EAA Lecture Notes Editors H.Bühlmann A.Pelsser W.Schachermayer H.Waters D.Filipovic,Chair EAA Lecture Notes is a series supported by the European 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 includingexamination,permanenteducationforcertifiedactuariesand consultingonactuarialeducation. actuarial-academy.com EAA Lecture Notes Wüthrich,M.V.;Bühlmann,H.;Furrer,H.Market-ConsistentActuarialValuation2007 Mario Valentin Wüthrich Hans Bühlmann • • Hansjörg Furrer Market-Consistent Actuarial Valuation With13 Figuresand17 Tables Authors MarioValentinWüthrich DepartmentofMathematics ETHZürich CH-8092Zürich Switzerland E-mail:[email protected] HansBühlmann HansjörgFurrer DepartmentofMathematics SwissLife ETHZürich General-Guisan-Quai40 CH-8092Zürich CH-8022Zürich Switzerland Switzerland E-mail:[email protected] E-mail:[email protected] LibraryofCongressControlNumber:2007931823 MathematicsSubjectClassification(2000):91B30,91B28 ISSN 1865-2174 ISBN 978-3-540-73642-4SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMediaGmbH springer.com (cid:2)c Springer-VerlagBerlinHeidelberg2008 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorandVTEXusingaSpringerLATEXmacropackage Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper SPIN:151163 40/3180/VTEX 543210 Preface The balance sheet of an insurance company is often difficult to interprete. Thisderivesfromthefactthatassetsandliabilitiesaremeasuredbydifferent yardsticks. Assets are mostly valued at market prices; liabilities – as far as they relate to contractual obligations to the insured – are measured by es- tablished actuarial methods. Since, in general, there is no trading market for insurancepolicies,thequestionariseshowtheseactuarialmethodsneedtobe changed to give values – as if these markets existed. The answer to this ques- tion is “Market-Consistent Actuarial Valuation”. These lecture notes explain thelogicalmathematicalframeworkthatleadstomarket-consistentvaluesfor insurance liabilities. In Chapter 1 we motivate the use of market-consistent values. Solvency requirements by regulators are one major reason for it. Chapter 2 introduces stochastic discounting, which in a market-consistent actuarial valuation replaces discounting with the classical technical interest rate. In this chapter we introduce the notion of “Financial Variables”, (which follow the laws of financial markets) and the notion of “Technical Variables”, (which are purely depending on insurance events). InChapter3theconceptofthe“ValuationPortfolio”(VaPo)isintroduced and explained in the life insurance context. The basic idea is not to calculate in monetary values but in units, which are appropriately chosen financial instruments. For life insurance products this choice is quite natural. The risk due to technical variables is included in the protected (against technical risk) VaPo, denoted by VaPoprot. FinancialRiskistreatedinChapter4.Itderivesfromthefactthattheac- tualinvestmentportfolio ofthe insurance company differsfromthe VaPoprot. Ways to control the financial risk are Margrabe Options and/or (additional) Risk Bearing Capital. In Chapter 5 the notion of the Valuation Portfolio (VaPo) and the pro- tected (against technical risk) Valuation Portfolio (VaPoprot) is extended to the non-life insurance sector. The basic difference to life insurance derives from the fact that in property-casualty insurance the technical risk is much VI Preface more important. The discussion of appropriate risk measures (in particular the quadratic prediction error) is therefore a central issue. The final Chapter 6 contains selected topics. We mention only the treat- ment of the “Legal Quote” in life insurance. These lecture notes stem from a course on Market-Consistent Actuarial Valuation, so far given twice at ETH Zu¨rich, namely in 2004/05 by HB and HJF and in 2006 by MW and HJF. MW has greatly improved on the first version of these notes. But obviously also this version is not to be considered as final. For this reason we are grateful that the newly created EAA Lecture Notes series gives us the opportunity to share these notes with many friends and colleagues, whom we invite to participate in the process of discussions and further improvement of the present text as well as of further clarification of our way of understanding and modeling. The authors wish to thank Professor Paul Embrechts for his interest and constantencouragementwhiletheywereworkingonthisproject.Hissupport has been a great stimulus for us. It is also a great honour for us that our text appears as the first volume of the newly founded EAA Lecture Notes series. We are greatful to Peter Diethelm, who as Managing Director has been the driving force in getting this series started. Zu¨rich, Mario Wu¨thrich May 2007 Hans Bu¨hlmann Hansjo¨rg Furrer Contents 1 Introduction............................................... 1 1.1 Three pillar approach .................................... 1 1.2 Solvency ............................................... 2 1.3 From the past to the future............................... 4 1.4 Full balance sheet approach............................... 4 1.5 Recent financial failures and difficulties..................... 6 2 Stochastic discounting ..................................... 7 2.1 Basic discrete time model ................................ 7 2.2 Market-consistent valuation in the basic discrete time model .. 9 2.2.1 Task of modelization............................... 12 2.2.2 Understanding deflators ............................ 12 2.2.3 Toy example for deflators........................... 14 2.3 Valuation at time t>0 .................................. 16 2.4 The meaning of basic reserves............................. 18 2.5 Equivalent martingale measures ........................... 20 2.6 Technical and financial variables........................... 24 2.7 Conclusions on Chapter 2 ................................ 26 3 Valuation portfolio in life insurance........................ 29 3.1 Deterministic life insurance model ......................... 29 3.2 Valuation portfolio for deterministic life model .............. 30 3.3 General valuation procedure for deterministic technical risks .. 32 3.4 Self-financing property of the VaPo (deterministic technical risk) ................................................... 33 3.5 VaPo protected against technical risks...................... 34 3.6 Back to the basic model.................................. 39 3.7 Conclusion on Chapter 3 ................................. 40 3.8 Examples............................................... 