Thesis (MComm (Statistics and Actuarial Science))--University of Stellenbosch, 2009. / When determining the chemical composition of a specimen, traditional laboratory techniques are often both expensive and time consuming. It is therefore preferable to employ more cost effective spectroscopic techniques such as near infrared (NIR). Traditionally, the calibration problem has been solved by means of multiple linear regression to specify the model between X and Y. Traditional regression techniques, however, quickly fail when using spectroscopic data, as the number of wavelengths can easily be several hundred, often exceeding the number of chemical samples. This scenario, together with the high level of collinearity between wavelengths, will necessarily lead to singularity problems when calculating the regression coefficients. Ways of dealing with the collinearity problem include principal component regression (PCR), ridge regression (RR) and PLS regression. Both PCR and RR require a significant amount of computation when the number of variables is large. PLS overcomes the collinearity problem in a similar way as PCR, by modelling both the chemical and spectral data as functions of common latent variables. The quality of the employed reference method greatly impacts the coefficients of the regression model and therefore, the quality of its predictions. With both X and Y subject to random error, the quality the predictions of Y will be reduced with an increase in the level of noise. Previously conducted research focussed mainly on the effects of noise in X. This paper focuses on a method proposed by Dardenne and Fernández Pierna, called Noise Addition Partial Least Squares (NAPLS) that attempts to deal with the problem of poor reference values. Some aspects of the theory behind PCR, PLS and model selection is discussed. This is then followed by a discussion of the NAPLS algorithm. Both PLS and NAPLS are implemented on various datasets that arise in practice, in order to determine cases where NAPLS will be beneficial over conventional PLS. For each dataset, specific attention is given to the analysis of outliers, influential values and the linearity between X and Y, using graphical techniques. Lastly, the performance of the NAPLS algorithm is evaluated for various

Thesis (MComm (Statistics and Actuarial Science))--University of Stellenbosch, 2005. / Financial institutions invest in financial securities like equities, options and government bonds. Two measures, namely return and risk, are associated with each investment position. Return is a measure of the profit or loss of the investment, whilst risk is defined as the uncertainty about return. A financial institution that holds a portfolio of securities is exposed to different types of risk. The most well-known types are market, credit, liquidity, operational and legal risk. An institution has the need to quantify for each type of risk, the extent of its exposure. Currently, standard risk measures that aim to quantify risk only exist for market and credit risk. Extensive calculations are usually required to obtain values for risk measures. The investments positions that form the portfolio, as well as the market information that are used in the risk measure calculations, change during each trading day. Hence, the financial institution needs a business tool that has the ability to calculate various standard risk measures for dynamic market and position data at the end of each trading day. SAS Risk Dimensions is a software package that provides a solution to the calculation problem. A risk management system is created with this package and is used to calculate all the relevant risk measures on a daily basis. The purpose of this document is to explain and illustrate all the steps that should be followed to create a suitable risk management system with SAS Risk Dimensions.

Assignment (MComm)--Stellenbosch University, 2005. / ENGLISH ABSTRACT: Volatilty was originally seen to be constant and deterministic, but it was later realised that return series are non-stationary. Owing to this non-stationarity nature of returns, there were no reliable ex-post volatility measurements. Subsequently, researchers focussed on ex-ante volatility models. It was only then realised that before good volatility models can be created, reliable ex-post volatility measuremetns need to be defined. In this study we examine non-parametric ex-post volatility measurements in order to obtain approximations of the variances of non-stationary return series. A detailed mathematical derivation and discussion of the already developed volatility measurements, in particular the realised volatility- and DST measurements, are given In theory, the higher the sample frequency of returns is, the more accurate the measurements are. These volatility measurements referred to above, however, all have short-comings in that the realised volatility fails if the sample frequency becomes to high owing to microstructure effects. On the other hand, the DST measurement cannot handle changing instantaneous volatility. In this study we introduce a new volatility measurement, termed microstructure realised volatility, that overcomes these shortcomings. This measurement, as with realised volatility, is based on quadratic variation theory, but the underlying return model is more realistic. / AFRIKAANSE OPSOMMING: Volatiliteit is oorspronklik as konstant en deterministies beskou, dit was eers later dat besef is dat opbrengste nie-stasionêr is. Betroubare volatiliteits metings was nie beskikbaar nie weens die nie-stasionêre aard van opbrengste. Daarom het navorsers gefokus op vooruitskattingvolatiliteits modelle. Dit was eers op hierdie stadium dat navorsers besef het dat die definieering van betroubare volatiliteit metings 'n voorvereiste is vir die skepping van goeie vooruitskattings modelle. Nie-parametriese volatiliteit metings word in hierdie studie ondersoek om sodoende benaderings van die variansies van die nie-stasionêre opbrengste reeks te beraam. 'n Gedetaileerde wiskundige afleiding en bespreking van bestaande volatiliteits metings, spesifiek gerealiseerde volatiliteit en DST- metings, word gegee. In teorie salopbrengste wat meer dikwels waargeneem word tot beter akkuraatheid lei. Bogenoemde volatilitieits metings het egter tekortkominge aangesien gerealiseerde volatiliteit faal wanneer dit te hoog raak, weens mikrostruktuur effekte. Aan die ander kant kan die DST meting nie veranderlike oombliklike volatilitiet hanteer nie. Ons stel in hierdie studie 'n nuwe volatilitieits meting bekend, naamlik mikro-struktuur gerealiseerde volatiliteit, wat nie hierdie tekortkominge het nie. Net soos met gerealiseerde volatiliteit sal hierdie meting gebaseer wees op kwadratiese variasie teorie, maar die onderliggende opbrengste model is meer realisties.

