Capital Allocation \ue0 La Aumann-Shapley for Non Differentiable Risk Measures

We study capital allocation rules satisfying suitable properties for convex and quasi-convex risk measures, by focusing in particular on a family of capital allocation rules based on the dual representation for risk measures and inspired by the Aumann\u2013Shapley allocation principle. These rules extend some well known methods of capital allocation for coherent and convex risk measures to the case of non-Gateauxdifferentiable risk measures. We also analyze the properties of the allocation principles here introduced and discuss their suitability in the quasi-convex context

Law-Invariant Functionals that Collapse to the Mean: Beyond Convexity

We establish general "collapse to the mean" principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the "collapse to the mean" for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the "collapse to the mean" to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis

Characterizing, optimizing and backtesting metrics of risk

Measures of risk and riskmetrics were proposed to quantify the risks people are faced with in financial, statistical, and economic practice. They are widely discussed and studied by literature in the context of financial regulation, insurance, operations research, and statistics. Several major research topics on riskmetrics remain to be important in both academic study and industrial practice. First, characterization, especially axiomatic characterization of riskmetrics, lays essential theoretical foundation of specific classes of riskmetrics about why they are widely adopted in practice and research. It usually involves challenging mathematical approaches and deep practical insights. Second, riskmetrics are used by researchers in optimization as the objective functionals of decision makers. This links riskmetrics to the literature of operations research and decision theory, and leads to wide applications of riskmetrics to portfolio management, robust optimization, and insurance design. Third, relevant statistical models of estimation and hypothesis tests for riskmetrics need to be established to serve for practical risk management and financial regulation. In particular, risk forecasts and backtests of different riskmetrics are always the main concern and challenge for risk managers and financial regulators. In this thesis, we investigate several important questions in characterization, optimization, and backtest for measures of risk with different focuses on establishing theoretical framework and solving practical problems. To offer a comprehensive theoretical toolkit for future study, in Chapter 2, we propose the class of distortion riskmetrics defined through signed Choquet integrals. Distortion riskmetrics include many classic risk measures, deviation measures, and other functionals in the literature of finance and actuarial science. We obtain characterization, finiteness, convexity, and continuity results on general model spaces, extending various results in the existing literature on distortion risk measures and signed Choquet integrals. To explore deeper applications of distortion riskmetrics in optimization problems, in Chapter 3, we study optimization of distortion riskmetrics with distributional uncertainty. One of our central findings is a unifying result that allows us to convert an optimization of a non-convex distortion riskmetric with distributional uncertainty to a convex one, leading to practical tractability. A sufficient condition to the unifying equivalence result is the novel notion of closedness under concentration, a variation of which is also shown to be necessary for the equivalence. Our results include many special cases that are well studied in the optimization literature, including but not limited to optimizing probabilities, Value-at-Risk, Expected Shortfall, Yaari's dual utility, and differences between distortion risk measures, under various forms of distributional uncertainty. We illustrate our theoretical results via applications to portfolio optimization, optimization under moment constraints, and preference robust optimization. In Chapter 4, we study characterization of measures of risk in the context of statistical elicitation. Motivated by recent advances on elicitability of risk measures and practical considerations of risk optimization, we introduce the notions of Bayes pairs and Bayes risk measures. Bayes risk measures are the counterpart of elicitable risk measures, extensively studied in the recent literature. The Expected Shortfall (ES) is the most important coherent risk measure in both industry practice and academic research in finance, insurance, risk management, and engineering. One of our central results is that under a continuity condition, ES is the only class of coherent Bayes risk measures. We further show that entropic risk measures are the only risk measures which are both elicitable and Bayes. Several other theoretical properties and open questions on Bayes risk measures are discussed. In Chapter 5, we further study characterization of measures of risk in insurance design. We study the characterization of risk measures induced by efficient insurance contracts, i.e., those that are Pareto optimal for the insured and the insurer. One of our major results is that we characterize a mixture of the mean and ES as the risk measure of the insured and the insurer, when contracts with deductibles are efficient. Characterization results of other risk measures, including the mean and distortion risk measures, are also presented by linking them to different sets of contracts. In Chapter 6, we focus on a larger class of riskmetrics, cash-subadditive risk measures. We study cash-subadditive risk measures without quasi-convexity. One of our major results is that a general cash-subadditive risk measure can be represented as the lower envelope of a family of quasi-convex and cash-subadditive risk measures. Representation results of cash-subadditive risk measures with some additional properties are also examined. The notion of quasi-star-shapedness, which is a natural analogue of star-shapedness, is introduced and we obtain a corresponding representation result. In Chapter 7, we discuss backtesting riskmetrics. One of the most challenging tasks in risk modeling practice is to backtest ES forecasts provided by financial institutions. To design a model-free backtesting procedure for ES, we make use of the recently developed techniques of e-values and e-processes. Model-free e-statistics are introduced to formulate e-processes for risk measure forecasts, and unique forms of model-free e-statistics for VaR and ES are characterized using recent results on identification functions. For a given model-free e-statistic, optimal ways of constructing the e-processes are studied. The proposed method can be naturally applied to many other risk measures and statistical quantities. We conduct extensive simulation studies and data analysis to illustrate the advantages of the model-free backtesting method, and compare it with the ones in the literature

Portfolio optimization with quasiconvex risk measures

In this paper, we focus on the portfolio optimization problem associated with a quasiconvex risk measure (satisfying some additional assumptions). For coherent/convex risk measures, the portfolio optimization problem has been already studied in the literature. Following the approach of Ruszczyński and Shapiro [Ruszczyński A, Shapiro A (2006) Optimization of convex risk functions. Math. Oper. Res. 31(3):433–452.], but by means of quasiconvex analysis and notions of subdifferentiability, we characterize optimal solutions of the portfolio problem associated with quasiconvex risk measures. The shape of the efficient frontier in the mean-risk space and some particular cases are also investigated. </jats:p