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

Transcription

1 Springer Finance Mario V. Wüthrich Michael Merz Financial Modeling, Actuarial Valuation and Solvency in Insurance

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

3 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:

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

5 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 (

6 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

7 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

8 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

9 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

10 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

11 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

12 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

13 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)

14 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

15 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 (I+1) X nb (I+1) X ac (I+1) X lc (I+1) X lc,ri (I+1) X (I+1) X costs (I+1) X incept (I+1) claims handling X(I+1) X liability (I+1) 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

16 Chapter 1 Introduction In the past few decades the financial industry has experienced several economic cycles. There were periods of rapid economic growth interspersed with periods of economic stagnation. Europe and America experienced a stagnation in the 1970s, which was followed by a growth period in the 1980s and 1990s. A severe financial crisis hit the (financial) industry around the year 2000 followed by a subsequent one between 2007 and These economic cycles are manifested in economic activity and production. They influence economic growth, purchasing power of money, supply and demand of goods and services as well as prices of insurance cover and of financial instruments. The latter two are affected by economic stability, supply and demand of products, liquidity, default and credit risk of financial players, money supply and interest rate policy of central banks, exchange rates between different economies, etc. Since the financial industry produces these financial products and guarantees it is at its core to understand their price formation in the economic environment. Moreover, it is at its heart to have good risk management policies in order to provide financial stability also in distress periods. This has not always been the case in the past couple of years and therefore supervision by government has increased in order to make sure that the financial industry really serves the community. The purpose of this book is to introduce sound risk measurement practices which form the bases of good risk management policies and solvency regulation. We define a comprehensive mathematical framework that adequately describes price formation and captures the corresponding risk factors that influence the stability of the financial industry. In particular, we develop quantitative risk management models for insurance companies. These can be used for risk assessment, supervision and management of the companies. The models that we describe are at the heart of quantitative solvency considerations of insurance companies and belong to the wider area of enterprise risk management. The essence of this book is exemplified in Sect where we derive a full risk measurement model for an insurance company that has a non-life insurance portfolio and a life-time annuity portfolio, which are exposed to financial risk, reinsurance default risk, etc. This model can be viewed as an important cornerstone towards a comprehensive enterprise risk management model. M.V. Wüthrich, M. Merz, Financial Modeling, Actuarial Valuation and Solvency in Insurance, Springer Finance, DOI / _1, Springer-Verlag Berlin Heidelberg

17 2 1 Introduction Modelers always face the (difficult) trade-off between complexity and simplicity. On the one hand the model should be sufficiently sophisticated so that it can appropriately capture real world behavior. Since real world behavior is very involved, modelers often arrive at rather complex models that can only (if at all) be solved numerically. The disadvantage of such complex models is that they lack proper understanding and interpretation because very often the risk factors are merged and hidden in the model such that one cannot identify the important risk drivers. On the other hand, simple models are well understood but they often do not describe real world behavior sufficiently well. Therefore, good quantitative risk management models are somewhere between very complex and rather simple models. Often, they cannot model all the real world features, however, they should capture the essential risk drivers in an appropriate way, so that these can be analyzed, understood and managed. When we try to solve real world problems using models, we should always keep in mind that we need to translate the real world problem to a question in an appropriate model world and that we need to re-translate the model world answer to an answer for the question stemming from the real world. In practice, we often forget that the quality of the real world answer then heavily relies on the quality of these two translations! If the translations are weak then the answer is not of any value for the real world problem. Therefore, good modelers should have a deep understanding for the real world problem, they should have strong modeling skills, for example, in probability theory, but they also need to have good skills in statistics which evaluates the quality of the translations (model testing, parameter fitting, etc.). In a nutshell, construction of good enterprise risk management models for insurance companies requires the possession of multiple skill sets. First of all, one needs to have a good understanding of the insurance and financial products. This includes a good understanding of external factors like economic developments (monetary policy, economic growth, insurance and financial markets, interest rate behavior, inflation, unemployment rate, legal and political changes), environmental factors (natural hazards, scientific and medical developments, longevity, etc.), the insurance contracts itself, policyholder behavior, management actions, etc. Also important is that all these considerations respect the valid accounting rules. At the end of the day, all (financial) values are determined using the relevant accounting rules, therefore one needs to know very well how assets and liabilities are displayed. In the next section we are going to describe the balance sheet of an insurance company which is the basis of solvency considerations. Secondly, one needs to have strong modeling skills and strong skills in statistics that bring the practical experience in line with the model world. Finally, one needs to have a strong analytical capability that allows to analyze the crucial features and to draw the right conclusions. Having all these skills one should not forget about the human factor. That is, all relevant systems are designed and run by human beings, and it is natural that errors occur. These can often not be captured in a quantitative model and therefore risk managers and regulators should also make sure that good qualitative control systems are in place.