Thesis (MComm)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Researchers and investors have been attempting to predict stock market activity for years. The possible financial gain that accurate predictions would offer lit a flame of greed and drive that would inspire all kinds of researchers. However, after many of these researchers have failed, they started to hypothesize that a goal such as this is not only improbable, but impossible. Previous predictions were based on historical data of the stock market activity itself and would often incorporate different types of auxiliary data. This auxiliary data ranged as far as imagination allowed in an attempt to find some correlation and some insight into the future, that could in turn lead to the figurative pot of gold. More often than not, the auxiliary data would not prove helpful. However, with the birth of the internet, endless amounts of new sources of auxiliary data presented itself. In this thesis I propose that the near in finite amount of data available on the internet could provide us with information that would improve stock market predictions. With this goal in mind, the different sources of information available on the internet are considered. Previous studies on similar topics presented possible ways in which we can measure internet activity, which might relate to stock market activity. These studies also gave some insights on the advantages and disadvantages of using some of these sources. These considerations are investigated in this thesis. Since a lot of this work is therefore based on the prediction of a time series, it was necessary to choose a prediction algorithm. Previously used linear methods seemed too simple for prediction of stock market activity and a new non-linear method, called Singular Spectrum Analysis, is therefore considered. A detailed study of this algorithm is done to ensure that it is an appropriate prediction methodology to use. Furthermore, since we will be including auxiliary information, multivariate extensions of this algorithm are considered as well. Some of the inaccuracies and inadequacies of these current multivariate extensions are studied and an alternative multivariate technique is proposed and tested. This alternative approach addresses the inadequacies of existing methods. With the appropriate methodology chosen and the appropriate sources of auxiliary information chosen, a concluding chapter is done on whether predictions that includes auxiliary information (obtained from the internet) improve on baseline predictions that are simply based on historical stock market data. / AFRIKAANSE OPSOMMING: Navorsers en beleggers is vir jare al opsoek na maniere om aandeelpryse meer akkuraat te voorspel. Die moontlike finansiële implikasies wat akkurate vooruitskattings kan inhou het 'n vlam van geldgierigheid en dryf wakker gemaak binne navorsers regoor die wêreld. Nadat baie van hierdie navorsers onsuksesvol was, het hulle begin vermoed dat so 'n doel nie net onwaarskynlik is nie, maar onmoontlik. Vorige vooruitskattings was bloot gebaseer op historiese aandeelprys data en sou soms verskillende tipes bykomende data inkorporeer. Die tipes data wat gebruik was het gestrek so ver soos wat die verbeelding toegelaat het, in 'n poging om korrelasie en inligting oor die toekoms te kry wat na die guurlike pot goud sou lei. Navorsers het gereeld gevind dat hierdie verskillende tipes bykomende inligting nie van veel hulp was nie, maar met die geboorte van die internet het 'n oneindige hoeveelheid nuwe bronne van bykomende inligting bekombaar geraak. In hierdie tesis stel ek dus voor dat die data beskikbaar op die internet dalk vir ons kan inligting gee wat verwant is aan toekomstige aandeelpryse. Met hierdie doel in die oog, is die verskillende bronne van inligting op die internet gebestudeer. Vorige studies op verwante werk het sekere spesifieke maniere voorgestel waarop ons internet aktiwiteit kan meet. Hierdie studies het ook insig gegee oor die voordele en die nadele wat sommige bronne inhou. Hierdie oorwegings word ook in hierdie tesis bespreek. Aangesien 'n groot gedeelte van hierdie tesis dus gebasseer word op die vooruitskatting van 'n tydreeks, is dit nodig om 'n toepaslike vooruitskattings algoritme te kies. Baie navorsers het verkies om eenvoudige lineêre metodes te gebruik. Hierdie metodes het egter te eenvoudig voorgekom en 'n relatiewe nuwe nie-lineêre metode (met die naam "Singular Spectrum Analysis") is oorweeg. 'n Deeglike studie van hierdie algoritme is gedoen om te verseker dat die metode van toepassing is op aandeelprys data. Verder, aangesien ons gebruik wou maak van bykomende inligting, is daar ook 'n studie gedoen op huidige multivariaat uitbreidings van hierdie algoritme en die probleme wat dit inhou. 'n Alternatiewe multivariaat metode is toe voorgestel en getoets wat hierdie probleme aanspreek. Met 'n gekose vooruitskattingsmetode en gekose bronne van bykomende data is 'n gevolgtrekkende hoofstuk geskryf oor of vooruitskattings, wat die bykomende internet data inkorporeer, werklik in staat is om te verbeter op die eenvoudige vooruitskattings, wat slegs gebaseer is op die historiese aandeelprys data.

