SSRN Author: Ruodu WangRuodu Wang SSRN Content
https://privwww.ssrn.com/author=2190264
https://privwww.ssrn.com/rss/en-usSat, 01 Aug 2020 01:00:38 GMTeditor@ssrn.com (Editor)Sat, 01 Aug 2020 01:00:38 GMTwebmaster@ssrn.com (WebMaster)SSRN RSS Generator 1.0REVISION: Risk Functionals With Convex Level SetsWe analyze the ``convex level sets" (CxLS) property of risk functionals, which is a necessary condition for the notions of elicitability, identifiability and backtestability, popular in the recent statistics and risk management literature. We put the CxLS property in the multi-dimensional setting, with a special focus on signed Choquet integrals, a class of risk functionals that are generally not monotone or convex. We obtain two main analytical results in dimension one and dimension two, by characterizing the CxLS property of all one-dimensional signed Choquet integrals, and that of all two-dimensional signed Choquet integrals with a quantile component. Using these results, we proceed to show that under some continuity assumption, a comonotonic-additive coherent risk measure is co-elicitable with Value-at-Risk if and only if it is the corresponding Expected Shortfall. The new findings generalize several results in the recent literature, and partially answer an open question on the ...
https://privwww.ssrn.com/abstract=3292661
https://privwww.ssrn.com/1926962.htmlFri, 31 Jul 2020 08:55:45 GMTREVISION: A Theory for Measures of Tail RiskThe notion of "tail risk" has been a crucial consideration in modern risk management. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures which quantify the tail risk, that is, the behavior of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, the Value-at-Risk (VaR) and the Expected Shortfall (ES), are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk ...
https://privwww.ssrn.com/abstract=2841909
https://privwww.ssrn.com/1926953.htmlFri, 31 Jul 2020 08:52:47 GMTREVISION: Trade-off between validity and efficiency of merging p-values under arbitrary dependenceVarious methods of combining individual p-values into one p-value are widely used in many areas of statistical applications. We say that a combining method is valid for arbitrary dependence (VAD) if it does not require any assumption on the dependence structure of the p-values, whereas it is valid for some dependence (VSD) if it requires some specific, perhaps realistic but unjustifiable, dependence structures. The trade-off between validity and efficiency of these methods is studied via analyzing the choices of critical values under different dependence assumptions. We introduce the notions of independence-comonotonicity balance (IC-balance)<br>and the price for validity. In particular, IC-balanced methods always produce an identical critical value for independent and perfectly positively dependent p-values, thus showing insensitivity to dependence assumptions. We show that, among two very general classes of merging methods commonly used in practice, the Cauchy combination ...
https://privwww.ssrn.com/abstract=3569329
https://privwww.ssrn.com/1924238.htmlThu, 23 Jul 2020 08:26:47 GMTREVISION: Characterizing Optimal Allocations in Quantile-Based Risk SharingUnlike classic risk sharing problems based on expected utilities or convex risk measures, quantile-based risk sharing games exhibit two special features. First, quantile-based risk measures (such as the Value-at-Risk) are often not convex, and second, they ignore some part of the distribution of the risk. These features create technical challenges in establishing a full characterization of optimal allocations, a question left unanswered in the literature. In this paper, we fully address the issues on the existence and the characterization of optimal allocations in quantile-based risk sharing games. It turns out that negative dependence plays an important role in the optimal allocations, in contrast to positive dependence appearing in classic risk sharing problems. As a by-product of our main finding, we obtain some results on the optimization of the Value-at-Risk and the Expected Shortfall.
https://privwww.ssrn.com/abstract=3173503
https://privwww.ssrn.com/1903997.htmlWed, 03 Jun 2020 08:24:40 GMTREVISION: Distortion Riskmetrics on General SpacesThe class of distortion riskmetrics is defined through signed Choquet integrals, and it includes 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. This paper offers a comprehensive toolkit of theoretical results on distortion riskmetrics which are ready for use in applications.
