Abstract Unlike classic risk sharing problems based on expected utilities or convex risk measures, quantile-based risk sharing problems exhibit two special features. First, quantile-based risk measures (such as the Value-at-Risk) are often not convex, and second, they ignore some part of the distribution of the risk. These features create technical challenges in establishing a full characterization of optimal allocations, a question left unanswered in the literature. In this paper, we address the issues on the existence and the characterization of (Pareto-)optimal allocations in risk sharing problems for the Range-Value-at-Risk family. It turns out that negative dependence, mutual exclusivity in particular, plays an important role in the optimal allocations, in contrast to positive dependence appearing in classic risk sharing problems. As a by-product of our main finding, we obtain some results on the optimization of the Value-at-Risk (VaR) and the Expected Shortfall, as well as a new result on the inf-convolution of VaR and a general distortion risk measure.
{"title":"Characterizing Optimal Allocations in Quantile-Based Risk Sharing","authors":"Ruodu Wang, Yunran Wei","doi":"10.2139/ssrn.3173503","DOIUrl":"https://doi.org/10.2139/ssrn.3173503","url":null,"abstract":"Abstract Unlike classic risk sharing problems based on expected utilities or convex risk measures, quantile-based risk sharing problems exhibit two special features. First, quantile-based risk measures (such as the Value-at-Risk) are often not convex, and second, they ignore some part of the distribution of the risk. These features create technical challenges in establishing a full characterization of optimal allocations, a question left unanswered in the literature. In this paper, we address the issues on the existence and the characterization of (Pareto-)optimal allocations in risk sharing problems for the Range-Value-at-Risk family. It turns out that negative dependence, mutual exclusivity in particular, plays an important role in the optimal allocations, in contrast to positive dependence appearing in classic risk sharing problems. As a by-product of our main finding, we obtain some results on the optimization of the Value-at-Risk (VaR) and the Expected Shortfall, as well as a new result on the inf-convolution of VaR and a general distortion risk measure.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125228964","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 model for conduct risk losses, in which conduct risk losses are characterized by having a small number of extremely large losses (perhaps only one) with more numerous smaller losses. It is assumed that the largest loss is actually a provision from which payments to customers are made periodically as required. We use the pseudo-marginal (PM) Markov chain Monte Carlo method to decompose the largest loss into smaller partitions in order to estimate 99.9% value-at-risk. The partitioning is done in a way that makes no assumption about the size of the partitions. The advantages and problems of using this method are discussed. The PM procedures were run on several representative data sets. The results indicate that, in cases where using approaches such as calculating a Monte Carlo-derived loss distribution yields a result that is not consistent with the risk profile expressed by the data, using the PM method yields results that have the required consistency.
{"title":"Estimation of Value-at-Risk for Conduct Risk Losses Using Pseudo-Marginal Markov Chain Monte Carlo","authors":"P. Mitic, Jiaqiao Hu","doi":"10.21314/jop.2019.232","DOIUrl":"https://doi.org/10.21314/jop.2019.232","url":null,"abstract":"We propose a model for conduct risk losses, in which conduct risk losses are characterized by having a small number of extremely large losses (perhaps only one) with more numerous smaller losses. It is assumed that the largest loss is actually a provision from which payments to customers are made periodically as required. We use the pseudo-marginal (PM) Markov chain Monte Carlo method to decompose the largest loss into smaller partitions in order to estimate 99.9% value-at-risk. The partitioning is done in a way that makes no assumption about the size of the partitions. The advantages and problems of using this method are discussed. The PM procedures were run on several representative data sets. The results indicate that, in cases where using approaches such as calculating a Monte Carlo-derived loss distribution yields a result that is not consistent with the risk profile expressed by the data, using the PM method yields results that have the required consistency.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124978039","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 work, we propose a non-parametric density estimation technique for measuring the risk in a credit portfolio, aiming at efficiently computing the marginal risk contributions. The novel method is based on wavelets, and we derive closed-form expressions to calculate the Value-at-Risk (VaR), the Expected Shortfall (ES) as well as the individual risk contributions to VaR (VaRC) and ES (ESC). We consider the multi-factor Gaussian and t-copula models for driving the defaults. The results obtained along the numerical experiments show the impressive accuracy and speed of this method when compared with crude Monte Carlo simulation. The presented methodology applies in the same manner regardless of the used model, and the computational performance is invariant under a considerable change in the dimension of the selected model. The speed-up with respect to the classical Monte Carlo approach ranges from twenty-five to one-thousand depending on the used model.
