T. Bielecki, Igor Cialenco, Marcin Pitera, Thorsten Schmidt
Abstract In this paper, we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide explicit formulae for fair capital allocations in case when the constituents of the risky portfolio are jointly normally distributed. The main focus of the paper is on the problem of approximating fair portfolio allocations in the case of not fully known law of the portfolio constituents. We define and study the concepts of fair allocation estimators and asymptotically fair allocation estimators. A substantial part of our study is devoted to the problem of estimating fair risk allocations for expected shortfall. We study this problem under normality as well as in a nonparametric setup. We derive several estimators, and prove their fairness and/or asymptotic fairness. Last, but not least, we propose two backtesting methodologies that are oriented at assessing the performance of the allocation estimation procedure. The paper closes with a substantial numerical study of the subject and an application to market data.
{"title":"Fair estimation of capital risk allocation","authors":"T. Bielecki, Igor Cialenco, Marcin Pitera, Thorsten Schmidt","doi":"10.1515/strm-2019-0011","DOIUrl":"https://doi.org/10.1515/strm-2019-0011","url":null,"abstract":"Abstract In this paper, we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide explicit formulae for fair capital allocations in case when the constituents of the risky portfolio are jointly normally distributed. The main focus of the paper is on the problem of approximating fair portfolio allocations in the case of not fully known law of the portfolio constituents. We define and study the concepts of fair allocation estimators and asymptotically fair allocation estimators. A substantial part of our study is devoted to the problem of estimating fair risk allocations for expected shortfall. We study this problem under normality as well as in a nonparametric setup. We derive several estimators, and prove their fairness and/or asymptotic fairness. Last, but not least, we propose two backtesting methodologies that are oriented at assessing the performance of the allocation estimation procedure. The paper closes with a substantial numerical study of the subject and an application to market data.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2019-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2019-0011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44263585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The debate of what quantitative risk measure to choose in practice has mainly focused on the dichotomy between Value at Risk (VaR) -- a quantile -- and Expected Shortfall (ES) -- a tail expectation. Range Value at Risk (RVaR) is a natural interpolation between these two prominent risk measures, which constitutes a tradeoff between the sensitivity of the latter and the robustness of the former, turning it into a practically relevant risk measure on its own. As such, there is a need to statistically validate RVaR forecasts and to compare and rank the performance of different RVaR models, tasks subsumed under the term 'backtesting' in finance. The predictive performance is best evaluated and compared in terms of strictly consistent loss or scoring functions. That is, functions which are minimised in expectation by the correct RVaR forecast. Much like ES, it has been shown recently that RVaR does not admit strictly consistent scoring functions, i.e., it is not elicitable. Mitigating this negative result, this paper shows that a triplet of RVaR with two VaR components at different levels is elicitable. We characterise the class of strictly consistent scoring functions for this triplet. Additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares in robust regression.
关于在实践中选择何种定量风险度量的争论主要集中在风险价值(VaR)(分位数)和预期缺口(ES)(尾部期望)之间的二分法上。风险极差值(Range Value at Risk, RVaR)是这两种主要风险度量之间的自然插值，它在后者的敏感性和前者的稳健性之间进行了权衡，使其本身成为一种实际相关的风险度量。因此，有必要对RVaR预测进行统计验证，并对不同RVaR模型的表现进行比较和排名，这些任务在金融领域被称为“回测”。预测性能最好根据严格一致的损失函数或评分函数进行评估和比较。也就是说，通过正确的RVaR预测使期望最小化的函数。就像ES一样，最近的研究表明，RVaR不承认严格一致的评分函数，也就是说，它是不可得到的。为了缓解这一负面结果，本文证明了具有两个不同水平VaR分量的RVaR三元组是可以产生的。我们描述了这个三元组的严格一致评分函数的性质。检查了这些评分函数的附加属性，包括墨菲图的诊断工具。通过模拟研究说明了结果，并且我们将我们的方法与鲁棒回归中裁剪最小二乘的经典方法相比较。
{"title":"Evaluating Range Value at Risk Forecasts.","authors":"Tobias Fissler, Johanna F. Ziegel","doi":"10.1515/strm-2020-0037","DOIUrl":"https://doi.org/10.1515/strm-2020-0037","url":null,"abstract":"The debate of what quantitative risk measure to choose in practice has mainly focused on the dichotomy between Value at Risk (VaR) -- a quantile -- and Expected Shortfall (ES) -- a tail expectation. Range Value at Risk (RVaR) is a natural interpolation between these two prominent risk measures, which constitutes a tradeoff between the sensitivity of the latter and the robustness of the former, turning it into a practically relevant risk measure on its own. As such, there is a need to statistically validate RVaR forecasts and to compare and rank the performance of different RVaR models, tasks subsumed under the term 'backtesting' in finance. The predictive performance is best evaluated and compared in terms of strictly consistent loss or scoring functions. That is, functions which are minimised in expectation by the correct RVaR forecast. Much like ES, it has been shown recently that RVaR does not admit strictly consistent scoring functions, i.e., it is not elicitable. Mitigating this negative result, this paper shows that a triplet of RVaR with two VaR components at different levels is elicitable. We characterise the class of strictly consistent scoring functions for this triplet. Additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares in robust regression.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2019-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67315146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g., in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks of such positions based on the idea of considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of risk measures of non-convex portfolios are presented.