Thesis (MComm)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The main topics of study in this thesis are the Principal Component Analysis (PCA) and Canonical Variate Analysis (CVA) biplots, with the primary focus falling on the quality measures associated with these biplots. A detailed study of different routes along which PCA and CVA can be derived precedes the study of the PCA biplot and CVA biplot respectively. Different perspectives on PCA and CVA highlight different aspects of the theory that underlie PCA and CVA biplots respectively and so contribute to a more solid understanding of these biplots and their interpretation. PCA is studied via the routes followed by Pearson (1901) and Hotelling (1933). CVA is studied from the perspectives of Linear Discriminant Analysis, Canonical Correlation Analysis as well as a two-step approach introduced in Gower et al. (2011). The close relationship between CVA and Multivariate Analysis of Variance (MANOVA) also receives some attention. An explanation of the construction of the PCA biplot is provided subsequent to the study of PCA. Thereafter follows an in depth investigation of quality measures of the PCA biplot as well as the relationships between these quality measures. Specific attention is given to the effect of standardisation on the PCA biplot and its quality measures. Following the study of CVA is an explanation of the construction of the weighted CVA biplot as well as two different unweighted CVA biplots based on the two-step approach to CVA. Specific attention is given to the effect of accounting for group sizes in the construction of the CVA biplot on the representation of the group structure underlying a data set. It was found that larger groups tend to be better separated from other groups in the weighted CVA biplot than in the corresponding unweighted CVA biplots. Similarly it was found that smaller groups tend to be separated to a greater extent from other groups in the unweighted CVA biplots than in the corresponding weighted CVA biplot. A detailed investigation of previously defined quality measures of the CVA biplot follows the study of the CVA biplot. It was found that the accuracy with which the group centroids of larger groups are approximated in the weighted CVA biplot is usually higher than that in the corresponding unweighted CVA biplots. Three new quality measures that assess that accuracy of the Pythagorean distances in the CVA biplot are also defined. These quality measures assess the accuracy of the Pythagorean distances between the group centroids, the Pythagorean distances between the individual samples and the Pythagorean distances between the individual samples and group centroids in the CVA biplot respectively. / AFRIKAANSE OPSOMMING: Die hoofonderwerpe van studie in hierdie tesis is die Hoofkomponent Analise (HKA) bistipping asook die Kanoniese Veranderlike Analise (KVA) bistipping met die primêre fokus op die kwaliteitsmaatstawwe wat daarmee geassosieer word. ’n Gedetailleerde studie van verskillende roetes waarlangs HKA en KVA afgelei kan word, gaan die studie van die HKA en KVA bistippings respektiewelik vooraf. Verskillende perspektiewe op HKA en KVA belig verskillende aspekte van die teorie wat onderliggend is tot die HKA en KVA bistippings respektiewelik en dra sodoende by tot ’n meer breedvoerige begrip van hierdie bistippings en hulle interpretasies. HKA word bestudeer volgens die roetes wat gevolg is deur Pearson (1901) en Hotelling (1933). KVA word bestudeer vanuit die perspektiewe van Linieêre Diskriminantanalise, Kanoniese Korrelasie-analise sowel as ’n twee-stap-benadering soos voorgestel in Gower et al. (2011). Die noue verwantskap tussen KVA en Meerveranderlike Analise van Variansie (MANOVA) kry ook aandag. ’n Verduideliking van die konstruksie van die HKA bistipping word voorsien na afloop van die