Abstract Asymptotic expansions for the tails of sums of random variables with regularly varying tails are mainly derived in the case of identically distributed random variables or in the case of random variables with the same tail index. Moreover, the higher-order terms are often given under the condition of existence of a moment of the distribution. In this paper, we obtain infinite series expansions for convolutions of Pareto distributions with non-integer tail indices. The Pareto random variables may have different tail indices and different scale parameters. We naturally find the same constants for the first terms as given in the previous asymptotic expansions in the case of identically distributed random variables, but we are now able to give the next additional terms. Since our series expansion is not asymptotic, it may be also used to compute the values of quantiles of the distribution of the sum as well as other risk measures such as the Tail Value at Risk. Examples of values are provided for the sum of at least five Pareto random variables and are compared to those determined via previous asymptotic expansions or via simulations.
{"title":"Series expansions for convolutions of Pareto distributions","authors":"Q. Nguyen, C. Robert","doi":"10.1515/strm-2014-1168","DOIUrl":"https://doi.org/10.1515/strm-2014-1168","url":null,"abstract":"Abstract Asymptotic expansions for the tails of sums of random variables with regularly varying tails are mainly derived in the case of identically distributed random variables or in the case of random variables with the same tail index. Moreover, the higher-order terms are often given under the condition of existence of a moment of the distribution. In this paper, we obtain infinite series expansions for convolutions of Pareto distributions with non-integer tail indices. The Pareto random variables may have different tail indices and different scale parameters. We naturally find the same constants for the first terms as given in the previous asymptotic expansions in the case of identically distributed random variables, but we are now able to give the next additional terms. Since our series expansion is not asymptotic, it may be also used to compute the values of quantiles of the distribution of the sum as well as other risk measures such as the Tail Value at Risk. Examples of values are provided for the sum of at least five Pareto random variables and are compared to those determined via previous asymptotic expansions or via simulations.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2015-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2014-1168","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313080","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}
S. Battiston, G. Caldarelli, M. d’Errico, S. Gurciullo
Abstract We develop a novel stress-test framework to monitor systemic risk in financial systems. The modular structure of the framework allows to accommodate for a variety of shock scenarios, methods to estimate interbank exposures and mechanisms of distress propagation. The main features are as follows. First, the framework allows to estimate and disentangle not only first-round effects (i.e. shock on external assets) and second-round effects (i.e. distress induced in the interbank network), but also third-round effects induced by possible fire sales. Second, it allows to monitor at the same time the impact of shocks on individual or groups of financial institutions as well as their vulnerability to shocks on counterparties or certain asset classes. Third, it includes estimates for loss distributions, thus combining network effects with familiar risk measures such as VaR and CVaR. Fourth, in order to perform robustness analyses and cope with incomplete data, the framework features a module for the generation of sets of networks of interbank exposures that are coherent with the total lending and borrowing of each bank. As an illustration, we carry out a stress-test exercise on a dataset of listed European banks over the years 2008–2013. We find that second-round and third-round effects dominate first-round effects, therefore suggesting that most current stress-test frameworks might lead to a severe underestimation of systemic risk.
{"title":"Leveraging the network: A stress-test framework based on DebtRank","authors":"S. Battiston, G. Caldarelli, M. d’Errico, S. Gurciullo","doi":"10.1515/strm-2015-0005","DOIUrl":"https://doi.org/10.1515/strm-2015-0005","url":null,"abstract":"Abstract We develop a novel stress-test framework to monitor systemic risk in financial systems. The modular structure of the framework allows to accommodate for a variety of shock scenarios, methods to estimate interbank exposures and mechanisms of distress propagation. The main features are as follows. First, the framework allows to estimate and disentangle not only first-round effects (i.e. shock on external assets) and second-round effects (i.e. distress induced in the interbank network), but also third-round effects induced by possible fire sales. Second, it allows to monitor at the same time the impact of shocks on individual or groups of financial institutions as well as their vulnerability to shocks on counterparties or certain asset classes. Third, it includes estimates for loss distributions, thus combining network effects with familiar risk measures such as VaR and CVaR. Fourth, in order to perform robustness analyses and cope with incomplete data, the framework features a module for the generation of sets of networks of interbank exposures that are coherent with the total lending and borrowing of each bank. As an illustration, we carry out a stress-test exercise on a dataset of listed European banks over the years 2008–2013. We find that second-round and third-round effects dominate first-round effects, therefore suggesting that most current stress-test frameworks might lead to a severe underestimation of systemic risk.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2015-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2015-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313495","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 propose an offline change detection method for the famous Cox–Ingersoll–Ross model based on a continuous sample. We develop one- and two-sided testing procedures for both drift parameters of the process. The test process is based on estimators that are motivated by the discrete time least-squares estimators, and its asymptotic distribution under the no-change hypothesis is that of a Brownian bridge. We prove the asymptotic weak consistence of the test, and derive the asymptotic properties of the change-point estimator under the alternative hypothesis of change at one point in time.
