scholarly journals Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory

1997 ◽  
Vol 27 (1) ◽  
pp. 117-137 ◽  
Author(s):  
Alexander J. McNeil
Keyword(s):  
Extreme Value Theory ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Parametric Curve ◽  
Excess Loss ◽  
Generalized Pareto ◽  
Fire Insurance ◽  
Loss Severity ◽  
Parametric Curve Fitting

AbstractGood estimates for the tails of loss severity distributions are essential for pricing or positioning high-excess loss layers in reinsurance. We describe parametric curve-fitting methods for modelling extreme historical losses. These methods revolve around the generalized Pareto distribution and are supported by extreme value theory. We summarize relevant theoretical results and provide an extensive example of their application to Danish data on large fire insurance losses.

Using Extreme Value Theory to Model Electricity Price Risk with an Application to the Alberta Power Market

2005 ◽  
Vol 23 (5) ◽  
pp. 375-403 ◽  
Author(s):  
W. D. Walls ◽  
Wei. Zhang
Keyword(s):  
Extreme Value Theory ◽  
Value At Risk ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Financial Assets ◽  
Historical Simulation ◽  
Generalized Pareto ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed

Value-at-risk (VaR) is a measure of the maximum potential change in value of a portfolio of financial assets with a given probability over a given time horizon. VaR has become a standard measure of market risk and a common practice is to compute VaR by assuming that changes in value of the portfolio are conditionally normally distributed. However, assets returns usually come from heavy-tailed distributions, so computing VaR under the assumption of conditional normality can be an important source of error. We illustrate in our application to competitive electric power prices in Alberta, Canada, that VaR estimates based on extreme value theory models, in particular the generalized Pareto distribution are, more accurate than those produced by alternative models such as normality or historical simulation.

scholarly journals Using Statistical Approaches to Model Natural Disasters

2016 ◽  
Vol 13 (2) ◽  
Author(s):  
Audrene Edwards ◽  
Kumer Das
Keyword(s):  
Extreme Events ◽  
Extreme Value Theory ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Threshold Values ◽  
Data Set ◽  
Generalized Pareto ◽  
Estimated Parameters

The study of extremes has attracted the attention of scientists, engineers, actuaries, policy makers, and statisticians for many years. Extreme value theory (EVT) deals with the extreme deviations from the median of probability distributions and is used to study rare but extreme events. EVT’s main results characterize the distribution of the sample maximum or the distribution of values above a given threshold. In this study, EVT has been used to construct a model on the extreme and rare earthquakes that have happened in the United States from 1700 to 2011.The primary goal of fitting such a model is to estimate the amount of losses due to those extreme events and the probabilities of such events. Several diagnostic methods (for example, QQ plot and Mean Excess Plot) have been used to justify that the data set follows generalized Pareto distribution (GPD). Three estimation techniques have been employed to estimate parameters. The consistency and reliability of estimated parameters have been observed for different threshold values. The purpose of this study is manifold: first, we investigate whether the data set follows GPD, by using graphical interpretation and hypothesis testing. Second, we estimate GPD parameters using three different estimation techniques. Third, we compare consistency and reliability of estimated parameters for different threshold values. Last, we investigate the bias of estimated parameters using a simulation study. The result is particularly useful because it can be used in many applications (for example, disaster management, engineering design, insurance industry, hydrology, ocean engineering, and traffic management) with a minimal set of assumptions about the true underlying distribution of a data set. KEYWORDS: Extreme Value Theory; QQ Plot; Mean Excess Plot; Mean Residual Plot; Peak Over Threshold; Generalized Pareto Distribution; Maximum Likelihood Method; Method of Moments; Probability-Weighted Moments; Shapiro-Wilk test; Anderson- Darling Test

scholarly journals Swimming performance index based on extreme value theory

2018 ◽  
Vol 14 (1) ◽  
pp. 51-62 ◽  
Author(s):  
Daniel T Gomes ◽  
Lígia Henriques-Rodrigues
Keyword(s):  
Extreme Value Theory ◽  
Performance Index ◽  
Pareto Distribution ◽  
Swimming Performance ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Selection Procedures ◽  
Generalized Pareto ◽  
Distribution Index

The International Swimming Federation has developed a points system that allows comparisons of results between different events. Such system is important for several reasons, since it is used as a criterion to rank swimmers in awards and selection procedures of national teams. The points system is based entirely on the world record of the correspondent event. Since it is based on only one observation, this work aims to suggest a new system, based on the probability distribution of the best performances in each event. Using extreme value theory, such distribution, under certain conditions, converges to a generalized Pareto distribution. The new performance index, based on the peaks over threshold methodology, is obtained based on the exceedance probabilities correspondent to the swimmers’ times that exceed a given threshold. We work with 17 officially recognized events in 50 m pool, for each women and men, and considered all-time rankings for all events until 31 December 2016. A study on the adequacy of the proposed generalized Pareto distribution index and a comparison between the performances of Usain Bolt and Michael Phelps are also conducted.

