Risk Measures for the 21st Century

GIORGIO SZEGO graduated in Physics at the University of Pavia. After postgraduate studies at the Technische Hochschule Darmstadt, he held teaching and research positions (Research Institute of Advanced Studies) in the USA. From 1964 to 1970 he had an appointment at the Faculty of Sciences of the University of Milan, and held visitng poisitions in various US universities. He was then granted a chair in Mathematics for Economics at the University of Venice and in 1977 he co-founded and became Managing Director of the Journal of Banking and Finance. Giorgio Szego holds a chair in "Economics of Financial Markets" at the University of Rome and in January 2000 the University of Bergamo awarded him an honorary degree in Economics. His current line of research is on Risk Measures and their regulatory impact. He is the author of more than 200 research papers and books on many subjects ranging from the theory of dynamical systems, to optimisation and portfolio theory.

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• About the Contributors. 1 On the (Non)Acceptance of Innovations (Giorgio Szegö).1.1 Introduction.1.2 The path towards acceptance of previous innovations.1.3 How to answer.1.4 Conclusions.References.PART I: RISK MEASURES AND REGULATION.2 The Emperor has no Clothes: Limits to Risk Modelling (Jón Daníelsson).2.1 Introduction.2.2 Risk modelling and endogenous response.2.3 Empirical properties of risk models.2.3.1 Background.2.3.2 Robustness of risk forecasts.2.3.3 Risk volatility.2.3.4 Model estimation horizon.2.3.5 Holding periods and loss horizons.2.3.6 Non-linear dependence.2.4 The concept of (regulatory) risk.2.4.1 Volatility.2.4.2 Value-at-risk.2.4.3 Coherent risk measures.2.4.4 Moral hazard – massaging VaR numbers.2.4.5 The regulatory 99% risk level.2.5 Implications for regulatory design.2.6 Conclusion.Acknowledgements.Appendix A: Empirical study.References.3 Upgrading Value-at-Risk from Diagnostic Metric to Decision Variable: A Wise Thing to Do? (Henk Grootveld and Winfried G. Hallerbach).3.1 Introduction.3.2 Preliminaries.3.2.1 VaR and downside risk.3.2.2 Downside risk portfolio selection.3.2.3 Incomplete risk meaure.3.2.4 Computational issues.3.3 The mean-value-at-risk portfolio selection model.3.3.1 Deriving the mean-VaR portfolio selection model.3.3.2 Distinctive properties of the mean-VaR portfolio selection model.3.3.3 Solving the mean-VaR portfolio selection problem.3.4 The mean-value-at-risk portfolio selection model in practice.3.4.1 Data.3.4.2 Methodology.3.4.3 Results.3.5 Conclusions.Acknowledgements.References.4 Concave Risk Measures in International Capital Regulation (Imre Kondor, András Szepessy and Tünde Ujvárosi).4.1 Introduction.4.2 Risk measures implied by the trading book regulation.4.2.1 Specific risk of bonds.4.2.2 Foreign exchange.4.2.3 Equity risk.4.2.4 The general risk of bonds.4.3 Conclusion.Acknowledgements.References.5 Value-at-Risk, Expected Shortfall and Marginal Risk Contribution (Hans Rau-Bredow).5.1 Introduction.5.2 Value-at-risk as a problematic risk measure.5.3 Derivatives of value-at-risk and expected shortfall.5.3.1 Preliminary remarks.5.3.2 First and second derivative of value-at-risk.5.3.3 First and second derivative of expected shortfall.5.4 Outlook.Appendix.References.6 Risk Measures for Asset Allocation Models (Rosella Giacometti and Sergio Ortobelli Lozza).6.1 Introduction.6.2 Portfolio risk measures.6.2.1 Safety risk measures.6.2.2 Dispersion measures.6.3 Portfolio choice comparison based on historical data.6.4 Portfolio choice comparison based on simulated returns.6.4.1 Portfolio choice comparison with jointly Gaussian returns.6.4.2 Portfolio choice comparison with jointly stable non-Gaussian returns.6.5 Conclusions.Acknowledgements.References.7 Regulation and Incentives for Risk Management in Incomplete Markets (J´on Daníelsson, Bjørn N. Jorgensen and Casper G. de Vries).7.1 Introduction.7.1.1 Complete and