{"title":"Change detection in the Cox–Ingersoll–Ross model","authors":"G. Pap, Tamás T. Szabó","doi":"10.1515/strm-2015-0008","DOIUrl":"https://doi.org/10.1515/strm-2015-0008","url":null,"abstract":"Abstract We propose an offline change detection method for the famous Cox–Ingersoll–Ross model based on a continuous sample. We develop one- and two-sided testing procedures for both drift parameters of the process. The test process is based on estimators that are motivated by the discrete time least-squares estimators, and its asymptotic distribution under the no-change hypothesis is that of a Brownian bridge. We prove the asymptotic weak consistence of the test, and derive the asymptotic properties of the change-point estimator under the alternative hypothesis of change at one point in time.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2015-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2015-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313463","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 It is standard in quantitative risk management to model a random vector 𝐗:={X t k } k=1,...,d ${mathbf {X}:=lbrace X_{t_k}rbrace _{k=1,ldots ,d}}$ of consecutive log-returns to ultimately analyze the probability law of the accumulated return X t 1 +⋯+X t d ${X_{t_1}+cdots +X_{t_d}}$ . By the Markov regression representation (see [25]), any stochastic model for 𝐗${mathbf {X}}$ can be represented as X t k =f k (X t 1 ,...,X t k-1 ,U k )${X_{t_k}=f_k(X_{t_1},ldots ,X_{t_{k-1}},U_k)}$ , k=1,...,d${k=1,ldots ,d}$ , yielding a decomposition into a vector 𝐔:={U k } k=1,...,d ${mathbf {U}:=lbrace U_{k}rbrace _{k=1,ldots ,d}}$ of i.i.d. random variables accounting for the randomness in the model, and a function f:={f k } k=1,...,d ${f:=lbrace f_krbrace _{k=1,ldots ,d}}$ representing the economic reasoning behind. For most models, f is known explicitly and Uk may be interpreted as an exogenous risk factor affecting the return Xtk in time step k. While existing literature addresses model uncertainty by manipulating the function f, we introduce a new philosophy by distorting the source of randomness 𝐔${mathbf {U}}$ and interpret this as an analysis of the model's robustness. We impose consistency conditions for a reasonable distortion and present a suitable probability law and a stochastic representation for 𝐔${mathbf {U}}$ based on a Dirichlet prior. The resulting framework has one parameter c∈[0,∞]${cin [0,infty ]}$ tuning the severity of the imposed distortion. The universal nature of the methodology is illustrated by means of a case study comparing the effect of the distortion to different models for 𝐗${mathbf {X}}$ . As a mathematical byproduct, the consistency conditions of the suggested distortion function reveal interesting insights into the dependence structure between samples from a Dirichlet prior.
