Springer Finance. For further volumes:

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1 Springer Finance Editorial Board Marco Avellaneda Giovanni Barone-Adesi Mark Broadie Mark H.A. Davis Emanuel Derman Claudia Klüppelberg Walter Schachermayer

2 Springer Finance Springer Finance is a programme of books addressing students, academics and practitioners working on increasingly technical approaches to the analysis of financial markets. It aims to cover a variety of topics, not only mathematical finance but foreign exchanges, term structure, risk management, portfolio theory, equity derivatives, and financial economics. For further volumes:

3 Mario V. Wüthrich Michael Merz Financial Modeling, Actuarial Valuation and Solvency in Insurance

4 Mario V. Wüthrich RiskLab Department of Mathematics ETH Zurich Zurich, Switzerland Michael Merz Faculty for Economic and Social Studies Department of Business Administration University of Hamburg Hamburg, Germany ISSN ISSN (electronic) ISBN ISBN (ebook) DOI / Springer Heidelberg New York Dordrecht London Library of Congress Control Number: Mathematics Subject Classification: 62P05, 91G30 JEL Classification: G22, D52, D53, D82, E43, G12, G17, G32, G38 Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part 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 or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar 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 purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright 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 Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Acknowledgements This book is the product of an ongoing project we have been working on for several years. As such it was not really defined as a project but it is rather the result of many activities we have been involved in. These include our own practical experience; discussions with regulators, scientists, practitioners, politicians, other decision-makers, colleagues and students; continuing education at conferences, workshops, working groups and our own lectures. We are deeply grateful to ETH Zürich and to University of Hamburg. During all these times we were very generously supported by our departments at these universities, and we have and continue to experience these environments as stimulating and motivating. Special thank-you s are reserved for Prof. Hans Bühlmann and Prof. Paul Embrechts for their continued support. We greatly appreciate that the present manuscript profits from various inspiring discussions, continuative thoughts, helpful contributions and critical comments with and by several people: Hansjörg Albrecher, Peter Antal, Philipp Arbenz, Manuela Baumann, Hans Bühlmann, Bikramjit Das, Catherine Donnelly, Karl- Theodor Eisele, Paul Embrechts, Peter England, Vicky Fasen, Damir Filipović, Alois Gisler, Sebastian Happ, Enkelejd Hashorva, Frank Häusler, John Hibbert, Laurent Huber, Philipp Keller, Roger Laeven, Alexander McNeil, Christoph Möhr, Antoon Pelsser, Enrico Perotti, Eckhard Platen, Simon Rentzmann, Robert Salzmann, Marc Sarbach, Urs Schubiger, Pavel Shevchenko, Werner Stahel, David Stefanovits, Josef Teichmann, Andreas Tsanakas, Richard Verrall, Frank Weber, Armin Wolf and Hans Peter Würmli. We especially thank Manuela Baumann for coding Example Moreover, we appreciate that several anonymous reviewers have read previous versions of this manuscript very carefully. They have approached the subject from several different angles which has led us to provide a more comprehensive and complete description of the topic and helped us bridge a few gaps in previous versions of the manuscript. Special thanks go to Alessia, Luisa, Anja, Rosmarie, Valo, Coral, Jürg, Giorgio, Matthias, Stephan, Ted, Juvy, Ursin, Francesco, Peter and Peter, Fritz, Reini. v

6 vi Acknowledgements Last but not least we thank Dave, Martin and Andy for endlessly enjoying the silence. Zurich, Switzerland Hamburg, Germany February 2013 Mario V. Wüthrich Michael Merz

