Decision making under risk with spectral risk measures - Concepts and applications in Financial Theory. Habilitationsschrift

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1 Decision making under risk with spectral risk measures - Concepts and applications in Financial Theory Habilitationsschrift vorgelegt dem Rat der Wirtschaftswissenschaftlichen Fakultät der Friedrich-Schiller-Universität Jena am von Dr. Mario Brandtner

2 Inhaltsverzeichnis I 1 Zusammenfassung 1 Literaturverzeichnis 16 2 Spektrale Risikomaße: Konzeption, betriebswirtschaftliche Anwendungen und Fallstricke Einleitung Konzeption Axiomatik und Repräsentation spektraler Risikomaße Wichtige Eigenschaften spektraler Risikomaße Beispiele spektraler Risikomaße Verwandte Konzepte Anwendungsfeld 1: Portfolio Selection Optimale Vermögensaufteilung riskant vs. risikofrei Zwei riskante Wertpapiere Anwendungsfeld 2: Optimale Rückversicherung Optimaler Vertragstyp Optimale Selbstbehalte und Plunging Anwendungsfeld 3: Optimale Bestellmengen im Newsvendor-Modell Zusammenfassung 57 Anhang Beweis: Spektrale Risikomaße repräsentieren Arrow-Pratt-Risikoaversion Beweis von Satz Beweis von Satz Beweis von Satz Literaturverzeichnis 69 I

3 3 Conditional Value-at-Risk, spectral risk measures, and portfolio selection Introduction Theoretical framework Spectral risk measures and the regulatory concept of diversification Portfolio selection problems (/x, <j 2 )-efficient frontiers versus (p, p^-efficient frontiers Comonotonic subsets of alternatives Two risky assets One risk free and two risky assets Extensions Optimal portfolios Determination of optimal portfolios The mean-variance utility function and füll diversification Spectral utility functions and non-diversification Discussion Conclusions 97 Appendix Extensions of the (/i, cr 2 )-framework 99 References Conditional Value-at-Risk, spectral risk measures, and optimal reinsurance Introduction Setting Reinsurance problem Spectral risk measures Examples Optimality of stop-loss reinsurance General result Unique optimality of stop-loss reinsurance Optimal deductibles General result Comparative statics of the optimal deductibles Conclusion 125 Appendix Proof of Proposition II

4 4.7 Proof of Proposition Proof of Proposition Proof of Proposition Proof of Proposition References Expected Shortfall, spectral risk measures, and the problem of comparative risk aversion Introduction Setting: Spectral risk measures Representation and properties Three populär subclasses Comparative AP-risk aversion: The case of deterministic initial wealth AP-risk aversion Comparative AP-risk aversion for SRM-decision makers Comparative AP-risk aversion and Standard financial decision problemsl Comparative AP-risk aversion and the spectral AP-measure Comparative R-risk aversion: The case of random initial wealth R-risk aversion Comparative R-risk aversion for SRM-decision makers Comparative AP-risk aversion vs. comparative R-risk aversion Comparative R-risk aversion and Standard financial decision problems Conclusions 162 Appendix Proof of Proposition Proof of Theorem Proof of Proposition Proof of Proposition Proof of Corollary References Spectral risk measures, risk vulnerability, and the aggravating effect of background risk Introduction Spectral risk measures: representation and properties Risk vulnerability and spectral risk measures 174 III

5 6.4 An application: portfolio selection Portfolio selection with deterministic initial wealth Portfolio selection with random background wealth Portfolio selection with random background wealth: the Gaussian case Conclusions 184 Appendix Proof of Proposition Proof of Corollary Example Proof of Proposition References "Spectral Risk Measures: Properties and Limitations": Comment Introduction Spectral risk measures Representation Exponential and power spectral risk measures Correction I: Comparative risk aversion of spectral risk measures The argument of Dowd et al. (2008) Correcting Dowd et al.'s argument Correction II: Counterintuitive characteristics in the risk parameter limits The argument of Dowd et al. (2008) Correcting Dowd et al.'s argument Power spectral risk measures of type I versus type II Correction III: Constant absolute versus constant relative risk aversion The argument of Dowd et al. (2008) Correcting Dowd et al.'s argument Conclusion 200 Appendix Dowd et al.'s proof on ambiguous comparative risk aversion Proof of Theorem References In-consistent modeling of risk averse behavior with spectral risk measures Introduction 207 IV

6 8.2 Framework Preliminaries: Decision making with SRMs and EU-theory Consistency W/M's linking-procedure In-consistency of W/M's linking-procedure The W/M-risk measure vs. SRMs and EU-theory Is the W/M-risk measure a SRM? Is the W/M-utility function in line with EU-theory? Can a linking-procedure provide consistency at all? Conclusions 218 Appendix SRMs, W/M-risk measure and DC-theory 219 References 222 V