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Thesis (MBA)--Stellenbosch University, 2001. / ENGLISH ABSTRACT: Commercial banks, investment banks, insurance companies, non-financial firms, and pension funds hold portfolios of assets that may include stocks, bonds, currencies, and derivatives. Each institution needs to quantify the amount of risk its portfolio is exposed to in the course of a day, week, month, or year. Extreme events in financial markets, such as the stock market crash of October 1987, are central issues in finance and particularly in risk management and financial regulation. A method called value at risk (VaR) can be used to estimate market risk. Value at risk is a powerful measure of risk that is gaining wide acceptance amongst institutions for the management of market risk. Value at Risk is an estimate of the largest lost that a portfolio is likely to suffer during all but truly exceptional periods. More precisely, the VaR is the maximum loss that an institution can be confident it would lose a certain fraction of the time over a particular period. The power of the concept is its generality. VaR measures are applicable to entire portfolios - encompassing many asset categories and multiple sources of risk. As with its power, the challenge of calculating VaR also stems from its generality. In order to measure risk in a portfolio using VaR, some means must be found for determining a return distribution for the portfolio. There exists a wide range of literature on different methods of implementing VaR. But, when one attempts to apply the results, several questions remain open. For example, given a VaR measure, how can the risk manager test that the particular measure at hand is appropriately specified? And secondly, given two different VaR measures, how can the risk manager pick the best measure? Despite the popularity of VaR for measuring market risk, no consensus has yet been reach as to the best method to implement this risk measure. The absence of consensus is in part derived from the realization that each method currently in use has some significant drawbacks. The aim of this project is threefold: to introduce the reader to the concept of VaR; present the theoretical basis for the general approaches to VaR computations; and to introduce and apply Extreme Value Theory to VaR calculations. The general approaches to VaR computation falls into three categories, namely, Analytic (Parametric) Approach, Historical Simulation Approach, and Monte Carlo Simulation Approach. Each of these approaches has its strengths and weaknesses, which will study more closely. The extreme value approach to VaR calculation is a relatively new approach. Since most observed returns are central ones, traditional VaR methods tend to ignore extreme events and focus on risk measures that accommodate the whole empirical distribution of central returns. The danger of this approach is that these models are prone to fail just when they are needed most - in large market moves, when institutions can suffer very large losses. The extreme value approach is a tool that attempts to provide the user with the best possible estimate of the tail area of the distribution. Even in the absence of useful historical data, extreme value theory provides guidance on the kind of distribution that should be selected so that extreme risks are handled conservatively. As an illustration, the extreme value method will be applied to a foreign exchange futures contract. The validity of EVT to VaR calculations will be tested by examining the data of the Rand/Dollar One Year Futures Contracts. An extended worked example will be provided wherein which attempts to highlight the considerable strengths of the methods as well as the pitfalls and limitations. These results will be compared to VaR measures calculated using a GARCH(l,l) model. / AFRIKAANSE OPSOMMING: Handelsbanke, aksepbanke, assuransiemaatskappye, nie-finansiële instellings en pensioenfondse beskik oor portefeuljes van finansiële bates soos aandele, effekte, geldeenhede en afgeleides. Elke instelling moet die omvang kan bepaal van die risiko waaraan die portefeulje blootgestel is in die loop van 'n dag, week, maand of jaar. Uitsonderlike gebeure op finansiële markte, soos die ineenstorting van die aandelemark in Oktober 1987, is van besondere belang vir finansies en veral vir risikobestuur en finansiële regulering. 