The coherent risk measure expected shortfall is a popular alternative to value-at-risk. However, the estimated value may miscommunicate the actual risk, especially when huge losses are present in the return series. This may force the financial institution to keep extra capital to meet the requirement set by the regulators. We propose a new robust coherent risk measure called modified expected shortfall, which quantifies the authentic risk of a portfolio. In comparison with the expected shortfall, the magnitude of the suggested risk measure is found to be lower. We propose non-parametric estimators of the modified expected shortfall and establish their statistical properties such as consistency and asymptotic normality.
{"title":"Modified Expected Shortfall: A New Robust Coherent Risk Measure","authors":"Deepak K Jadhav, R. V, U. Naik-Nimbalkar","doi":"10.21314/JOR.2013.269","DOIUrl":"https://doi.org/10.21314/JOR.2013.269","url":null,"abstract":"The coherent risk measure expected shortfall is a popular alternative to value-at-risk. However, the estimated value may miscommunicate the actual risk, especially when huge losses are present in the return series. This may force the financial institution to keep extra capital to meet the requirement set by the regulators. We propose a new robust coherent risk measure called modified expected shortfall, which quantifies the authentic risk of a portfolio. In comparison with the expected shortfall, the magnitude of the suggested risk measure is found to be lower. We propose non-parametric estimators of the modified expected shortfall and establish their statistical properties such as consistency and asymptotic normality.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127728027","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 purpose of this paper is to introduce an evolution of estimation of ex-ante VaR of the Monte Carlo Filtered Bootstrap. We define the "modus operandi" borrowing from Bayesian statistic the idea of prior, likelihood and posterior distribution to have a mixture distribution of future returns. We perform three tests, Unconditional Coverage, Independence and Conditional Coverage, according to Christoffersen (1998). We present results on both VaR1% and VaR5% on a one day horizon for the following indices: Standard&Poors 500, Topix, Dax, MSCI United Kingdom, MSCI France, Italy Comit Globale, MSCI Canada, MSCI Emerging Markets, RJ/CRB. We also test the model on a ten equities portfolios and over four commodity sector indices. Our results show that the improved Filtered Bootstrap approach satisfies Conditional Coverage for all tested indices and porfolios while the standard Filtered Bootstrap has more rejection cases. We also test the models in a regulatory framework (rolling window of 250 daily observations) and discuss the advantages of each method in the risk management process.
{"title":"An Heuristic Improvement of a Filtered Bootstrap Approach","authors":"Stefano Colucci","doi":"10.2139/ssrn.2328211","DOIUrl":"https://doi.org/10.2139/ssrn.2328211","url":null,"abstract":"The purpose of this paper is to introduce an evolution of estimation of ex-ante VaR of the Monte Carlo Filtered Bootstrap. We define the \"modus operandi\" borrowing from Bayesian statistic the idea of prior, likelihood and posterior distribution to have a mixture distribution of future returns. We perform three tests, Unconditional Coverage, Independence and Conditional Coverage, according to Christoffersen (1998). We present results on both VaR1% and VaR5% on a one day horizon for the following indices: Standard&Poors 500, Topix, Dax, MSCI United Kingdom, MSCI France, Italy Comit Globale, MSCI Canada, MSCI Emerging Markets, RJ/CRB. We also test the model on a ten equities portfolios and over four commodity sector indices. Our results show that the improved Filtered Bootstrap approach satisfies Conditional Coverage for all tested indices and porfolios while the standard Filtered Bootstrap has more rejection cases. We also test the models in a regulatory framework (rolling window of 250 daily observations) and discuss the advantages of each method in the risk management process.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"2015 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125641054","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 dependency structure of credit risk parameters is a key driver for capital consumption and receives regulatory and scientific attention. The impact of parameter imperfections on the quality of expected loss (EL) in the sense of a fair, unbiased estimate of risk expenses, however, is barely covered. So far there are no established backtesting procedures for EL that quantify its impact with regards to pricing or risk-adjusted profitability measures. In this paper, a practically oriented, top-down approach to assessing the quality of EL by backtesting with a properly defined risk measure is introduced. In a first step, the concept of risk expenses ("Cost of Risk") has to be extended beyond the classical provisioning view, toward a more adequate capital consumption approach ("Impact of Risk"). On this basis, the difference between parameter-based EL and actually reported Impact of Risk is decomposed into its key components. The proposed method will deepen the understanding of the practical properties of EL, reconcile the EL with a clearly defined and observable risk measure and provide a link between upcoming IFRS 9 accounting standards for loan loss provisioning and the regulatory capital requirements under the internal ratings-based approach (IRBA). The method is robust irrespective of whether parameters are simple, expert-based values or highly predictive and perfectly calibrated IRBA-compliant methods, as long as the parameters and default identification procedures are stable. The attached pdf is aligned with respect to numbering of formulas, lemmas etc to the published version to simplify referencing. The terminology got slightly updated to provide a more consistent basis for further empirical research: PL/NPL Backtest is replaced by PL/NPL Dashboard.
