Modified Value-at-Risk (mVaR) and Modified Expected Shortfall (mES) are risk estimators that can be calculated without modelling the distribution of asset returns. These modified estimators use skewness and kurtosis corrections to normal distribution parametric VaR and ES formulas to reduce bias in risk measurement for non-normal return distributions. However, the use of skewness and kurtosis estimators that are needed to implement mVaR and mES can lead to highly inflated mVaR and mES estimator standard errors. To assess the magnitude of standard error inflation we derive formulas for the large sample standard errors of mVaR and mES using multivariate delta method and compare them against standard errors of parametric VaR and ES estimators, under both normal and t-distributions. Our asymptotic results show that mVaR and mES estimators can have standard errors considerably larger than those of parametric VaR and ES estimators, and small-sample Monte Carlo confirms that the asymptotic results are approximately correct in sample sizes commonly used in practice.
{"title":"Inefficiency of Modified VaR and ES","authors":"Doug Martin, Rohit Arora","doi":"10.2139/ssrn.2692543","DOIUrl":"https://doi.org/10.2139/ssrn.2692543","url":null,"abstract":"Modified Value-at-Risk (mVaR) and Modified Expected Shortfall (mES) are risk estimators that can be calculated without modelling the distribution of asset returns. These modified estimators use skewness and kurtosis corrections to normal distribution parametric VaR and ES formulas to reduce bias in risk measurement for non-normal return distributions. However, the use of skewness and kurtosis estimators that are needed to implement mVaR and mES can lead to highly inflated mVaR and mES estimator standard errors. To assess the magnitude of standard error inflation we derive formulas for the large sample standard errors of mVaR and mES using multivariate delta method and compare them against standard errors of parametric VaR and ES estimators, under both normal and t-distributions. Our asymptotic results show that mVaR and mES estimators can have standard errors considerably larger than those of parametric VaR and ES estimators, and small-sample Monte Carlo confirms that the asymptotic results are approximately correct in sample sizes commonly used in practice.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117235942","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}
This paper develops earlier work on the impairment to the value of investment portfolios from global warming later this century. The growth in renewables and electric vehicles may be enough to strand fossil fuel assets from the late 2020s onwards, but will not alone bring emissions down fast enough to prevent high warming. A consequence is increasing systemic risk in investment portfolios. Using a probability-weighted family of climate damage functions it is estimated that the chance that future climate damage reaches one half of global gdp by 2100 is of the order of 3%. This outcome implies an equity portfolio value impairment of 10% currently, equivalent to $7 trillion in aggregate, increasing at 50 basis points a year. Development towards a high damage outcome of this kind could create a specific risk for the financial sector.
{"title":"The Value at Risk from Climate Change","authors":"Howard E. Covington","doi":"10.2139/ssrn.2681035","DOIUrl":"https://doi.org/10.2139/ssrn.2681035","url":null,"abstract":"This paper develops earlier work on the impairment to the value of investment portfolios from global warming later this century. The growth in renewables and electric vehicles may be enough to strand fossil fuel assets from the late 2020s onwards, but will not alone bring emissions down fast enough to prevent high warming. A consequence is increasing systemic risk in investment portfolios. Using a probability-weighted family of climate damage functions it is estimated that the chance that future climate damage reaches one half of global gdp by 2100 is of the order of 3%. This outcome implies an equity portfolio value impairment of 10% currently, equivalent to $7 trillion in aggregate, increasing at 50 basis points a year. Development towards a high damage outcome of this kind could create a specific risk for the financial sector.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"16 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114117263","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 present a two-factor option-pricing model, which parsimoniously captures the difference in volatility persistences under the historical and risk-neutral probabilities. The model generates an S-shaped pricing kernel that exhibits time-varying risk aversion. We apply our model for two purposes. First, we analyze the risk preference implied by S&P500 index options during 2001–2009 and find that risk-aversion level strongly increases during stressed market conditions. Second, we apply our model for Value-at-Risk (VaR) forecasts during the subprime crisis period and find that it outperforms several leading VaR models.