https://privwww.ssrn.com/abstract=3510363
https://privwww.ssrn.com/1901017.htmlTue, 26 May 2020 10:21:43 GMTREVISION: PELVE: Probability Equivalent Level of VaR and ESIn the recent Fundamental Review of the Trading Book (FRTB), the Basel Committee on Banking Supervision proposed the shift from the 99% Value-at-Risk (VaR) to the 97.5% Expected Shortfall (ES) for internal models in market risk assessment. Inspired by the above transition, we introduce a new distributional index, the probability equivalence level of VaR and ES (PELVE), which identifies the balancing point for the equivalence between VaR and ES. PELVE enjoys many desirable theoretical properties and it distinguishes empirically heavy-tailed distributions from light-tailed ones via a threshold of 2.72. Convergence properties and asymptotic normality of the empirical PELVE estimators are established. Applying PELVE to financial asset and portfolio data leads to interesting observations that are not captured by VaR or ES alone. We find that, in general, the transition from VaR to ES in the FRTB yields an increase in risk capital for single-asset portfolios, but for well-diversified ...
https://privwww.ssrn.com/abstract=3489566
https://privwww.ssrn.com/1901007.htmlTue, 26 May 2020 10:19:00 GMTREVISION: Combining e-values and p-valuesMultiple testing of a single hypothesis and testing multiple hypotheses are usually done in terms of p-values. In this paper we replace p-values with their natural competitor, e-values, which are closely related to betting, Bayes factors, and likelihood ratios. We demonstrate that e-values are often mathematically more tractable; in particular, in multiple testing of a single hypothesis, e-values can be merged simply by averaging them. This allows us to develop ecient procedures using e-values for testing multiple hypotheses.
https://privwww.ssrn.com/abstract=3504009
https://privwww.ssrn.com/1900999.htmlTue, 26 May 2020 10:17:12 GMTREVISION: Trade-Off between Anytime- and Sometime-Valid Methods for Merging P-ValuesVarious methods of combining individual p-values into one p-value are widely used in many areas of statistical applications. We say that a combining method is anytime-valid if it does not require any assumption on the dependence structure of the p-values, whereas it is sometime-valid if it requires some specific, perhaps realistic but unjustifiable, dependence structures. The trade-off between validity and efficiency of these methods is studied via analyzing the choices of critical values under different dependence assumptions. We introduce the notions of independence-comonotonicity balance (IC-balance) and the price for validity. In particular, IC-balanced methods always produce an identical critical value for independent and perfectly positively dependent p-values, thus showing insensitivity to dependence assumptions. We show that, among two very general classes of merging methods commonly used in practice, the Cauchy combination method and the Simes method are the only ...
https://privwww.ssrn.com/abstract=3569329
https://privwww.ssrn.com/1892248.htmlFri, 01 May 2020 17:45:32 GMTREVISION: Risk Functionals With Convex Level SetsWe analyze the ``convex level sets" (CxLS) property of risk functionals, which is a necessary condition for the notions of elicitability, identifiability and backtestability, popular in the recent statistics and risk management literature. We put the CxLS property in the multi-dimensional setting, with a special focus on signed Choquet integrals, a class of risk functionals that are generally not monotone or convex. We obtain two main analytical results in dimension one and dimension two, by characterizing the CxLS property of all one-dimensional signed Choquet integrals, and that of all two-dimensional signed Choquet integrals with a quantile component. Using these results, we proceed to show that under some continuity assumption, a comonotonic-additive coherent risk measure is co-elicitable with Value-at-Risk if and only if it is the corresponding Expected Shortfall. The new findings generalize several results in the recent literature, and partially answer an open question on the ...
https://privwww.ssrn.com/abstract=3292661
https://privwww.ssrn.com/1890622.htmlTue, 28 Apr 2020 08:27:40 GMTREVISION: Distortion Riskmetrics on General SpacesThe class of distortion riskmetrics is defined through signed Choquet integrals, and it includes 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. This paper offers a comprehensive toolkit of theoretical results on distortion riskmetrics which are ready for use in applications.
https://privwww.ssrn.com/abstract=3510363
https://privwww.ssrn.com/1884840.htmlMon, 13 Apr 2020 08:18:07 GMTREVISION: A Model-Free Continuum of Degrees of Risk AversionWe establish a theory for a continuum of degrees of risk aversion and risk seeking, referred to as fractional risk aversion and risk seeking. The proposed degrees are well defined for any distribution-based monotone preference on any set of prospects; no particular model assumption is required on the preference. The two degrees combined together intuitively reflect a decision-maker's attitude towards risk. We characterize fractional risk aversion and risk seeking in the models of rank-dependent utility (RDU) and cumulative prospect theory (CPT) for four different classes of sets of prospects. For instance, the degree of risk aversion for an RDU preference is equal to the ratio between its index of pessimism and its index of greediness; both indices depend on the set of prospects at consideration.