{"title":"Expected Loss and Impact of Risk: Backtesting Parameter-Based Expected Loss in a Basel II Framework","authors":"W. Reitgruber","doi":"10.21314/JRMV.2013.107","DOIUrl":"https://doi.org/10.21314/JRMV.2013.107","url":null,"abstract":"The dependency structure of credit risk parameters is a key driver for capital consumption and receives regulatory and scientific attention. The impact of parameter imperfections on the quality of expected loss (EL) in the sense of a fair, unbiased estimate of risk expenses, however, is barely covered. So far there are no established backtesting procedures for EL that quantify its impact with regards to pricing or risk-adjusted profitability measures. In this paper, a practically oriented, top-down approach to assessing the quality of EL by backtesting with a properly defined risk measure is introduced. In a first step, the concept of risk expenses (\"Cost of Risk\") has to be extended beyond the classical provisioning view, toward a more adequate capital consumption approach (\"Impact of Risk\"). On this basis, the difference between parameter-based EL and actually reported Impact of Risk is decomposed into its key components. The proposed method will deepen the understanding of the practical properties of EL, reconcile the EL with a clearly defined and observable risk measure and provide a link between upcoming IFRS 9 accounting standards for loan loss provisioning and the regulatory capital requirements under the internal ratings-based approach (IRBA). The method is robust irrespective of whether parameters are simple, expert-based values or highly predictive and perfectly calibrated IRBA-compliant methods, as long as the parameters and default identification procedures are stable. The attached pdf is aligned with respect to numbering of formulas, lemmas etc to the published version to simplify referencing. The terminology got slightly updated to provide a more consistent basis for further empirical research: PL/NPL Backtest is replaced by PL/NPL Dashboard.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114830350","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 paper introduces a new, moment-based representation of version independent, coherent risk functionals for distributions with a finite second moment. The representation is based on L-moments. We analyze the second- and the third-order approximations and provide a method for constructing coherent approximations with the first few moments of the distribution. The method can be applied to coherent and non-coherent risk functionals and is interpreted in terms of a weighted average of particular Bayesian versions of Conditional Value-at-Risk. We formulate a conservative risk functional and a minimax portfolio construction problem which is non-parametric, convex, and exhibits a relative statistical robustness of the optimal solution compared to the classical utility-based approach. The developed approach bridges the gap between the intuitive utility-based higher-order moment portfolio construction and the formal construct of coherent risk functionals.
{"title":"Coherent Moment-Based Approximations of Risk Functionals","authors":"Stoyan Stoyanov","doi":"10.2139/ssrn.2346011","DOIUrl":"https://doi.org/10.2139/ssrn.2346011","url":null,"abstract":"The paper introduces a new, moment-based representation of version independent, coherent risk functionals for distributions with a finite second moment. The representation is based on L-moments. We analyze the second- and the third-order approximations and provide a method for constructing coherent approximations with the first few moments of the distribution. The method can be applied to coherent and non-coherent risk functionals and is interpreted in terms of a weighted average of particular Bayesian versions of Conditional Value-at-Risk. We formulate a conservative risk functional and a minimax portfolio construction problem which is non-parametric, convex, and exhibits a relative statistical robustness of the optimal solution compared to the classical utility-based approach. The developed approach bridges the gap between the intuitive utility-based higher-order moment portfolio construction and the formal construct of coherent risk functionals.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124733532","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}
We propose a portfolio approach for operational risk quantification based on a class of analytical models from which we derive new results on the correlation problem. In particular, we show that uniform correlation is a robust assumption for measuring capital charges in these models.
