Abstract Distortion functions are employed to define measures of risk. Receiver operating characteristic (ROC) curves are used to describe the performance of parametrized test families in testing a simple null hypothesis against a simple alternative. This paper provides a connection between distortion functions on the one hand, and ROC curves on the other. This leads to a new interpretation of some well-known classes of distortion risk measures, and to a new notion of divergence between probability measures.
{"title":"Distortion risk measures, ROC curves, and distortion divergence","authors":"J. Schumacher","doi":"10.1515/strm-2017-0012","DOIUrl":"https://doi.org/10.1515/strm-2017-0012","url":null,"abstract":"Abstract Distortion functions are employed to define measures of risk. Receiver operating characteristic (ROC) curves are used to describe the performance of parametrized test families in testing a simple null hypothesis against a simple alternative. This paper provides a connection between distortion functions on the one hand, and ROC curves on the other. This leads to a new interpretation of some well-known classes of distortion risk measures, and to a new notion of divergence between probability measures.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2017-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2017-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41794182","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 study parameter estimation via the Expectation Maximization (EM) algorithm for a continuous-time hidden Markov model with diffusion and point process observation. Inference problems of this type arise for instance in credit risk modelling. A key step in the application of the EM algorithm is the derivation of finite-dimensional filters for the quantities that are needed in the E-Step of the algorithm. In this context we obtain exact, unnormalized and robust filters, and we discuss their numerical implementation. Moreover, we propose several goodness-of-fit tests for hidden Markov models with Gaussian noise and point process observation. We run an extensive simulation study to test speed and accuracy of our methodology. The paper closes with an application to credit risk: we estimate the parameters of a hidden Markov model for credit quality where the observations consist of rating transitions and credit spreads for US corporations.
{"title":"EM algorithm for Markov chains observed via Gaussian noise and point process information: Theory and case studies","authors":"Camilla Damian, Zehra Eksi, R. Frey","doi":"10.1515/strm-2017-0021","DOIUrl":"https://doi.org/10.1515/strm-2017-0021","url":null,"abstract":"Abstract In this paper we study parameter estimation via the Expectation Maximization (EM) algorithm for a continuous-time hidden Markov model with diffusion and point process observation. Inference problems of this type arise for instance in credit risk modelling. A key step in the application of the EM algorithm is the derivation of finite-dimensional filters for the quantities that are needed in the E-Step of the algorithm. In this context we obtain exact, unnormalized and robust filters, and we discuss their numerical implementation. Moreover, we propose several goodness-of-fit tests for hidden Markov models with Gaussian noise and point process observation. We run an extensive simulation study to test speed and accuracy of our methodology. The paper closes with an application to credit risk: we estimate the parameters of a hidden Markov model for credit quality where the observations consist of rating transitions and credit spreads for US corporations.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2017-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2017-0021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41506071","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 A proposal for applying network analysis to a foreign exchange (FX) settlement system is considered. In particular, network centrality metrics are used to analyse payments of financial institutions which settle through CLS Bank (CLS). Network centrality metrics provide a way to study settlement members’ connectivity, obtain a sense of their payments evolution with time, and measure their network topology variability. The analysis shows that although the continuous link settlement (CLS) network structure can be approximated with a power law degree distribution for many trade days, this is not always the case. A network community detection algorithm is applied to the FX settlement network to explore relationships between communities and to detect classification patterns in the FX trading net payments. A metric called SinkRank is used to build a ranking of the most systemic settlement risk important financial institutions trading on the FX system, and to understand how the metric depends on network’s connectivity. Since network metrics do not fully explain the dynamics of the settlement process, the CLS’ settlement system is simulated to measure the contagion of unsettled trades and its spread among network members. The effect of settlement failure and contagion on the settlement members is also explored.
