Peaks Over Threshold Model Research Articles

Double Risk Catastrophe Reinsurance Premium Based on Houses Damaged and Deaths

The peaks over threshold (POT) model for catastrophe (CAT) reinsurance pricing has been widely used, but has mainly focused on univariate CAT reinsurance pricing. We provide further justification and support for the model by considering the addition of more than one type of CAT risk in the context of extreme value theory. We further extend the applicability of the CAT reinsurance premium model by considering house damage and deaths as CAT risk. Using the proposed model, we present a simulation framework for pricing double risk CAT reinsurance, based on excess-of-loss reinsurance contract. Furthermore, we fit the POT model to the earthquake loss data in Indonesia. Finally, we provide the price of the double risk CAT reinsurance premium under the standard deviation premium principle. The framework results obtained show that the pricing formulas in this study are appropriate for the double risk claim and may be used as a basis for the pricing of double risk CAT excess-of-loss reinsurance contracts.

Determination of Monitoring Control Value for Concrete Gravity Dam Spatial Deformation Based on POT Model

Deformation can directly reflect the working behavior of the dam, so determining the deformation monitoring control value can effectively monitor the safety of dam operation. The traditional dam deformation monitoring control value only considers the single measuring point. In order to overcome the limitation, this paper presents a new method to determine the monitoring control value for concrete gravity dam based on the deformations of multi-measuring points. A dam’s comprehensive deformation displacement is determined by the measured values at different measuring points on the positive inverted vertical line and the corresponding weight of each measuring point. The projection pursuit method (PPM) combined with the grey wolf optimization (GWO) algorithm is used to determine the weight of each measuring point according to the spatial correlation distribution characteristics of dam deformation. The peaks over threshold (POT) model based on the extreme value theory is adopted to determine the monitoring control value with the obtained dam comprehensive deformation displacement. In addition, the POT model is improved with the automatic threshold determination method based on the criterion in probability theory and the GWO algorithm, which can avoid subjectivity and randomness in determining the threshold. The results of the engineering application show the feasibility and applicability of the proposed method.

Crude oil futures tail risk measurement based on extreme value theory

In this paper, we use three common tail risk measurements of Value-at-Risk (VaR), Expected Shortfall (ES) and Spectral Risk Measure (SRM) to calculate the tail risk of crude oil futures based on extreme value theory. Specifically, we propose a method to determine the optimal threshold in the extreme value theory, and further to calculate the values of VaR, ES and SRM based on Peak Over Threshold (POT) model. Empirical results show that the extreme value POT model can be used to characterise the tail risk of the price return under extreme fluctuations in Brent crude oil futures market. Moreover, the risk of VaR, ES and SRM in the Brent crude oil futures market based on extreme value theory is higher than that under the normal distribution assumption, which indicates that the traditional normal distribution assumption underestimates the tail risk. Owing the flexibility and the accuracy, we suggest that investors use ERM to measure the extreme risk of crude oil futures.

Seismic risk in the east of the Bayan Har block based on the POT model

This article elaborates on a peaks over threshold (POT) model with earthquake samples based on the Pareto distribution. We analyze the earthquakes using the POT model in the eastern Bayan Har block. The result shows that the area experiences an earthquake above 7.5 every 30 years and a large earthquake above 7.8 every 100 years. The average magnitude of earthquakes above 5.2 that occur in the area every 5 years is 7.1 or greater. The upper limit of the theoretical magnitude predicted by the model is 8.7. Therefore, the risk of geological hazards is high in this area.

Reproduction method of time-domain load of construction machinery internal combustion engine

Bench test has been proved to be an effective method in the field of fatigue life prediction and reliability design for construction machinery internal combustion engine (CMICE), and the reasonable load reproduction is a promising supplement to the experimental data. To reproduce the load reasonably and eliminate the limitation that traditional peak over threshold (POT) extrapolation only focuses on extreme load, a method to reproduce the time-domain load of CMICE is proposed in this paper. Here, the extreme load is reproduced with POT model, where the upper and lower thresholds are selected through the mean excess function graph, while the intermediate load is reproduced by fitting the Gaussian mixed distribution. The load before and after reproduction is compared with the results of through-level counting and pseudo-damage calculation, which verifies the rationality of the proposed time-domain load reproduction method.