18 1.1 Full Balance Sheet Approach Full Balance Sheet Approach There is a general agreement in risk measurement and risk management that assets and liabilities need to be considered simultaneously in order to analyze the financial strength of a company. Therefore, for auditing an insurance company we need to perform a so-called full balance sheet approach, which means that we display all asset and liability positions simultaneously. Typically, a balance sheet of an insurance company has the following positions: Assets cash and cash equivalents debt securities bonds loans mortgages real estate equity equity securities private equity investments in associates hedge funds derivatives futures, swaptions, equity options insurance and other receivables reinsurance assets property and equipment intangible assets goodwill deferred acquisition costs income tax assets other assets Liabilities deposits policyholder deposits reinsurance deposits borrowings money market hybrid debt convertible debt insurance liabilities mathematical reserves claims reserves premium reserves derivatives insurance and other payables reinsurance liabilities employee benefit plan provisions income tax liabilities other liabilities The main difficulty in the simultaneous analysis of these balance sheet positions is that every item should be valued with the same measure. This is quite a difficult task because there is no natural market (or valuation system) in which all the positions are traded (or valued, respectively). Therefore, there is no obvious best measure for the simultaneous valuation of all these balance sheet positions. State-of-the-art valuation and accounting uses market values for traded instruments and so-called market-consistent values for the remaining instruments. Market-consistent values means that prices are calculated in a marked-to-model approach. This marked-to-model approach should mimic a market and should give prices for non-traded instruments as if they were traded and which are consistent with values of traded instruments. For example, Solvency II guidelines state, see Article 75 in [63]: assets shall be valued at the amount for which they could be exchanged between knowledgeable willing parties in an arm s length transaction and liabilities shall be valued at the amount for which they could be transferred, or settled, between knowledgeable willing parties in an arm s length transaction.

19 4 1 Introduction The first quantity of importance with regard to the full balance sheet approach is then the difference between the values of assets and liabilities. If the resulting asset value exceeds the corresponding liability value then we say that all liabilities are covered by asset values. This is the so-called accounting condition, see Sect , and says that the company has sufficient asset values to cover today s liabilities. We should also mention that depending on the purpose one could also use values different from market and market-consistent values. For example, traditional actuarial valuation uses constant interest rates. If we apply this constant interest rate valuation framework to all balance sheet positions we also obtain a consistent full balance sheet valuation approach. There may be good reasons for using such an actuarial valuation approach, for instance, in many situations it provides more stability over time. The full balance sheet valuation approach records the current situation. However, for solvency considerations we need to go beyond this snapshot because we would also like to know that the liabilities are covered by asset values in the future. In particular, this means that we should have sufficient asset values when the liabilities are due so that we can fulfill them when they need to be paid out to the policyholders. This is described in further detail in the next section. 1.2 Solvency Considerations In the last section we have described the full balance sheet valuation approach which requires that all balance sheet positions are valued with the same measure. As already mentioned in the last section, a first requirement then is that under the current valuation approach all liabilities are covered by asset values, which provides the socalled accounting condition. However, solvency considerations go much beyond this view. Namely, liabilities should not only be covered by asset values today, but this should hold true for all time points in the future. In particular, the asset values should be available when the liability payouts are due according to the insurance contract terms. This second view provides the so-called insurance contract condition, see Sect An insurance company is solvent if it fulfills both the accounting condition and the insurance contract condition, see Definition The evaluation of the insurance contract condition is rather involved. Since the future development of the values (assets and liabilities) is not known today (involves uncertainties), we need to model them stochastically. That is, we need to build stochastic models that are able to give a description of the (random) development of the future values of all balance sheet positions. This stochastic description (i) needs to be done simultaneously for all balance sheet positions because their random future values may interact, and (ii) needs to be done consistently such that the valuation system does not allow for arbitrage. Solvency is then achieved if, under these stochastic considerations, liabilities are covered by asset values for all/most of the possible future developments. The International Association of Insurance Supervisors IAIS [89] states this as follows:

