1,721,028 research outputs found
How Superadditive Can a Risk Measure Be?
No full textIn this paper, we study the extent to which any risk measure can lead to superadditive risk assessments, implying the potential for penalizing portfolio diversification. For this purpose we introduce
the notion of extreme-aggregation risk measures. The extreme-aggregation measure characterizes the most superadditive behavior of a risk measure by yielding the worst-possible diversification ratio across dependence structures. One of the main contributions is demonstrating that, for a wide range of risk measures, the extreme-aggregation measure corresponds to the smallest dominating coherent risk measure. In our main result, it is shown that the extreme-aggregation measure induced by a distortion risk measure is a coherent distortion risk measure. In the case of convex risk measures, a general robust representation of coherent extreme-aggregation measures is provided. In particular, the extreme-aggregation measure induced by a convex shortfall risk measure is a coherent expectile. These results show that, in the presence of dependence uncertainty, quantification of a coherent risk measure is often necessary, an observation that lends further support to the use of coherent risk measures in portfolio risk management
Star-shaped risk measures
No full textIn this paper, monetary risk measures that are positively superhomogeneous, called star-shaped risk measures, are characterized and their properties are studied. The measures in this class, which arise when the subadditivity property of coherent risk measures is dispensed with and positive homogeneity is weakened, include all practically used risk measures, in particular, both convex risk measures and value-at-risk. From a financial viewpoint, our relaxation of convexity is necessary to quantify the capital requirements for risk exposure in the presence of liquidity risk, competitive delegation, or robust aggregation mechanisms. From a decision theoretical perspective, star-shaped risk measures emerge from variational preferences when risk mitigation strategies can be adopted by a rational decision maker
Diversification limit of quantiles under dependence uncertainty
No full textIn this paper, we investigate the asymptotic behavior of the portfolion diversification ratio based on Value-at-Risk (quantile) under dependence uncertainty, which we refer to as “worst-case diversification limit”. We show that the worst-case diversification limit is equal to the upper limit of the worst-case diversification ratio under mild conditions on the portfolio marginal distributions. In the case of regularly
varying margins, we provide explicit values for the worst-case diversification limit. Under the framework of dependence uncertainty the worst-case diversification limit is significantly higher compared to classic results obtained in the literature of multivariate regularly varying distributions. The results carried out in this paper bring together extreme value theory and dependence uncertainty, two popular topic in the recent study of risk aggregation
A theory of multivariate stress testing
No full textWe present a theoretical framework for stressing multivariate stochastic models. We consider a stress to be a change of measure, placing a higher weight on multivariate scenarios of interest. In particular, a stressing mechanism is a mapping from random vectors to Radon-Nikodym densities. We postulate desirable properties for stressing mechanisms addressing alternative objectives. Consistently with our focus on dependence, we require throughout invariance to monotonic transformations of risk factors. We study in detail the properties of two families of stressing mechanisms, based respectively on mixtures of univariate stresses and on transformations of statistics we call Spearman and Kendall’s cores. Furthermore, we characterize the aggregation properties of those stressing mechanisms, which motivate their use in deriving new capital allocation methods, with properties different to those typically found in the literature. The proposed methods are applied to stress testing and capital allocation, using the simulation model of a UK-based non-life insurer
Adjusted Expected Shortfall
Full text linkWe introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position X to ensure that Expected Shortfall ESp (X) does not exceed a pre-specified threshold g(p) for every probability level p is an element of [0, 1]. Through the choice of the benchmark risk profile gone can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance
Elicitable distortion risk measures: A concise proof
Full text linkElicitability has recently been discussed as a desirable property for risk measures. Kou and Peng (2014) showed that an elicitable distortion risk measure is either a Value-at-Risk or the mean. We give a concise alternative proof of this result, and discuss the conflict between comonotonic additivity and elicitability
Characterizing, optimizing and backtesting metrics of risk
Full text linkMeasures 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
Centers of probability measures without the mean
Full text linkIn the recent years, the notion of mixability has been developed with applications to operations research, optimal transportation, and quantitative finance. An n-tuple of distributions is said to be jointly mixable if there exist n random variables following these distributions and adding up to a constant, called center, with probability one. When the n distributions are identical, we speak of complete mixability. If each distribution has finite mean, the center is obviously the sum of the means. In this paper, we investigate the set of centers of completely and jointly mixable distributions not having a finite mean. In addition to several results, we show the (possibly counterintuitive) fact that, for each (Formula presented.), there exist n standard Cauchy random variables adding up to a constant C if and only if (Formula presented.
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