Forecasting tail risk for Bitcoin: A dynamic peak over threshold approach

This paper employs a dynamic peak over threshold (PoT) model to measure and forecast both the lower and upper tail Value at Risks (VaRs) of Bitcoin returns, which offers a new perspective to investigate the tail risk dynamics for Bitcoin. We evaluate the VaR forecasting accuracy of this model compared with that of the GARCH-EVT models based on Student-t, skewed Student-t and Generalized error distribution. The empirical results illustrate that the dynamic PoT model exhibits superior out-of-sample VaR predictive ability, specifically for the lower tail VaR. Thus, this model can be a useful and reliable alternative for forecasting tail risk.

Pricing catastrophe reinsurance under the standard deviation premium principle

<abstract><p>Catastrophe reinsurance is an important way to prevent and resolve catastrophe risks. As a consequence, the pricing of catastrophe reinsurance becomes a core problem in catastrophic risk management field. Due to the severity of catastrophe loss, the Peak Over Threshold (POT) model in extreme value theory (EVT) is extensively applied to capture the tail characteristics of catastrophic loss distribution. However, there is little research available on the pricing formula of catastrophe excess of loss (Cat XL) reinsurance when the catastrophe loss is modeled by POT. In the context of POT model, we distinguish three different relations between retention and threshold, and then prove the explicit pricing formula respectively under the standard deviation premium principle. Furthermore, we fit POT model to the earthquake loss data in China during 1990–2016. Finally, we give the prices of earthquake reinsurance for different retention cases. The computational results illustrate that the pricing formulas obtained in this paper are valid and can provide basis for the pricing of Cat XL reinsurance contracts.</p></abstract>

Comparison of annual maximum and peaks-over-threshold methods with automated threshold selection in flood frequency analysis: a case study for Australia

Flood frequency analysis (FFA) involves fitting of a probability distribution to observed flood data. Two main models, annual maximum (AM) and peaks over threshold (POT), are generally adopted in FFA. The POT model is underemployed due to its complexity and uncertainty associated with threshold selection and meeting independence criteria in selecting POT data series. This study evaluates the POT and AM models using data from 188 gauged stations in southeast Australia. The POT model adopted in this study applies different average numbers of events per year fitted with generalized Pareto (GP) distribution with an automated threshold detection method. For the AM model, the GP distribution is also adopted, and flood quantiles estimated by the two models are compared. It has been found that there are notable differences in design flood estimates between the AM and POT models. The study uses catchment characteristics data to understand the differences in quantile estimates between the AM and POT models. The percentage differences between the AM and POT models can be explained by mean annual rainfall (MAR) and mean annual evapotranspiration (MAE) (ARIs of 1.01–5 years), MAR (10-year ARI), stream density (SDEN) (for 20- and 50-year ARIs) and SDEN, main stream slope (S1085) and MAE (for 100-year ARI). Stations showing smaller % differences have relatively higher MAR and smaller MAE (indicating wetter condition); conversely, stations showing higher % differences have smaller MAR and higher MAE (drier condition).

Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights

The peaks-over-threshold (POT) method has a long tradition in modelling extremes in environmental variables. However, it has originally been introduced under the assumption of independently and identically distributed (iid) data. Since environmental data often exhibits a time series structure, this assumption is likely to be violated due to short- and long-term dependencies in practical settings, leading to clustering of high-threshold exceedances. In this paper, we first review popular approaches that either focus on modelling short- or long-range dynamics explicitly. In particular, we consider conditional POT variants and the Mittag–Leffler distribution modelling waiting times between exceedances. Further, we propose a new two-step approach capturing both short- and long-range correlations simultaneously. We suggest the autoregressive fractionally integrated moving average peaks-over-threshold (ARFIMA-POT) approach, which in a first step fits an ARFIMA model to the original series and then in a second step utilises a classical POT model for the residuals. Applying these models to an oceanographic time series of significant wave heights measured on the Sefton coast (UK), we find that neither solely modelling short- nor long-range dependencies satisfactorily explains the clustering of extremes. The ARFIMA-POT approach, however, provides a significant improvement in terms of model fit, underlining the need for models that jointly incorporate short- and long-range dependence to address extremal clustering, and their theoretical justification.