studie van HKA. Daarna volg ’n indiepte-ondersoek van die HKA bistipping kwaliteitsmaatstawwe sowel as die onderlinge verhoudings tussen hierdie kwaliteitsmaatstawe. Spesifieke aandag word gegee aan die effek van die standaardisasie op die HKA bistipping en sy kwaliteitsmaatstawe. Opvolgend op die studie van KVA is ’n verduideliking van die konstruksie van die geweegde KVA bistipping sowel as twee veskillende ongeweegde KVA bistippings gebaseer op die twee-stap-benadering tot KVA. Spesifieke aandag word gegee aan die effek wat die inagneming van die groepsgroottes in die konstruksie van die KVA bistipping op die voorstelling van die groepstruktuur onderliggend aan ’n datastel het. Daar is gevind dat groter groepe beter geskei is van ander groepe in die geweegde KVA bistipping as in die oorstemmende ongeweegde KVA bistipping. Soortgelyk daaraan is gevind dat kleiner groepe tot ’n groter mate geskei is van ander groepe in die ongeweegde KVA bistipping as in die oorstemmende geweegde KVA bistipping. ’n Gedetailleerde ondersoek van voorheen gedefinieerde kwaliteitsmaatstawe van die KVA bistipping volg op die studie van die KVA bistipping. Daar is gevind dat die akkuraatheid waarmee die groepsgemiddeldes van groter groepe benader word in die geweegde KVA bistipping, gewoonlik hoër is as in die ooreenstemmende ongeweegde KVA bistippings. Drie nuwe kwaliteitsmaatstawe wat die akkuraatheid van die Pythagoras-afstande in die KVA bistipping meet, word gedefinieer. Hierdie kwaliteitsmaatstawe beskryf onderskeidelik die akkuraatheid van die voorstelling van die Pythagoras-afstande tussen die groepsgemiddeldes, die Pythagoras-afstande tussen die individuele observasies en die Pythagoras-afstande tussen die individuele observasies en groepsgemiddeldes in die KVA bistipping.

Assignment (MSc) -- University of Stellenbosch, 2004. / ENGLISH ABSTRACT: In an integrated crop-livestock production farm, the profitability and sustainability of farm production is dependent on the crop rotation strategy applied. Crop rotations have historically been applied to maintain long-term profitability and sustainabiliry of farming production by exploiting the jointly beneficial interrelationships existing among different crop types and the animal production activity. Monocrop (specifically wheat) growers in the Swartland area of the Western Cape are struggling to maintain long-term profitability and sustainability of the crop production, challenging them to rethink about the introduction crop rotation in the production planning. By making proper assumptions, this paper develops a mixed integer linear programming model to suggest a decision planning for the farm planning problem faced by an integratedcrop- livestock production farmer. The mathematical model developed includes crop production, dairy production and wool sheep production activities, which permitted the consideration of five crop types within a crop rotation system. By assuming that a farmer uses a cycle of at most three years, the crop rotation model was incorporated in the composite mixed integer linear farm planning model. In order to demonstrate the application of the mathematical farm planning model formulated, a case study is presented. Relevant data from the Koeberg area of the Swartland region of the Western Cape was applied. For each planning period, the model assumed that the farm has the option of selecting from any of 15 cropping strategies. A land which is not allocated to any of the 15 crop rotation strategies due to risky production situation is left as grass land for roughage purposes of the animal production. Results of the mathematical model indicated that farm profit is dependent on the cropping strategy selected. Additionally, animal production level was also dependent on the crop strategy appl ied. Furthermore, study results suggest that the profit generated from the integrated crop-livestock farm production by adopting crop rotation was superior to profit generated 1'1'0111 the farm activities which are based on monocrop wheat strategy. Empirical results also indicated that the complex interrelationship involved in a mixed crop-livestock farm operation play a major role in determining optimal farm plans. This complex interrelationships favour the introduction of crop rotation in the crop production activities of the farm under investigation. Crop production risk is the major risk component of risk the farmer faces in the farm production. In this study, risk is incorporated in the mixed integer programrnmg