Thesis (MBA)--Stellenbosch University, 2004. / 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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/53743 |
| Date | 03 1900 |
| Creators | Baldwin, Sheena |
| Contributors | Smith, E. VdM, Stellenbosch University. Faculty of Economic and Management Sciences. Graduate School of Business. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 147 p. |
| Rights | Stellenbosch University |
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