##
**Mathematical risk analysis. Dependence, risk bounds, optimal allocations and portfolios.**
*(English)*
Zbl 1266.91001

Springer Series in Operations Research and Financial Engineering. Berlin: Springer (ISBN 978-3-642-33589-1/hbk; 978-3-642-33590-7/ebook). xii, 408 p. (2013).

This book by Prof. Ludger Rüschendorf is a rare example in the literature on financial and insurance mathematics that tries to highlight the similarities between these two areas of practical importance and to provide a unified view towards risk management. When trying to evaluate the riskiness of large financial positions or insurance contract portfolios, one is inevitably led to modelling dependence between various entities (firms, individuals, countries, etc.), financial assets at the heart of numerous financial instruments, or risky events described in insurance contracts. Once a model is specified, one has to choose an appropriate risk measure to evaluate potential risk and then choose an optimal hedging strategy or investment portfolio to withstand the worst case scenario.

To be able to accomplish these tasks, one has to master a good deal of probability theory, functional analysis, optimization theory as well as a fair share of financial and insurance mathematics. This book is definitely a good, though perhaps not the easiest possible guide towards this goal. The book contains four parts: stochastic dependence and extremal risk, risk measures and worst case portfolios, optimal risk allocation, and optimal portfolios and extreme risk.

Part I on stochastic dependence and extremal risk comprises six chapters and starts by introducing copulas, distributional transform and Sklar’s theorem. Then various copula models and constructions are described, many copula examples are given. The author next presents multivariate distributional and quantile transforms and their applications to pair copula construction of copula models, stochastic ordering optimal couplings, identification and goodness of fit tests, empirical copula processes and dependence functions. The first chapter concludes with a discussion about multivariate and overlapping marginals and corresponding copulas.

Chapter 2 discusses Fréchet classes of multivariate distributions generated by given sets of marginals, risk bounds and connections to duality theory. Starting from the classical Fréchet-Hoeffding bounds describing the influence of dependence structure of the risk vector \(X=(X_1,\dots,X_d)\) on the expectation \(\mathbb{E}f(X)\) for a given function \(f\), the author presents several duality theorems for the generalized mass-transportation problem and the problem of generalized Fréchet bounds. Applications of these theorems to bounds on risk functionals, e.g., excess of loss, value at risk, and maximal risk, are given next. Transferring Fréchet bounds to random variables leads to the notion of comonotonicity. Several results at the end of Chapter 2 illustrate the sharpness of Fréchet bounds.

Chapter 3 is devoted to various notions of order (convex, Schur, integral, etc.) and their relations to the excess of loss and comonotonicity. Two approaches to show that comonotonicity is the worst case dependence structure with respect to the convex order of the joint portfolio \(\sum_{i=1}^dX_i\), or equivalently, of the excess loss, are given.

Chapter 4 deals with an important problem of financial and insurance risk management, that is, determining sharp bounds for the distribution function of the portfolio \(\sum_{i=1}^dX_i\) of the risk vector \(X=(X_1,\dots,X_d)\) where the marginal distribution functions \(F_i\sim X_i\) are known, but the dependence between the components \(X_i\) is not specified. The so-called standard bounds as well as dual bounds are presented and their sharpness is discussed.

The general bounds of Chapter 4 are shown to be improvable in Chapter 5 if one restricts to subclasses of the Fréchet class of all dependence structures with given marginals.

Chapter 6 discusses dependence orderings of risk vectors and portfolios.

Part II of the book deals with risk measures and worst case portfolios. The material here is typically found scattered throughout many papers so it is indeed nice to have it collected in a book. Starting with real (one-dimensional) risks in Chapter 7, the author introduces various axioms that a good risk measure should satisfy, leading to the notions and examples of various classes of risk measures (monetary, convex, coherent, law invariant, etc.). The setting is rather general in terms of probability spaces and classes of risks considered. So a good deal of functional analysis is necessary to fully appreciate the results. Robust representation theorems for convex and coherent risk measures on \(L^\infty\) and on \(L^p\) are discussed from the view point of convex analysis and the Fenchel-Moreau theorem. A collection of continuity results for various risk measures encountered in practice concludes this chapter.

Chapter 8 is devoted to risk measures for portfolio vectors. Many one-dimensional risk measures are naturally extended to the multivariate setting by exploiting various notions of order described earlier in the book. Representation theorems and results on continuity properties of the described multivariate risk measures conclude this chapter.