{"title":"Network analysis and systemic FX settlement risk","authors":"José Henry León-Janampa","doi":"10.1515/strm-2015-0006","DOIUrl":"https://doi.org/10.1515/strm-2015-0006","url":null,"abstract":"Abstract A proposal for applying network analysis to a foreign exchange (FX) settlement system is considered. In particular, network centrality metrics are used to analyse payments of financial institutions which settle through CLS Bank (CLS). Network centrality metrics provide a way to study settlement members’ connectivity, obtain a sense of their payments evolution with time, and measure their network topology variability. The analysis shows that although the continuous link settlement (CLS) network structure can be approximated with a power law degree distribution for many trade days, this is not always the case. A network community detection algorithm is applied to the FX settlement network to explore relationships between communities and to detect classification patterns in the FX trading net payments. A metric called SinkRank is used to build a ranking of the most systemic settlement risk important financial institutions trading on the FX system, and to understand how the metric depends on network’s connectivity. Since network metrics do not fully explain the dynamics of the settlement process, the CLS’ settlement system is simulated to measure the contagion of unsettled trades and its spread among network members. The effect of settlement failure and contagion on the settlement members is also explored.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2015-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313826","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 discuss the problem of calculating the primary risk measures in the variance-gamma model. A portfolio of investments in a one-period setting is considered. It is supposed that the investment returns are dependent on each other. In terms of the variance-gamma model, we assume that there are relations in both groups of the normal random variables and the gamma stochastic volatilities. The value at risk, the expected shortfall and the entropic monetary risk measures are discussed. The obtained analytical expressions are based on values of hypergeometric functions.
{"title":"On risk measuring in the variance-gamma model","authors":"R. Ivanov","doi":"10.1515/strm-2017-0008","DOIUrl":"https://doi.org/10.1515/strm-2017-0008","url":null,"abstract":"Abstract In this paper, we discuss the problem of calculating the primary risk measures in the variance-gamma model. A portfolio of investments in a one-period setting is considered. It is supposed that the investment returns are dependent on each other. In terms of the variance-gamma model, we assume that there are relations in both groups of the normal random variables and the gamma stochastic volatilities. The value at risk, the expected shortfall and the entropic monetary risk measures are discussed. The obtained analytical expressions are based on values of hypergeometric functions.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2017-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67315018","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 financial crisis of 2007–2009 has underlined the importance of interconnectedness among financial institutions andmarkets [1], the insufficiency of monitoring the balance sheet of individual financial institutions in isolation, and the necessity of adopting a system-wide view of financial stability. In the wake of the crisis, regulators have sought well-grounded and forward-looking indicators for monitoring the development of systemic risks in the financial system. The construction and interpretation of indicators and the identification and collection of relevant data for computing such indicators have proven to be major and ongoing challenges. The design of indicators for monitoring systemic risk requires the prior identification of contagionmechanisms and calls for an interplay between theory and empirical research. Many researchers have attempted to tackle the challenge of understanding the mechanisms underlying systemic risk. This two-part special issue grew out of a one-week workshop on Monitoring Systemic Risk: Data, Models and Metrics, organized by Rama Cont (Imperial College), Michael Gordy (Federal Reserve Board) and Christian Gourieroux (CREST and University of Toronto). The workshop, held in September 2014, was hosted by the Isaac Newton Institute of Mathematical Sciences (Cambridge, UK) as part of a semester-long program on “SystemicMathematicalmodelling and interdisciplinary approaches” (www.newton.ac.uk/event/syr). The workshop gathered together more than 100 researchers from various disciplines – mathematical finance, economics, econometrics and operations research – together with regulators, central bankers and industry risk professionals, to discuss how mathematical modeling may contribute to the modeling and monitoring of systemic risk. Further material and video recordings of all lectures are available for download from the website of the workshop at www.newton.ac.uk/event/syrw02. The contributions to this Special Issue underline some key issues that arose during the discussions at the workshop: estimation and validation of risk measures for capital adequacy, models of interconnectedness and centrality in banking networks, fire sales spillovers and portfolio overlaps. We thank the Isaac Newton Institute of Mathematical Sciences (Cambridge) for hosting and supporting theworkshop andOldMutual for its financial support of the program“Systemic Risk:MathematicalModeling and Interdisciplinary Approaches”.
