Application Of Extreme Value Theory Research Articles

Extreme value analysis of daily Canadian crude oil prices

Crude oil markets are highly volatile and risky. Extreme Value Theory (EVT), an approach to modelling and measuring risks under rare events, has seen a more prominent role in risk management in recent years. This article presents an application of EVT to the daily returns of crude oil prices in the Canadian spot market between 1998 and 2006. We focus on the Peak Over Threshold (POT) method by analysing the generalized Pareto-distributed exceedances over some high threshold. This method provides an effective means for estimating tail risk measures such as Value-at-Risk (VaR) and Expected Shortfall (ES). The estimates of risk measures computed under different high quantile levels exhibit strong stability across a range of the selected thresholds. At the 99th quantile, the estimates of VaR are approximately 6.3% and 6.8% for daily positive and negative returns, respectively.

Weather regime dependence of extreme value statistics for summer temperature and precipitation

Abstract. Extreme Value Theory (EVT) is a useful tool to describe the statistical properties of extreme events. Its underlying assumptions include some form of temporal stationarity in the data. Previous studies have been able to treat long-term trends in datasets, to obtain the time dependence of EVT parameters in a parametric form. Since there is also a dependence of surface temperature and precipitation to weather patterns obtained from pressure data, we determine the EVT parameters of those meteorological variables over France conditional to the occurrence of North Atlantic weather patterns in the summer. We use a clustering algorithm on geopotential height data over the North Atlantic to obtain those patterns. This approach refines the straightforward application of EVT on climate data by allowing us to assess the role of atmospheric variability on temperature and precipitation extreme parameters. This study also investigates the statistical robustness of this relation. Our results show how weather regimes can modulate the different behavior of mean climate variables and their extremes. Such a modulation can be very different for the mean and extreme precipitation.

Philippe Jorion: Value at Risk – The New Benchmark for Managing Financial Risk

Professor Jorion’s book provides a comprehensive treatment of the concept of Value at Risk, tailored towards the needs of professionals and MBA students. Much emphasis is put on the implementation and use of Value at Risk in a risk management framework as well as the Basel regulations. The book also points the reader toward more advanced topics. The new sections outlining applications of extreme value theory, principal component analysis, copulas and the Basel II rule sets are very welcome additions to the previous editions. The author has divided the treatment of Value at Risk into six parts, all building upon each other. Part I provides an overview of markets and risk factors for financial institutions. The author takes a historical approach to motivate Value at Risk while highlighting several high profile failures and shortcomings of risk management. He also introduces the reader to the evolution of the Basel frameworks. The second and third parts make up the core pieces of the book. Part II, “Building Blocks,” provides the reader with techniques to estimate Value at Risk, including nonparametric and parametric methods. The emphasis thereby is always on application—something the target audience will know to appreciate. Professor Jorion also discusses estimation uncertainty and the need for backtesting obtained Value at Risk figures. Drawing from portfolio theory, the impact of correlation/diversification and the modeling of such are addressed. New to the third edition is a section on multivariate models. In particular, dimension reduction techniques such as factor models, principal components and copulas are also outlined. The third part, called “Value-at-Risk Systems,” focuses directly on the implementation of Value at Risk as part of risk management. Analytical, marginal and sim-

An Extreme Value Analysis Of Advanced Age Mortality Data

Extreme value theory describes the behavior of random variables at extremely high or low levels. The application of extreme value theory to statistics allows us to fit models to data from the upper tail of a distribution. This paper presents a statistical analysis of advanced age mortality data, using extreme value models to quantify the upper tail of the distribution of human life spans. Our analysis focuses on mortality data from two sources. Statistics Canada publishes the annual number of deaths in Canada, broken down by angender and age. We use the deaths data from 1949 to 1997 in our analysis. The Japanese Ministry of Health, Labor, and Welfare also publishes detailed annual mortality data, including the 10 oldest reported ages at death in each year. We analyze the Japanese data over the period from 1980 to 2000. Using the r-largest and peaks-over-threshold approaches to extreme value modeling, we fit generalized extreme value and generalized Pareto distributions to the life span data. Changes in distribution by birth cohort or over time are modeled through the use of covariates. We then evaluate the appropriateness of the fitted models and discuss reasons for their shortcomings. Finally, we use our findings to address the existence of a finite upper bound on the life span distribution and the behavior of the force of mortality at advanced ages.