https://privwww.ssrn.com/abstract=2907499
https://privwww.ssrn.com/1881298.htmlWed, 01 Apr 2020 16:43:36 GMTREVISION: Risk Functionals With Convex Level SetsWe analyze the ``convex level sets" (CxLS) property of risk functionals, which is a necessary condition for the notions of elicitability, identifiability and backtestability, popular in the recent statistics and risk management literature. We put the CxLS property in the multi-dimensional setting, with a special focus on signed Choquet integrals, a class of risk functionals that are generally not monotone or convex. We obtain two main analytical results in dimension one and dimension two, by characterizing the CxLS property of all one-dimensional signed Choquet integrals, and that of all two-dimensional signed Choquet integrals with a quantile component. Using these results, we proceed to show that under some continuity assumption, a comonotonic-additive coherent risk measure is co-elicitable with Value-at-Risk if and only if it is the corresponding Expected Shortfall. The new findings generalize several results in the recent literature, and partially answer an open question on the ...
https://privwww.ssrn.com/abstract=3292661
https://privwww.ssrn.com/1878311.htmlTue, 24 Mar 2020 07:40:47 GMTREVISION: Distributional Transforms, Probability Distortions, and Their ApplicationsIn this paper we provide a general mathematical framework for distributional transforms, which allows for many examples that are used extensively in the literature of finance, economics and optimization. We put a special focus on the class of probability distortions, which is a fundamental tool in decision theory. As our main results, we characterize distributional transforms satisfying various properties and this includes an equivalent set of conditions which forces a distributional transform to be a probability distortion. As the first application, we construct new risk measures using distributional transforms. Sufficient and necessary conditions are given to ensure the convexity or coherence of the generated risk measures. In the second application, we introduce a new method for sensitivity analysis of risk measures based on composition groups of probability distortions. Finally, we construct probability distortions describing change of measures with an example in ...
https://privwww.ssrn.com/abstract=3419388
https://privwww.ssrn.com/1877946.htmlMon, 23 Mar 2020 09:56:06 GMTREVISION: Distortion Riskmetrics on General SpacesThe class of distortion riskmetrics is defined through signed Choquet integrals, and it includes 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. This paper offers a comprehensive toolkit of theoretical results on distortion riskmetrics which are ready for use in applications.
https://privwww.ssrn.com/abstract=3510363
https://privwww.ssrn.com/1877246.htmlThu, 19 Mar 2020 09:38:14 GMTREVISION: An Axiomatic Foundation for the Expected ShortfallIn the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is - in addition to many other nice properties - a coherent risk measure, it does not yet have an axiomatic foundation. In this paper we put forward four intuitive economic axioms for portfolio risk assessment - which are monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail. As a most important feature, ES rewards portfolio diversification and penalizes risk concentration in a ...
https://privwww.ssrn.com/abstract=3423042
https://privwww.ssrn.com/1876380.htmlTue, 17 Mar 2020 07:58:38 GMTREVISION: An Axiomatic Foundation for the Expected ShortfallIn the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is - in addition to many other nice properties - a coherent risk measure, it does not yet have an axiomatic foundation. In this paper we put forward four intuitive economic axioms for portfolio risk assessment - which are monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail. As a most important feature, ES awards portfolio diversification and penalizes risk concentration in a ...
https://privwww.ssrn.com/abstract=3423042
https://privwww.ssrn.com/1875751.htmlSat, 14 Mar 2020 15:53:49 GMTREVISION: An Axiomatic Foundation for the Expected ShortfallIn the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is - in addition to many other nice properties - a coherent risk measure, it does not yet have an axiomatic foundation. In this paper we put forward four intuitive economic axioms for portfolio risk assessment - which are monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail. As a most important feature, ES awards portfolio diversification and penalizes risk concentration in a ...
https://privwww.ssrn.com/abstract=3423042
https://privwww.ssrn.com/1869135.htmlSun, 23 Feb 2020 17:58:15 GMTREVISION: An Axiomatic Foundation for the Expected ShortfallIn the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is - in addition to many other nice properties - a coherent risk measure, it does not yet have an axiomatic foundation. In this paper we put forward four intuitive economic axioms for portfolio risk assessment - which are monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail.