{"title":"Multivariate risk measures in the non-convex setting","authors":"A. Haier, I. Molchanov","doi":"10.1515/strm-2019-0002","DOIUrl":"https://doi.org/10.1515/strm-2019-0002","url":null,"abstract":"Abstract The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g., in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks of such positions based on the idea of considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of risk measures of non-convex portfolios are presented.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2019-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2019-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48083770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we derive a semiparametric efficient adaptive estimator for the GJR-GARCH ( 1 , 1 ) {(1,1)} model. We first show that the quasi-maximum likelihood estimator is consistent and asymptotically normal for the model used in analysis, and we secondly derive a semiparametric estimator that is more efficient than the quasi-maximum likelihood estimator. Through Monte Carlo simulations, we show that the semiparametric estimator is adaptive for the parameters included in the conditional variance of the GJR-GARCH ( 1 , 1 ) {(1,1)} model with respect to the unknown distribution of the innovation.
{"title":"Semiparametric efficient adaptive estimation of the GJR-GARCH model","authors":"Nicola Ciccarelli","doi":"10.1515/strm-2017-0015","DOIUrl":"https://doi.org/10.1515/strm-2017-0015","url":null,"abstract":"Abstract In this paper we derive a semiparametric efficient adaptive estimator for the GJR-GARCH ( 1 , 1 ) {(1,1)} model. We first show that the quasi-maximum likelihood estimator is consistent and asymptotically normal for the model used in analysis, and we secondly derive a semiparametric estimator that is more efficient than the quasi-maximum likelihood estimator. Through Monte Carlo simulations, we show that the semiparametric estimator is adaptive for the parameters included in the conditional variance of the GJR-GARCH ( 1 , 1 ) {(1,1)} model with respect to the unknown distribution of the innovation.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2017-0015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43450368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)).
{"title":"Time consistency for scalar multivariate risk measures","authors":"Zachary Feinstein, Birgit Rudloff","doi":"10.1515/strm-2019-0023","DOIUrl":"https://doi.org/10.1515/strm-2019-0023","url":null,"abstract":"Abstract In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)).","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2018-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43347474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Multivariate expectiles, a new family of vector-valued risk measures, were recently introduced in the literature. [22]. Here we investigate the asymptotic behavior of these measures in a multivariate regular variation context. For models with equivalent tails, we propose an estimator of extreme multivariate expectiles in the Fréchet domain of attraction case with asymptotic independence, or for comonotonic marginal distributions.
{"title":"Extremes for multivariate expectiles","authors":"V. Maume-Deschamps, D. Rullière, K. Said","doi":"10.1515/strm-2017-0014","DOIUrl":"https://doi.org/10.1515/strm-2017-0014","url":null,"abstract":"Abstract Multivariate expectiles, a new family of vector-valued risk measures, were recently introduced in the literature. [22]. Here we investigate the asymptotic behavior of these measures in a multivariate regular variation context. For models with equivalent tails, we propose an estimator of extreme multivariate expectiles in the Fréchet domain of attraction case with asymptotic independence, or for comonotonic marginal distributions.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2017-0014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43152877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Understanding how people make decisions from risky choices has attracted increasing attention of researchers in economics, psychology and neuroscience. While economists try to evaluate individual’s risk preference through mathematical modeling, neuroscientists answer the question by exploring the neural activities of the brain. We propose a model-free method, 3-dimensional image functional principal component analysis (3DIF), to provide a connection between active risk related brain region detection and individual’s risk preference. The 3DIF methodology is directly applicable to 3-dimensional image data without artificial vectorization or mapping and simultaneously guarantees the contiguity of risk related brain regions rather than discrete voxels. Simulation study evidences an accurate and reasonable region detection using the 3DIF method. In real data analysis, five important risk related brain regions are detected, including parietal cortex (PC), ventrolateral prefrontal cortex (VLPFC), lateral orbifrontal cortex (lOFC), anterior insula (aINS) and dorsolateral prefrontal cortex (DLPFC), while the alternative methods only identify limited risk related regions. Moreover, the 3DIF method is useful for extraction of subjective specific signature scores that carry explanatory power for individual’s risk attitude. In particular, the 3DIF method perfectly classifies both strongly and weakly risk averse subjects for in-sample analysis. In out-of-sample experiment, it achieves 73 -88 overall accuracy, among which 90 -100 strongly risk averse subjects and 49 -71 weakly risk averse subjects are correctly classified with leave-k-out cross validations.