{"title":"Special Issue: Monitoring Systemic Risk: Data, Models and Metrics","authors":"R. Cont, Michael B. Gordy","doi":"10.1515/strm-2016-0024","DOIUrl":"https://doi.org/10.1515/strm-2016-0024","url":null,"abstract":"The financial crisis of 2007–2009 has underlined the importance of interconnectedness among financial institutions andmarkets [1], the insufficiency of monitoring the balance sheet of individual financial institutions in isolation, and the necessity of adopting a system-wide view of financial stability. In the wake of the crisis, regulators have sought well-grounded and forward-looking indicators for monitoring the development of systemic risks in the financial system. The construction and interpretation of indicators and the identification and collection of relevant data for computing such indicators have proven to be major and ongoing challenges. The design of indicators for monitoring systemic risk requires the prior identification of contagionmechanisms and calls for an interplay between theory and empirical research. Many researchers have attempted to tackle the challenge of understanding the mechanisms underlying systemic risk. This two-part special issue grew out of a one-week workshop on Monitoring Systemic Risk: Data, Models and Metrics, organized by Rama Cont (Imperial College), Michael Gordy (Federal Reserve Board) and Christian Gourieroux (CREST and University of Toronto). The workshop, held in September 2014, was hosted by the Isaac Newton Institute of Mathematical Sciences (Cambridge, UK) as part of a semester-long program on “SystemicMathematicalmodelling and interdisciplinary approaches” (www.newton.ac.uk/event/syr). The workshop gathered together more than 100 researchers from various disciplines – mathematical finance, economics, econometrics and operations research – together with regulators, central bankers and industry risk professionals, to discuss how mathematical modeling may contribute to the modeling and monitoring of systemic risk. Further material and video recordings of all lectures are available for download from the website of the workshop at www.newton.ac.uk/event/syrw02. The contributions to this Special Issue underline some key issues that arose during the discussions at the workshop: estimation and validation of risk measures for capital adequacy, models of interconnectedness and centrality in banking networks, fire sales spillovers and portfolio overlaps. We thank the Isaac Newton Institute of Mathematical Sciences (Cambridge) for hosting and supporting theworkshop andOldMutual for its financial support of the program“Systemic Risk:MathematicalModeling and Interdisciplinary Approaches”.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2016-0024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67315237","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 proposes a rating methodology that is based on a non-linear classification method, a support vector machine, and a non-parametric isotonic regression for mapping rating scores into probabilities of default. We also propose a four data set model validation and training procedure that is more appropriate for credit rating data commonly characterised with cyclicality and panel features. Tests on representative data covering fifteen years of quarterly accounts and default events for 10,000 US listed companies confirm superiority of non-linear PD estimation. Our methodology demonstrates the ability to identify companies of diverse credit quality from Aaa to Caa–C.
{"title":"Company rating with support vector machines","authors":"Russ A. Moro, W. Härdle, Dorothea Schäfer","doi":"10.1515/strm-2012-1141","DOIUrl":"https://doi.org/10.1515/strm-2012-1141","url":null,"abstract":"Abstract This paper proposes a rating methodology that is based on a non-linear classification method, a support vector machine, and a non-parametric isotonic regression for mapping rating scores into probabilities of default. We also propose a four data set model validation and training procedure that is more appropriate for credit rating data commonly characterised with cyclicality and panel features. Tests on representative data covering fifteen years of quarterly accounts and default events for 10,000 US listed companies confirm superiority of non-linear PD estimation. Our methodology demonstrates the ability to identify companies of diverse credit quality from Aaa to Caa–C.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2012-1141","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312109","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 Financial product prices often depend on unknown parameters. Their estimation introduces the risk that a better informed counterparty may strategically pick mispriced products. Understanding estimation risk, and how to properly price it, is essential. We discuss how total estimation risk can be minimized by selecting a probability model of appropriate complexity. We show that conditional estimation risk can be measured only if the probability model predictions have little bias. We illustrate how a premium for conditional estimation risk may be determined when one counterparty is better informed than the other, but a market collapse is to be avoided, using a simple example from pricing regime credit scoring. We empirically examine the approach on a panel data set from a German credit bureau, where we also study dynamic dependencies such as prior rating migrations and defaults.
{"title":"Loan pricing under estimation risk","authors":"Richard Neuberg, Lauren Hannah","doi":"10.1515/strm-2016-0005","DOIUrl":"https://doi.org/10.1515/strm-2016-0005","url":null,"abstract":"Abstract Financial product prices often depend on unknown parameters. Their estimation introduces the risk that a better informed counterparty may strategically pick mispriced products. Understanding estimation risk, and how to properly price it, is essential. We discuss how total estimation risk can be minimized by selecting a probability model of appropriate complexity. We show that conditional estimation risk can be measured only if the probability model predictions have little bias. We illustrate how a premium for conditional estimation risk may be determined when one counterparty is better informed than the other, but a market collapse is to be avoided, using a simple example from pricing regime credit scoring. We empirically examine the approach on a panel data set from a German credit bureau, where we also study dynamic dependencies such as prior rating migrations and defaults.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2016-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67314126","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 We introduce a functional factor model to investigate the dependence of cumulative return curves of individual assets on the market and other factors. We propose a new statistical test to determine whether the dependence in two sample periods are equal. The statistical properties of the test are established by asymptotic theory and simulations. We apply this test to study the impact of the recent financial crisis and trends in oil price on individual stock and sector ETFs. Our analysis reveals the significance of the daily oil futures curves and their different impact on individual stocks and sector ETFs. It also shows that the functional approach has an information content different from that obtained from scalar factor models for point-to-point returns.
