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Annegret Weng - One of the best experts on this subject based on the ideXlab platform.
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Practical aspects of modelling Parameter Uncertainty for risk capital calculation
Zeitschrift für die gesamte Versicherungswissenschaft, 2019Co-Authors: David Blanco, Annegret WengAbstract:We assume that an insurance undertaking models its risk by a random variable $$\boldsymbol{X}=\boldsymbol{X}(\theta_{0})$$ with a fixed Parameter (vector) $$\theta_{0}$$ . If the undertaking does not know $$\theta_{0}$$ and can only estimate it from historical data, it faces Parameter Uncertainty. Neglecting Parameter Uncertainty can lead to an underestimation of the true risk capital requirement (see e.g. Gerrard and Tsanakas 2011; Frohlich and Weng 2015). In this contribution we address some practical questions. To illustrate the relevance of the Parameter risk we determine the probability of solvency for a risk capital model not taking Parameter Uncertainty into account for different distributions and samples sizes. We then follow the “inversion method” introduced in Frohlich and Weng (2015) known to model an appropriate risk capital requirement respecting Parameter Uncertainty for a wide class of distributions and common estimation methods. We extend the idea to distribution families and estimation methods that have not been considered so far in this context but are frequently used to model the losses of an insurance undertaking.
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Parameter Uncertainty and reserve risk under Solvency II
Insurance: Mathematics and Economics, 2018Co-Authors: Andreas Fröhlich, Annegret WengAbstract:Abstract In this article we consider the Parameter risk in the context of internal modelling of the reserve risk under Solvency II. We point out that the objectives of risk capital calculations differ from those of classical reserving and conclude that standard methods of classical reserving focusing on the estimation error of claims reserves are in general not appropriate to model the full impact of Parameter Uncertainty for the reserve risk. Referring to the requirements of Solvency II we assess different methods to model Parameter Uncertainty for the reserve risk by comparing the attained probability of solvency with the required confidence level. We provide evidence that the popular bootstrapping approach is not appropriate to model Parameter Uncertainty for the reserve risk according to this quantitative assessment. For the normal model we present an adaption of the approach proposed in Frohlich and Weng (2015) based on an analytical result and derive a risk capital model attaining the required confidence level in good approximation. Furthermore, for the lognormal model experimental results suggest that even a direct application of the method proposed in Frohlich and Weng (2015) clearly outperforms the bootstrapping approach according to the quantitative criterion mentioned above.
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Parameter Uncertainty and reserve risk under Solvency II
2017Co-Authors: Andreas Fr "ohlich, Annegret WengAbstract:In this article we consider the Parameter risk in the context of internal modelling of the reserve risk under Solvency II. We discuss two opposed perspectives on Parameter Uncertainty and point out that standard methods of classical reserving focusing on the estimation error of claims reserves are in general not appropriate to model the impact of Parameter Uncertainty upon the actual risk of economic losses from the undertakings's perspective. Referring to the requirements of Solvency II we assess methods to model Parameter Uncertainty for the reserve risk by comparing the probability of solvency actually attained when modelling the solvency risk capital requirement based on the respective method to the required confidence level. Using the simple example of a normal model we show that the bootstrapping approach is not appropriate to model Parameter Uncertainty according to this criterion. We then present an adaptation of the approach proposed in \cite {froehlich2014}. Experimental results demonstrate that this new method yields a risk capital model for the reserve risk achieving the required confidence level in good approximation.
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Modelling Parameter Uncertainty for risk capital calculation
European Actuarial Journal, 2015Co-Authors: Andreas Fröhlich, Annegret WengAbstract:For risk capital calculation within the framework of Solvency II the possible loss of basic own funds over the next business year of an insurance undertaking is usually interpreted as a random variable \( \varvec{X} \). If we assume that the parametric distribution family \(\left\{ \varvec{X} (\theta )|\theta \in I\subseteq \mathbb {R}^d \right\} \) is known, but the Parameter \(\theta \) is unknown and has to be estimated from the available historical data, the undertaking faces Parameter Uncertainty. To assess methods to model Parameter Uncertainty for risk capital calculations we apply a criterion going back to the theory of predictive inference which has already been used in the context of Solvency II. In particular, we show that the bootstrapping approach is not appropriate to model Parameter Uncertainty from the undertaking’s perspective. Based on ideas closely related to the concept of fiducial inference we introduce a new approach to model Parameter Uncertainty. For a wide class of distributions and for common estimators including the maximum likelihood method we prove that this approach is appropriate to model Parameter Uncertainty according to the criterion mentioned above. Several examples demonstrate that our method can easily be applied in practice.
Dennis S. Bernstein - One of the best experts on this subject based on the ideXlab platform.