https://privwww.ssrn.com/abstract=3423042
https://privwww.ssrn.com/1865825.htmlTue, 11 Feb 2020 13:43:15 GMTREVISION: Self-Consistency, Subjective Pricing, and a Theory of Credit RatingWe propose a theory for rating financial securities based on a concept of self-consistency, which does not allow issuers to gain, by tranching financial securities, from investors who rely on the rating criterion. While the expected loss criterion used by Moody's satisfies self-consistency, the probability of default criterion used by S&P does not. We find empirical evidences in the post-Dodd-Frank period (i.e., after July 2010) that the issuers may take advantage of the absence of self-consistency. We further propose a concept of scenario-relevance which reflects practical evaluation procedures of potential losses from defaultable securities. Our main theoretical results show that a self-consistent rating measure admits a Choquet integral representation, and this representation is also analytically tractable if one further takes economic scenarios into account. We suggest new examples of self-consistent and scenario-based rating criteria, such as ones based on the VaR and ...
https://privwww.ssrn.com/abstract=3504065
https://privwww.ssrn.com/1860515.htmlSun, 26 Jan 2020 10:12:21 GMTREVISION: Is the Inf-convolution of Law-invariant Preferences Law-invariant?We analyze the question of whether the inf-convolution of law-invariant risk functionals (preferences) is still law-invariant. In economic terms, this question means if all agents in a risk sharing system only care about the distributions of risks, whether the resulting (after risk redistribution) representative agent also only cares about the distribution of the total risk, regardless of how the total risk is defined as a random variable. We first illustrate with some examples that such an assertion is generally false. Although the answer to the above question seems to be affirmative for many examples of commonly used risk functionals in the literature, the situation becomes delicate without assuming specific forms and properties of the individual functionals. We illustrate with examples the surprising fact that the answer to the main question is generally negative, even in an atomless probability space. Furthermore, we establish a few very weak conditions under which the answer ...
https://privwww.ssrn.com/abstract=3371642
https://privwww.ssrn.com/1858977.htmlFri, 17 Jan 2020 16:50:52 GMTREVISION: Self-Consistency, Subjective Pricing, and a Theory of Credit RatingWe propose a theory for rating financial securities based on a concept of self-consistency, which does not allow issuers to gain, by tranching financial securities, from investors who rely on the rating criterion. While the expected loss criterion used by Moody's satisfies self-consistency, the probability of default criterion used by S&P does not. We find empirical evidences in the post-Dodd-Frank period (i.e., after July 2010) that the issuers may take advantage of the absence of self-consistency. We further propose a concept of scenario-relevance which reflects practical evaluation procedures of potential losses from defaultable securities. Our main theoretical results show that a self-consistent rating measure admits a Choquet integral representation, and this representation is also analytically tractable if one further takes economic scenarios into account. We suggest new examples of self-consistent and scenario-based rating criteria, such as ones based on the VaR and ...
https://privwww.ssrn.com/abstract=3504065
https://privwww.ssrn.com/1857535.htmlTue, 14 Jan 2020 09:46:30 GMTREVISION: An Axiomatic Approach to Credit RatingWe propose an axiomatic framework for rating financial securities based on an axiom of self-consistency, which does not allow issuers to gain, by tranching financial securities, from investors who rely on the rating criterion. While the expected loss criterion used by Moody's satisfies the self-consistency axiom, the probability of default criterion used by S\&P does not. We find empirical evidences in the post-Dodd-Frank period (i.e., after July 2010) that the issuers may take advantage of the absence of self-consistency. We further propose an axiom of scenario-relevance which reflects practical evaluation procedures of potential losses from defaultable securities. Our main theoretical results show that a self-consistent rating measure admits a Choquet integral representation, and this representation becomes analytically tractable if one further takes economic scenarios into account. We also suggest new examples of self-consistent and scenario-based rating criteria, such as ...
https://privwww.ssrn.com/abstract=3504065
https://privwww.ssrn.com/1854108.htmlWed, 01 Jan 2020 07:15:47 GMTREVISION: Combining e-values and p-valuesMultiple testing of a single hypothesis and testing multiple hypotheses are usually done in terms of p-values. In this paper we replace p-values with their Bayesian counterpart, e-values, which are, essentially, Bayes factors stripped of their Bayesian context. We demonstrate that e-values are often mathematically more tractable and develop procedures using e-values for multiple testing of a single hypothesis and testing multiple hypotheses.