Pricing catastrophe reinsurance risk premium using Peaks Over Threshold (POT) model

An incident that cost more lives than a certain threshold is called a catastrophic event. In the world of insurance, the incidence of catastrophe causes many policies to make a claim together, so it can cause big losses for insurance companies. The risk of catastrophic events can be minimized by purchasing a catastrophic reinsurance contract. The reinsurance company then calculates the amount of risk premium for catastrophic reinsurance to be paid by the insurance company. The model used to calculate the risk premium is the peaks over threshold (POT) model. The POT model is generally used to model extreme events. In this model, the number of catastrophic events is modeled using the Poisson distribution. Then, the number of casualties is modeled using the discrete generalized Pareto distribution (DGPD). To estimate the parameters of the model, the maximum likelihood method is used and the data on the number of fatalities resulting from a particular event in Indonesia is collected and compiled. Furthermore, the number of claims is modeled with the Beta-Binomial distribution and the size of claims resulting from a catastrophic event is modeled by Exponential distribution. Numerical simulations are then performed to obtain the total size of claims resulting from all catastrophic events within 1 year of catastrophic reinsurance contracts. Ultimately the risk premium can be calculated using the standard deviation premium principle by utilizing the expected value and the variance of the total amount of the claim.

Generation of 100-year-return value maps of maximum significant wave heights with automated threshold value estimation

Spectral numerical wave models are unable to predict extreme sea conditions accurately, and the probability of occurrence of a given wave parameter for fixed return periods (RPs) remains unexplored by such an approach. In this study, the significant wave height (SWH) parameter that plays a major role in ocean engineering and marine and naval activities was estimated spatially for the entire Bay of Bengal (BOB) region for the given RPs. Spatially gridded 44 years of SWH data at a spatial resolution of 1° × 1° were considered, and return value maps were generated for the BOB for 10-, 15-, 25-, 50-, and 100-year RPs. The peak over threshold (POT) model was chosen over a generalized Pareto distribution for data fitting and calculating the return values. As there is no existing method to determine the threshold value for use in the POT model, in this study, for the first time, an algorithm was designed to calculate the threshold value, which determined the accuracy of the estimation. Along with the conventional time-series analysis of the SWH data using POT, the novelty of the study lies in the spatial return period maps generated for the entire BOB region leading to the basin scale estimation.

Time domain load extrapolation method for CNC machine tools based on GRA-POT model

The load extrapolation is to obtain load data that may occur during the whole life cycle of product based on the limited measured load data, especially extreme load. Aiming at the two problems of traditional time domain extrapolation: without considering the interval time between extreme adjacent values and the high sensitivity of the peak over threshold (POT) model to the extreme threshold, a time domain load extrapolation method for computerized numerical control (CNC) machine tools based on Gray relational analysis-peak over threshold model (GRA-POT) is proposed. Firstly, for the first problem, an improved Markov chain Monte Carlo (MCMC) time domain load reconstruction method is proposed and analysis of reconstructed time domain load; secondly, according to the mean excess function (MEF) method, the extreme threshold range of POT model is determined and divided into multiple sets. The generalized Pareto distribution (GPD) fitting is performed for multiple sets of candidate load extreme samples. Finally, the Gray relational degrees (GRD) between each group of GPD and the median rank distribution are calculated by the Gray relational analysis (GRA), and the extreme threshold corresponding to the maximum GRD is selected as the optimal threshold of the POT model. The dynamic cutting force signal of CNC machine tool under constant speed cutting condition was reconstructed and extrapolated using the proposed method, and the dynamic cutting force spectrum of CNC machine tool was compiled. Case analysis results show that the POT extrapolation model with high fitting precision can be obtained using the proposed method. Furthermore, the precision of load spectrum of CNC machine tool is improved.

Determination of the Threshold for Extreme Load Extrapolation Based on Multi-Criteria Decision-Making Technology

The peak over threshold (POT) model is commonly used in extreme load extrapolation. Due to the important role that the threshold plays in establishing a POT model, a method to select the suitable threshold is designed based on multi-criteria decision-making (MCDM) technology. The mean deviation in probability distribution function, Kolmogorov-Smirnov test, and Chi-Square test are employed as the test criteria. An entropy method is applied to obtain the weight values of these tests. The VIKOR method is adopted to obtain compromise solutions according to the corresponding criteria and weight values. The POT requirement is used to determine a suitable threshold. The effectiveness and feasibility of this method are validated by the load time history measured in experiment and generated by simulation. A comparative analysis between this method and two other common methods is also conducted. The proposed method based on MCDM shows better performance.