20 1.3 Further Modeling Issues 5 Solvency: 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 practicable definition, it is necessary to make clear under which circumstances the appropriateness of the assets to cover claims is to be considered,.... This means that we not only need to be able to give present values to all balance sheet positions today but we also need to describe their stochastic behavior in the future. This includes: (i) process risk because we study stochastic processes; (ii) model uncertainty because we do not know the exact model description of all stochastic factors that drive the price processes; (iii) parameter uncertainty because once we have specified the stochastic model we also need to determine the explicit values of the model parameters. Concluding, there are many factors of uncertainty and randomness that the modeler has to take care of in order to describe the future values, and, finally, he should quantify how reliable his predictions of future values are. This then determines the necessary financial strength (solvency requirement) of the insurance company so that the policyholder can feel confident that the insurance company is able to respect and fulfill his insurance contract terms. An excellent historical review of the development of modern solvency guidelines that are based on market and market-consistent values in a full balance sheet approach is provided in Part A of Sandström [141]. 1.3 Further Modeling Issues General modeling approaches split the total future balance sheet uncertainties into different building blocks, so-called solvency modules or risk classes. In a first step, these risk classes are modeled in the best possible way. In a second step, the risk classes are aggregated using appropriate dependence structures. Often the second step is rather difficult and therefore this questions such a risk classes approach. Our building blocks will be such that the process of aggregation becomes as simple as possible, i.e. a first analysis should study the ideal decoupling into building blocks so that we have good modeling approaches for the building blocks but at the same time good aggregation properties between the building blocks. This is going to be discussed in Part II of this book. We have already stated that the accounting rules are crucial for the valuation process. Accounting rules are not only relevant for the values itself, they also rule the time frame of the valuation process. In the present book we assume that we have year-end balance sheet closings and therefore solvency questions are studied over a time horizon of one accounting year. This is quite different from classical ruin theory (e.g. in the Cramér Lundberg model) where one studies the ruin probability in continuous time over an infinite time horizon. Here, we consider the solvency question over a time horizon of one year, and we ask the question whether liabilities are covered by asset values in one year from today. Thereby, we need to specify how values develop within the next accounting

21 6 1 Introduction year. However, this specification also includes the definition of the insurance business model beyond the next accounting year because the values (especially the ones of liabilities) will depend on the future business strategy. For example, it is important to know whether we consider the run-off situation of the existing insurance liability portfolio or whether the existing portfolio is merged with new insurance business. In the latter case there is a risk and cost diversification between old and new insurance business which will lead to different (market-consistent) values compared to the first situation. Hence, in order to determine values and solvency the the whole story beyond the next accounting year is important, because the circumstances of the evaluation essentially influence the values. At the end, for solvency regulation, it is the supervisor who defines the general setup under which the insurance market needs to prove sufficient financial strength. This is accomplished in Part II of this book. 1.4 Outline of This Book This book is divided into two parts: Part I considers financial valuation principles and Part II studies actuarial valuation and solvency. In the appendix (labeled as Part III) we present mathematical technicalities that are of wider interest. In the following paragraphs we describe the first two parts in more detail. Part I In Part I of this book we introduce the financial valuation framework. The crucial property of this framework is that all assets and liabilities are valued consistently at any time. This eliminates arbitrage. The crucial tools are equivalent martingale measures and state price deflators. Equivalent martingale measures are favored in financial mathematics, state price deflators are often used in actuarial mathematics. We introduce these concepts in Chap. 2, describe their interrelationship and apply them to the valuation of cash flows. At this stage these concepts are introduced as abstract mathematical (and economically sensible) tools. In Chaps. 3 and 4 we give explicit models and we calibrate these to market data. In the former chapter the models are based on so-called spot rate processes which describe the short term behavior of interest rates. These include, for instance, the Vasicek model. The latter chapter is based on the Heath Jarrow Morton framework which describes the arbitrage-free development of the entire interest rate curve. Finally, Chap. 5 concludes Part I of this book. This chapter provides explicit examples of cash flow valuation, introduces the financial market and the price processes of its financial instruments and the derivatives thereof. Moreover, the Vasicek financial model (see Model 5.7) is introduced which will serve as a toy model in our case studies on solvency. It is worth noting that all these considerations are done in discrete time whereas classical financial mathematics literature studies price processes in continuous time. As already mentioned in Sect. 1.3, accounting and solvency is studied on a discrete time grid, therefore in most situations in this book it suffices to study discrete time models. We will see that this has both advantages and disadvantages.