'n Metode wat genoem word Waarde op Risiko (WoR), kan gebruik word om markverliese te meet. WoR is 'n kragtige maatstaf vir risiko en word deur vele instellings gebruik vir die bestuur van mark-risiko. Waarde op Risiko is 'n raming van die grootste verlies wat 'n portefeulje moontlik kan ly gedurende enige tydperk, met uitsluiting van werklik uitsonderlike tydperke. Van nader beskou, is WoR die maksimum verlies wat 'n instelling kan verwag om gedurende 'n sekere tydperk binne 'n bepaalde periode te ly. Die waarde van die konsep lê in die algemene aard daarvan. WoR metings is van toepassing op portefeuljes in dié geheel en dit omvat baie kategorieë bates en veelvuldige bronne van risiko. Soos met die waarde van die konsep, hou die uitdaging om WoR te bereken ook verband met die algemene aard van die konsep. Ten einde die risiko te bepaal in 'n portefeulje waar WoR gebruik word, moet metodes gevind word waarvolgens 'n opbrengsverdeling vir die portefeulje vasgestel kan word. Daar bestaan 'n groot verskeidenheid literatuur oor die verskillende metodes om WoR te implementeer. Wanneer dit egter kom by die toepassing van die resultate, bly verskeie vrae onbeantwoord. Byvoorbeeld, hoe kan die risikobestuurder aan die hand van 'n gegewe WoR-maatstaf toets of die spesifieke maatstaf reg gespesifiseer is? Tweedens, hoe kan die risikobestuurder die beste maatstaf kies in die geval van twee verskillende WoR-maatstawwe? Ondanks die feit dat WoR algemeen gebruik word vir die meting van markrisiko, is daar nog nie konsensus bereik oor die beste metode om hierdie benadering tot risikometing te implementeer nie. Die feit dat daar nie konsensus bestaan nie, kan deels daaraan toegeskryf word dat elkeen van die metodes wat tans gebruik word, ernstige leemtes het. Die doel van hierdie projek is om die konsep WoR bekend te stel, om die teoretiese grondslag te lê vir die algemene benadering tot die berekening van WoR en om die Ekstreme Waarde-teorie bekend te stel en toe te pas op WoR-berekenings. Die algemene benadering tot die berekening van WoR word in drie kategorieë verdeel naamlik die Analitiese (Parametriese) benadering, die Historiese simulasiebenadering en die Monte Carlo-simulasiebenadering. Elkeen van die benaderings het sterk- en swakpunte wat van nader ondersoek sal word. Die Ekstreme Waarde-benadering tot WoR is 'n relatief nuwe benadering. Aangesien die meeste opbrengste middelwaarde-gesentreer is, is tradisionele WoR-metodes geneig om uitsonderlike gebeure buite rekening te laat en te fokus op risiko-maatstawwe wat die hele empiriese verdeling van middelwaarde-gesentreerde opbrengste akkommodeer. Die gevaar bestaan dan dat hierdie modelle geneig is om te faal juis wanneer dit die meeste benodig word, byvoorbeeld in die geval van groot markverskuiwings waartydens organisasies baie groot verliese kan ly. Daar word beoog om met behulp van die Ekstreme Waarde-benadering aan die gebruiker die beste moontlike skatting van die stert-area van die verdeling te gee. Selfs in die afwesigheid van bruikbare historiese data verskaf die Ekstreme Waarde-teorie riglyne ten opsigte van die aard van die verdeling wat gekies moet word, sodat uiterste risiko's versigtig hanteer kan word. Ten einde hierdie metode te illustreer, word dit in hierdie studie toegepas op 'n termynkontrak ten opsigte van buitelandse wisselkoerse. Die geldigheid van die Ekstreme Waarde-teorie ten opsigte van WoR berekenings word getoets deur die data van die Rand/Dollar Eenjaartermynkontrak te bestudeer. 'n Volledig uitgewerkte voorbeeld word verskaf waarin die slaggate en beperkings asook die talle sterkpunte van die model uitgewys word. Hierdie resultate sal vergelyk word met 'n WoR-meting wat bereken is met die GARCH (1,1) model.

