Hanspeter Schmidli Risk Theory 123 Hanspeter Schmidli Institute of Mathematics University of Cologne Germany SpringerActuarial Lecture Notes ISBN978-3-319-72004-3 ISBN978-3-319-72005-0 (eBook) https://doi.org/10.1007/978-3-319-72005-0 LibraryofCongressControlNumber:2017962068 Mathematics Subject Classification (2010): 91B30, 60F10, 60G42, 60G44, 60G51, 60G55, 60J10, 60J25,60K05,60K20,91B16 ©SpringerInternationalPublishingAG,partofSpringerNature2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAG partofSpringerNature Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Thisbookaimstogiveanintroductiontothemethodsusedinnon-lifeinsurance.In additiontoprovidinganoverviewofclassicalactuarialmethods,themainpartdeals with ruin models, which is particularly interesting from a mathematical point of view.However,ruintheoryalsogivesadeeperinsightintoandunderstandinghow losses or an insolvency happens, if it happens at all, and what precautions may be taken to avoid an undesirable situation. Even though “ruin in infinite time” is not considered by the solvency rules, the theory gives an understanding of the risks taken. Istarted writing thisbookbackin1994,when Igavealectureonrisk theoryat Heriot–Watt University in Edinburgh. The lecture was based on notes by my colleagues, in particular by Howard Waters. The main part of the book has its origins in a two-semester course I gave at the University of Aarhus from 1994 to 2000. Parts of the notes to this course were also used inthe book project [110]. In the last year, I added the chapter on claims reserving and the discussions on some aspects of solvency. I hope that all the theoretical background that an actuary may need can now be found in this book. Many colleagues have directly or indirectly contributed to this book. I want to thank my Ph.D. supervisor Paul Embrechts, who started my interest in this topic; my former colleagues in Edinburgh, in particular Howard Waters, who supported meduringmystayinScotland’scapital;JanGrandellandSørenAsmussen,whose experience has increased my research skills; Hansjörg Albrecher, who drew my attention to several interesting references; and Mario Wüthrich, whose fruitful discussions with me on claims reserving has improved the corresponding chapter. Lastbutnotleast,Ithankmywife,whohasalwayssupportedmyscientificcareer, even though it was often quite hard for her, in particular, when the children were small. Cologne, Germany Hanspeter Schmidli September 2017 v Contents 1 Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Compound Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Compound Poisson Model . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 The Compound Mixed Poisson Model. . . . . . . . . . . . . . . . . . . . 6 1.5 The Compound Negative Binomial Model . . . . . . . . . . . . . . . . . 6 1.6 A Note on the Individual Model . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7 A Note on Reinsurance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7.1 Proportional Reinsurance . . . . . . . . . . . . . . . . . . . . . . 9 1.7.2 Excess of Loss Reinsurance . . . . . . . . . . . . . . . . . . . . 9 1.8 Computation of the Distribution of S in the Discrete Case . . . . . 11 1.9 Approximations to S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.9.1 The Normal Approximation . . . . . . . . . . . . . . . . . . . . 15 1.9.2 The Translated Gamma Approximation . . . . . . . . . . . . 16 1.9.3 The Edgeworth Approximation . . . . . . . . . . . . . . . . . . 18 1.9.4 The Normal Power Approximation . . . . . . . . . . . . . . . 20 1.10 Premium Calculation Principles. . . . . . . . . . . . . . . . . . . . . . . . . 21 1.10.1 The Expected Value Principle. . . . . . . . . . . . . . . . . . . 22 1.10.2 The Variance Principle . . . . . . . . . . . . . . . . . . . . . . . . 22 1.10.3 The Standard Deviation Principle . . . . . . . . . . . . . . . . 22 1.10.4 The Modified Variance Principle. . . . . . . . . . . . . . . . . 23 1.10.5 The Principle of Zero Utility. . . . . . . . . . . . . . . . . . . . 23 1.10.6 The Mean Value Principle . . . . . . . . . . . . . . . . . . . . . 25 1.10.7 The Exponential Principle. . . . . . . . . . . . . . . . . . . . . . 26 1.10.8 The Esscher Principle. . . . . . . . . . . . . . . . . . . . . . . . . 27 1.10.9 The Distortion Principle . . . . . . . . . . . . . . . . . . . . . . . 28 1.10.10 The Percentage Principle. . . . . . . . . . . . . . . . . . . . . . . 