Conditional VaR using EVT – Towards a planned margin scheme

This paper constructs a robust Value-at-Risk (VaR) measure for the Indian stock markets by combining two well-known facts about equity return time series — dynamic volatility resulting in the well-recognized phenomenon of volatility clustering, and non-normality giving rise to fat tails of the return distribution. While the phenomenon of volatility dynamics has been extensively studied using GARCH model and its many relatives, the application of Extreme Value Theory (EVT) is relatively recent in tracking extreme losses in the study of risk measurement. There are recent applications of Extreme Value Theory to estimate the unexpected losses due to extreme events and hence modify the current methodology of VaR. Extreme value theory (EVT) has been used to analyze financial data showing clear non-normal behavior. We combine the two methodologies to come up with a robust model with much enhanced predictive abilities. A robust model would obviate the need for imposing special ad hoc margins by the regulator in times of extreme volatility. A rule based margin system would increase efficiency of the price discovery process and also the market integrity with the regulator no longer seen as managing volatility.

Extreme value analysis of wet and dry periods in Sicily

For assessing risk of highly unusual events extreme value statistics needs to be applied, which plays an important role in engineering practices for water resources design and management. In hydrology, the typical application of extreme value theory concerns floods in river basins or landslides. The present paper is, instead, focused on the analysis of extreme wet and dry periods in a sample area (Sicily). First, we have studied monthly precipitation extremes both using the annual maximum and partial duration methods, and return times have been estimated by standard statistical techniques. Next, we studied the extremes of the Standardized Precipitation Index (SPI), which has been proposed as an indicator for monitoring wet and dry conditions. We found considerable differences both in the return periods and in the time location of the extremes. From our study it appears that the SPI better describes wet and dry periods than the precipitation does. Maps of return times for extreme conditions in Sicily are also presented, which cluster the territory into areas of different extreme return periods. Finally, the occurrence of extremes in Sicily has been related to large-scale atmospheric circulation.

Microdosimetry of the ultraviolet erasable programmable read-only memory experiment on the microelectronics and photonics test bed: recent advances in small-volume analysis

A commercial ultraviolet erasable programmable read-only memory (UVPROM) was used to demonstrate a dosimetry technique for both ground and space applications. An equivalent amount of UV to reproduce the same amount of erasure was used to calibrate dose. The new method of readout, unlike other methods, does not require the evidence of exposure to be destroyed in the course of a measurement. It requires power only during readout. Results from an experiment using this technique aboard the Microelectronics and Photonics Test Bed (MPTB) satellite are discussed. Application of Extreme Value Theory is used to analyze whether early failures in the device were statistically feasible or more likely due to large, rare energy depositions. A new dosimeter approach using a change injection method that will eliminate the need for UV as a metric is also presented as well new experiments for the devices under test aboard MPTB.

Efficient filtering of financial time series and extreme value theory

We describe an efficient filtering process for a univariate time series, yt, for subsequent application of extreme value theory and the estimation of extreme quantiles, ie, value-at-risk. Our filtering model is based on a stationary ARMA-asymmetric Garch (1, 1) process. After filtering, the sequence of conditional residuals is approximately independent and identically distributed and therefore appropriate for the application of extreme value theory. The efficiency of the filtering process is of importance in this approach, and we apply an efficient estimator based on the generalized method of moments. The estimator is simple to compute but - more importantly - it is, in case of skewness or excess kurtosis of the conditional residuals, asymptotically efficient relative to the commonly applied quasi-maximum likelihood estimator. We further evaluate the robustness and efficiency of extreme value theory tail index estimators through a Monte Carlo study. An application to the daily returns of the ABB (Asea Brown Boveri) equity illustrates the methods, and a comparison is made with methods based on conditional normality, the conditional t-distribution and the empirical distribution function.

Quantifying Operational Risk in General Insurance Companies. Developed by a Giro Working Party

ABSTRACTThe paper overviews the application of existing actuarial techniques to operational risk. It considers how, working in conjunction with other experts, actuaries can develop a new framework to monitor/review, establish context, identify, understand and decide what to do in terms of the management and mitigation of operational risk. It suggests categorisations of risk to help analyses and proposes how new risk indicators may be needed, in conjunction with more normal quantification approaches.Using a case study, it explores the application of stress and scenario testing, statistical curve fitting (including the application of extreme value theory), causal (Bayesian) modelling and the extension of dynamic financial analysis to include operational risk. It suggests there is no one correct approach and that the choice of parameters and modelling assumptions is critical. It lists a number of other techniques for future consideration.There is a section about how ‘soft issues’ including dominance risk, the impact of belief systems and culture, the focus of performance management systems and the psychology of organisations affect operational risk. An approach to rating the people aspects of risk in parallel with quantification may help give a better overall assessment of risk and improve the understanding for capital implications.The paper concludes with a brief review of implications for reporting and considers what future work will help develop the actuarial contribution. It is hoped the paper will sow seeds for the development of best practice in dealing with operational risk and increase the interest of actuaries in this emerging new topic.