Chapter 9 discusses law invariant convex risk measures on \(L_d^p\) and connections to optimal mass transportation. It also explains why there is no general notion of multivariate comonotonicity what would describe the worst case dependence structure in such a setting. The worst case dependence structure crucially depends on the convex law invariant risk measure taken. And this is the topic for the end of the chapter.

Part III deals with optimal risk allocation or risk sharing problem which has a long history in mathematical economics and insurance. In Chapter 10, the author considers the unrestricted optimal risk allocation problem of distributing a risk \(X\) into components \(X_i\in L^p\) (\(X=\sum_{i=1}^nX_i\)) so that the sum \(\sum_{i=1}^n\rho_i(X_i)\) is minimized, where \(\rho_i\) are possibly different risk measures of the \(n\) traders. Solutions are shown to be not unique, given only under an equilibrium condition by the set of Pareto optimal allocations. The necessary Pareto equilibrium condition is later characterized for several classes of risk measures. Then this equilibrium condition is dropped by considering several restrictions of the class of all possible allocations. Chapter 11 discusses some generalizations of the classical characterization results of optimal allocations due to Borch and others; it is concluded with extensions to the risk allocation problem for vectors \(X\in\mathbb{R}^d\). Chapter 12 applies the dependence bounds from Part I to the construction of optimal contingent claims, portfolios, and (re-)insurance contracts.

Part IV considers extensions of the famous Markowitz mean-variance theory to the diversification problem for portfolios with heavy-tailed components where variances and/or means are infinite and so mean–variance or convex risk measures are not applicable. The problem is discussed in Chapter 13, in the framework of the extreme value theory, and the portfolios are compared with respect to sensitivity (measured by the extreme risk index) to extremal risk events. Empirical estimators of the optimal portfolio and the above-mentioned extreme risk index are introduced and their consistency as well as asymptotic normality is established. Later, in Chapter 14, the author compares different stochastic models with respect to the asymptotic portfolio losses. This is quantified by introducing the corresponding asymptotic portfolio loss order which is later illustrated in examples. Connections to other stochastic orders are also elaborated.

Overall, the book will be definitely interesting to researchers and graduate students in the areas of insurance, financial mathematics, risk management, etc., as it gives a clear picture which research directions have been pursued and to what extent. Bright undergraduates will find it challenging to study from as there are no problems to solve, a lot of proofs are omitted for ease of reading, thus requiring other sources. Also one cannot expect undergraduates to be well familiar with advanced functional analysis, empirical processes, convex optimization. On the other hand, the book might provide a good reason to study these rather abstract theories.

To be able to accomplish these tasks, one has to master a good deal of probability theory, functional analysis, optimization theory as well as a fair share of financial and insurance mathematics. This book is definitely a good, though perhaps not the easiest possible guide towards this goal. The book contains four parts: stochastic dependence and extremal risk, risk measures and worst case portfolios, optimal risk allocation, and optimal portfolios and extreme risk.

Part I on stochastic dependence and extremal risk comprises six chapters and starts by introducing copulas, distributional transform and Sklar’s theorem. Then various copula models and constructions are described, many copula examples are given. The author next presents multivariate distributional and quantile transforms and their applications to pair copula construction of copula models, stochastic ordering optimal couplings, identification and goodness of fit tests, empirical copula processes and dependence functions. The first chapter concludes with a discussion about multivariate and overlapping marginals and corresponding copulas.

Chapter 2 discusses Fréchet classes of multivariate distributions generated by given sets of marginals, risk bounds and connections to duality theory. Starting from the classical Fréchet-Hoeffding bounds describing the influence of dependence structure of the risk vector \(X=(X_1,\dots,X_d)\) on the expectation \(\mathbb{E}f(X)\) for a given function \(f\), the author presents several duality theorems for the generalized mass-transportation problem and the problem of generalized Fréchet bounds. Applications of these theorems to bounds on risk functionals, e.g., excess of loss, value at risk, and maximal risk, are given next. Transferring Fréchet bounds to random variables leads to the notion of comonotonicity. Several results at the end of Chapter 2 illustrate the sharpness of Fréchet bounds.

Chapter 3 is devoted to various notions of order (convex, Schur, integral, etc.) and their relations to the excess of loss and comonotonicity. Two approaches to show that comonotonicity is the worst case dependence structure with respect to the convex order of the joint portfolio \(\sum_{i=1}^dX_i\), or equivalently, of the excess loss, are given.