{"title":"Analytical Models of Operational Risk and New Results on the Correlation Problem","authors":"V. Brunel","doi":"10.2139/ssrn.2281340","DOIUrl":"https://doi.org/10.2139/ssrn.2281340","url":null,"abstract":"We propose a portfolio approach for operational risk quantification based on a class of analytical models from which we derive new results on the correlation problem. In particular, we show that uniform correlation is a robust assumption for measuring capital charges in these models.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117097096","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}
We develop a self-contained pedagogical introduction to monetary measures of risk. This part deals with general definitions and a treatment of Value-at-Risk.
我们开发了一个独立的教学介绍风险的货币措施。这一部分涉及一般定义和风险价值的处理。
{"title":"Monetary Measurement of Risk: A Critical Overview - Part I: General Definitions and Value-at-Risk","authors":"Lionel Lecesne, Andrea Roncoroni","doi":"10.2139/ssrn.2689131","DOIUrl":"https://doi.org/10.2139/ssrn.2689131","url":null,"abstract":"We develop a self-contained pedagogical introduction to monetary measures of risk. This part deals with general definitions and a treatment of Value-at-Risk.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"192 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116522443","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 Cornish-Fisher expansion is a popular method to adjust value-at-risk calculations for the skewness and kurtosis of non-normal return distribution. On the other hand, it is an open secret that “modified value-at-risk” calculations produce “strange” results from time to time, under certain parameter constellations. But the phenomenon was poorly understood, and no guidance was available from academia. In this research note, we illustrate the shortcomings of the traditional Cornish-Fisher expansion, by analyzing the distribution of S&P 500 price returns. We apply insights from recent research, which turns the Cornish-Fisher expansion into a well-behaved and accurate tool for modelling empirical non-normal return distributions.
{"title":"Fitting Non-Normal Distributions With Calibrated Cornish-Fisher Expansions","authors":"Andreas Steiner","doi":"10.2139/ssrn.2287543","DOIUrl":"https://doi.org/10.2139/ssrn.2287543","url":null,"abstract":"The Cornish-Fisher expansion is a popular method to adjust value-at-risk calculations for the skewness and kurtosis of non-normal return distribution. On the other hand, it is an open secret that “modified value-at-risk” calculations produce “strange” results from time to time, under certain parameter constellations. But the phenomenon was poorly understood, and no guidance was available from academia. In this research note, we illustrate the shortcomings of the traditional Cornish-Fisher expansion, by analyzing the distribution of S&P 500 price returns. We apply insights from recent research, which turns the Cornish-Fisher expansion into a well-behaved and accurate tool for modelling empirical non-normal return distributions.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121420662","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}
As, in light of the recent financial crises, stress tests have become an integral part of risk management and banking supervision, the analysis and understanding of risk model behaviour under stress has become ever more important. In this paper, we present a general approach to implementing stress scenarios in a multi-factor credit portfolio model and analyse asset correlations, default probabilities and default correlations under stress. We use our results to study the implications for credit reserves and capital requirements and illustrate the proposed methodology by stressing a large investment banking portfolio. Although our stress testing approach is developed in a particular credit portfolio model, the main concept - stressing risk factors through a truncation of their distributions - is independent of the model specification and can be applied to other risk types as well.