Risk management under extreme events

This article presents two applications of extreme value theory (EVT) to financial markets: computation of value at risk (VaR) and cross-section dependence of extreme returns (i.e., tail dependence). We use a sample comprised of the United States, Europe, Asia, and Latin America. Our main findings are the following. First, on average, EVT gives the most accurate estimate of VaR. Second, tail dependence of paired returns decreases substantially when both heteroscedasticity and serial correlation are filtered out by a multivariate GARCH model. Both findings are in agreement with previous research in this area for other financial markets.

Drinking to extremes: theoretical and empiricalanalyses of peak drinking levels among college students.

Heavy drinking among college students continues to be a substantial problem on campuses across the United States. Attempts to predict these drinking events have been restricted to assessments of the correlates of heavy drinking (measured at 4 or 5 drinks) and have not examined the peak drinking levels that can be fatal to students. This article presents a theoretical analysis of college drinking patterns that provides a basis for estimating peak drinking levels and predicts future risks related to peak drinking events. Survey data were collected on sociodemographics and drinking patterns of 2,102 college students from two college campuses in California. A mathematical model of drinking patterns was used to characterize the stochastic distribution of drinking events among 1,273 students who drank five or more times and consumed more than one drink on some occasion since the beginning of the school year. An application of extreme value theory enabled the estimation of peak drinking levels for every college drinker. These estimates were related to self-reported maximum drinking levels and sociodemographic characteristics of respondents. Among these drinkers, the distribution of self-reported maximum drinking levels ranged from 2 to 43 drinks per occasion. Estimated peak drinking levels ranged from 3 to 49. Maximum drinking levels were well characterized by peak drinking estimates (R2 = 0.503). Variations in peak drinking levels were large and specifically related to particular sociodemographic groups (i.e., white male freshmen). The theoretical model of peak drinking events effectively characterizes maximum drinking levels among college students. High levels of peak drinking are to be expected among specific sociodemographic subgroups. These risks can be assessed on an individual basis. At the population level, risks for harm related to peak drinking events are predictable.

Using extreme value theory to estimate value‐at‐risk

This study examines problems that may occur when conventional Value‐at‐Risk (VaR) estimators are used to quantify market risks in an agricultural context. For example, standard VaR methods, such as the variance‐covariance method or historical simulation, can fail when the return distribution is fat tailed. This problem is aggravated when long‐term VaR forecasts are desired. Extreme Value Theory (EVT) is proposed to overcome these problems. The application of EVT is illustrated by an example from the German hog market. Multi‐period VaR forecasts derived by EVT are found to deviate considerably from standard forecasts. We conclude that EVT is a useful complement to traditional VaR methods.

Extreme Value of Response to Nonstationary Excitation

An efficient method is presented for approximate computation of extreme value characteristics of the response of a linear structure subjected to nonstationary Gaussian excitation. The characteristics considered are the mean and standard deviation of the extreme value and fractile levels having specific probabilities of not being exceeded by the random process within a specified time interval. The approximate procedure can significantly facilitate the utilization of nonstationary models in engineering practice, since it avoids computational difficulties associated with direct application of extreme value theory. The method is based on the approximation of the cumulative distribution function (CDF) of the extreme value of a nonstationary process by the CDF of a corresponding “equivalent” stationary process. Approximate procedures are developed for both the Poisson and Vanmarcke approaches to the extreme value problem, and numerical results are obtained for an example problem. These results demonstrate that ...

The bootstrap methodology in statistics of extremes: Choice of the optimal sample fraction

The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the tail index γ, usually performed on the basis of the largest k order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive estimation of γ. We shall be here mainly interested in the use of the bootstrap methodology to estimate γ adaptively, and although the methods provided may be applied, with adequate modifications, to the general domain of attraction of Gγ, γ ∈ ℝ, we shall here illustrate the methods for heavy right tails, i.e. for γ > 0. Special relevance will be given to the use of an auxiliary statistic that is merely the difference of two estimators with the same functional form as the estimator under study, computed at two different levels. We shall also compare, through Monte Carlo simulation, these bootstrap methodologies with other data-driven choices of the optimal sample fraction available in the literature.