22 1.4 Outline of This Book 7 Part II Part II of this book refines the valuation framework introduced in the first part. The aim of this refinement is to clearly distinguish between hedgeable part and non-hedgeable part (of assets and liabilities) in an actuarial context. We separate financial instruments modeling and insurance technical events within the state price deflator setup. Essentially, we isolate all risk drivers that cannot be explained by financial market movements. In particular, this also requires that we decouple the state price deflator into the financial deflator and the probability distortion. The financial deflator describes the price formation at the financial market, whereas the probability distortion is used for the calculation of the so-called risk margin which supports the risk bearing of non-hedgeable risks. These concepts are introduced in Chap. 6. In Chap. 7 we introduce the valuation portfolio and the best-estimate reserves. The valuation portfolio is a systematic approach that decouples (insurance) liabilities into the component that can be replicated by instruments of the financial market and into the insurance technical component (residual component). This provides a clear understanding of the insurance liabilities and makes them comparable to the assets side of the balance sheet. The best-estimate reserves are then obtained by simply replacing the insurance technical component (random variable) by its expected value. This construction, which is described in three steps in Chap. 7, is demonstrated for three explicit insurance portfolios. However, best-estimate reserves do not reflect appropriate prices for insurance liabilities. By replacing random variables with expected values we are (only) covered up to the average outcome of the liability. Every risk averse risk bearer will charge an additional (risk) margin for compensating possible shortfalls in the outcomes. In Chap. 8 we introduce the protected valuation portfolio. This construction relies on probability distortions which (under appropriate assumptions) provide positive risk margins and the corresponding risk-adjusted reserves that also compensate for risk bearing. In Sect. 8.3 we give explicit numerical examples: a non-life insurance run-off example, a life-time annuity example and an endowment policy example are studied. In Chap. 9 we introduce the core of risk measurement and solvency assessment. Based on the valuation principles introduced in the previous chapters we determine whether the financial assets are sufficient to cover the liabilities also in the case of (well-specified) stress situations. Well-specified stress situations are described by the introduction of a risk measure, that is, we consider whether the asset and liability positions in the full balance sheet approach are sufficiently safe according to the chosen risk measure. This adds the dynamic aspect to the balance sheets of insurance companies. In this chapter, we introduce the notions of asset deficit and free capital that play a crucial role in solvency (and acceptability) considerations. This is supported by many examples and outlines of particular balance sheet choices (business plans). Moreover, we discuss the limited liability option, the Margrabe option, dividend payments, cost-of-capital loadings, risk spreading and the law of large numbers (which is the basis of insurance). Chapter 10 is our final chapter. In this chapter we give insight in selected topics and further developments. The heart of this chapter (and maybe of the whole

23 8 1 Introduction book) is Sect (Solvency Toy Model in Action) where we build our own insurance company. We then study solvency of our company for different business plans, which shows how all the risk factors enter the risk management and solvency analysis. Furthermore, we study in this chapter the important topics of parameter uncertainty, applied cost-of-capital concepts, modeling of accounting and calender year dependence in non-life insurance as well as premium liability and re-insurance modeling. The latter provides risk mitigation techniques for insurance technical risks. The concluding remarks in Sect complete Parts I and II of the book.