{"title":"Option Pricing Under Time-Varying Risk-Aversion with Applications to Risk Forecasting","authors":"Ruediger Kiesel, F. Rahe","doi":"10.2139/ssrn.2668542","DOIUrl":"https://doi.org/10.2139/ssrn.2668542","url":null,"abstract":"We present a two-factor option-pricing model, which parsimoniously captures the difference in volatility persistences under the historical and risk-neutral probabilities. The model generates an S-shaped pricing kernel that exhibits time-varying risk aversion. We apply our model for two purposes. First, we analyze the risk preference implied by S&P500 index options during 2001–2009 and find that risk-aversion level strongly increases during stressed market conditions. Second, we apply our model for Value-at-Risk (VaR) forecasts during the subprime crisis period and find that it outperforms several leading VaR models.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123003658","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}
This article presents a simple methodology for computing Value at Risk (VaR) for a portfolio of financial instruments that is sensitive to market risk, rating change, and default risk. An integrated model for market and credit risks is developed. The Jarrow, Lando and Turnbull model (the Markov chain model) is used to represent the dynamics of the credit rating. Procedures for calculating VaR are presented. Numerical illustration results are included.
{"title":"An Integrated Risk Management Method: VaR Approach","authors":"Hailiang Yang","doi":"10.17578/4-3/4-4","DOIUrl":"https://doi.org/10.17578/4-3/4-4","url":null,"abstract":"This article presents a simple methodology for computing Value at Risk (VaR) for a portfolio of financial instruments that is sensitive to market risk, rating change, and default risk. An integrated model for market and credit risks is developed. The Jarrow, Lando and Turnbull model (the Markov chain model) is used to represent the dynamics of the credit rating. Procedures for calculating VaR are presented. Numerical illustration results are included.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125060493","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 (marginal) risk contribution is very useful for analyzing the concentration risk in a portfolio. However, it is difficult to estimate the risk contributions for value-at-risk (VaR) and expected shortfall (ES) precisely, especially using a Monte Carlo simulation. We applied a saddlepoint approximation to estimate the distribution function, so that the difficulty of estimating the risk contributions for VaR was dissolved. In this paper, we propose new estimation methods for ES and the risk contributions for ES based on the conditional independence and a saddlepoint approximation. Numerical studies confirm that these new methods are much better than existing ones.
{"title":"Improved Estimation Methods for Value-at-Risk, Expected Shortfall and Risk Contributions with High Precision","authors":"Yukio Muromachi","doi":"10.21314/JOR.2015.314","DOIUrl":"https://doi.org/10.21314/JOR.2015.314","url":null,"abstract":"The (marginal) risk contribution is very useful for analyzing the concentration risk in a portfolio. However, it is difficult to estimate the risk contributions for value-at-risk (VaR) and expected shortfall (ES) precisely, especially using a Monte Carlo simulation. We applied a saddlepoint approximation to estimate the distribution function, so that the difficulty of estimating the risk contributions for VaR was dissolved. In this paper, we propose new estimation methods for ES and the risk contributions for ES based on the conditional independence and a saddlepoint approximation. Numerical studies confirm that these new methods are much better than existing ones.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121043867","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 Traffic Light approach to backtesting Expected Shortfall which is completely consistent and analogous to the Traffic Light approach to backtesting VaR initially proposed by the Basel Committee on Banking Supervision in their 1996 consultative document. The approach relies on the generalized coverage test for Expected Shortfall developed in.
{"title":"A Simple Traffic Light Approach to Backtesting Expected Shortfall","authors":"Nick Costanzino, Michael Curran","doi":"10.2139/ssrn.2603976","DOIUrl":"https://doi.org/10.2139/ssrn.2603976","url":null,"abstract":"We propose a Traffic Light approach to backtesting Expected Shortfall which is completely consistent and analogous to the Traffic Light approach to backtesting VaR initially proposed by the Basel Committee on Banking Supervision in their 1996 consultative document. The approach relies on the generalized coverage test for Expected Shortfall developed in.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121616168","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}
This paper studies the approximation of extreme quantiles of random sums of heavy-tailed random variables, or more specifically, subexponential random variables. A key application of this approximation is the calculation of operational VaR (value at risk) for financial institutions, to determine operational risk capital requirements. The paper follows work by Bocker & Kluppelberg (2005) & Bocker and Sprittulla (2006) and makes several advances. These include two new approximations of VaR and an extension to multiple loss types where the VaR relates to a sum of random sums, each of which is defined by different distributions. The proposed approximations are assessed via a simulation study.