From value at risk to stress testing: The extreme value approach

This article presents an application of extreme value theory to compute the value at risk of a market position. In statistics, extremes of a random process refer to the lowest observation (the minimum) and to the highest observation (the maximum) over a given time-period. Extreme value theory gives some interesting results about the distribution of extreme returns. In particular, the limiting distribution of extreme returns observed over a long time-period is largely independent of the distribution of returns itself. In financial markets, extreme price movements correspond to market corrections during ordinary periods, and also to stock market crashes, bond market collapses or foreign exchange crises during extraordinary periods. An approach based on extreme values to compute the VaR thus covers market conditions ranging from the usual environment considered by the existing VaR methods to the financial crises which are the focus of stress testing. Univariate extreme value theory is used to compute the VaR of a fully aggregated position while multivariate extreme value theory is used to compute the VaR of a position decomposed on risk factors.

Application of extreme value theory to level estimation in nonlinearly distorted hidden Markov models

This paper is concerned with the application of extreme value theory (EVT) to the state level estimation problem for discrete-time, finite-state Markov chains hidden in additive colored noise and subjected to unknown nonlinear distortion. If the nonlinear distortion affects only those observations with small magnitudes or those that lie outside a finite interval, we show that the level estimation problem can be reduced to a curve fitting problem with a unique global minimum. Compared with optimum maximum likelihood estimation algorithms, the developed level estimation algorithms are computationally inexpensive and are not affected by the unknown nonlinearity as long as the extreme values of observations are not distorted. This work has been motivated by unknown deadzone and saturation nonlinearities introduced by sensors in data measurement systems. We illustrate the effectiveness of the new EVT-based level estimation algorithms with computer simulations.

Pitfalls and Opportunities in the Use of Extreme Value Theory in Risk Management

Extreme value theory (EVT) holds promise for advancing the assessment and management of extreme financial risks. Recent literature suggests that the application of EVT generally results in more precise estimates of extreme quantiles and tail probabilities of financial asset returns. This article assesses EVT from the perspective of financial risk management. The authors believe that the recent optimism regarding EVT may be appropriate but exaggerated, and that much of its potential remains latent. They support their claim by describing various pitfalls associated with the current use of EVT techniques, and illustrate how these can be avoided. In conclusion, the article defines several specific research directions that may further the practical and effective application of EVT to risk management.

Application of extreme value theory for predicting violations of air quality standards for an urban road intersection

Extreme value theory (EVT) has been used for making predictions of the expected number of violations of the National Ambient Air Quality Standards (NAAQS) as prescribed by the Central Pollution Control Board (CPCB), India, for hourly- and eight-hourly average carbon monoxide(CO) concentration for an Air Quality Control Region (AQCR) comprising of an urban road intersection. Almost 100% contribution to the CO pollution in the region is due to vehicular traffic. The parameters of the extreme value distribution (EVD) have been estimated by four methods viz. method of moments, maximum likelihood, least square fit and Gumbel's fitting method. Least square fit and Gumbel's method gave the best fit and predictions. A comparison of the predicted violations of NAAQS and the exceedence of the maximum pollution concentration with that of the observed data suggests that the theory performs satisfactorily for the extreme air pollution events.

On the estimation of extreme tail probabilities

Applications of extreme value theory to problems of statistical inference typically involve estimating tail probabilities well beyond the range of the data, without the benefit of a concise mathematical model for the sampling distribution. The available model is generally only an asymptotic one. That is, an approximation to probabilities of extreme deviation is supposed, which is assumed to become increasingly accurate as one moves further from the range of the data, but whose concise accuracy is unknown. Quantification of the level of accuracy is essential for optimal estimation of tail probabilities. In the present paper we suggest a practical device, based on a nonstandard application of the bootstrap, for determining empirically the accuracy of the approximation and thereby constructing appropriate estimators.

Applications of Extreme Value Theory in Corrosion Engineering.

In the context of corrosion engineering it is often natural to be concerned with extreme events. This is because, firstly, it is these extreme events that often lead to failure and, secondly, it may only be possible to measure the extremes, with much of the underlying measurements by their very nature unobservable Statistical methods relating to extreme value theory can be used to model and predict the statistical behaviour of extremes such as the largest pit, thinnest wall, maximum penetration or similar assessment of a corrosion phenomenon. These techniques can be applied to the single largest value, or to a given number of the largest values, measured over individual areas or coupons; or to all values exceeding a given threshold. The data can be modeled to account for dependence on environmental conditions, surface area examined, and the duration of exposure or of experimentation. The application of a selection of these techniques is demonstrated on data from industry and from laboratory experiments.