Chapter 4 deals with an important problem of financial and insurance risk management, that is, determining sharp bounds for the distribution function of the portfolio \(\sum_{i=1}^dX_i\) of the risk vector \(X=(X_1,\dots,X_d)\) where the marginal distribution functions \(F_i\sim X_i\) are known, but the dependence between the components \(X_i\) is not specified. The so-called standard bounds as well as dual bounds are presented and their sharpness is discussed.

The general bounds of Chapter 4 are shown to be improvable in Chapter 5 if one restricts to subclasses of the Fréchet class of all dependence structures with given marginals.

Chapter 6 discusses dependence orderings of risk vectors and portfolios.

Part II of the book deals with risk measures and worst case portfolios. The material here is typically found scattered throughout many papers so it is indeed nice to have it collected in a book. Starting with real (one-dimensional) risks in Chapter 7, the author introduces various axioms that a good risk measure should satisfy, leading to the notions and examples of various classes of risk measures (monetary, convex, coherent, law invariant, etc.). The setting is rather general in terms of probability spaces and classes of risks considered. So a good deal of functional analysis is necessary to fully appreciate the results. Robust representation theorems for convex and coherent risk measures on \(L^\infty\) and on \(L^p\) are discussed from the view point of convex analysis and the Fenchel-Moreau theorem. A collection of continuity results for various risk measures encountered in practice concludes this chapter.

Chapter 8 is devoted to risk measures for portfolio vectors. Many one-dimensional risk measures are naturally extended to the multivariate setting by exploiting various notions of order described earlier in the book. Representation theorems and results on continuity properties of the described multivariate risk measures conclude this chapter.

Chapter 9 discusses law invariant convex risk measures on \(L_d^p\) and connections to optimal mass transportation. It also explains why there is no general notion of multivariate comonotonicity what would describe the worst case dependence structure in such a setting. The worst case dependence structure crucially depends on the convex law invariant risk measure taken. And this is the topic for the end of the chapter.

Part III deals with optimal risk allocation or risk sharing problem which has a long history in mathematical economics and insurance. In Chapter 10, the author considers the unrestricted optimal risk allocation problem of distributing a risk \(X\) into components \(X_i\in L^p\) (\(X=\sum_{i=1}^nX_i\)) so that the sum \(\sum_{i=1}^n\rho_i(X_i)\) is minimized, where \(\rho_i\) are possibly different risk measures of the \(n\) traders. Solutions are shown to be not unique, given only under an equilibrium condition by the set of Pareto optimal allocations. The necessary Pareto equilibrium condition is later characterized for several classes of risk measures. Then this equilibrium condition is dropped by considering several restrictions of the class of all possible allocations. Chapter 11 discusses some generalizations of the classical characterization results of optimal allocations due to Borch and others; it is concluded with extensions to the risk allocation problem for vectors \(X\in\mathbb{R}^d\). Chapter 12 applies the dependence bounds from Part I to the construction of optimal contingent claims, portfolios, and (re-)insurance contracts.

Part IV considers extensions of the famous Markowitz mean-variance theory to the diversification problem for portfolios with heavy-tailed components where variances and/or means are infinite and so mean–variance or convex risk measures are not applicable. The problem is discussed in Chapter 13, in the framework of the extreme value theory, and the portfolios are compared with respect to sensitivity (measured by the extreme risk index) to extremal risk events. Empirical estimators of the optimal portfolio and the above-mentioned extreme risk index are introduced and their consistency as well as asymptotic normality is established. Later, in Chapter 14, the author compares different stochastic models with respect to the asymptotic portfolio losses. This is quantified by introducing the corresponding asymptotic portfolio loss order which is later illustrated in examples. Connections to other stochastic orders are also elaborated.

Overall, the book will be definitely interesting to researchers and graduate students in the areas of insurance, financial mathematics, risk management, etc., as it gives a clear picture which research directions have been pursued and to what extent. Bright undergraduates will find it challenging to study from as there are no problems to solve, a lot of proofs are omitted for ease of reading, thus requiring other sources. Also one cannot expect undergraduates to be well familiar with advanced functional analysis, empirical processes, convex optimization. On the other hand, the book might provide a good reason to study these rather abstract theories.

Reviewer: Jonas Šiaulys (Vilnius)

### MSC:

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

91-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to game theory, economics, and finance |

91B30 | Risk theory, insurance (MSC2010) |

91G40 | Credit risk |