SSRN Author: Ruodu WangRuodu Wang SSRN Content
https://www.ssrn.com/author=2190264
https://www.ssrn.com/rss/en-usSun, 01 May 2022 01:19:51 GMTeditor@ssrn.com (Editor)Sun, 01 May 2022 01:19:51 GMTwebmaster@ssrn.com (WebMaster)SSRN RSS Generator 1.0New: A Reverse Expected Shortfall Optimization FormulaThe celebrated Expected Shortfall (ES) optimization formula implies that ES at a fixed probability level is the minimum of a linear real function plus a scaled mean excess function. We establish a reverse ES optimization formula, which says that a mean excess function at any fixed threshold is the maximum of an ES curve minus a linear function. Despite being a simple result, this formula reveals elegant symmetries between the mean excess function and the ES curve, as well as their optimizers. The reverse ES optimization formula is closely related to the Fenchel-Legendre transforms, and our formulas are generalized from ES to optimized certainty equivalents, a popular class of convex risk measures. We analyze worst-case values of the mean excess function under two popular settings of model uncertainty to illustrate the usefulness of the reverse ES optimization formula, and this is further demonstrated with an application using insurance datasets.
https://www.ssrn.com/abstract=4092939
https://www.ssrn.com/2133432.htmlFri, 29 Apr 2022 11:59:08 GMTREVISION: Optimizing Distortion Riskmetrics With Distributional UncertaintyOptimization of distortion riskmetrics with distributional uncertainty has wide applications in finance and operations research. Distortion riskmetrics include many commonly applied risk measures and deviation measures, which are not necessarily monotone or convex. 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 great 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 ...
https://www.ssrn.com/abstract=3728638
https://www.ssrn.com/2108689.htmlThu, 24 Feb 2022 23:03:34 GMTREVISION: One Axiom To Rule Them All: A Minimalist Axiomatization of QuantilesWe offer a minimalist axiomatization of quantiles among all real-valued mappings on a general set of distributions through only one axiom. This axiom is called ordinality: quantiles are the only mappings that commute with all increasing and continuous transforms. Other convenient properties of quantiles, monotonicity, semicontinuity, comonotonic additivity and elicitability in particular, follow from this axiom. Furthermore, on the set of convexly supported distributions, the median is the only mapping that commutates with all monotone and continuous transforms. On a general set of distributions, the median interval is pinned down as the unique minimal interval-valued mapping that commutes with all monotone and continuous transforms. Finally, our main result, put in a decision-theoretic setting, leads to a minimalist axiomatization of quantile preferences. In banking and insurance, quantiles are known as the standard regulatory risk measure Value-at-Risk (VaR), and thus, an ...
https://www.ssrn.com/abstract=3944312
https://www.ssrn.com/2103718.htmlWed, 09 Feb 2022 21:52:13 GMTREVISION: Variance Comparison between Infinitesimal Perturbation Analysis and Likelihood Ratio Estimators to Stochastic GradientWe theoretically compare variances between the Infinitesimal Perturbation Analysis (IPA) estimator and the Likelihood Ratio (LR) estimator to Monte Carlo gradient for stochastic systems. The results presented in [1] on variance comparison between these two estimators are substantially improved. We also prove a practically interesting result that the IPA estimators to European vanilla and arithmetic Asian options' Delta, respectively, have smaller variance when the underlying asset's return process is independent with the initial price and square integrable.
https://www.ssrn.com/abstract=3876623
https://www.ssrn.com/2102009.htmlSat, 05 Feb 2022 21:11:19 GMTREVISION: Inf-convolution, Optimal Allocations, and Model Uncertainty 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. Optimal allocations are obtained in the setting of elliptical models,<br>and several results are established for tail risk measures and risk sharing problems in the presence of model uncertainty. The technical conclusions are quite general without assuming any form of convexity of the tail risk measures. Our analysis generalizes in several directions the recent literature on quantile-based risk ...
https://www.ssrn.com/abstract=3490348
https://www.ssrn.com/2095032.htmlTue, 18 Jan 2022 12:15:53 GMTREVISION: Risk Measures Induced by Efficient Insurance ContractsThe Expected Shortfall (ES) is one of the most important regulatory risk measures in finance, insurance, and statistics, which has recently been characterized via sets of axioms from perspectives of portfolio risk management and statistics. Meanwhile, there is large literature on insurance design with ES as an objective or a constraint. A visible gap is to justify the special role of ES in insurance and actuarial science. To fill this gap, 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.