{"title":"Stressed Testing in Credit Portfolio Models","authors":"M. Kalkbrener, L. Overbeck","doi":"10.2139/ssrn.2269536","DOIUrl":"https://doi.org/10.2139/ssrn.2269536","url":null,"abstract":"As, in light of the recent financial crises, stress tests have become an integral part of risk management and banking supervision, the analysis and understanding of risk model behaviour under stress has become ever more important. In this paper, we present a general approach to implementing stress scenarios in a multi-factor credit portfolio model and analyse asset correlations, default probabilities and default correlations under stress. We use our results to study the implications for credit reserves and capital requirements and illustrate the proposed methodology by stressing a large investment banking portfolio. Although our stress testing approach is developed in a particular credit portfolio model, the main concept - stressing risk factors through a truncation of their distributions - is independent of the model specification and can be applied to other risk types as well.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126302817","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}
In a move towards effective management of interest rate risk in Indian banking, in addition to the existing return on Interest Rate Sensitivity under Traditional Gap Analysis, a new return is being introduced to monitor the interest rate risk using Duration Gap Analysis (DGA), called Interest Rate Sensitivity under Duration Gap Analysis (IRSD). The DGA involves bucketing of all Risk Sensitive Assets (RSA) and Risk Sensitive Liabilities (RSL) as per residual maturity/re-pricing dates in various time bands and computing the Modified Duration Gap (MDG). One of the important things to note is that the RSA and RSL include the rate-sensitive off-balance sheet assets and liabilities as well. MDG can be used to evaluate the impact on the Market Value of Equity (MVE) of the bank under different interest rate scenarios. The past few years have seen banks’ foray into financing long-term assets, such as home loans and infrastructure projects. Banks have been allowed to raise funds through long-term bonds with a minimum maturity of five years to the extent of their exposure of residual maturity of more than five years to the infrastructural sector. This article attempts to illustrate the significance of interest rate risk management and approaches towards its management in the Indian context.
{"title":"Management of Interest Rate Risk in Indian Banking","authors":"Vighneswara Swamy","doi":"10.2139/ssrn.2265288","DOIUrl":"https://doi.org/10.2139/ssrn.2265288","url":null,"abstract":"In a move towards effective management of interest rate risk in Indian banking, in addition to the existing return on Interest Rate Sensitivity under Traditional Gap Analysis, a new return is being introduced to monitor the interest rate risk using Duration Gap Analysis (DGA), called Interest Rate Sensitivity under Duration Gap Analysis (IRSD). The DGA involves bucketing of all Risk Sensitive Assets (RSA) and Risk Sensitive Liabilities (RSL) as per residual maturity/re-pricing dates in various time bands and computing the Modified Duration Gap (MDG). One of the important things to note is that the RSA and RSL include the rate-sensitive off-balance sheet assets and liabilities as well. MDG can be used to evaluate the impact on the Market Value of Equity (MVE) of the bank under different interest rate scenarios. The past few years have seen banks’ foray into financing long-term assets, such as home loans and infrastructure projects. Banks have been allowed to raise funds through long-term bonds with a minimum maturity of five years to the extent of their exposure of residual maturity of more than five years to the infrastructural sector. This article attempts to illustrate the significance of interest rate risk management and approaches towards its management in the Indian context.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130114591","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}
We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion. Optimal strategies in the adapted class under various risk criteria, namely value-at-risk, expected shortfall and a new criterion called "squared asset expectation" (SAE), related to a version of the cost variance measure, are derived and compared. It is well known that displaced diffusions (DD) exhibit dynamics which are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter. Furthermore, DD allows for changes in the support of the mid asset price distribution, allowing one to include a minimum permitted value for the mid price, either positive or negative. We study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although differences are not negligible for high levels of risk aversion and low market impact assets. This is illustrated with numerical examples.
{"title":"Optimal Execution Comparison Across Risks and Dynamics, with Solutions for Displaced Diffusions","authors":"D. Brigo, Giuseppe Di Graziano","doi":"10.2139/ssrn.2247951","DOIUrl":"https://doi.org/10.2139/ssrn.2247951","url":null,"abstract":"We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion. Optimal strategies in the adapted class under various risk criteria, namely value-at-risk, expected shortfall and a new criterion called \"squared asset expectation\" (SAE), related to a version of the cost variance measure, are derived and compared. It is well known that displaced diffusions (DD) exhibit dynamics which are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter. Furthermore, DD allows for changes in the support of the mid asset price distribution, allowing one to include a minimum permitted value for the mid price, either positive or negative. We study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although differences are not negligible for high levels of risk aversion and low market impact assets. This is illustrated with numerical examples.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123750243","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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