Extreme value theory : from a financial risk management perspective

Baldwin, Sheena (2004-03)

Thesis (MBA)--Stellenbosch University, 2004.

Thesis

ENGLISH ABSTRACT: Risk managers and regulators are primarily concerned with ensuring that there is sufficient capital to withstand the effects of adverse movements in market prices. The accurate prediction of the maximum amount that a financial institution can expect to Jose over a specified period is essential to guard against catastrophic losses that can threaten the viability of an individual finn or the stability of entire markets. Value-at-risk (VaR) is a quantile-based measure of risk that is widely used for calculating the capital adequacy requirements of banks and other financial institutions. However, the current models for price risk tend to underestimate the risk of catastrophic losses because the entire return distribution is used to calculate the value-at-risk. By contrast, Extreme Value" Theory uses only the largest observations to model the tails of a distribution, which should provide a better fit for estimates of extreme quantiles and probabilities. The semi-parametric Hill (1975) estimator has often been used to fit the tails of financial returns, but its performance is heavily dependent on the number k" of order statistics used in the estimation process and the estimator can be very biased if this choice is suboptimal. Since k" depends on unknown properties of the tail, it has to be estimated from the sample. The first truly data-driven method for choosing an optimal number of order statistics adaptively was introduced by Beirlant, Dierckx. Goegebeur and Matthys (1999) and modified by Beirlanl. Dierckx and Stmca (2000) and Matthys and Beirlanl (2000b). Their methods are based on an exponential regression model developed independently by Beirlant et a/. (1999) and Feuerverger and Hall (1999) to reduce the bias found in the Hill estimator. The reduced bias of these adaptive estimators and the associated estimator for extreme quantiles developed by Matthys and Beirlant (2000b) makes these estimators attractive from a risk management point of view, but more work needs to be done on characterising their finite sample properties before they can be used in practice. In particular, it is crucially important to establish the smallest sample size that will yield reliable estimates of extreme quantiles and probabilities and to determine the widths and coverage probabilities of confidence intervals. This study project reviews the probability and statistical theory of univariate Extreme Value Theory from a financial risk management perspective. It is clear from a survey of the literature that the most worthwhile direction to pursue in terms of practical research will be intimately connected with developments in the fast-moving field of EVT with a future emphasis not only on fully evaluating the existing models, but indeed on creating even less biased and more precise models. Keywords and phrases: Extreme value index, Pareto-type distributions, maximum likelihood estimation, bias reduction, exponential regression model, market risk.

AFRIKAANSE OPSOMMING: Risikobestuurders en -reguleerders is hoofsaaklik gemoeid met die versekering dat genoegsame kapitaal beskikbaar is om die effek van ongunstige beweging in markpryse die hoof te kan bied. Die akkurate vooruitskatting van die maksimum verlies wat 'n finansiele instelling oor 'n spesifieke tydperk kan ly, is noodsaaklik as beskerming teen katastrofiese verliese wat die voortbestaan van 'n individuele firma, of die stabiliteit van die totale mark, mag bedreig. Waarde-op-Risiko (WoR) is 'n kwantiel gebaseerde maatstaaf van risiko wat algemeen vir die berekening van kapitaaltoereikendheid van banke en ander finansiele instellings benut word. Die huidige prys risikomodelle neig om die risiko van katastrofiese verliese te onderskat, omdat die totale opbrengs verspreiding gebruik word om WoR te bereken. In teenstelling benut die Ekstreme Waarde Teorie (EWT), slegs die grootste waarnemings om die eindverdelings te modelleer en is as sulks meer geskik om ekstreme kwantiele en waarskynlikhede te bepaal. Die semi-parametriese Hill (1975) skatter word gereeld gebruik om die stertgedeeltes van finansiele opbrengste te beraam, maar sy verrigting is swaar afhanklik van die getal k~ van rangstatistieke wat in die skattingsproses gebruik word en die skatting kan baie sydig wees indien die keuse suboptimaal is. Weens die afhanklikheid van kn van onbekende eienskappe van die stertgedeeltes, moet dit geskat word vanuit die steekproefdata. Die eerste data-gedrewe metode vir die keuse van die optimale rangordestatistieke, is deur Beiriant, Dierckx, Goegebeur en Matthys (1999) ontwikkel en aangepas deur Beirlant, Dierckx and Starica (2000), asook Matthys en Beirlant (2000b). Hul metodes is op 'n eksponensiele regressiemodel gebaseer, en is onafhanklik deur Beirlant et at. (1999), en Feuerverger en Hall (1999) ontwikkel met die doel om die sydigheid van die Hill skatter te verminder. Die verminderde sydigheid van hierdie adaptiewe skatters en die verwante skatter vir ekstreme kwantiele, ontwikkel deur Matthys en Beirlant (2000b), maak hierdie skatters aantreklik vanuit 'n risikobestuur oogpunt, maar meer werk word benodig met die karakterisering van hul eindige steekproefeienskappe, alvorens dit in die praktyk benut kan word. In besonder is dit van uiterste belang dat die kleinste steekproefgrootte bepaal sal word wat die betroubare skattings van ekstreme kwantiele en moontlikhede sal verseker, en wat ook benut kan word om betroubaarheidsintervalle op te ste!. Hierdie studie bied 'n oorsig van die moontlikhede en statistiese teorie van die eenveranderlike EWT vanuit 'n finansiele risikobestuur perspektief. Dit is duidelik vanuit die literatuurstudie dat die mees nuttige rigting om voort te gaan met praktiese navorsing, verband hou met die ontwikkeling in die vinnig ontwikkelende veld van EWT met toekomstige fokus, nie slegs op die volle evaluering van die bestaande modelle nie, maar ook op die ontwikkeling van minder sydige en meer akkurate modelle.

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