https://privwww.ssrn.com/abstract=3504009
https://privwww.ssrn.com/1854101.htmlWed, 01 Jan 2020 06:59:46 GMTREVISION: Distortion Riskmetrics on General SpacesWe study distortion riskmetrics on general model spaces, which are 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. This paper offers a comprehensive toolkit of theoretical results on distortion riskmetrics which are ready for use in applications.
https://privwww.ssrn.com/abstract=3510363
https://privwww.ssrn.com/1853525.htmlMon, 30 Dec 2019 16:13:36 GMTREVISION: Risk Aversion in Regulatory Capital PrinciplesWe incorporate a notion of risk aversion favoring prudent decisions from financial institutions into regulatory capital calculation principles. In the context of Basel III, IV as well as Solvency II, regulatory capital calculation is carried out through the tools of monetary risk measures. The notion of risk aversion that we focus on has four equivalent formulations: through consistency with second-order stochastic dominance, or with conditional expectations, or with portfolio diversification, and finally through expected social impact. The class of monetary risk measures representing this notion of risk aversion is referred to as consistent risk measures. We characterize the class of consistent risk measures by establishing an Expected Shortfall-based representation, and as a by-product, we obtain new results on the representation of convex risk measures. We present several examples where consistent risk measures naturally appear. Using the obtained representation results, we study ...
https://privwww.ssrn.com/abstract=2658669
https://privwww.ssrn.com/1852842.htmlThu, 26 Dec 2019 16:03:14 GMTREVISION: PELVE: Probability Equivalent Level of VaR and ESAs part of the revised Fundamental Review of the Trading Book (FRTB), the Basel Committee on Banking Supervision proposed the shift from the 99% Value-at-Risk (VaR) to the 97.5% Expected Shortfall (ES) for internal models in market risk assessment. Inspired by the above transition, we introduce a new distributional index, the probability equivalence level of VaR and ES (PELVE), which identifies the balancing point for the equivalence between VaR and ES. As a measure of variability, PELVE enjoys many desirable properties such as location-scale invariance, monotonicity under convex transformations, and quasi-convexity/concavity for unhedged porfolios. Analyzing PELVE for commonly used models in risk management, we find that, in general, PELVE distinguishes heavy-tailed distributions from light-tailed ones via a threshold e=2.72. Convergence properties and asymptotic normality of the empirical PELVE estimators are established. Applying PELVE to financial asset and portfolio data ...
https://privwww.ssrn.com/abstract=3489566
https://privwww.ssrn.com/1849503.htmlThu, 12 Dec 2019 18:00:20 GMTNew: Inf-convolution and Optimal Allocations for Tail Risk MeasuresInspired by the recent developments in risk sharing problems for the Value-at-Risk (VaR), the Expected Shortfall (ES), or the Range-Value-at-Risk (RVaR), we study the optimization of risk sharing for general tail risk measures. Explicit formulas of the inf-convolution and Pareto optimal allocations are obtained in the case of a mixed collection of left and right VaRs, and in that of a VaR and another tail risk measure. The inf-convolution of tail risk measures is shown to be a tail risk measure with an aggregated tail parameter, a phenomenon very similar to the cases of VaR , ES and RVaR. The technical conclusions are quite general without assuming any form of convexity of the tail risk measures. Moreover, we find, via several results, that the roles of left and right VaRs are generally asymmetric in the optimization problems. Our analysis generalizes in several directions the recent work on quantile-based risk sharing.
https://privwww.ssrn.com/abstract=3490348
https://privwww.ssrn.com/1847184.htmlThu, 05 Dec 2019 15:18:28 GMTREVISION: An Axiomatic Foundation for the Expected ShortfallIn the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is - in addition to many other nice properties - a coherent risk measure, it does not yet have an axiomatic foundation. In this paper we put forward four intuitive economic axioms for portfolio risk assessment - which are monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail.
https://privwww.ssrn.com/abstract=3423042
https://privwww.ssrn.com/1846267.htmlTue, 03 Dec 2019 10:18:22 GMTREVISION: Is the Inf-convolution of Law-invariant Preferences Law-invariant?We analyze the question of whether the inf-convolution of law-invariant risk functionals (preferences) is still law-invariant. In economic terms, this question means if all agents in a risk sharing system only care about the distributions of risks, whether the resulting (after risk redistribution) representative agent also only cares about the distribution of the total risk, regardless of how the total risk is defined as a random variable. We first illustrate with some examples that such an assertion is generally false. Although the answer to the above question seems to be affirmative for many examples of commonly used risk functionals in the literature, the situation becomes delicate without assuming specific forms and properties of the individual functionals. We illustrate with examples the surprising fact that the answer to the main question is generally negative, even in an atomless probability space. Furthermore, we establish a few very weak conditions under which the answer ...