{"title":"Risk related brain regions detection and individual risk classification with 3D image FPCA","authors":"Ying Chen, W. Härdle, Qiang He, Piotr Majer","doi":"10.1515/strm-2017-0011","DOIUrl":"https://doi.org/10.1515/strm-2017-0011","url":null,"abstract":"Abstract Understanding how people make decisions from risky choices has attracted increasing attention of researchers in economics, psychology and neuroscience. While economists try to evaluate individual’s risk preference through mathematical modeling, neuroscientists answer the question by exploring the neural activities of the brain. We propose a model-free method, 3-dimensional image functional principal component analysis (3DIF), to provide a connection between active risk related brain region detection and individual’s risk preference. The 3DIF methodology is directly applicable to 3-dimensional image data without artificial vectorization or mapping and simultaneously guarantees the contiguity of risk related brain regions rather than discrete voxels. Simulation study evidences an accurate and reasonable region detection using the 3DIF method. In real data analysis, five important risk related brain regions are detected, including parietal cortex (PC), ventrolateral prefrontal cortex (VLPFC), lateral orbifrontal cortex (lOFC), anterior insula (aINS) and dorsolateral prefrontal cortex (DLPFC), while the alternative methods only identify limited risk related regions. Moreover, the 3DIF method is useful for extraction of subjective specific signature scores that carry explanatory power for individual’s risk attitude. In particular, the 3DIF method perfectly classifies both strongly and weakly risk averse subjects for in-sample analysis. In out-of-sample experiment, it achieves 73 -88 overall accuracy, among which 90 -100 strongly risk averse subjects and 49 -71 weakly risk averse subjects are correctly classified with leave-k-out cross validations.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2017-0011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43538156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The present paper analyzes an optimal consumption and investment problem of a retiree with a constant relative risk aversion (CRRA) who faces parameter uncertainty about the financial market. We solve the optimization problem under partial information by making the market observationally complete and consequently applying the martingale method to obtain closed-form solutions to the optimal consumption and investment strategies. Further, we provide some comparative statics and numerical analyses to deeply understand the consumption and investment behavior under partial information. Bearing partial information has little impact on the optimal consumption level, but it makes retirees with an RRA smaller than one invest more riskily, while it makes retirees with an RRA larger than one invest more conservatively.
{"title":"Optimal retirement planning under partial information","authors":"N. Bäuerle, A. Chen","doi":"10.1515/strm-2018-0027","DOIUrl":"https://doi.org/10.1515/strm-2018-0027","url":null,"abstract":"Abstract The present paper analyzes an optimal consumption and investment problem of a retiree with a constant relative risk aversion (CRRA) who faces parameter uncertainty about the financial market. We solve the optimization problem under partial information by making the market observationally complete and consequently applying the martingale method to obtain closed-form solutions to the optimal consumption and investment strategies. Further, we provide some comparative statics and numerical analyses to deeply understand the consumption and investment behavior under partial information. Bearing partial information has little impact on the optimal consumption level, but it makes retirees with an RRA smaller than one invest more riskily, while it makes retirees with an RRA larger than one invest more conservatively.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2018-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2018-0027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47195313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper introduces optimal expected utility (OEU) risk measures, investigates their main properties and puts them in perspective to alternative risk measures and notions of certainty equivalents. By taking the investor’s point of view, OEU maximizes the sum of capital available today and the certainty equivalent of capital in the future. To the best of our knowledge, OEU is the only existing utility-based risk measure that is (non-trivial and) coherent if the utility function u has constant relative risk aversion. We present several different risk measures that can be derived with special choices of u and illustrate that OEU is more sensitive than value at risk and average value at risk with respect to changes of the probability of a financial loss.
{"title":"Optimal expected utility risk measures","authors":"S. Geissel, Jörn Sass, F. Seifried","doi":"10.1515/strm-2017-0027","DOIUrl":"https://doi.org/10.1515/strm-2017-0027","url":null,"abstract":"Abstract This paper introduces optimal expected utility (OEU) risk measures, investigates their main properties and puts them in perspective to alternative risk measures and notions of certainty equivalents. By taking the investor’s point of view, OEU maximizes the sum of capital available today and the certainty equivalent of capital in the future. To the best of our knowledge, OEU is the only existing utility-based risk measure that is (non-trivial and) coherent if the utility function u has constant relative risk aversion. We present several different risk measures that can be derived with special choices of u and illustrate that OEU is more sensitive than value at risk and average value at risk with respect to changes of the probability of a financial loss.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2017-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2017-0027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48908544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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