incomplete markets.7.2 Moral hazard regarding project choice.7.2.1 Deposit insurance and moral hazard.7.2.2 Threat of an alternative project choice.7.3 Moral hazard regarding risk management.7.3.1 The basic principal–agent model.7.3.2 Supervision.7.4 Risk monitoring and risk management.7.4.1 Coarser risk monitoring without regulation.7.4.2 Indirect risk monitoring with regulation.7.4.3 Finer risk monitoring: no regulation.7.4.4 Direct risk monitoring with regulation.7.4.5 Evaluation.7.5 Conclusion.References.8 Granularity Adjustment in Portfolio Credit Risk Measurement (Michael B. Gordy).8.1 Introduction.8.2 Granularity adjustment of VaR for homogeneous portfolios.8.3 Granularity adjustment of ES for homogeneous portfolios.8.4 Application to heterogeneous portfolios.Appendix: Wilde’s formula for ?.Acknowledgements.References.9 A Comparison of Value-at-Risk Models in Finance (Simone Manganelli and Robert F. Engle).9.1 Introduction.9.2 Value-at-risk methodologies.9.2.1 Parametric models.9.2.2 Nonparametric models.9.2.3 Semiparametric models.9.3 Expected shortfall.9.4 Monte Carlo simulation.9.4.1 Simulation study of the threshold choice for EVT.9.4.2 Comparison of quantile methods performance.9.5 Conclusion.References.Appendix: Tables.PART II: NEW RISK MEASURES.10 Coherent Representations of Subjective Risk-Aversion (Carlo Acerbi).10.1 Forewords and motivations.10.1.1 In defense of axiomatics.10.1.2 Scope and objectives.10.1.3 Outline of the work.10.2 Building a risk measure: the expected shortfall.10.2.1 A close look into VaR’s definition.10.2.2 A natural remedy to probe the tail: the expected shortfall.10.2.3 Coherency of ES.10.2.4 Estimation of ES.10.3 Spectral measures of risk.10.3.1 Estimation of spectral measures of risk.10.3.2 Characterization of spectral measures via additional conditions.10.3.3 Spectral measures and capital adequacy.10.4 Optimization of spectral measures of risk.10.4.1 Coherent measures and convex risk surfaces.10.4.2 Minimization of expected shortfall.10.4.3 Minimization of general spectral measures.10.4.4 Risk–reward optimization.10.5 Statistical errors of spectral measures of risk.10.5.1 Variance of the estimator.10.5.2 Some meaningful examples.Acknowledgements.References.11 Spectral Risk Measures for Credit Portfolios (Claudio Albanese and Stephan Lawi).11.1 Introduction.11.2 Test-portfolios with market risk and entity-specific risk.11.3 Properties of risk measures.11.4 Discussion of test-portfolios.11.5 Concluding remarks.Acknowledgements.References.Appendix: Tables.12 Dynamic Convex Risk Measures (Marco Frittelli and Emanuela Rosazza Gianin).12.1 Introduction.12.1.1 Notation.12.1.2 Axioms.12.1.3 Coherent risk measures.12.2 Convex risk measures.12.2.1 Representation of convex risk measures.12.2.2 Law-invariant convex risk measures.12.3 Indifferent prices and risk measures.12.4 Dynamic risk measures.12.5 Appendix.References.13 A Risk Measure for Income Processes (Georg Ch. Pflug and Andrzej Ruszczyński)13.1 Introduction.13.2 The one-period case.13.3 Risk of multi-period income streams.13.4 Finite filtrations.13.5 Properties of the risk measure.13.6 Mean–risk models.13.7 Examples.13.8 A comparison with the ADEHK approach.13.9 The discounted martingale property for final processes.References.PART III: COPULA FUNCTIONS FOR THE ANALYSIS OF DEPENDENCE STRUCTURES.14 Financial Applications of Copula Functions (Jean-Frédéric Jouanin, Gaëulet and Thierry Roncalli).14.1 Introduction.14.2 Copula functions.14.3 Market risk management.14.3.1 Non-Gaussian value-at-risk.14.3.2 Stress testing.14.3.3 Monitoring the risk of the dependence in basket derivatives.14.4 Credit risk management.14.4.1 Measuring the risk of a credit portfolio.14.4.2 Modelling basket credit derivatives.14.5 Operational risk management.14.5.1 The loss distribution approach.14.5.2 The diversification effect.References.15 Hedge Funds: A Copula Approach for Risk Management (Hélyette Geman and Cécile Kharoubi).15.1 Introduction.15.2 