7 Contents 1 Introduction Full Balance Sheet Approach SolvencyConsiderations FurtherModelingIssues Outline of This Book... 6 Part I Financial Valuation Principles 2 State Price Deflators and Stochastic Discounting Zero Coupon Bonds and Term Structure of Interest Rates Motivation for Discounting Spot Rates and Term Structure of Interest Rates EstimatingtheYieldCurve Basic Discrete Time Stochastic Model ValuationatTime Interpretation of State Price Deflators Valuation at Time t> EquivalentMartingaleMeasure Bank Account Numeraire MartingaleMeasureandtheFTAP MarketPriceofRisk Spot Rate Models General Gaussian Spot Rate Models One-Factor Gaussian Affine Term Structure Models Discrete Time One-Factor Vasicek Model Spot Rate Dynamics on a Yearly Grid Spot Rate Dynamics on a Monthly Grid Parameter Calibration in the One-Factor Vasicek Model Conditionally Heteroscedastic Spot Rate Models Auto-Regressive Moving Average (ARMA) Spot Rate Models AR(1) Spot Rate Model vii

8 viii Contents AR(p) Spot Rate Model General ARMA Spot Rate Models Parameter Calibration in ARMA Models Discrete Time Multifactor Vasicek Model Motivation for Multifactor Spot Rate Models Multifactor Vasicek Model (with Independent Factors) Parameter Estimation and the Kalman Filter One-Factor Gamma Spot Rate Model Gamma Affine Term Structure Model Parameter Calibration in the Gamma Spot Rate Model Discrete Time Black Karasinski Model Log-Normal Spot Rate Dynamics Parameter Calibration in the Black Karasinski Model ARMA Extended Black Karasinski Model Stochastic Forward Rate and Yield Curve Modeling General Discrete Time HJM Framework GaussianDiscreteTimeHJMFramework General Gaussian Discrete Time HJM Framework Two-Factor Gaussian HJM Model Nelson Siegel and Svensson HJM Framework YieldCurveModeling DerivationsfromtheForwardRateFramework Stochastic Yield Curve Modeling Appendix Proofs of Chap Pricing of Financial Assets PricingofCashFlows General Cash Flow Valuation in the Vasicek Model Defaultable Coupon Bonds Financial Market A Log-Normal Example in the Vasicek Model A First Asset-and-Liability Management Problem PricingofDerivativeInstruments Appendix Proofs of Chap Part II Actuarial Valuation and Solvency 6 Actuarial and Financial Modeling Financial Market and Financial Filtration Basic Actuarial Model Improved Actuarial Model Valuation Portfolio ConstructionoftheValuationPortfolio Financial Portfolios and Cash Flows ConstructionoftheVaPo...171

9 Contents ix Best-EstimateReserves Examples Examples in Life Insurance Example in Non-life Insurance ClaimsDevelopmentResultandALM ClaimsDevelopmentResult Hedgeable Filtration and ALM ExamplesRevisited ApproximateValuationPortfolio Protected Valuation Portfolio ConstructionoftheProtectedValuationPortfolio Market-ValueMargin Risk-AdjustedReserves Claims Development Result of Risk-Adjusted Reserves Fortuin Kasteleyn Ginibre (FKG) Inequality Examples in Life Insurance Example in Non-life Insurance Further Probability Distortion Examples NumericalExamples Non-life Insurance Run-Off Life Insurance Examples Solvency RiskMeasures Definition of (Conditional) Risk Measures ExamplesofRiskMeasures Solvency and Acceptability Definition of Solvency and Acceptability FreeCapitalandSolvencyTerminology Insolvency No Insurance Technical Risk Theoretical ALM Solution and Free Capital General Asset Allocations Limited Liability Option MargrabeOption Hedging Margrabe Options Inclusion of Insurance Technical Risk Insurance Technical and Financial Result Theoretical ALM Solution and Solvency General ALM Problem and Insurance Technical Risk Cost-of-Capital Loading and Dividend Payments Risk Spreading and Law of Large Numbers Limitations of the Vasicek Financial Model PortfolioOptimization Standard Deviation Based Risk Measure Estimation of the Covariance Matrix