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The octomorphic criterion for real Parameter Uncertainty: real-m bounds without circles and D,N -scales
Systems & Control Letters, 1995Co-Authors: Wassim M. Haddad, Dennis S. BernsteinAbstract:Abstract In this paper we introduce new bounds for robust stability analysis with real Parameter Uncertainty. The approach is based on an absolute stability criterion that excludes the Nyquist plot from a paraboloidal region containing the point −1 + J0. Transformation of this criterion to the case of norm-bounded Uncertainty leads to a stability criterion in terms of the octomorphic, or figure-eight shaped, region. The requirement that the Nyquist plot lie inside the octomorphic region thus yields a bound on the allowable real Parameter Uncertainty. This stability criterion is distinct from recent bounds for real-μ which involve frequency-dependent scales having a frequency-dependent, off-axis circle interpretation. Since the octomorphic region includes both upper and lower halves, it is able to encompass the entire Nyquist plot without using frequency-dependent scales.
Andreas Fröhlich - One of the best experts on this subject based on the ideXlab platform.
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Parameter Uncertainty and reserve risk under Solvency II
Insurance: Mathematics and Economics, 2018Co-Authors: Andreas Fröhlich, Annegret WengAbstract:Abstract In this article we consider the Parameter risk in the context of internal modelling of the reserve risk under Solvency II. We point out that the objectives of risk capital calculations differ from those of classical reserving and conclude that standard methods of classical reserving focusing on the estimation error of claims reserves are in general not appropriate to model the full impact of Parameter Uncertainty for the reserve risk. Referring to the requirements of Solvency II we assess different methods to model Parameter Uncertainty for the reserve risk by comparing the attained probability of solvency with the required confidence level. We provide evidence that the popular bootstrapping approach is not appropriate to model Parameter Uncertainty for the reserve risk according to this quantitative assessment. For the normal model we present an adaption of the approach proposed in Frohlich and Weng (2015) based on an analytical result and derive a risk capital model attaining the required confidence level in good approximation. Furthermore, for the lognormal model experimental results suggest that even a direct application of the method proposed in Frohlich and Weng (2015) clearly outperforms the bootstrapping approach according to the quantitative criterion mentioned above.
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Modelling Parameter Uncertainty for risk capital calculation
European Actuarial Journal, 2015Co-Authors: Andreas Fröhlich, Annegret WengAbstract:For risk capital calculation within the framework of Solvency II the possible loss of basic own funds over the next business year of an insurance undertaking is usually interpreted as a random variable \( \varvec{X} \). If we assume that the parametric distribution family \(\left\{ \varvec{X} (\theta )|\theta \in I\subseteq \mathbb {R}^d \right\} \) is known, but the Parameter \(\theta \) is unknown and has to be estimated from the available historical data, the undertaking faces Parameter Uncertainty. To assess methods to model Parameter Uncertainty for risk capital calculations we apply a criterion going back to the theory of predictive inference which has already been used in the context of Solvency II. In particular, we show that the bootstrapping approach is not appropriate to model Parameter Uncertainty from the undertaking’s perspective. Based on ideas closely related to the concept of fiducial inference we introduce a new approach to model Parameter Uncertainty. For a wide class of distributions and for common estimators including the maximum likelihood method we prove that this approach is appropriate to model Parameter Uncertainty according to the criterion mentioned above. Several examples demonstrate that our method can easily be applied in practice.
Christian Bontemps - One of the best experts on this subject based on the ideXlab platform.
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Moment-Based Tests under Parameter Uncertainty
Review of Economics and Statistics, 2019Co-Authors: Christian BontempsAbstract:This paper considers moment-based tests applied to estimated quantities. We propose a general class of transforms of moments to handle the Parameter Uncertainty problem. The construction requires only a linear correction that can be implemented in sample and remains valid for some extended families of nonsmooth moments. We reemphasize the attractiveness of working with robust moments, which lead to testing procedures that do not depend on the estimator. Furthermore, no correction is needed when considering the implied test statistic in the out-of-sample case. We apply our methodology to various examples with an emphasis on the backtesting of value-at-risk forecasts.
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Moment-Based Tests under Parameter Uncertainty
The Review of Economics and Statistics, 2019Co-Authors: Christian BontempsAbstract:Abstract This paper considers moment-based tests applied to estimated quantities. We propose a general class of transforms of moments to handle the Parameter Uncertainty problem. The construction r...
Sridhar Tayur - One of the best experts on this subject based on the ideXlab platform.
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Winter Simulation Conference - Demand fulfillment probability under Parameter Uncertainty
2016 Winter Simulation Conference (WSC), 2016Co-Authors: Canan G. Corlu, Bahar Biller, Sridhar TayurAbstract:We study a multi-item inventory system with normally distributed demands in the presence of demand Parameter Uncertainty - the Uncertainty that stems from the estimation of the unknown demand Parameters necessitated by limited amounts of historical demand data. Using an asymptotic normality approximation, we quantify the variance of the demand fulfillment probability (i.e., the probability that all item demands will be satisfied from stock) that is due to demand Parameter Uncertainty. We use this quantification to understand the impact of demand Parameter Uncertainty on the demand fulfillment probability and investigate the sensitivity of the variance of the demand fulfillment probability to selected inventory model Parameters.