摘要风险定量管理的标准方法是建立随机向量 ={X t k } k=1，…，d ${mathbf {X}:=lbrace X_{t_k}rbrace _{k=1,ldots ,d}}$ 连续对数收益的概率律，最终分析累计收益X t 1 +⋯+X t d的概率律 ${X_{t_1}+cdots +X_{t_d}}$ ． 通过马尔可夫回归表示(见[25])，任意的随机模型${mathbf {X}}$ 可以表示为X t k =f k (X t 1，…，X t k-1,U k)${X_{t_k}=f_k(X_{t_1},ldots ,X_{t_{k-1}},U_k)}$ ， k=1，…，d${k=1,ldots ,d}$ ，得到向量的分解:={英国 } k=1，…，d ${mathbf {U}:=lbrace U_{k}rbrace _{k=1,ldots ,d}}$ 的i.i.d随机变量，表示模型中的随机性，函数f:={F。 } k=1，…，d ${f:=lbrace f_krbrace _{k=1,ldots ,d}}$ 代表背后的经济推理。对于大多数模型，f是明确已知的，而Uk可能被解释为影响时间步长k的回报Xtk的外生风险因素。虽然现有文献通过操纵函数f来解决模型不确定性，但我们通过扭曲随机性来源引入了一种新的哲学${mathbf {U}}$ 并将其解释为对模型稳健性的分析。我们对合理的失真施加了一致性条件，并给出了一个合适的概率律和一个随机表示${mathbf {U}}$ 基于狄利克雷先验。得到的框架有一个参数c∈[0，∞]${cin [0,infty ]}$ 调整施加的扭曲的严重程度。通过一个案例研究，比较了失真对不同模型的影响，说明了该方法的普遍性${mathbf {X}}$ ． 作为数学上的副产品，所建议的失真函数的一致性条件揭示了对Dirichlet先验样本之间依赖结构的有趣见解。
{"title":"Analyzing model robustness via a distortion of the stochastic root: A Dirichlet prior approach","authors":"Jan-Frederik Mai, Steffen Schenk, M. Scherer","doi":"10.1515/strm-2015-0009","DOIUrl":"https://doi.org/10.1515/strm-2015-0009","url":null,"abstract":"Abstract It is standard in quantitative risk management to model a random vector 𝐗:={X t k } k=1,...,d ${mathbf {X}:=lbrace X_{t_k}rbrace _{k=1,ldots ,d}}$ of consecutive log-returns to ultimately analyze the probability law of the accumulated return X t 1 +⋯+X t d ${X_{t_1}+cdots +X_{t_d}}$ . By the Markov regression representation (see [25]), any stochastic model for 𝐗${mathbf {X}}$ can be represented as X t k =f k (X t 1 ,...,X t k-1 ,U k )${X_{t_k}=f_k(X_{t_1},ldots ,X_{t_{k-1}},U_k)}$ , k=1,...,d${k=1,ldots ,d}$ , yielding a decomposition into a vector 𝐔:={U k } k=1,...,d ${mathbf {U}:=lbrace U_{k}rbrace _{k=1,ldots ,d}}$ of i.i.d. random variables accounting for the randomness in the model, and a function f:={f k } k=1,...,d ${f:=lbrace f_krbrace _{k=1,ldots ,d}}$ representing the economic reasoning behind. For most models, f is known explicitly and Uk may be interpreted as an exogenous risk factor affecting the return Xtk in time step k. While existing literature addresses model uncertainty by manipulating the function f, we introduce a new philosophy by distorting the source of randomness 𝐔${mathbf {U}}$ and interpret this as an analysis of the model's robustness. We impose consistency conditions for a reasonable distortion and present a suitable probability law and a stochastic representation for 𝐔${mathbf {U}}$ based on a Dirichlet prior. The resulting framework has one parameter c∈[0,∞]${cin [0,infty ]}$ tuning the severity of the imposed distortion. The universal nature of the methodology is illustrated by means of a case study comparing the effect of the distortion to different models for 𝐗${mathbf {X}}$ . As a mathematical byproduct, the consistency conditions of the suggested distortion function reveal interesting insights into the dependence structure between samples from a Dirichlet prior.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2015-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313574","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 propose a model for the computation of the loss probability distribution allowing to take into account the not-exchangeable behavior of a portfolio clustered into several classes of homogeneous loans. These classes are classified as `large' or `small' depending on their cardinality. The hierarchical hybrid copula-based model (HHC for short) follows the idea of the clusterized homogeneous copula-based approach (CHC) and its limiting version or the limiting clusterized copula-based model (LCC) proposed in our earlier work. This model allows us to recover a possible risk hierarchy. We suggest an algorithm to compute the HHC loss distribution and we compare this cdf with that computed through the CHC and LCC approaches (in the Gaussian and Archimedean limit) and also with the pure limiting approaches which are commonly used for high-dimensional problems. We study the scalability of the algorithm.