Thesis (MComm (Statistics and Actuarial Science))--University of Stellenbosch, 2005. / Extremes are unusual or rare events. However, when such events – for example earthquakes, tidal waves and market crashes - do take place, they typically cause enormous losses, both in terms of human lives and monetary value. For this reason, it is of critical importance to accurately model extremal events. Extreme value theory entails the development of statistical models and techniques in order to describe and model such rare observations. In this document we discuss aspects of extreme value theory. This theory consists of two approaches: The classical maxima method, based on the properties of the maximum of a sample and the more popular threshold theory, based upon the properties of exceedances of a specified threshold value. This document provides the practitioner with the theoretical and practical tools for both these approaches. This will enable him/her to perform extreme value analyses with confidence. Extreme value theory – for both approaches - is based upon asymptotic arguments. For finite samples, the limiting result for the sample maximum holds approximately only. Similarly, for finite choices of the threshold, the limiting distribution for exceedances of that threshold holds only approximately. In this document we investigate the quality of extreme value based inferences with regard to the unknown underlying distribution when the sample size or threshold is finite. Estimation of extreme tail quantiles of the underlying distribution, as well as the calculation of confidence intervals, are typically the most important objectives of an extreme analysis. For that reason, we evaluate the accuracy of extreme based inferences in terms of these estimates. This investigation was carried out using a simulation study, performed with the software package S-Plus.

The central issue of this thesis is investigating the eventuality of systemic break- downs in the international financial system through examining systemic depen- dence between bank and insurance sectors. Standard models of systemic risk often use correlation of stock returns to evaluate the magnitude of intercon- nectedness between financial institutions. One of the main drawbacks of this approach is that it is oriented towards observations occurring along the central part of the distribution and it does not capture the dependence structure of outlying observations. To account for that, we use methodology which builds on the Extreme Value Theory and is solely focused on capturing dependence in extremes. The analysis is performed using the data on stock prices of the EU largest banks and insurance companies. We study dependencies in the pre- crisis and post-crisis period. The objective is to discover which sector poses a higher systemic threat to the international financial stability. Also, we try to find empirical evidence about an increase in interconnections in recent post- crisis years. We find that in both examined periods systemic dependence in the banking sector is higher than in the insurance sector. Our results also in- dicate that extremal interconnections in the respective sectors increased,...

[pt] A partir da década de 90, a metodologia de Valor em Risco (VaR) se difundiu pelo mundo, tanto em instituições financeiras quanto em não financeiras, como uma boa prática de mensuração de riscos. Um dos fatos estilizados mais pronunciados acerca das distribuições de retornos financeiros diz respeito à presença de caudas pesadas. Isso torna os modelos paramétricos tradicionais de cálculo de Valor em Risco (VaR) inadequados para a estimação de VaR de baixas probabilidades, dado que estes se baseiam na hipótese de normalidade para as distribuições dos retornos. Sendo assim, o objetivo do presente trabalho é investigar o desempenho de modelos baseados na Teoria dos Valores Extremos para o cálculo do VaR. Os resultados indicam que os modelos baseados na Teoria dos Valores Extremos são adequados para a modelagem das caudas, e consequentemente para a estimação de Valor em Risco quando os níveis de probabilidade de interesse são baixos. / [en] Since the 90 decade, the use of Value at Risk (VaR) methodology has been disseminated among both financial and non-financial institutions around the world, as a good practice in terms of risks management. The existence of fat tails is one of the striking stylized facts of financial returns distributions. This fact makes the use of traditional parametric models for Value at Risk (VaR) estimation unsuitable for the estimation of low probability events. This is because traditional models are based on the conditional normality assumption for financial returns distributions. The main purpose of this dissertation is to investigate the performance of VaR models based on Extreme Value Theory. The results indicates that Extreme Value Theory based models are suitable for low probability VaR estimation.