28 1.10.11 Desirable Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 28 vii viii Contents 1.11 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.11.2 Representation of Convex and Coherent Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1 The Expected Utility Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 The Zero Utility Premium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 Optimal Insurance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 The Position of the Insurer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Pareto-Optimal Risk Exchanges. . . . . . . . . . . . . . . . . . . . . . . . . 42 3 Credibility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Bayesian Credibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.1 The Poisson-Gamma Model . . . . . . . . . . . . . . . . . . . . 50 3.2.2 The Normal-Normal Model. . . . . . . . . . . . . . . . . . . . . 51 3.2.3 Is the Credibility Premium Formula Always Linear? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Empirical Bayes Credibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.1 The Bühlmann Model. . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 The Bühlmann–Straub Model . . . . . . . . . . . . . . . . . . . 57 3.3.3 The Bühlmann–Straub Model with Missing Data . . . . . 64 3.4 General Bayes Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.5 Hilbert Space Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6 Bonus-Malus Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4 Claims Reserving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Classical Claims Reserving Methods . . . . . . . . . . . . . . . . . . . . . 73 4.2.1 The Chain-Ladder Method . . . . . . . . . . . . . . . . . . . . . 73 4.2.2 The Loss-Development Method. . . . . . . . . . . . . . . . . . 74 4.2.3 The Additive Method . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.4 The Cape Cod Method . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.5 The Bornhuetter–Ferguson Method . . . . . . . . . . . . . . . 76 4.2.6 The Cross-Classified Model . . . . . . . . . . . . . . . . . . . . 77 4.3 The Dirichlet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5 The Cramér–Lundberg Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1 Definition of the Cramér–Lundberg Process . . . . . . . . . . . . . . . . 83 5.2 A Note on the Model and Reality . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 A Differential Equation for the Ruin Probability. . . . . . . . . . . . . 86 5.4 The Adjustment Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.5 Lundberg’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.6 The Cramér–Lundberg Approximation. . . . . . . . . . . . . . . . . . . . 94 Contents ix 5.7 Reinsurance and Ruin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.7.1 Proportional Reinsurance . . . . . . . . . . . . . . . . . . . . . . 97 5.7.2 Excess of Loss Reinsurance . . . . . . . . . . . . . . . . . . . . 99 5.8 The Severity of Ruin, the Capital Prior to Ruin and the Distribution of inffCt :t(cid:2)0g . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.9 The Laplace Transform of w . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.10 Approximations to w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.10.1 Diffusion Approximations. . . . . . . . . . . . . . . . . . . . . . 108 5.10.2 The deVylder Approximation . . . . . . . . . . . . . . . . . . . 110 5.10.3 The Beekman–Bowers Approximation. . . . . . . . . . . . . 112 5.11 Subexponential Claim Size Distributions . . . . . . . . . . . . . . . . . . 113 5.12 The Time to Ruin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.13 Seal’s Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.14 Finite Time Lundberg Inequalities . . . . . . . . . . . . . . . . . . . . . . . 124 5.15 Capital Injections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6 The Renewal Risk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.1 Definition of the Renewal Risk Model. . . . . . . . . . . . . . . . . . . . 131 6.2 The Adjustment Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.3 Lundberg’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.3.1 The Ordinary Case. . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.3.2 The General Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.4 The Cramér–Lundberg Approximation. . . . . . . . . . . . . . . . . . . . 140 6.4.1 The Ordinary Case. . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.4.2 The General Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.5 Diffusion Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.6 Subexponential Claim Size Distributions . . . . . . . . . . . . . . . . . . 