41 VIII Contents 4 Financial risks ............................................. 47 4.1 Asset liability management ............................... 47 4.2 Procedure to control financial risks ........................ 49 4.3 Financial modeling ...................................... 51 4.3.1 Stochastic discounting ............................. 51 4.3.2 Modeling Margrabe options......................... 52 4.3.3 Conclusions....................................... 53 4.4 Pricing Margrabe options................................. 53 4.4.1 Pricing using Esscher transforms .................... 54 4.4.2 Application of the Esscher transform to the multi-dimensional Wiener process ................... 57 4.4.3 Hedging Margrabe options.......................... 59 4.4.4 Risk bearing capital ............................... 61 5 Valuation portfolio in non-life insurance ................... 63 5.1 Introduction ............................................ 63 5.2 Construction of the VaPo................................. 66 5.3 VaPo protected against technical risks, pragmatic approach ... 68 5.4 VaPo protected against technical risks, theoretical considerations........................................... 70 5.5 Loss development triangles ............................... 73 5.5.1 Definitions ....................................... 73 5.5.2 Chain-ladder method .............................. 76 5.5.3 Estimation of the technical risks in the chain-ladder model, single accident year ......................... 83 5.5.4 Aggregation of parameter risks across different accident years..................................... 93 5.6 Unallocated loss adjustment expenses ...................... 97 5.6.1 Motivation ....................................... 97 5.6.2 Pure claims payments.............................. 98 5.6.3 ULAE charges .................................... 99 5.6.4 New York-method .................................100 5.6.5 Example .........................................104 5.7 Conclusion on the non-life VaPo...........................104 6 Selected Topics ............................................107 6.1 Sources of losses and profits, profit sharing..................107 6.2 Remarks on the self-financing property.....................109 6.3 Legal quote in life insurance ..............................111 References.....................................................115 Index..........................................................119 1 Introduction 1.1 Three pillar approach The recent years have shown that (financial) companies need to have a good management, a good business strategy, a good financial strength and a good risk management in order to survive. It is essential that the risks are known, specified and controlled by the management. Especially in the past few years, we have observed several failures of fi- nancialcompanies(forexampleBaringsBank,HIHInsuranceAustralia,etc.). From1996until2002manycompanieshaddifferentdifficultiesaboutsolvency and liquidity. As a consequence, supervision and politics have started several initiatives to analyze these problems and to improve qualitative and quanti- tative risk management within the companies (Basel II, Solvency 2 and local initiatives like the Swiss Solvency Test [SST06], for an overview we refer to Sandstro¨m [Sa06]). Concerning insurance companies: The goal behind all these initiatives is to protect the policyholder (and the injured, respectively) from the conse- quences of an insolvency of an insurance company. Hence, in most cases it is not primarily the object of the regulator to avoid insolvencies of insurance companies, but given an insolvency of an insurance company, the regulator has to ensure that all liabilities are covered with assets and can be fulfilled in an appropriate way (this is not the shareholder’s point of view). One special project was carried out by the “London working group”. The London working group has analyzed 21 cases of solvency problems (ac- tual failures and ’near misses’) in 17 European countries. Their findings can be found in the famous Sharma Report [Sha02]. The main lessons learned are: In most cases bad management was the source of the problem. • Another central problem was that often head office had designed business • strategies which were not adapted to local situations. From this perspective, what can we really do? 2 1 Introduction Sharmasays:“Capitalisonlythesecondstrategyofdefenseinacompany, the first is a good risk management”. Supervisionhasstartedseveralinitiativestostrengthenthefinancialbasis and to improve risk managment thinking within the companies. Most of the new approaches and requirements (e.g. Basel II, Solvency 2, Swiss Solvency Test [SST06]) are formulated in three pillars: 1. Pillar 1: Minimum financial requirements (quantitative requirements) 2. Pillar 2: Supervisory review process, adaquate risk management (qualita- tive requirements) 3. Pillar 3: Market discipline and public transparency Consequences: Regulators as well as actuaries, mathematicians and risk managers in financial companies and universities search for new solvency guidelines. These guidelines should be risk-adjusted. Moreover they should be based on a market-consistent valuation of the balance sheet (full bal- ance sheet approach). From these perspectives we derive the valuation portfolio which reflects a market-consistent actuarial valuation of our balance sheet. Moreover, we describe the uncertainties within this portfolio which corresponds to a risk- adjusted analysis of our assets and liabilities. 1.2 Solvency TheInternationalAssociationofInsuranceSupervisiorsIAIS[IAIS05]defines solvency as follows ”the ability of an insurer to meet its obligations (liabilities) under all contracts at any time. Due to the very nature of insurance business, it is impossible to guarantee solvency with certainty. In order to come to a practi- cable definition, it is necessary to make clear under which circumstances the appropriateness of the assets to cover claims is to be considered, ...”. Hencetheaimofsolvencyistoprotectthepolicyholderorinjured,respec- tively.AsitisformulatedinSwisslaw:itisnotthemaingoaloftheregulator toavoidbankruptciesofinsurancecompanies,butincaseofabankruptcythe regulator tries to guarantee that the company is still able to meet its obliga- tions. Avoiding bankruptcies must be the main task of the management and the board of an insurance company. A side effect of the aim of solvency is to ensure a certain stability of the financial market. In this lecture we give a mathematical approach and interpretation to the solvency definition of the IAIS [IAIS05].