{"title":"Approximations of Value-at-Risk As an Extreme Quantile of a Random Sum of Heavy-Tailed Random Variables","authors":"L. Hannah, B. Puza","doi":"10.21314/JOP.2015.154","DOIUrl":"https://doi.org/10.21314/JOP.2015.154","url":null,"abstract":"This paper studies the approximation of extreme quantiles of random sums of heavy-tailed random variables, or more specifically, subexponential random variables. A key application of this approximation is the calculation of operational VaR (value at risk) for financial institutions, to determine operational risk capital requirements. The paper follows work by Bocker & Kluppelberg (2005) & Bocker and Sprittulla (2006) and makes several advances. These include two new approximations of VaR and an extension to multiple loss types where the VaR relates to a sum of random sums, each of which is defined by different distributions. The proposed approximations are assessed via a simulation study.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131632606","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}
A central problem for regulators and risk managers concerns the risk assessment of an aggregate portfolio defined as the sum of d individual dependent risks Xi. This problem is mainly a numerical issue once the joint distribution of X1,X2,…,Xd is fully specified. Unfortunately, while the marginal distributions of the risks Xi are often known, their interaction (dependence) is usually either unknown or only partially known, implying that any risk assessment of the portfolio is subject to model uncertainty.
当X1,X2，…，Xd的联合分布完全确定后，这个问题主要是一个数值问题。
{"title":"A New Approach to Assessing Model Risk in High Dimensions","authors":"C. Bernard, S. Vanduffel","doi":"10.2139/ssrn.2393054","DOIUrl":"https://doi.org/10.2139/ssrn.2393054","url":null,"abstract":"A central problem for regulators and risk managers concerns the risk assessment of an aggregate portfolio defined as the sum of d individual dependent risks Xi. This problem is mainly a numerical issue once the joint distribution of X1,X2,…,Xd is fully specified. Unfortunately, while the marginal distributions of the risks Xi are often known, their interaction (dependence) is usually either unknown or only partially known, implying that any risk assessment of the portfolio is subject to model uncertainty.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134571680","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 this paper, we propose a copula-free approach for modeling correlated frequency distributions using an Erlang-based multivariate mixed Poisson distribution. We investigate some of the properties possessed by this class of distributions and derive a tailormade expectation-maximization algorithm for fitting purposes. The applicability of the proposed distribution is illustrated in an operational risk management context, where this class is used to model the operational loss frequencies and their complex dependence structure in a high-dimensional setting. Furthermore, by assuming that operational loss severities follow the mixture of Erlang distributions, our approach leads to a closed-form expression for the total aggregate loss distribution and its value-at-risk can be calculated easily by any numerical method. The efficiency and accuracy of the proposed approach are analyzed using a modified real operational loss data set.
{"title":"Modeling Correlated Frequencies with Application in Operational Risk Management","authors":"A. Badescu, Gong Lan, X. Lin, Dameng Tang","doi":"10.21314/JOP.2015.157","DOIUrl":"https://doi.org/10.21314/JOP.2015.157","url":null,"abstract":"In this paper, we propose a copula-free approach for modeling correlated frequency distributions using an Erlang-based multivariate mixed Poisson distribution. We investigate some of the properties possessed by this class of distributions and derive a tailormade expectation-maximization algorithm for fitting purposes. The applicability of the proposed distribution is illustrated in an operational risk management context, where this class is used to model the operational loss frequencies and their complex dependence structure in a high-dimensional setting. Furthermore, by assuming that operational loss severities follow the mixture of Erlang distributions, our approach leads to a closed-form expression for the total aggregate loss distribution and its value-at-risk can be calculated easily by any numerical method. The efficiency and accuracy of the proposed approach are analyzed using a modified real operational loss data set.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125377953","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 this paper, we modify the Constant Conditional Correlation (CCC) model and its dynamic counterpart, the Dynamic Conditional Correlation (DCC) model by combining them with a pairwise test for constant correlations, a test for a constant correlation matrix, and a test for a constant covariance matrix. We compare these models to their plain counterparts with respect to the accuracy for forecasting the Value-at-Risk of financial portfolios by a set of distinct backtests. In an empirical horse race of these models based on multivariate portfolios, our study shows that correlation models can be improved by approaches modified by tests for structural breaks in co-movements in several settings.
{"title":"Testing for Structural Breaks in Correlations: Does it Improve Value-at-Risk Forecasting?","authors":"Tobias Berens, Gregor N. F. Weiß, Dominik Wied","doi":"10.2139/ssrn.2265488","DOIUrl":"https://doi.org/10.2139/ssrn.2265488","url":null,"abstract":"In this paper, we modify the Constant Conditional Correlation (CCC) model and its dynamic counterpart, the Dynamic Conditional Correlation (DCC) model by combining them with a pairwise test for constant correlations, a test for a constant correlation matrix, and a test for a constant covariance matrix. We compare these models to their plain counterparts with respect to the accuracy for forecasting the Value-at-Risk of financial portfolios by a set of distinct backtests. In an empirical horse race of these models based on multivariate portfolios, our study shows that correlation models can be improved by approaches modified by tests for structural breaks in co-movements in several settings.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115948012","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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