POT model for operational risk: Experience with the analysis of the data collected from Chinese commercial banks

This paper takes 533 operational risk loss events publicly announced by Chinese commercial banks in the period of 1995-2012 as the sample, using Peaks over Threshold (POT) model to quantify the operational risk. The statistical data classification indicates the internal fraud is the main type of operational risk in Chinese commercial banks. This paper explains its causes from the perspective of behavioral finance. The results are as follows: first, Chinese commercial banks' operational risk loss events show an upward trend, then downward trend beginning in 2003 and currently an upward trend again; second, through the empirical analysis, this paper simulates the extreme value distribution function, finds the optimal threshold, and calculates the VaR and ES of the operational risk of Chinese commercial banks and compare them at different confidence levels; and third, in view of behavioral finance theory, overconfidence and loss aversion contribute to high internal fraud incidence.

Non-stationary frequency analysis of heavy rainfall events in southern France

Heavy rainfall events often occur in southern French Mediterranean regions during the autumn, leading to catastrophic flood events. A non-stationary peaks-over-threshold (POT) model with climatic covariates for these heavy rainfall events is developed herein. A regional sample of events exceeding the threshold of 100 mm/d is built using daily precipitation data recorded at 44 stations over the period 1958–2008. The POT model combines a Poisson distribution for the occurrence and a generalized Pareto distribution for the magnitude of the heavy rainfall events. The selected covariates are the seasonal occurrence of southern circulation patterns for the Poisson distribution parameter, and monthly air temperature for the generalized Pareto distribution scale parameter. According to the deviance test, the non-stationary model provides a better fit to the data than a classical stationary model. Such a model incorporating climatic covariates instead of time allows one to re-evaluate the risk of extreme precipitation on a monthly and seasonal basis, and can also be used with climate model outputs to produce future scenarios. Existing scenarios of the future changes projected for the covariates included in the model are tested to evaluate the possible future changes on extreme precipitation quantiles in the study area. Editor Z.W. Kundzewicz; Associate editor K. Hamed Citation Tramblay, Y., Neppel, L., Carreau, J., and Najib, K., 2013. Non-stationary frequency analysis of heavy rainfall events in southern France. Hydrological Sciences Journal, 58 (2), 280–294.

变化环境下超定量与年最大值洪水序列频率分析模型对比研究

POT模型比AMS模型能更完整灵活地描述洪水及其产生过程，适用于变化情况下的洪水频率分析。本文以武江流域犁市(二)站为例进行超定量(POT)与年最大值(AMS)洪水频率对比分析。应用门限值选取标准确定门限值，根据POT模型分布参数及设计洪水估计结果随门限值变化情况验证所选门限值的合理性；以泊松分布为超定量发生次数分布，用广义Pareto(GP)分布拟合POT样本。对比POT模型和基于P-III分布的AMS模型，探讨大重现期洪水频发是否受频率分析方法不当影响。结果表明：根据门限值选取标准确定的门限值，经验证能够保证POT模型的稳定性以及样本独立性；估计大重现期设计洪峰时，P-Ⅲ分布的AMS模型的估计结果小于GP分布的POT模型，100年一遇洪水估计结果差异度大于6%，可见AMS模型存在低估大重现期设计洪峰的现象，这是近年来计算出大重现期洪水频发的原因之一。 Taking the daily average discharge of Lishi (2) station in the Wujiang river basin for an example, two flood frequency analysis models, respectively based on peaks-over-threshold (POT) and annual maximum series (AMS), are compared. POT model has better flexibility describing floods and the flood generating process than AMS model; thus is suitable for flood frequency analysis under changing environment. Threshold selection approaches are applied to choose the threshold value, which is tested by the relationship of estimated results of parameters of the POT model and designed floods with changing thresholds afterwards. The POT samples are fitted by the generalized Pareto (GP) distribution with a Poisson model for arrival. Compare POT model based on the GP distribution and AMS model based on the P-III distribution and discuss whether the frequent occurrence of large return period flood is influenced by undue frequency analysis method(the large return period flood-prone by the undue influenced of the frequency analysis). The results show that, threshold selected by combined approaches is able to ensure the sample independence and stability of the POT model. When estimating the designed flood peak of large return period, result estimated by the AMS model based on P-III distribution is smaller than the POT model based on GP distribution, the 100-year flood is estimated that the results of differences greater than 6%, visible AMS model underestimate of the phenomenon of large return period designed flood peak, which is one of the reasons of the frequent occurrence of large return period floods in recent years.