{"title":"Model-Free Computation of Risk Contributions in Credit Portfolios","authors":"Álvaro Leitao, L. Ortiz-Gracia","doi":"10.2139/ssrn.3273894","DOIUrl":"https://doi.org/10.2139/ssrn.3273894","url":null,"abstract":"Abstract In this work, we propose a non-parametric density estimation technique for measuring the risk in a credit portfolio, aiming at efficiently computing the marginal risk contributions. The novel method is based on wavelets, and we derive closed-form expressions to calculate the Value-at-Risk (VaR), the Expected Shortfall (ES) as well as the individual risk contributions to VaR (VaRC) and ES (ESC). We consider the multi-factor Gaussian and t-copula models for driving the defaults. The results obtained along the numerical experiments show the impressive accuracy and speed of this method when compared with crude Monte Carlo simulation. The presented methodology applies in the same manner regardless of the used model, and the computational performance is invariant under a considerable change in the dimension of the selected model. The speed-up with respect to the classical Monte Carlo approach ranges from twenty-five to one-thousand depending on the used model.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121562334","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 show that maximizing distortion risk measures over the set of distributions with given mean is equivalent to maximizing their concave counterpart. In the case of Value-at-Risk and Tail Value-at-Risk the equivalence also holds when adding information on higher moments.
{"title":"Equivalent Distortion Risk Measures on Moment Spaces","authors":"D. Cornilly, S. Vanduffel","doi":"10.2139/ssrn.3175936","DOIUrl":"https://doi.org/10.2139/ssrn.3175936","url":null,"abstract":"Abstract We show that maximizing distortion risk measures over the set of distributions with given mean is equivalent to maximizing their concave counterpart. In the case of Value-at-Risk and Tail Value-at-Risk the equivalence also holds when adding information on higher moments.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122825184","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 investigates the prediction of Value-at-Risk (VaR) using option-implied information obtained by the maximum entropy method. The maximum entropy method provides an estimate of the risk-neutral distribution based on option prices. Besides commonly used implied volatility, we obtain implied skewness, kurtosis and quantile from the estimated risk-neutral distribution. We find that using the implied volatility and implied quantile as explanatory variables significantly outperforms considered benchmarks in predicting the VaR, including the commonly used GARCH(1,1)-model. This holds for all considered VaR prediction models and VaR probability levels. Overall, a simple quantile regression model performs best for all considered VaR probability levels and forecast horizons.
{"title":"Value-at-Risk Prediction Using Option-Implied Risk Measures","authors":"Kai Schindelhauer, Chen Zhou","doi":"10.2139/ssrn.3279398","DOIUrl":"https://doi.org/10.2139/ssrn.3279398","url":null,"abstract":"This paper investigates the prediction of Value-at-Risk (VaR) using option-implied information obtained by the maximum entropy method. The maximum entropy method provides an estimate of the risk-neutral distribution based on option prices. Besides commonly used implied volatility, we obtain implied skewness, kurtosis and quantile from the estimated risk-neutral distribution. We find that using the implied volatility and implied quantile as explanatory variables significantly outperforms considered benchmarks in predicting the VaR, including the commonly used GARCH(1,1)-model. This holds for all considered VaR prediction models and VaR probability levels. Overall, a simple quantile regression model performs best for all considered VaR probability levels and forecast horizons.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126970271","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 focuses on risk over long time horizons and within extreme percentiles, which have attracted considerable recent interest in numerous subfields of finance. Value at risk (VaR) aggregates several components of asset risk into a single quantitative measurement and is commonly used in tail risk management. Due to realistic data limits, many practitioners might use the square-root-of-time rule (SRTR) to compute long-term VaR. However, serial dependence and heavy-tailedness can bias the SRTR. This paper addresses two deficiencies of the study by Wang et al. (2011), who propose the modified-SRTR (MSRTR) to partially correct the serial dependence and use subsampling estimation as the benchmark to verify the performance of MSRTR. First, we investigate the validity of the subsampling approach through numerical simulations. Second, to reduce the heavy-tailedness bias, we propose a new MSRTR approach (MSRTR∗) in light of the Central Limit Theorem (CLT). In the empirical study, 28 country-level exchange-traded funds (ETFs) from 2010 to 2015 are considered to estimate the 30-day VaR. After modifying both serial dependence and heavy-tailedness, our approach reduces the bias from 26.46% to 5.97%, on average, compared to the SRTR. We also provide a backtesting analysis to verify the robustness of the MSRTR∗. This new approach should be considered when estimating long-term VaR using short-term VaR.