Global warming is affecting the Earth climate year by year, the biggest difference being observable in increasing temperatures in the World Ocean. Following the long- term global ocean warming trend, average sea surface temperatures across the global tropics and subtropics have increased by 0.4–1◦C in the last 40 years. These rates become even higher in semi-enclosed southern seas, such as the Red Sea, threaten- ing the survival of thermal-sensitive species. As average sea surface temperatures are projected to continue to rise, careful study of future developments of extreme temper- atures is paramount for the sustainability of marine ecosystem and biodiversity. In this thesis, we use Extreme-Value Theory to study sea surface temperature extremes from a gridded dataset comprising 16703 locations over the Red Sea. The data were provided by Operational SST and Sea Ice Analysis (OSTIA), a satellite-based data system designed for numerical weather prediction. After pre-processing the data to account for seasonality and global trends, we analyze the marginal distribution of ex- tremes, defined as observations exceeding a high spatially varying threshold, using the Generalized Pareto distribution. This model allows us to extrapolate beyond the ob- served data to compute the 100-year return levels over the entire Red Sea, confirming the increasing trend of extreme temperatures. To understand the dynamics govern- ing the dependence of extreme temperatures in the Red Sea, we propose a flexible local approach based on R-Pareto processes, which extend the univariate Generalized Pareto distribution to the spatial setting. Assuming that the sea surface temperature varies smoothly over space, we perform inference based on the gradient score method over small regional neighborhoods, in which the data are assumed to be stationary in space. This approach allows us to capture spatial non-stationarity, and to reduce the overall computational cost by taking advantage of distributed computing resources. Our results reveal an interesting extremal spatial dependence structure: in particular, from our estimated model, we conclude that significant extremal dependence prevails for distances up to about 2500 km, which roughly corresponds to the Red Sea length.

The current study empirically analyzes the extreme behavior and the impact of deregulation policies as well as financial turmoil on the extreme behavior of changes of Taiwan short term interest rate. A better knowledge of short-term interest rate properties, such as heavy tails, asymmetry, and uneven tail fatness between right and left tails, provide an insight to the extreme behavior of short-term interest rate as well as a more accurate estimation of interest risk. The predicting performances of filtered and unfiltered VaR (Value at risk) models are also examined to suggest the proper models for management of interest rate risk. By applying Extreme Value theory (EVT), tail behavior is analyzed and tested and the VaR based on parametric and non-parametric EVT models are calculated.The empirical findings show that, first, the distribution of change of rate are heavy-tailed indicating that the actual risk would be underestimated based on normality assumption. Second, the unconditional distribution is consistent with the heavier-tailed distributions such as ARCH process or Student¡¦t. Third, the right tail of distribution of change of rate are significantly heavier than the left one pointing out that the probabilities and magnitudes of rise in rate could be higher than those of drop in rate. Fourth, the amount of tail-fatness in tail of distribution of change of rate increase after 1999 and the vital factors to cause structural break in tail index are the interest rate policies taken by central bank of Taiwan instead of the deregulation policies in money market. Fifth, based on the two break points found in tail index of right and left tail, long sample of CP rates should not be treated as samples from a single distribution. Sixth, the dependent and heteroscedastic properties of data series should be considered in applying EVT to improve accuracy of VaR forecasts. Finally, EVT models predict VaR accurately before 2001 and the benchmark model, HS and GARCH, generally are superior to EVT models after 2001. Among EVT models, MRE and CHE are relative consistent and reliable in VaR prediction.