https://www.ssrn.com/abstract=3915592
https://www.ssrn.com/2092099.htmlMon, 10 Jan 2022 14:37:26 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://www.ssrn.com/abstract=3489566
https://www.ssrn.com/2089332.htmlThu, 30 Dec 2021 21:47:55 GMTREVISION: An Impossibility Theorem on Capital AllocationTwo natural and potentially desirable properties for capital allocation rules are top-down<br>consistency and shrinking independence. Top-down consistency means that the total capital is<br>determined by the aggregate portfolio risk. Shrinking independence means that the risk capital<br>allocated to a given business line should not be affected by a proportional reduction of exposure<br>in another business line. These two properties are satised by, respectively, the Euler allocation<br>rule and the stress allocation rule. We prove an impossibility theorem which states that these two<br>properties jointly lead to the trivial capital allocation based on the mean. When a subadditive<br>risk measure is used, the same result holds for weaker versions of shrinking independence, which<br>prevents the increase in risk capital in one line, when exposure to another is reduced. The<br>impossibility theorem remains valid even if one assumes strong positive dependence among the<br>risk vectors.
https://www.ssrn.com/abstract=3964775
https://www.ssrn.com/2086955.htmlTue, 21 Dec 2021 18:35:33 GMTREVISION: Inf-convolution, Optimal Allocations, and Model Uncertainty 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. Optimal allocations are obtained in the setting of elliptical models,<br>and several results are established for tail risk measures and risk sharing problems in the presence of model uncertainty. The technical conclusions are quite general without assuming any form of convexity of the tail risk measures. Our analysis generalizes in several directions the recent literature on quantile-based risk ...
https://www.ssrn.com/abstract=3490348
https://www.ssrn.com/2082761.htmlWed, 08 Dec 2021 04:31:29 GMTREVISION: An Impossibility Theorem on Capital AllocationTwo natural and potentially desirable properties for capital allocation rules are top-down<br>consistency and shrinking independence. Top-down consistency means that the total capital is<br>determined by the aggregate portfolio risk. Shrinking independence means that the risk capital<br>allocated to a given business line should not be affected by a proportional reduction of exposure<br>in another business line. These two properties are satised by, respectively, the Euler allocation<br>rule and the stress allocation rule. We prove an impossibility theorem which states that these two<br>properties jointly lead to the trivial capital allocation based on the mean. When a subadditive<br>risk measure is used, the same result holds for weaker versions of shrinking independence, which<br>prevents the increase in risk capital in one line, when exposure to another is reduced. The<br>impossibility theorem remains valid even if one assumes strong positive dependence among the<br>risk vectors.
https://www.ssrn.com/abstract=3964775
https://www.ssrn.com/2082499.htmlTue, 07 Dec 2021 01:08:13 GMTREVISION: A Theory of Credit Rating CriteriaWe 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 and structural maximization, from investors who rely on the rating criterion for pricing. While the expected loss criterion used by Moody's satisfies self-consistency, the probability of default criterion used by S\&P does not. Empirical evidences in the post-Dodd-Frank period are consistent with the theoretical implication. We show that a set of axioms based on self-consistency leads a tractable representation for all self-consistent rating criteria, which can also be extended to incorporate economic scenarios. New examples of self-consistent and scenario-based rating criteria are suggested.
https://www.ssrn.com/abstract=3504065
https://www.ssrn.com/2082461.htmlTue, 07 Dec 2021 00:56:03 GMTREVISION: A Theory of Multivariate Stress TestingWe 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 ...
https://www.ssrn.com/abstract=3966204
https://www.ssrn.com/2081656.htmlFri, 03 Dec 2021 01:54:37 GMTREVISION: An Impossibility Theorem on Capital AllocationTwo natural and desirable properties for capital allocation rules are top-down consistency and shrinking independence. Top-down consistency means that the total capital is determined by the aggregate portfolio risk. Shrinking independence means that the risk capital allocated to a given business line should not be affected by a proportional reduction of exposure in another business line. These two properties are satisfied by, respectively, the Euler allocation rule and the stress allocation rule. We prove an impossibility theorem which states that these two properties jointly lead to the trivial capital allocation based on the mean. When a subadditive risk measure is used, the same result holds for a weaker version of shrinking independence, which prevents the increase in risk capital in one line, when exposure to another is reduced.