https://privwww.ssrn.com/abstract=3371642
https://privwww.ssrn.com/1840255.htmlSun, 10 Nov 2019 07:21:14 GMTREVISION: A Theory for Measures of Tail RiskThe notion of "tail risk" has been a crucial consideration in modern risk management. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures which quantify the tail risk, that is, the behavior of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, the Value-at-Risk (VaR) and the Expected Shortfall (ES), are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk ...
https://privwww.ssrn.com/abstract=2841909
https://privwww.ssrn.com/1839239.htmlWed, 06 Nov 2019 15:39:09 GMTREVISION: Combining P-Values Via AveragingThis paper proposes general methods for the problem of multiple testing of a single hypothesis, with a standard goal of combining a number of p-values without making any assumptions about their dependence structure. An old result by Ruschendorf and, independently, Meng implies that the p-values can be combined by scaling up their arithmetic mean by a factor of 2 (and no smaller factor is sufficient in general). Based on more recent developments in mathematical finance, specifically, robust risk aggregation techniques, we show that K p-values can be combined by scaling up their geometric mean by a factor of e (for all K) and by scaling up their harmonic mean by a factor of ln K (asymptotically as K -> infinity). These and other results lead to a generalized version of the Bonferroni-Holm method. A simulation study compares the performance of various averaging methods.
https://privwww.ssrn.com/abstract=3166304
https://privwww.ssrn.com/1837916.htmlFri, 01 Nov 2019 09:09:13 GMTREVISION: An Axiomatic Foundation for the Expected ShortfallIn the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is - in addition to many other nice properties - a coherent risk measure, it does not yet have an axiomatic foundation. In this paper we put forward four intuitive economic axioms for portfolio risk assessment - which are monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail.
https://privwww.ssrn.com/abstract=3423042
https://privwww.ssrn.com/1836493.htmlSun, 27 Oct 2019 09:11:23 GMTREVISION: Convex Risk Functionals: Representation and ApplicationsWe introduce the family of law-invariant convex risk functionals, which includes a wide majority of practically used convex risk measures and deviation measures. We obtain a unified representation theorem for this family of functionals. Two related optimization problems are studied. In the first application, we determine worst-case values of a law-invariant convex risk functional when the mean and a higher moment such as the variance of a risk are known. Second, we consider its application in optimal reinsurance design for an insurer. With the help of the representation theorem, we can show the existence and the form of optimal solutions.
https://privwww.ssrn.com/abstract=3216336
https://privwww.ssrn.com/1835106.htmlTue, 22 Oct 2019 09:36:24 GMTREVISION: Risk Functionals With Convex Level SetsWe analyze the “convex level sets” (CxLS) property of risk functionals, which is a necessary condition for the notions of elicitability, identifiability and backtestability, popular in the recent statistics and risk management literature. We put the CxLS property in the context of multi-dimensional risk functionals with a special focus on signed Choquet integrals, a class of risk functionals that are generally not monotone or convex. We obtain two main analytical results in dimension one and dimension two, by characterizing the CxLS property of all one-dimensional signed Choquet integrals, and that of all two-dimensional signed Choquet integrals with a quantile component. Using these results, we proceed to show that under some continuity assumption, a comonotonic-additive coherent risk measure is co-elicitable with a Value-at-Risk if and only if it is the corresponding Expected Shortfall. The new findings generalize several results in the recent literature, and partially answer an ...
https://privwww.ssrn.com/abstract=3292661
https://privwww.ssrn.com/1831121.htmlMon, 07 Oct 2019 15:16:41 GMTREVISION: Combining P-Values Via AveragingThis paper proposes general methods for the problem of multiple testing of a single hypothesis, with a standard goal of combining a number of p-values without making any assumptions about their dependence structure. An old result by Ruschendorf and, independently, Meng implies that the p-values can be combined by scaling up their arithmetic mean by a factor of 2 (and no smaller factor is sufficient in general). Based on more recent developments in mathematical finance, specifically, robust risk aggregation techniques, we show that K p-values can be combined by scaling up their geometric mean by a factor of e (for all K) and by scaling up their harmonic mean by a factor of ln K (asymptotically as K -> infinity). These and other results lead to a generalized version of the Bonferroni-Holm method. A simulation study compares the performance of various averaging methods.
