . . . . . . . . . . . . . . . . . . . . . . . 28
4.4.1 Small Claims. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
vii
viii Contents
4.4.2 Large Claims. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4.3 Intermediate Cases . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 Improved Asymptotics for Ruin Probabilities . . . . . . . . . . . . . . . 37
Dmitrii Silvestrov
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Nonlinearly Perturbed Renewal Equation. . . . . . . . . . . . . . . . 38
5.3 Asymptotic Expansions in the Cramér–Lundberg
Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4 First-Rare-Event Times for Semi-Markov Processes . . . . . . . . 52
5.5 Stable Approximation for Non-ruin Probabilities . . . . . . . . . . 55
5.6 Coupling for Risk Processes . . . . . . . . . . . . . . . . . . . . . . . . 59
5.7 Explicit Estimates for the Rate of Convergence
in the Cramér–Lundberg Approximation
for Ruin Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Exponential Asymptotical Expansions for Ruin Probability
in a Classical Risk Process with Non-polynomial
Perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Ying Ni
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 The Perturbed Risk Process. . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2.1 Diffusion Approximation and the Cramér–Lundberg
Approximation for the Ruin Probability . . . . . . . . . . 73
6.3 The Perturbed Risk Process with Non-polynomial
Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.4 Exponential Asymptotic Expansions
for the Ruin Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.5 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.5.1 Ruin Probability Under the Diffusion
Approximation Setting . . . . . . . . . . . . . . . . . . . . . . 79
6.5.2 Ruin Probability Under the Cramér–Lundberg
Approximation Setting . . . . . . . . . . . . . . . . . . . . . . 83
6.6 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.6.1 Proof of Corollary 6.1. . . . . . . . . . . . . . . . . . . . . . . 86
6.6.2 Proof of Proposition 6.1 . . . . . . . . . . . . . . . . . . . . . 86
6.6.3 Proof of Proposition 6.2 . . . . . . . . . . . . . . . . . . . . . 86
6.6.4 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . 87
6.6.5 Proof of Corollary 6.2. . . . . . . . . . . . . . . . . . . . . . . 91
6.6.6 Proof of Corollary 6.3. . . . . . . . . . . . . . . . . . . . . . . 92
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Contents ix
7 Asymptotics of Ruin Probabilities for Perturbed Discrete
Time Risk Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Mikael Petersson
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2 Perturbed Renewal Equations. . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 Perturbed Risk Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4 Asymptotic Expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.4.1 Asymptotic Expansion for the Root
of the Characteristic Equation . . . . . . . . . . . . . . . . . 102
7.4.2 Asymptotic Expansion for the Renewal Limit . . . . . . 103
7.4.3 Approximations of Ruin Probabilities. . . . . . . . . . . . 104
7.4.4 Modification of Perturbation Conditions . . . . . . . . . . 105
7.5 Proof of Theorem 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8 Coherent Risk Measures Under Dominated Variation . . . . . . . . . 113
Dimitrios G. Konstantinides and Christos E. Kountzakis
8.1 Distributions, Wedges and Risk Measures. . . . . . . . . . . . . . . 113
8.2 Adjusted Expected Shortfall. . . . . . . . . . . . . . . . . . . . . . . . . 118
8.3 Optimisation in L1þe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.3.1 Estimation of AES ðZ Þ . . . . . . . . . . . . . . . . . . . . 124
a;b T
8.4 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
9 Estimation of the Ruin Probability in Infinite Time
for Heavy Right-Tailed Losses . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Abdelaziz Rassoul
9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.2 Defining the Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9.3 Main Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.3.1 Confidence Bounds. . . . . . . . . . . . . . . . . . . . . . . . . 145
9.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.5 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Part III Stochastic Modelling of Insurance Business
10 A Simulation-Based ALM Model in Practical Use
by a Norwegian Life Insurance Company . . . . . . . . . . . . . . . . . . 155
Kjersti Aas, Linda R. Neef, Dag Raabe and Ingeborg D. Vårli
10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.2 Solvency Capital Requirement and Market
Value of Liabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
x Contents
10.3 Liability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10.4 Balance Sheet for a Norwegian Life Insurance Company . . . . 160
10.5 Asset Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
10.5.1 Interest Rate Model . . . . . . . . . . . . . . . . . . . . . . . . 161
10.5.2 Credit Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.5.3 Other Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.5.4 Management Actions . . . . . . . . . . . . . . . . . . . . . . . 163
10.6 Balance Sheet Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.7.1 Portfolio Characteristics . . . . . . . . . . . . . . . . . . . . . 165
10.7.2 Asset Model Parameters . . . . . . . . . . . . . . . . . . . . . 165
10.7.3 Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . 167
10.7.4 Selected Results. . . . . . . . . . . . . . . . . . . . . . . . . . . 167
10.8 Summary and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . 170
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
11 Predicting Future Claims Among High Risk Policyholders
Using Random Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Clara-Cecilie Günther, Ingunn Fride Tvete, Kjersti Aas,
Jørgen Andreas Hagen, Lars Kvifte and Ørnulf Borgan
11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
11.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
11.3 Models, Estimation and Prediction . . . . . . . . . . . . . . . . . . . . 174
11.3.1 Fixed and Mixed Effects Models . . . . . . . . . . . . . . . 174
11.3.2 A Posteriori Risk Distributions and Expected
Number of Claims . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.3.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . 176
11.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
11.4.1 In-Sample Evaluation . . . . . . . . . . . . . . . . . . . . . . . 177
11.4.2 Out-of-Time Validation. . . . . . . . . . . . . . . . . . . . . . 180
11.5 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
12 Disability Insurance Claims Study by a Homogeneous
Discrete Time Alternating Renewal Process. . . . . . . . . . . . . . . . . 187
Guglielmo D’Amico, Fulvio Gismondi, Jacques Janssen
and Raimondo Manca
12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
12.2 Discrete Time Homogeneous Alternating
Renewal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
12.3 The Discrete Time Homogeneous Alternating
Renewal Process Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 192
12.4 The Temporary Disability Insurance Studied
by a Discrete Time Alternating Renewal Model. . . . . . . . . . . 193