144 6.7 Finite Time Lundberg Inequalities . . . . . . . . . . . . . . . . . . . . . . . 146 7 The Ammeter Risk Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.1 Mixed Poisson Risk Processes. . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.2 Definition of the Ammeter Risk Model . . . . . . . . . . . . . . . . . . . 151 7.3 Lundberg’s Inequality and the Cramér–Lundberg Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.3.1 The Ordinary Case. . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.3.2 The General Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.4 The Subexponential Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.5 Finite Time Lundberg Inequalities . . . . . . . . . . . . . . . . . . . . . . . 165 8 Change of Measure Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.1 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.2 The Cramér–Lundberg Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 x Contents 8.3 The Renewal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.3.1 Markovisation via the Time Since the Last Claim . . . . 175 8.3.2 Markovisation via the Time till the Next Claim . . . . . . 177 8.4 The Ammeter Risk Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 9 The Markov Modulated Risk Model. . . . . . . . . . . . . . . . . . . . . . . . . 183 9.1 Definition of the Markov Modulated Risk Model . . . . . . . . . . . . 183 9.2 The Lundberg Exponent and Lundberg’s Inequality . . . . . . . . . . 186 9.3 The Cramér–Lundberg Approximation. . . . . . . . . . . . . . . . . . . . 190 9.4 Subexponential Claim Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.5 Finite Time Lundberg Inequalities . . . . . . . . . . . . . . . . . . . . . . . 194 Appendix A: Stochastic Processes .. .... .... .... .... .... ..... .... 197 Appendix B: Martingales.... ..... .... .... .... .... .... ..... .... 199 Appendix C: Renewal Processes ... .... .... .... .... .... ..... .... 201 Appendix D: Brownian Motion .... .... .... .... .... .... ..... .... 211 Appendix E: Random Walks and the Wiener–Hopf Factorisation . .... 213 Appendix F: Subexponential Distributions ... .... .... .... ..... .... 217 Appendix G: Concave and Convex Functions. .... .... .... ..... .... 225 Table of Distribution Functions.... .... .... .... .... .... ..... .... 229 References.... .... .... .... ..... .... .... .... .... .... ..... .... 233 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 239 Principal Notation 1IA Indicator function IEI Expected value INI The natural numbers, INI ¼f0; 1; 2; ...g IPI The basic probability measure IRI The real numbers IRI The positive real numbers, IRI ¼½0;1Þ þ þ Z The integers, Z¼f...; (cid:3)2; (cid:3)1; 0; 1; 2; ...g F r-algebra of the probability space fFtg Filtration fFXg Natural filtration of the process X t L1 Space of bounded random variables fCtg Surplus process c Premium rate F Distribution function or interclaim arrival distribution G Claim size distribution MYðrÞ Moment-generating function of the random variable Y Nð¼fNtgÞ Claim number or claim number process R Lundberg coefficient Ti ith occurrence time xþ ¼x_0 Positive part of x x(cid:3) ¼ð(cid:3)xÞ_0 Negative part of x bxc Integer part of x, bxc¼supfn2Z:n(cid:4)xg Yk Claim sizes X Event space on which probabilities are defined dðxÞ Survival probability U Standard normal distribution function k Claim arrival intensity l, l nth moment of the claim sizes n xi xii PrincipalNotation wðxÞ Ruin probability s Time of ruin Hi Risk characteristics hðrÞ Cumulant-generating function of the surplus Chapter 1 Risk Models In this chapter we will consider a risk in a single time period. We will see how to approximatethedistributionofacompoundsum,howtocalculatepremia,andwe willintroduceriskmeasures. 1.1 Introduction Letusconsidera(collective)insurancecontractinsomefixedtimeperiod(0,T],for instanceT =1year.LetN denotethenumberofclaimsin(0,T]andY ,Y ,...,Y 1 2 N thecorrespondingclaims.Then (cid:2)N S = Y i i=1 istheaccumulatedsumofclaims.Weassume (i) N and{Y ,Y ,...}areindependent. 1 2 (ii) Y ,Y ,...areindependent. 1 2 (iii) Y ,Y ,...havethesamedistributionfunction,G say. 1 2 WefurtherassumethatG(0)=0,i.e.theclaimamountsarepositive.Let M (r)= Y IE[erYi],μn =IE[Y1n]iftheexpressionsexistandμ=μ1.ThedistributionofScan bewrittenas (cid:2)∞ IP[S ≤ x] =IE[IP[S ≤ x | N]]= IP[S ≤ x | N =n]IP[N =n] n=0 (cid:2)∞ = IP[N =n]G∗n(x). n=0 ©SpringerInternationalPublishingAG,partofSpringerNature2017 1 H.Schmidli,RiskTheory,SpringerActuarial, DOI:10.1007/978-3-319-72005-0_1