https://privwww.ssrn.com/abstract=3166304
https://privwww.ssrn.com/1830704.htmlSun, 06 Oct 2019 15:00:51 GMTREVISION: Scenario-Based Risk EvaluationRisk measures such as Expected Shortfall (ES) and Value-at-Risk (VaR) have been prominent in banking regulation and financial risk management. Motivated by practical considerations in the assessment and management of risks, including tractability, scenario relevance and robustness, we consider theoretical properties of scenario-based risk evaluation. We propose several novel scenario-based risk measures, including various versions of Max-ES and Max-VaR, and study their properties. We establish axiomatic characterizations of scenario-based risk measures that are comonotonic-additive or coherent and an ES-based representation result is obtained. These results provide a theoretical foundation for the recent Basel III & IV market risk calculation formulas. We illustrate the theory with financial data examples.
https://privwww.ssrn.com/abstract=3235450
https://privwww.ssrn.com/1827639.htmlWed, 25 Sep 2019 12:02:43 GMTREVISION: Weak ComonotonicityThe classical notion of comonotonicity has played a pivotal role when solving diverse problems in economics, finance, and insurance. In various practical problems, however, this notion of extreme positive dependence structure is overly restrictive and sometimes unrealistic. In the present paper, we put forward a notion of weak comonotonicity, which contains the classical notion of comonotonicity as a special case, and gives rise to necessary and sufficient conditions for a number of optimization problems, such as those arising in portfolio diversification, risk aggregation, and premium calculation. In particular, we show that a combination of weak comonotonicity and weak antimonotonicity with respect to some choices of measures is sufficient for the maximization of Value-at-Risk aggregation, and weak comonotonicity is necessary and sufficient for the Expected Shortfall aggregation. Finally, with the help of weak comonotonicity acting as an intermediate notion of dependence between ...
https://privwww.ssrn.com/abstract=3300276
https://privwww.ssrn.com/1823710.htmlFri, 13 Sep 2019 12:00:09 GMTREVISION: A Theory for Measures of Tail RiskThe notion of "tail risk" has been a crucial consideration in modern risk management. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures which quantify the tail risk, that is, the behavior of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, the Value-at-Risk (VaR) and the Expected Shortfall (ES), are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk ...
https://privwww.ssrn.com/abstract=2841909
https://privwww.ssrn.com/1820328.htmlMon, 02 Sep 2019 12:13:28 GMTREVISION: An Axiomatic Foundation for the Expected ShortfallIn the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is - in addition to many other nice properties - a coherent risk measure, it does not yet have an axiomatic foundation. In this paper we put forward four intuitive economic axioms for portfolio risk assessment - which are monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail.
https://privwww.ssrn.com/abstract=3423042
https://privwww.ssrn.com/1818968.htmlTue, 27 Aug 2019 08:41:41 GMTREVISION: An Axiomatic Foundation for the Expected ShortfallIn the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is - in addition to many other nice properties - a coherent risk measure, it does not yet have an axiomatic foundation. In this paper we put forward four intuitive economic axioms for portfolio risk assessment - which are monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail.
https://privwww.ssrn.com/abstract=3423042
https://privwww.ssrn.com/1817536.htmlThu, 22 Aug 2019 10:03:34 GMTREVISION: Robustness in the Optimization of Risk MeasuresWe study issues of robustness in the context of Quantitative Risk Management and Optimization. Depending on the underlying objectives, we develop a general methodology for determining whether a given risk measurement related optimization problem is robust. Motivated by practical issues from financial regulation, we give special attention to the two most widely used risk measures in the industry, Value-at-Risk (VaR) and Expected Shortfall (ES). We discover that for many simple representative optimization problems, VaR generally leads to non-robust optimizers whereas ES generally leads to robust ones. Our results thus shed light from a new angle on the ongoing discussion about the comparative advantages of VaR and ES in banking and insurance regulation. Our notion of robustness is conceptually different from the field of robust optimization, to which some interesting links are discovered.
https://privwww.ssrn.com/abstract=3254587
https://privwww.ssrn.com/1815211.htmlTue, 13 Aug 2019 16:43:10 GMT