{"title":"A copula-based hierarchical hybrid loss distribution","authors":"E. Bernardi, S. Romagnoli","doi":"10.1515/strm-2012-1128","DOIUrl":"https://doi.org/10.1515/strm-2012-1128","url":null,"abstract":"Abstract We propose a model for the computation of the loss probability distribution allowing to take into account the not-exchangeable behavior of a portfolio clustered into several classes of homogeneous loans. These classes are classified as `large' or `small' depending on their cardinality. The hierarchical hybrid copula-based model (HHC for short) follows the idea of the clusterized homogeneous copula-based approach (CHC) and its limiting version or the limiting clusterized copula-based model (LCC) proposed in our earlier work. This model allows us to recover a possible risk hierarchy. We suggest an algorithm to compute the HHC loss distribution and we compare this cdf with that computed through the CHC and LCC approaches (in the Gaussian and Archimedean limit) and also with the pure limiting approaches which are commonly used for high-dimensional problems. We study the scalability of the algorithm.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2012-1128","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312259","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 Equity basket correlation can be estimated both using the physical measure from stock prices, and also using the risk neutral measure from option prices. The difference between the two estimates motivates a so-called “dispersion strategy”. We study the performance of this strategy on the German market and propose several profitability improvement schemes based on implied correlation (IC) forecasts. Modelling IC conceals several challenges. Firstly the number of correlation coefficients would grow with the size of the basket. Secondly, IC is not constant over maturities and strikes. Finally, IC changes over time. We reduce the dimensionality of the problem by assuming equicorrelation. The IC surface (ICS) is then approximated from the implied volatilities of stocks and the implied volatility of the basket. To analyze the dynamics of the ICS we employ a dynamic semiparametric factor model.
{"title":"Implied basket correlation dynamics","authors":"W. Härdle, Elena Silyakova","doi":"10.1515/strm-2014-1176","DOIUrl":"https://doi.org/10.1515/strm-2014-1176","url":null,"abstract":"Abstract Equity basket correlation can be estimated both using the physical measure from stock prices, and also using the risk neutral measure from option prices. The difference between the two estimates motivates a so-called “dispersion strategy”. We study the performance of this strategy on the German market and propose several profitability improvement schemes based on implied correlation (IC) forecasts. Modelling IC conceals several challenges. Firstly the number of correlation coefficients would grow with the size of the basket. Secondly, IC is not constant over maturities and strikes. Finally, IC changes over time. We reduce the dimensionality of the problem by assuming equicorrelation. The IC surface (ICS) is then approximated from the implied volatilities of stocks and the implied volatility of the basket. To analyze the dynamics of the ICS we employ a dynamic semiparametric factor model.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2014-1176","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313257","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 consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion. The aim is to derive a filter for the underlying continuous-time Markov chain. The recursion formula for the discrete-time filter is easy to derive, however involves densities which are very hard to obtain. In this paper we derive exact formulas for the necessary densities in the case the state space of the HMM consists of two elements only. This is done by relating the underlying integrated continuous-time Markov chain to the so-called asymmetric telegraph process and by using recent results on this process. In case the state space consists of more than two elements we present three different ways to approximate the densities for the filter. The first approach is based on the continuous filter problem. The second approach is to derive a PDE for the densities and solve it numerically. The third approach is a crude discrete time approximation of the Markov chain. All three approaches are compared in a numerical study.