24 Part I Financial Valuation Principles

25 Chapter 2 State Price Deflators and Stochastic Discounting In this chapter, we describe stochastic discounting and valuation of random cash flows in a discrete time setting. We therefore introduce a consistent multiperiod pricing framework. This consistent multiperiod pricing framework is either based on state price deflators or on equivalent martingale measures. The connection between these two pricing concepts is then described by the market price of risk idea introduced in Sect Before we start with these stochastic valuation models, we explain the fundamental notion and terminology from interest rate modeling. 2.1 Zero Coupon Bonds and Term Structure of Interest Rates To introduce the term structure of interest rates notion, we consider for the time being a continuous time setting. Thereafter we restrict to discrete time, see Sect. 2.2 onward. Throughout this book we work with one fixed reference currency Motivation for Discounting What is discounting and why do we discount? Discounting means attaching time values to assets and liabilities. Assume we put $100 on a bank account, i.e. we lend out money to the bank. We expect that the value of this bank deposit grows with an annual interest rate r,sayr = 3 %. Hence, we expect that in one year s time from today we can withdraw $103 from the bank account. If the bank account would not offer a positive interest rate r then we could as well store the $100 at home. Thus, banks attract deposits by offering positive interest rates. This example shows that we have the expectation that money grows over time and therefore currency has a time value. The amount and speed at which it grows depends on economic factors such as growth of the economy, state of the economy, M.V. Wüthrich, M. Merz, Financial Modeling, Actuarial Valuation and Solvency in Insurance, Springer Finance, DOI / _2, Springer-Verlag Berlin Heidelberg

26 12 2 State Price Deflators and Stochastic Discounting money supply and interest policy of the central bank, government expenditure, inflation rate, unemployment rate, foreign exchange rates, etc. All these factors interact in a non-trivial way and macro-economic theory tries to explain these interrelationships. One should also be aware of the fact that the growth of money by an annual interest rate r is very different from the real growth of money which determines the purchasing power of capital. Economists therefore consider the nominal interest rate r and the (expected) real interest rate which is the difference between the nominal interest rate and the (expected) inflation rate, see for instance Gärtner [73], p. 204, Romer [136], p. 73, or Fig. 7.2 in Ross et al. [137]. The aim of this book is to model growth of money and to value future (random) cash flows. For example, we model how the value of $100 is growing over time using stochastic interest rate models. In particular, if we put the deposit of $100 on a bank account and the bank guarantees a fixed (deterministic) annual interest rate of r = 3 %, then the final wealth of this investment in one year s time from today is $103. Therefore, we call $100 the discounted value of the final wealth $103, and (1 + r) 1 = 100/103 = % is termed the (deterministic) discount factor. As discount factors are not known for all future periods, the future economic factors being random variables based on our knowledge today, we are going to model future interest rates and discount factors stochastically. This will lead to stochastic discounting using so-called state price deflators which can be viewed as economic indicators for the time value of money in stochastic term structure models Spot Rates and Term Structure of Interest Rates Definition 2.1 A default-free zero coupon bond (ZCB) with maturity m 0isa contract that pays one unit of currency at time m. Its price at time t [0,m] is denoted by P(t,m). By convention we set P (m, m) = 1. Since money grows over time, see last subsection, we expect P(t,m)<1fort<m. A ZCB is a so-called default-free financial instrument. That is, its issuer cannot go bankrupt and hence always fulfills the ZCB contract (see also Example 2.8, below). In general, there is no default-free bond on the financial market, typically, bonds that are issued (by companies or governments) may default, i.e., there is a positive probability that the issuer is not able to fulfill the contract. In such cases, one speaks about credit risk that needs a special pricing component. This will be investigated in Sect , below. For the time being we will work in a continuous time setting and we will assume that ZCBs exist for all maturities m 0. These ZCBs will describe the underlying dynamics of time value of money.