https://www.ssrn.com/abstract=3964775
https://www.ssrn.com/2078190.htmlThu, 18 Nov 2021 20:46:17 GMTREVISION: A Framework for Measures of Risk under UncertaintyA risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure. We present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent ...
https://www.ssrn.com/abstract=3943660
https://www.ssrn.com/2070080.htmlThu, 21 Oct 2021 00:42:35 GMTREVISION: A Framework for Measures of Risk under UncertaintyA risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure. We present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent ...
https://www.ssrn.com/abstract=3943660
https://www.ssrn.com/2069764.htmlWed, 20 Oct 2021 03:23:27 GMTREVISION: Risk Measures Induced by Efficient Insurance ContractsThe Expected Shortfall (ES) is one of the most important regulatory risk measures in finance, insurance, and statistics, which has recently been characterized via sets of axioms from perspectives of portfolio risk management and statistics. Meanwhile, there is large literature on insurance design with ES as an objective or a constraint. A visible gap is to justify the special role of ES in insurance and actuarial science. To fill this gap, we study 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.
https://www.ssrn.com/abstract=3915592
https://www.ssrn.com/2066003.htmlWed, 06 Oct 2021 00:53:15 GMTREVISION: Adjusted Expected ShortfallWe 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 <i>X</i> to ensure that Expected Shortfall <i>ES<sub>p</sub>(X)</i> does not exceed a pre-specified threshold <i>g(p)</i> for every probability level <i>p\in[0,1]</i>. Through the choice of the benchmark risk profile <i>g</i> one 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.
https://www.ssrn.com/abstract=3650887
https://www.ssrn.com/2052342.htmlThu, 19 Aug 2021 11:32:15 GMTREVISION: Adjusted Expected ShortfallWe 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 <i>X</i> to ensure that Expected Shortfall <i>ES<sub>p</sub>(X)</i> does not exceed a pre-specified threshold <i>g(p)</i> for every probability level <i>p\in[0,1]</i>. Through the choice of the benchmark risk profile <i>g</i> one 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.
https://www.ssrn.com/abstract=3650887
https://www.ssrn.com/2052150.htmlWed, 18 Aug 2021 21:01:18 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.<br>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, a specific type of insensitivity to a family of dependence assumptions. We show that, among two very general classes of merging methods commonly used in practice, the ...
https://www.ssrn.com/abstract=3569329
https://www.ssrn.com/2051357.htmlMon, 16 Aug 2021 23:43:32 GMTREVISION: Variance Comparison between Infinitesimal Perturbation Analysis and Likelihood Ratio Estimators to Stochastic GradientWe theoretically compare variances between the Infinitesimal Perturbation Analysis (IPA) estimator and the Likelihood Ratio (LR) estimator to Monte Carlo gradient for stochastic systems. The conditions proposed in [Cui et al., 2020] when the IPA estimator has a smaller variance can yield sharper inequalities or be further relaxed. We also prove a practically interesting result that the IPA estimators to European vanilla and arithmetic Asian options' Delta, respectively, have smaller variance when the underlying asset's return process is independent of the initial price.
https://www.ssrn.com/abstract=3876623
https://www.ssrn.com/2039434.htmlFri, 02 Jul 2021 10:12:39 GMTREVISION: Dependence and Risk Attitudes: An EquivalenceSuppose that a decision maker faces a random outcome which is the sum of several risky components. If she is indifferent to the dependence structure of the risky components, then we say that she (or her preference) is dependence neutral. Obviously, if the decision maker is risk neutral, i.e., her preference is numerically represented by the expectation, then she is dependence neutral. We show the converse direction is also true: dependence neutrality and risk neutrality are indeed equivalent. Moreover, the decision maker may be averse to strong positive dependence representing extreme comovement (comovement aversion), and she may prefer less to more positive dependence (dependence monotonicity). Under a continuity assumption on the preference, we show that comovement aversion, dependence monotonicity, and strong risk aversion are all equivalent. Our results bridge the gap between dependence and risk attitudes, connecting two prominent concepts in statistics and decision theory.
https://www.ssrn.com/abstract=3707709
https://www.ssrn.com/2031776.htmlTue, 08 Jun 2021 11:45:17 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://www.ssrn.com/abstract=3235450
https://www.ssrn.com/2020976.htmlTue, 04 May 2021 09:12:58 GMT