A New Generation of Earthquake Recurrence Models Based on The Extreme Value Theory and Impact on Probabilistic Seismic Hazard Assessments

2021 ◽  
Author(s):  
Anne Dutfoy ◽  
Gloria Senfaute
Keyword(s):  
Seismic Hazard ◽  
Extreme Value Theory ◽  
Generalized Pareto Distribution ◽  
Random Variable ◽  
Value Theory ◽  
Extreme Value ◽  
Extreme Value Analysis ◽  
Probabilistic Seismic Hazard ◽  
Universal Method ◽  
Value Analysis

Abstract Probabilistic Seismic Hazard Analysis (PSHA) procedures require that at least the mean activity rate be known, as well as the distribution of magnitudes. Within the Gutenberg-Richter assumption, that distribution is an Exponential distribution, upperly truncated to a maximum possible magnitude denoted $m_{max}$. This parameter is often fixed from expert judgement under tectonics considerations, due to a lack of universal method. In this paper, we propose two innovative alternatives to the Gutenberg-Richter model, based on the Extreme Value Theory and that don't require to fix a priori the value of $m_{max}$: the first one models the tail distribution magnitudes with a Generalized Pareto Distribution; the second one is a variation on the usual Gutenberg-Richter model where $m_{max}$ is a random variable that follows a distribution defined from an extreme value analysis. We use the maximum likelihood estimators taking into account the unequal observation spans depending on magnitude, the incompleteness threshold of the catalog and the uncertainty in the magnitude value itself. We apply these new recurrence models on the data observed in the Alps region, in the south of France and we integrate them into a probabilistic seismic hazard calculation to evaluate their impact on the seismic hazard levels. The proposed new recurrence models introduce a reduction of the seismic hazard level compared to the common Gutenberg-Richter model conventionally used for PSHA calculations. This decrease is significant for all frequencies below 10 Hz, mainly at the lowest frequencies and for very long return periods. To our knowledge, both new models have never been used in a probabilistic seismic hazard calculation and constitute a new promising generation of recurrence models.

Recurrence Models based on Extreme Value Theory : Impacts on Probabilistic Seismic Hazard Assessments and Comparison with the Standard Approach

2021 ◽  
Author(s):  
Anne Dutfoy ◽  
Gloria Senfaute
Keyword(s):  
Seismic Hazard ◽  
Exponential Distribution ◽  
Extreme Value Theory ◽  
A Priori ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Maximum Likelihood Estimates ◽  
Probabilistic Seismic Hazard ◽  
Hazard Level

Abstract Probabilistic Seismic Hazard Analyses (PSHA) require that at least the mean activity rate be known, as well as the distribution of magnitudes. Within the Gutenberg-Richter assumption, the magnitudes follow an exponential distribution which is upperly truncated to a maximum possible magnitude denoted mmax. This parameter is often fixed from expert judgement under tectonics considerations, due to a lack of universal method.In this paper, we present two alternatives to the exponential distribution of the magnitudes, based on the extreme value theory and that don't require to fix a priori the value of mmax: the first model is based on a generalized Pareto distribution (GPD) to model the tail distribution of the magnitudes; the second model, the Randomized Gutenberg-Richter model, is a variation on the usual exponential distribution where mmax is randomized and follows a distribution defined from an extreme value analysis.We use the maximum likelihood estimates taking into account the time varying level of completeness of the catalog and the uncertainty in the magnitude value itself. We apply both modelizations to the Alps region. Then, we integrate the resulting magnitude-frequency relations into a probabilistic seismic hazard calculation to evaluate their impacts on the seismic hazard levels. These extreme-value-based recurrence models introduce a reduction of the seismic hazard level compared to the common Gutenberg-Richter model conventionally used for PSHA calculations. This decrease is significant mainly for long periods.

scholarly journals Extreme Value Theory and Large Fire Losses

1974 ◽  
Vol 7 (3) ◽  
pp. 293-310 ◽  
Author(s):  
G. Ramachandran
Keyword(s):  
Statistical Theory ◽  
Extreme Value Theory ◽  
Extreme Values ◽  
Value Theory ◽  
Extreme Value ◽  
Large Fire ◽  
Fire Insurance ◽  
Reinsurance Treaty

The statistical theory of extreme values well described by Gumbel [1] has been fruitfully applied in many fields, but only in recent times has it been suggested in connection with fire insurance problems. The idea originally stemmed from a consideration of the ECOMOR reinsurance treaty proposed by Thepaut [2]. Thereafter, a few papers appeared investigating the usefulness of the theory in the calculation of an excess of loss premium. Among these, Beard [3, 4], d'Hooge [5] and Jung [6] have made contributions which are worth studying. They have considered, however, only the largest claims during a succession of periods. In this paper, generalized techniques are presented which enable use to be made of all large losses that are available for analysis and not merely the largest. These methods would be particularly useful in situations where data are available only for large losses.

scholarly journals Dynamic VaR Measurement of Gold Market with SV-T-MN Model

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Fenglan Li ◽  
Jie Wang ◽  
Liyun Su ◽  
Bao Yang
Keyword(s):  
At Risk ◽  
Extreme Value Theory ◽  
Value At Risk ◽  
Generalized Pareto Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Distribution Model ◽  
Market Return ◽  
Extreme Risk ◽  
Gold Market

VaR (Value at Risk) in the gold market was measured and predicted by combining stochastic volatility (SV) model with extreme value theory. Firstly, for the fat tail and volatility persistence characteristics in gold market return series, the gold price return volatility was modeled by SV-T-MN (SV-T with Mixture-of-Normal distribution) model based on state space. Secondly, future sample volatility prediction was realized by using approximate filtering algorithm. Finally, extreme value theory based on generalized Pareto distribution was applied to measure dynamic risk value (VaR) of gold market return. Through the proposed model on the price of gold, empirical analysis was investigated; the results show that presented combined model can measure and predict Value at Risk of the gold market reasonably and effectively and enable investors to further understand the extreme risk of gold market and take coping strategies actively.

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