Predicting Malaysian palm oil price using Extreme Value Theory

This paper uses the extreme value theory (EVT) to predict extreme price events of Malaysian palm oil in the future, based on monthly futures price data for a 25 year period (mid-1986 to mid-2011). Model diagnostic has confirmed non-normal distribution of palm oil price data, thereby justifying the use of EVT. Two principal approaches to model extreme values – the Block Maxima (BM) and Peak-Over- Threshold (POT) models – were used. Both models revealed that the palm oil price will peak at an incremental rate in the next 5, 10, 25, 50 and 100 year periods. The price growth level in Year-5 is estimated at 17.6% and 44.6% in Year-100 using BM approach. Use of the POT approach indicated a growth rate of 37.6% in Year-5 and 50.8% in Year 100, respectively. The key conclusion is that although the POT model outperformed the BM model, both approaches are effective in providing predictions of growth in prices caused by extreme events. The results could serve as a useful guide to farmers, exporters, governments, and other stakeholders of the palm oil industry informing strategic planning for the future.

Assessing wavestorm hazard evolution in the NW Mediterranean with hindcast and buoy data

It has been suggested that climate change might modify the occurrence rate of large storms and their magnitude, due to a higher availability of energy in the atmosphere-ocean system. Forecasting physical models are commonly used to assess the effects. No one expects the physical model forecasts for one specific day to be accurate; we consider them to be good if they adequately describe the statistical characteristics of the climate. The Peak-Over-Threshold (POT) method is a common way to statistically treat the occurrence and magnitude of hazardous events: here, occurrence is modelled as a Poisson process and magnitude over a given threshold is assumed to follow a Generalized Pareto Distribution (GPD). We restrict our attention to Weibull-related GPDs, which exhibit an upper bound, to comply with the fact that any physical process has a finite upper limit. This contribution uses this framework to model time series of log-significant wave-height constructed joining quasi-collocated hindcast data and buoy measurements. Two of the POT model parameters (inhomogeneous Poisson rate and logarithm of the GPD shape parameter are considered to be a combination of a linear function of time and a series indicator function. The third parameter, logarithm of the GPD upper bound, is considered to have only a series indicator component. The resulting parameters are estimated using Bayesian methods. Using hincast and buoy series, the time span of the data set is extended, enhancing the precision of statistical results about potential linear changes. Simultaneously the statistical behaviour of hincast and buoy series are compared. At the same time, the step function allows to calibrate the statistical reproduction of storms by hindcasting.

A New Method to Choose the Threshold in the Pot Model

A new method which is based on the Hill plot is proposed to determine an appropriate threshold in the POT (Peaks Over Threshold) model. Different from the traditional Hill plot method, the proposed method can determine the threshold by quantification, and is easy to be implemented by a computer program. An empirical study using the prices of copper futures contracts traded in LME is also presented. It's shown that the proposed method can well capture the turning point of the Hill plot.

The POT model described by the generalized Pareto distribution with Poisson arrival rate

The use of the peaks over threshold (POT) model described by the generalized Pareto distribution with Poisson arrival rate, which results in the generalized extreme value (GEV) distribution for the annual maximum (AM) series for flood frequency analysis, is discussed. Estimation of the generalized Pareto distribution by the method of probability weighted moments (PWM) is formulated for both known and unknown thresholds. The effect of raising the threshold of the POT series on high quantile estimation is investigated. A comparison of the efficiencies of the POT and AM models in high quantile estimation is made. It is shown that the threshold of the POT series can be raised to a reasonably high level without affecting the efficiency of high quantile estimation. This property is useful in practice. When one cannot be certain that both small and large values in a sample are well described by a single simple distribution or that peaks are independent of each other, a relatively high threshold can be introduced to overcome these difficulties without reduction in efficiency. It is also shown that the POT and AM models are of similar efficiency even when a relatively high threshold is used for the POT series. It is suggested that there is no theoretical reason to use the AM model in preference to the POT model.