The recent elevated North Atlantic hurricane activity has generated considerable interests in the interaction between tropical cyclones (TCs) and climate change. The possible connection between TCs and the changing climate has been indicated by observational studies based on historical TC records; they indicate emerging trends in TC frequency and intensity in some TC basins, but the detection of trends has been hotly debated due to TC track data issues. Dynamical climate modeling has also been applied to the problem, but brings its own set of limitations owing to limited model resolution and uncertainties. The final goal of this study is to project the future changes of North Atlantic TC behavior with global warming for the next 50 years using the Nested Regional Climate Model (NRCM). Throughout the course of reaching this goal, various uncertainties and limitations in simulating TCs by the NRCM are identified and explored. First we examine the TC tracking algorithm to detect and track simulated TCs from model output. The criteria and thresholds used in the tracking algorithm control the simulated TC climatology, making it difficult to objectively assess the model's ability in simulating TC climatology. Existing tracking algorithms used by previous studies are surveyed and it is found that the criteria and thresholds are very diverse. Sensitivity of varying criteria and thresholds in TC tracking algorithm to simulated TC climatology is very high, especially with the intensity and duration thresholds. It is found that the commonly used criteria may not be strict enough to filter out intense extratropical systems and hybrid systems. We propose that a better distinction between TCs and other low-pressure systems can be achieved by adding the Cyclone Phase technique. Two sets of NRCM simulations are presented in this dissertation: One in the hindcasting mode, and the other with forcing from the Community Climate System Model (CCSM) to project into the future with global warming. Both of these simulations are assessed using the tracking algorithm with cyclone phase technique. The NRCM is run in a hindcasting mode for the global tropics in order to assess its ability to simulate the current observed TC climatology. It is found that the NRCM is capable of capturing the general spatial and temporal distributions of TCs, but tends to overproduce TCs particularly in the Northwest Pacific. The overpredction of TCs is associated with the overall convective tendency in the model added with an outstanding theory of wave energy accumulation leading to TC genesis. On the other hand, TC frequency in the tropical North Atlantic is under predicted due to the lack of moist African Easterly Waves. The importance of high-resolution is shown with the additional simulation with two-way nesting. The NRCM is then forced by the CCSM to project the future changes in North Atlantic TCs. An El Nino-like SST bias in the CCSM induced a high vertical wind shear in tropical North Atlantic, preventing TCs from forming in this region. A simple bias correction method is applied to remove this bias. The model projected an increase both in TC frequency and intensity owing to enhanced TC genesis in the main development region, where the model projects an increased favorability of large-scale environment for TC genesis. However, the model is not capable of explicitly simulating intense (Category 3-5) storms due to the limited model resolution. To extrapolate the prediction to intense storms, we propose a hybrid approach that combines the model results and a statistical modeling using extreme value theory. Specifically, the current observed TC intensity is statistically modeled with the General Pareto distribution, and the simulated intensity changes from the NRCM are applied to the statistical model to project the changes in intense storms. The results suggest that the occurrence of Category 5 storms may be increased by approximately 50% by 2055.

In this work were considered the maxima and minima structures. Where number of value is random and is distributed by negative binomial distribution. There were theorems that were improved in this work, that helped to find the limit distribute function of this standard structures. These theorems generalize propositions, when set size is geometric random number. Also, there was the concrete distribution analysis done and such distributions were chosen: exponential, general logistic and uniform.

自從90年代以來，許多機構因為金融商品的操縱不當或是金融風暴的衝擊數度造成全球金融市場的動盪，使得風險管理的重要性與日俱增，而量化風險模型的準確性也益受重視，基於財務資料的相關性質如異質變異、厚尾現象等，本文主要結合AR(1)-GARCH(1,1)模型、極值理論、copula函數三種模型應用在風險值的估算，且將報酬分配的假設區分為三類，一是無母數模型的歷史模擬法，二是基於常態分配假設下考量隨機波動度的有母數模型，三是利用歷史資料配適尾端分配的極值理論法來對聯電、鴻海、國泰金、中鋼四檔個股和台幣兌美元、日圓兌美元、英鎊兌美元三種外匯資料作一日風險值、十日風險值、組合風險值的測試。 實證結果發現，在一日風險值方面，95%信賴水準下以動態風險值方法表現相對較好，99%信賴水準下動態極值理論法和動態歷史模擬法皆有不錯的估計效果；就十日風險值而言，因為未來十日資產的報酬可能受到特定事件影響，所以估計上較為困難，整體看來在99%信賴水準下以條件GPD＋蒙地卡羅模擬的表現相對較理想；以組合風險值來說， copula、Clayton copula+GPD marginals模擬股票或外匯組合的聯合分配不論在95%或99%信賴水準下對其風險值的估計都獲得最好的結果；雖然台灣個股股價受到上下漲跌幅7%的限制，台幣兌美元的匯率也受到央行的干涉，但以極值理論來描述資產尾端的分配情形相較於假設其他兩種分配仍有較好的估計效果。