{"title":"Testing for asymmetry in betas of cumulative returns: Impact of the financial crisis and crude oil price","authors":"P. Kokoszka, Hong Miao, Ben Zheng","doi":"10.1515/strm-2016-0010","DOIUrl":"https://doi.org/10.1515/strm-2016-0010","url":null,"abstract":"Abstract We introduce a functional factor model to investigate the dependence of cumulative return curves of individual assets on the market and other factors. We propose a new statistical test to determine whether the dependence in two sample periods are equal. The statistical properties of the test are established by asymptotic theory and simulations. We apply this test to study the impact of the recent financial crisis and trends in oil price on individual stock and sector ETFs. Our analysis reveals the significance of the daily oil futures curves and their different impact on individual stocks and sector ETFs. It also shows that the functional approach has an information content different from that obtained from scalar factor models for point-to-point returns.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2016-0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67314831","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}
Financial institutions’ interconnectedness is a key component of systemic risk. However there is still no consensus on its measurement. Using a unique database of network of exposures of French financial institutions, we compare three strategies to measure interconnectedness: closeness of exposure distributions, identification of core-periphery structure and contagion models. The closeness of exposure distributions is adequate to identify outlier institutions. The “core-periphery” structure, usually applied to banking network, is still valid with insurance companies. However this approach is not immune to size effect. This result contrasts with previous analyses where size was not accounted for. Contagion-based stress-tests are the best suited to capture institutions’ systemic fragility, emphasizing their importance as a supervisory tool.
{"title":"How to measure interconnectedness between banks, insurers and financial conglomerates","authors":"Hauton Gaël, Héam Jean-Cyprien","doi":"10.1515/STRM-2014-1177","DOIUrl":"https://doi.org/10.1515/STRM-2014-1177","url":null,"abstract":"Financial institutions’ interconnectedness is a key component of systemic risk. However there is still no consensus on its measurement. Using a unique database of network of exposures of French financial institutions, we compare three strategies to measure interconnectedness: closeness of exposure distributions, identification of core-periphery structure and contagion models. The closeness of exposure distributions is adequate to identify outlier institutions. The “core-periphery” structure, usually applied to banking network, is still valid with insurance companies. However this approach is not immune to size effect. This result contrasts with previous analyses where size was not accounted for. Contagion-based stress-tests are the best suited to capture institutions’ systemic fragility, emphasizing their importance as a supervisory tool.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/STRM-2014-1177","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313331","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 goal of this paper is to study organized flocking behavior and systemic risk in heterogeneous mean-field interacting diffusions. We illustrate in a number of case studies the effect of heterogeneity in the behavior of systemic risk in the system, i.e., the risk that several agents default simultaneously as a result of interconnections. We also investigate the effect of heterogeneity on the “flocking behavior” of different agents, i.e., when agents with different dynamics end up following very similar paths and follow closely the mean behavior of the system. Using Laplace asymptotics, we derive an asymptotic formula for the tail of the loss distribution as the number of agents grows to infinity. This characterizes the tail of the loss distribution and the effect of the heterogeneity of the network on the tail loss probability.
{"title":"On the effect of heterogeneity on flocking behavior and systemic risk","authors":"Fei Fang, Yiwei Sun, K. Spiliopoulos","doi":"10.1515/strm-2016-0013","DOIUrl":"https://doi.org/10.1515/strm-2016-0013","url":null,"abstract":"Abstract The goal of this paper is to study organized flocking behavior and systemic risk in heterogeneous mean-field interacting diffusions. We illustrate in a number of case studies the effect of heterogeneity in the behavior of systemic risk in the system, i.e., the risk that several agents default simultaneously as a result of interconnections. We also investigate the effect of heterogeneity on the “flocking behavior” of different agents, i.e., when agents with different dynamics end up following very similar paths and follow closely the mean behavior of the system. Using Laplace asymptotics, we derive an asymptotic formula for the tail of the loss distribution as the number of agents grows to infinity. This characterizes the tail of the loss distribution and the effect of the heterogeneity of the network on the tail loss probability.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2016-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2016-0013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67314557","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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