10 x Contents 10 Selected Topics and Examples Extreme Value Distributions and Copulas Parameter Uncertainty Parameter Uncertainty for a Non-life Run-Off Modeling of Longevity Risk Cost-of-Capital Loading in Practice General Considerations Cost-of-Capital Loading Example Accounting Year Factors in Run-Off Triangles Model Assumptions PredictiveDistribution Premium Liability Modeling Modeling Attritional Claims ModelingLargeClaims Reinsurance RiskMeasurementandSolvencyModeling Insurance Liabilities Asset Portfolio and Premium Income Cost Process and Other Risk Factors Accounting Condition and Acceptability Solvency Toy Model in Action Concluding Remarks Part III Appendix 11 Auxiliary Considerations HelpfulResultswithGaussianDistributions Change of Numeraire Technique General Changes of Numeraire Forward Measures and European Options on ZCBs European Options with Log-Normal Asset Prices References Index...427

11 Notation 1 m 0 Maturity of zero coupon bonds (ZCBs) R(t,m) Continuously-compounded spot rate at time t for maturity m>t L(t, m) Simply-compounded spot rate at time t for maturity m>t Y(t,m) Annually-compounded spot rate at time t for maturity m>t r(t) Instantaneous spot rate (short rate) at time t 0 r t = R(t,t + 1) Continuously-compounded spot rate at time t for maturity t + 1 (one-year risk-free rollover) F(t,s+ 1) Forward interest rate at time t for s t f(t,m) Instantaneous forward interest rate at time t<m β Parameter of Svensson and Nelson Siegel modeling n N Final time horizon J ={0,...,n} Set of all points in time J ={0,...,n 1} Set of points in time F = (F t ) t J Filtration on measurable space (Ω, F ) with F 0 ={,Ω} and F n = F P Real world probability measure on measurable space (Ω, F ) (Ω, F, P, F) Filtered probability space P P Equivalent martingale measure on measurable space (Ω, F ) 1 We give some notational conventions we are using. We do however stress that it is not always easy to find good and consistent notation throughout the text. It may therefore happen that the same letter is used for different objects. This we cannot avoid completely because we join concepts and models from three different subject areas, namely actuarial science, financial mathematics and economic theory. We mainly work in a discrete time and finite time horizon model. The interval between two points in time typically is one year and t R + measures time in yearly units. xi

12 xii Notation (ξ t ) t J,(ζ t ) t J Density processes X = (X 0,...,X n ) Discrete time cash flow L 2 n+1 (Ω, F, P) Hilbert space of (n + 1)-dimensional square integrable cash flows X L 2 n+1 (Ω, F, P, F) Hilbert space of (n + 1)-dimensional square integrable, F-adapted cash flows X L 1 n+1 (Ω, F, P, F) Space of (n + 1)-dimensional integrable, F-adapted cash flows X L ϕ Set of priceable cash flows X for state price deflator ϕ k J Index for single cash flow X k t J Today s time point used for price processes A = (A t ) t J Financial filtration on measurable space (Ω, F ) T = (T t ) t J Insurance technical filtration on measurable space (Ω, F ) H = (H t ) t J Hedgeable filtration on measurable space (Ω, F ) ϕ = (ϕ t ) t J State price deflator ϕ = ( ϕ t ) t J Span-deflator ϕ A = (ϕt A ) t J Financial deflator ϕ T = (ϕt T ) t J Probability distortion I Financial market of basis financial instruments A (i) Basis financial instrument i I (A (i) t ) t J Price process of basis financial instrument A (i), i I Z (m) ZCB with maturity m Z (m) Cash flow of ZCB with maturity m P(t,m) Price of ZCB Z (m) at time t m U Financial portfolio (U t ) t J Price process of financial portfolio U U (k) Financial portfolio sold at time k J (U t (k) ) t J Price process of financial portfolio U (k) B Bank account (B t ) t J Price process of bank account B M (t) Margrabe option with maturity t (M s (t) ) s J Price process of Margrabe option Call t (A,K,T) Price at time t of European call option on instrument A, with strike K and maturity T Put t (A,K,T) Price at time t of European put option on instrument A, with strike K and maturity T Λ = (Λ (0),...,Λ (n) ) T-adapted insurance technical liability (Λ (k) t ) t J Probability distorted process of insurance liability Λ (k), k J S Asset side of balance sheet S t Value of asset side S of balance sheet at time t J Asset portfolio with allocation chosen at time t J S (t)