27 2.1 Zero Coupon Bonds and Term Structure of Interest Rates 13 Definition 2.2 Choose 0 t<m. The continuously-compounded spot rate for maturity m at time t is defined by R(t,m) = 1 log P(t,m). m t The simply-compounded spot rate for maturity m at time t is defined by L(t, m) = 1 m t 1 P(t,m) P(t,m) = 1 ( P(t,m) 1 1 ). m t The annually-compounded spot rate for maturity m at time t is defined by Y(t,m)= P(t,m) 1 m t 1. These are different notions to describe the ZCB price P(t,m)at time t [0,m). We have the identities P(t,m)= e (m t)r(t,m) = ( 1 + (m t)l(t,m) ) 1 = ( 1 + Y(t,m) ) (m t). (2.1) This provides the relationships R(t,m) = 1 m t log( 1 + (m t)l(t,m) ) = log ( 1 + Y(t,m) ). Our aim is to model these spot rates. This requires that we calibrate the spot rates to actual financial market data and that we describe their stochastic development in the future. For the calibration we will use two different sets of data. For long times to maturity m t (more than one year) we will use government bond prices for the calibration (this is further described in Sect and Example 2.6, below). For short times to maturity m t (less than one year) the simply-compounded spot rate L(t, m) is often calibrated with the LIBOR (London InterBank Offered Rate). The LIBOR is fixed daily at London market and is used for (unsecured) short term deposits that are exchanged between banks. That is, this is the rate at which highly credited financial institutions offer and borrow money at the interbank market. Therefore, in this book, we will use the LIBORs as approximation to short term risk-free rates for model calibration. We would like to mention that especially in periods of financial distress this needs to be done rather carefully. The spot rates should describe ZCB prices of default-free financial instruments. Therefore, these rates should not include any credit spread (default pricing component) and liquidity spread. However, credit and liquidity risks may have a major impact on prices during distress periods. The financial crisis of 2008 has demonstrated that also the interbank market can become almost illiquid and highly credited financial institutions may default. The high uncertainty at financial markets during distress periods can, for instance, be seen between the different rates of Repo-Overnight-Indexes (secured, see Example 3.9, below) and LIBOR curves (see also Figs. 3.1 and 3.2 below). We see a clear spread widening between these two curves between 2007

28 14 2 State Price Deflators and Stochastic Discounting and This indicates high default and liquidity risks and shows that secured versus unsecured funds may behave rather differently in distress periods. As a consequence, model calibration of default-free ZCBs needs to be done carefully, and it is not always clear which data should be chosen for the calibration because typical financial market data always contain default and liquidity components that need to be isolated appropriately. This segmentation is heavily debated both in the financial and in the actuarial community, see, for instance, Das et al. [50], Mercurio [ ], Danielsson et al. [48, 49] and Keller et al. [95]. We come back to these issues in terms of model calibration in Example 3.9 and Sect Moreover, there is an additional difficulty because typically we do not have observations for all maturity dates. The latter becomes relevant especially for the valuation of long-term guarantees in life insurance products, for more on this topic we refer to Sects , 6.1 and 9.4.3, below. Definition 2.3 The instantaneous spot rate (also called short rate) is, for t 0, defined by r(t) = lim m t R(t,m). Throughout this text we assume that the ZCB prices are sufficiently smooth functions so that all the necessary limits and derivatives exist. Note that we obtain from the power series expansion of R(t,m) r(t) = lim m t L(t, m). Therefore, if we use the LIBORs as approximation to L(t, m), we can calibrate the instantaneous spot rate r(t) by the study of the LIBOR for small time intervals [t,m]. Definition 2.4 The term structure of interest rates (yield curve) at time t 0is given by the graph of the function m R(t,m), m > t. The yield curve m R(t,m) at time t determines the ZCB prices P(t,m) for allmaturities m>t andvice versa, see (2.1). Atany pointin time u<t future ZCB prices P(t,m) are random and therefore need to be modeled stochastically. This stochastic term structure modeling of R(t,m) and P(t,m), respectively, is our aim in the subsequent sections and chapters. Definition 2.5 The forward interest rate at time t, fors t, is defined by F(t,s+ 1) = log P(t,s+ 1) + log P(t,s)= log P(t,s+ 1). P(t,s)