Hedge funds industry, strategies and data.15.2.1 Hedge funds industry: definitions and description.15.2.2 The different strategies.15.2.3 Biases in hedge funds data.15.2.4 Hedge funds indices: descriptive statistics.15.3 Copulas and hedge funds.15.4 Value-at-risk with copulas.15.4.1 Monte Carlo simulation.15.4.2 Value-at-risk computation.15.5 Conclusion.Acknowledgements.References.16 Change-point Analysis for Dependence Structures in Finance and Insurance (Alexandra Dias and Paul Embrechts).16.1 Introduction.16.2 Statistical change-point analysis.16.2.1 The test statistic.16.2.2 An example: the Gumbel case.16.2.3 The power of the test.16.2.4 The time of the change and corresponding confidence intervals.16.2.5 Multiple changes.16.3 A comment on pricing.16.4 An example with insurance data.16.5 Conclusion.Acknowledgements.References.PART IV: ADVANCED APPLICATIONS.17 Derivative Portfolio Hedging Based on CVaR (Siddharth Alexander, Thomas F. Coleman and Yuying Li).17.1 Introduction.17.2 Minimizing VaR and CVaR for derivative portfolios.17.2.1 How well is the minimum risk derivative portfolio defined?17.2.2 Difficulties due to ill-posedness.17.3 Regularizing the derivative CVaR optimization.17.3.1 Example 1: Hedging a short maturity at-the-money call.17.3.2 Example 2: Hedging a portfolio of binary options.17.4 Minimizing CVaR efficiently.17.4.1 Efficiency for CVaR minimization using an LP approach.17.4.2 A smoothing technique for CVaR minimization.17.5 Concluding remarks.Acknowledgements.References.18 Estimation of Tail Risk and Portfolio Optimisation with Respect to Extreme Measures (Giorgio Consigli).18.1 Introduction.18.2 From risk measurement to risk control: the setup.18.2.1 VaR control with non-normal return distributions.18.3 Beyond VaR: From non coherent to coherent measures.18.3.1 Risk measures in the tails: methods accuracy.18.3.2 A case study. Application 1: Risk measurement.18.3.3 Multidimensional Poisson–Gaussian model.18.4 Risk control based on portfolio optimization.18.4.1 Risk–return and trade-off optimisation: QP and LP solvability.18.4.2 Optimal portfolios during periods of market instability.18.5 Conclusions and future research.Acknowledgements.References.19 Risk Return Management Approach for the Bank Portfolio (Ursula A. Theiler).19.1 Introduction.19.2 Step 1 of the RRM approach: optimization model for the bank portfolio.19.2.1 Survey.19.2.2 Modeling the internal risk constraint.19.2.3 Integration of the regulatory risk constraint into the optimization model.19.2.4 Summary of the optimization model of step 1 of the RRM Approach.19.3 Step 2 of the RRM Approach: risk return keys for the optimum portfolio.19.3.1 Survey.19.3.2 Derivation of risk return keys on the asset level.19.3.3 Aggregation of risk return keys on the profit center level.19.3.4 Summary of the risk return ratios generated by the RRM Approach.19.4 Application example.19.4.1 Situation and problem statement.19.4.2 Results.19.5 Conclusion.References.PART V: LAST, BUT NOT LEAST.20 Capital Allocation, Portfolio Enhancement and Performance Measurement: A Unified Approach (Winfried G. Hallerbach).20.1 Introduction.20.2 Preliminaries.20.3 Portfolio optimization, RAROC and RAPM.20.3.1 Portfolio optimization without risk-free rate.20.3.2 Portfolio optimization allowing for risk-free activities.20.4 Conclusions.Appendix.Acknowledgements.References.21 Pricing in Incomplete Markets: From Absence of Good Deals to Acceptable Risk (H´elyette Geman and Dilip B. Madan).21.1 Introduction 45121.2 No-good-deal pricing in incomplete markets.21.2.1 Good-deal asset price bounds (Cochrane and Saá, 2000).21.2.2 Gain, loss and asset pricing (Bernardo and Ledoit, 2000).21.2.3 The theory of good-deal pricing (Cerny and Hodges, 2001).21.3 Pricing with acceptable risk.21.3.1 The economic model.21.3.2 The first fundamental theorem.21.3.3 The second fundamental theorem.21.3.4 Pricing under acceptable incompleteness.21.4 Conclusion.References.Index.