{"title":"Exact and approximate hidden Markov chain filters based on discrete observations","authors":"N. Bäuerle, Igor Gilitschenski, U. Hanebeck","doi":"10.1515/strm-2015-0004","DOIUrl":"https://doi.org/10.1515/strm-2015-0004","url":null,"abstract":"Abstract We consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion. The aim is to derive a filter for the underlying continuous-time Markov chain. The recursion formula for the discrete-time filter is easy to derive, however involves densities which are very hard to obtain. In this paper we derive exact formulas for the necessary densities in the case the state space of the HMM consists of two elements only. This is done by relating the underlying integrated continuous-time Markov chain to the so-called asymmetric telegraph process and by using recent results on this process. In case the state space consists of more than two elements we present three different ways to approximate the densities for the filter. The first approach is based on the continuous filter problem. The second approach is to derive a PDE for the densities and solve it numerically. The third approach is a crude discrete time approximation of the Markov chain. All three approaches are compared in a numerical study.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2015-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313428","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 concerns sequential computation of risk measures for financial data and asks how, given a risk measurement procedure, we can tell whether the answers it produces are ‘correct’. We draw the distinction between ‘external’ and ‘internal’ risk measures and concentrate on the latter, where we observe data in real time, make predictions and observe outcomes. It is argued that evaluation of such procedures is best addressed from the point of view of probability forecasting or Dawid’s theory of ‘prequential statistics’ [12]. We introduce a concept of ‘calibration’ of a risk measure in a dynamic setting, following the precepts of Dawid’s weak and strong prequential principles, and examine its application to quantile forecasting (VaR – value at risk) and to mean estimation (applicable to CVaR – expected shortfall). The relationship between these ideas and ‘elicitability’ [24] is examined. We show in particular that VaR has special properties not shared by any other risk measure. Turning to CVaR we argue that its main deficiency is the unquantifiable tail dependence of estimators. In a final section we show that a simple data-driven feedback algorithm can produce VaR estimates on financial data that easily pass both the consistency test and a further newly-introduced statistical test for independence of a binary sequence.
{"title":"Verification of internal risk measure estimates","authors":"Mark H. A. Davis","doi":"10.1515/strm-2015-0007","DOIUrl":"https://doi.org/10.1515/strm-2015-0007","url":null,"abstract":"Abstract This paper concerns sequential computation of risk measures for financial data and asks how, given a risk measurement procedure, we can tell whether the answers it produces are ‘correct’. We draw the distinction between ‘external’ and ‘internal’ risk measures and concentrate on the latter, where we observe data in real time, make predictions and observe outcomes. It is argued that evaluation of such procedures is best addressed from the point of view of probability forecasting or Dawid’s theory of ‘prequential statistics’ [12]. We introduce a concept of ‘calibration’ of a risk measure in a dynamic setting, following the precepts of Dawid’s weak and strong prequential principles, and examine its application to quantile forecasting (VaR – value at risk) and to mean estimation (applicable to CVaR – expected shortfall). The relationship between these ideas and ‘elicitability’ [24] is examined. We show in particular that VaR has special properties not shared by any other risk measure. Turning to CVaR we argue that its main deficiency is the unquantifiable tail dependence of estimators. In a final section we show that a simple data-driven feedback algorithm can produce VaR estimates on financial data that easily pass both the consistency test and a further newly-introduced statistical test for independence of a binary sequence.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2015-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313888","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 how regional prediction of car insurance risks can be improved for finer subregions by combining explanatory modeling with phenomenological models from industrial practice. Motivated by the control-variates technique, we propose a suitable combined predictor when claims data are available for regions but not for subregions. We provide explicit conditions which imply that the mean squared error of the combined predictor is smaller than the mean squared error of the standard predictor currently used in industry and smaller than predictors from explanatory modeling. We also discuss how a non-parametric random forest approach may be used to practically compute such predictors and consider an application to German car insurance data.
{"title":"Prediction of regionalized car insurance risks based on control variates","authors":"M. Christiansen, C. Hirsch, V. Schmidt","doi":"10.1515/strm-2013-1148","DOIUrl":"https://doi.org/10.1515/strm-2013-1148","url":null,"abstract":"Abstract We show how regional prediction of car insurance risks can be improved for finer subregions by combining explanatory modeling with phenomenological models from industrial practice. Motivated by the control-variates technique, we propose a suitable combined predictor when claims data are available for regions but not for subregions. We provide explicit conditions which imply that the mean squared error of the combined predictor is smaller than the mean squared error of the standard predictor currently used in industry and smaller than predictors from explanatory modeling. We also discuss how a non-parametric random forest approach may be used to practically compute such predictors and consider an application to German car insurance data.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2013-1148","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312928","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 By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).
{"title":"Stochastic orderings with respect to a capacity and an application to a financial optimization problem","authors":"Miryana Grigorova","doi":"10.1515/strm-2013-1151","DOIUrl":"https://doi.org/10.1515/strm-2013-1151","url":null,"abstract":"Abstract By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2013-1151","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313134","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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