13 Notation xiii S (t) s S (t) = n k=t+1 w (t) k Value of asset portfolio S (t) at time s J U (k) Cash flow representation of asset portfolio S (t) S (t) = i I w(t) i A (i) Instrument representation of asset portfolio S (t) VaPo t (X) Valuation portfolio of cash flow X at time t J VaPo prot t (X) Protected valuation portfolio of X at time t J VaPo approx t (X) Approximate valuation portfolio of X at time t J Q t (X) Value of X at time t J Q 0 t (X) Undistorted value of X at time t J X (t+1) Outstanding liabilities at time t J Rt 0(X (t+1)) Best-estimate reserves at time t J R t (X (t+1) ) Risk-adjusted reserves at time t J Rt nom (X (t+1) ) Nominal reserves at time t J MVM ϕ t (X (t+1) ) Market-value margin at time t with state price deflator ϕ CDR t+1 (X (t+1) ) Claims development result for best-estimate reserves at time t + 1 CDR + t+1 (X (t+1)) Claims development result for risk-adjusted reserves at time t + 1 I Last observed accident year (non-life insurance) i {1,...,I} Accident years J Last development year (in non-life insurance) j {0,...,J} Development years X i,j Claims payment in non-life insurance for accident year i and development year j, i.e. accounting year k = i + j C i,j Nominal cumulative payments in non-life insurance for accident year i and development year j Nominal ultimate claim in non-life insurance C i,j f j f j + f (t) j f (+t) j L x+k D x+k p x+k q x+k p + x+k q + x+k ρ Chain-ladder factor for development period j Risk-adjusted chain-ladder factor for development period j Posterior chain-ladder factor at time t Posterior risk-adjusted chain-ladder factor at time t Number of people alive aged x + k at time k Number of people aged x + k that die within (k 1,k] Second order survival probability within (k 1,k] for people aged x + k Second order death probability within (k 1,k] for people aged x + k First order survival probability within (k 1,k] for people aged x + k First order death probability within (k 1,k] for people aged x + k Risk measure

14 xiv Notation ρ t Conditional risk measure M Subset of a.s. finite random variables VaR 1 p (X) Value-at-Risk of X on security level 1 p ES 1 p (X) Expected shortfall of X on security level 1 p CTE 1 p (X) Conditional tail expectation of X on security level 1 p AD t+1 Asset deficit at time t + 1 F t Free capital at time t SC t Solvency capital at time t TC t Target capital at time t RBC t Risk bearing capital at time t λ Market price of risk δ Span of time grid (in yearly units) sp CoC Cost-of-capital spread r (t) CoC Cost-of-capital rate at time t r RoSC Return on solvency capital SR t Sharpe ratio at time t r 0:T Observations {r 0,...,r T } at time T Vco(X) Coefficient of variation of random variable X Ψ β1 ( ), Ψ β1,β 2 ( ) Risk reward functions Run-off liability cash flow at time I X run-off X nb X ac X lc X lc,ri X X costs X incept claims handling X X liability Cash flow new business (premium liability) of year I + 1 Cash flow attritional claims Cash flow large claims without reinsurance cover Cash flow large claims, including reinsurance cover Run-off life-time annuity cash flow at time I Costs cash flow Inception costs cash flow Claims handling costs cash flow Total liability cash flow after time I (Π 0,Π 1 ) Premium cash flow