{"title":"Measuring Long-Term Tail Risk: Evaluating the Performance of the Square-Root-of-Time Rule","authors":"Jying‐Nan Wang, Jiangze Du, Yuan‐Teng Hsu","doi":"10.2139/ssrn.3374288","DOIUrl":"https://doi.org/10.2139/ssrn.3374288","url":null,"abstract":"This paper focuses on risk over long time horizons and within extreme percentiles, which have attracted considerable recent interest in numerous subfields of finance. Value at risk (VaR) aggregates several components of asset risk into a single quantitative measurement and is commonly used in tail risk management. Due to realistic data limits, many practitioners might use the square-root-of-time rule (SRTR) to compute long-term VaR. However, serial dependence and heavy-tailedness can bias the SRTR. This paper addresses two deficiencies of the study by Wang et al. (2011), who propose the modified-SRTR (MSRTR) to partially correct the serial dependence and use subsampling estimation as the benchmark to verify the performance of MSRTR. First, we investigate the validity of the subsampling approach through numerical simulations. Second, to reduce the heavy-tailedness bias, we propose a new MSRTR approach (MSRTR∗) in light of the Central Limit Theorem (CLT). In the empirical study, 28 country-level exchange-traded funds (ETFs) from 2010 to 2015 are considered to estimate the 30-day VaR. After modifying both serial dependence and heavy-tailedness, our approach reduces the bias from 26.46% to 5.97%, on average, compared to the SRTR. We also provide a backtesting analysis to verify the robustness of the MSRTR∗. This new approach should be considered when estimating long-term VaR using short-term VaR.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133387521","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}
Tail dependence is a probability-based concept meant to address the challenge of detecting and modeling the extreme comovements that can be observed in many real-life situations. Huge financial losses for a bank, floods and epidemics are obvious instances of such extreme comovements. Like extreme value theory in the univariate case, tail dependence depends on asymptotic theory. Therefore, the statistical assessment of tail dependence faces exactly the same problem as extreme value theory: a scarcity of extreme event observations. In the field of dependence modeling, copulas have stood out as a tool of singular importance. They are widely used to account for the various dependence structures that can be encountered in real life. In 2009, Genest et al provided a series of tests to achieve copula selection but showed that these tests were not greatly powerful. This is all the more true when it comes to selecting a copula where tail dependence is crucial. In this paper, we suggest the use of tail indexes in order to detect the presence of tail dependence in a given data set and thus improve the process of selecting a copula. Because tail dependence often goes with data scarcity, we focus on this specific issue through an application to operational losses in the banking industry and propose a way to apply the benefits from theory in practice, while being conscious of the boundaries of such a notion.
{"title":"Tail Dependence in Small Samples: From Theory to Practice","authors":"Sophie Lavaud","doi":"10.21314/JOP.2017.196","DOIUrl":"https://doi.org/10.21314/JOP.2017.196","url":null,"abstract":"Tail dependence is a probability-based concept meant to address the challenge of detecting and modeling the extreme comovements that can be observed in many real-life situations. Huge financial losses for a bank, floods and epidemics are obvious instances of such extreme comovements. Like extreme value theory in the univariate case, tail dependence depends on asymptotic theory. Therefore, the statistical assessment of tail dependence faces exactly the same problem as extreme value theory: a scarcity of extreme event observations. In the field of dependence modeling, copulas have stood out as a tool of singular importance. They are widely used to account for the various dependence structures that can be encountered in real life. In 2009, Genest et al provided a series of tests to achieve copula selection but showed that these tests were not greatly powerful. This is all the more true when it comes to selecting a copula where tail dependence is crucial. In this paper, we suggest the use of tail indexes in order to detect the presence of tail dependence in a given data set and thus improve the process of selecting a copula. Because tail dependence often goes with data scarcity, we focus on this specific issue through an application to operational losses in the banking industry and propose a way to apply the benefits from theory in practice, while being conscious of the boundaries of such a notion.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121186705","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 study economic pricing of reinsurance contracts via competition of an insurer with multiple reinsurers. All firms are endowed with distortion risk measures or expected exponential utilities. We require that contracts are Pareto optimal, individually rational, and satisfy a competition constraint that we call coalition stability. Indemnities are characterized by imposing Pareto optimality, as studied in the literature. In this paper, we characterize the corresponding premiums. There is a gain for the insurer due to the competition constraint. When the firms use distortion risk measures, this constraint yields stability for subcoalitions, which is a condition akin to the core in cooperative game theory. We show this gain for the insurer in closed form. Then, we derive that the premium is represented by a distortion premium function. If the firms use expected exponential utilities, the premium is represented by an exponential premium. We illustrate this premium function with the Mean Conditional Value-at-Risk.
{"title":"Optimal Reinsurance with Multiple Reinsurers: Competitive Pricing and Coalition Stability","authors":"T. Boonen, K. S. Tan, S. Zhuang","doi":"10.2139/ssrn.3143224","DOIUrl":"https://doi.org/10.2139/ssrn.3143224","url":null,"abstract":"We study economic pricing of reinsurance contracts via competition of an insurer with multiple reinsurers. All firms are endowed with distortion risk measures or expected exponential utilities. We require that contracts are Pareto optimal, individually rational, and satisfy a competition constraint that we call coalition stability. Indemnities are characterized by imposing Pareto optimality, as studied in the literature. In this paper, we characterize the corresponding premiums. There is a gain for the insurer due to the competition constraint. When the firms use distortion risk measures, this constraint yields stability for subcoalitions, which is a condition akin to the core in cooperative game theory. We show this gain for the insurer in closed form. Then, we derive that the premium is represented by a distortion premium function. If the firms use expected exponential utilities, the premium is represented by an exponential premium. We illustrate this premium function with the Mean Conditional Value-at-Risk.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134553876","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 revisits the relationship between information risk and the cost of equity capital in the U.S. property-casualty (P/C) insurance industry. Eckles, Halek and Zhang (2014) find that information risk has no effect on the cost of equity using a sample of U.S. P/C insurers. Following their approach, we decompose information risk into innate and discretionary components. We find that innate information risk affects the cost of equity capital through two opposing channels. On the one hand, innate information risk directly increases an insurer’s cost of equity capital by increasing investors’ assessment of the riskiness of the insurer’s future cash flows. On the other hand, innate information risk indirectly decreases the insurer’s cost of equity capital by changing its production so that the assessed riskiness of the firm’s future cash flows are reduced. This (negative) indirect effect depends on factors that influence the insurer’s underwriting decisions. Our empirical results provide supporting evidence for a significant, positive direct effect of innate information risk, while the magnitude of the (negative) indirect effect increases with the insurer’s proportion of long-tail business and decreases with its affiliated reinsurance usage. As to the impact of discretionary information risk, our results are mixed. We also find that, on average, the overall effect of information risk on the cost of equity capital for property-casualty insurers is significant and negative.
{"title":"Information Risk and the Cost of Equity Capital Revisited: Evidence from the U.S. Property-Casualty Insurance Industry","authors":"Hua Chen, Yingrui Lu, Mary A. Weiss","doi":"10.2139/ssrn.3045472","DOIUrl":"https://doi.org/10.2139/ssrn.3045472","url":null,"abstract":"This paper revisits the relationship between information risk and the cost of equity capital in the U.S. property-casualty (P/C) insurance industry. Eckles, Halek and Zhang (2014) find that information risk has no effect on the cost of equity using a sample of U.S. P/C insurers. Following their approach, we decompose information risk into innate and discretionary components. We find that innate information risk affects the cost of equity capital through two opposing channels. On the one hand, innate information risk directly increases an insurer’s cost of equity capital by increasing investors’ assessment of the riskiness of the insurer’s future cash flows. On the other hand, innate information risk indirectly decreases the insurer’s cost of equity capital by changing its production so that the assessed riskiness of the firm’s future cash flows are reduced. This (negative) indirect effect depends on factors that influence the insurer’s underwriting decisions. Our empirical results provide supporting evidence for a significant, positive direct effect of innate information risk, while the magnitude of the (negative) indirect effect increases with the insurer’s proportion of long-tail business and decreases with its affiliated reinsurance usage. As to the impact of discretionary information risk, our results are mixed. We also find that, on average, the overall effect of information risk on the cost of equity capital for property-casualty insurers is significant and negative.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"113 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115747544","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 builds on the established two-stage modeling framework for retail mortgages in which loss given default is computed as the product of property possession given default probability and loss given possession. In deriving the former, previous studies have suffered from a lack of clarity in their definitions of the post default outcomes of “cure” (no loss) and “possession” (some loss). The present study remedies this through the use of competing risks survival analysis, where to cure requires completion of a probationary period in which accounts return to nondefault status only when the ability to make repayments is demonstrated for a certain number of consecutive months (a recent regulatory requirement of the European Banking Authority). For loss given possession the distribution of survival time until this event can be conveniently used to appreciate the discounting of future receivables from property sale.
{"title":"Addressing Probationary Period within a Competing Risks Survival Model for Retail Mortgage Loss Given Default","authors":"R. Wood, David Powell","doi":"10.21314/JCR.2017.228","DOIUrl":"https://doi.org/10.21314/JCR.2017.228","url":null,"abstract":"This paper builds on the established two-stage modeling framework for retail mortgages in which loss given default is computed as the product of property possession given default probability and loss given possession. In deriving the former, previous studies have suffered from a lack of clarity in their definitions of the post default outcomes of “cure” (no loss) and “possession” (some loss). The present study remedies this through the use of competing risks survival analysis, where to cure requires completion of a probationary period in which accounts return to nondefault status only when the ability to make repayments is demonstrated for a certain number of consecutive months (a recent regulatory requirement of the European Banking Authority). For loss given possession the distribution of survival time until this event can be conveniently used to appreciate the discounting of future receivables from property sale.","PeriodicalId":203996,"journal":{"name":"ERN: Value-at-Risk (Topic)","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133048721","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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