farm planning model as a deviation from the expected values of an activity of returns. Model solution with risk indicated that crop rotation strategy and animal production level is sensitive to risk levels considered. The Results also showed that the incorporation of risk in the model greatly affects the level of acreage allocation, crop rotation and animal production level of the farm. Finally, to improve the profitability and sustainability of the farm activity, the study results suggest that the introduction of crop rotation which consist cereals, oil crops and leguminous forages is of paramount importance. Furthermore, the inclusion of forage crops such as medics in the integrated crop livestock production is beneficial for sustained profitability from year to year. / AFRIKAANSE OPSOMMING: Wisselbou is baie belangrik om volhoubare winsgewindheid te verseker in 'n geintegreerde lewendehawe I gewasverbouing boerdery in die Swartland gebied van Wes-Kaap. "n Monokultuur van veral koring produksie het ernstige problerne vir produsente veroorsaak. In hierdie studie word 'n gemengde heeltallige liniere prograrnmerings-model gebruik om te help met besluitneming in sulke boerderye.Die wiskundige model beskou die produksie van kontant- en voer-gewasse (5 verskillende soorte) asook suiwel- en wol/vleis-produksie (beeste en skape) .Daar word aanvaar dat die boer "n siklus van hoogstens 3 jaar in die wisselbou rotasie model gebruik .. 'n Gevallestudie word gedoen met behulp van toepaslike data van 'n plaas in die Koeberg gebied. Die model aanvaar dat die produsent 'n keuse het uit 16 wisselbou strategic .Resultate toon dat winsgewindheid afhanklik is van die strategie gekies en dat wisselbou beter resultate lewer as in die geval van "n monokultuur.Dit wys ook dat die wisselwerking tussen diereproduksie en gewasproduksie baie belangrik is in die keuse van 'n optimale strategie. Die risiko in gewasverbouing is die belangrikste risiko factor vir die produsent.In hierdie studie word risiko ook ingesluit in die gemengde heeltallige model, naamlik as 'n afwyking van die verwagte opbrengs-waardes .Die model toon duidelik dat gewasproduksie en lewendehawe-produksie baie sensitief is ten opsigte van die gekose risiko vlak. Die studie toon ook dat 'n wisselbou program wat die produksie van graan (veral koring) .oliesade asook voere insluit belangrik is vir volhoubare winsgewindheid Die insluiting van klawers (bv "medics") is veral belangrik hier.

Thesis (PhD (Statistics and Actuarial Sciences))--Stellenbosch University, 2008. / We consider the problem of model assessment by risk estimation. Various approaches to risk estimation are considered in a uni ed framework. This a discussion of various complexity dimensions and approaches to obtaining bounds on covering numbers is also presented. The second type of training sample interval estimator discussed in the thesis is Rademacher bounds. These bounds use advanced concentration inequalities, so a chapter discussing such inequalities is provided. Our discussion of Rademacher bounds leads to the presentation of an alternative, slightly stronger, form of the core result used for deriving local Rademacher bounds, by avoiding a few unnecessary relaxations. Next, we turn to a discussion of PAC-Bayesian bounds. Using an approach developed by Olivier Catoni, we develop new PAC-Bayesian bounds based on results underlying Hoe ding's inequality. By utilizing Catoni's concept of \exchangeable priors", these results allowed the extension of a covering number-based result to averaging classi ers, as well as its corresponding algorithm- and data-dependent result. The last contribution of the thesis is the development of a more exible shell decomposition bound: by using Hoe ding's tail inequality rather than Hoe ding's relative entropy inequality, we extended the bound to general loss functions, allowed the use of an arbitrary number of bins, and introduced between-bin and within-bin \priors". Finally, to illustrate the calculation of these bounds, we applied some of them to the UCI spam classi cation problem, using decision trees and boosted stumps. framework is an extension of a decision-theoretic framework proposed by David Haussler. Point and interval estimation based on test samples and training samples is discussed, with interval estimators being classi ed based on the measure of deviation they attempt to bound. The main contribution of this thesis is in the realm of training sample interval estimators, particularly covering number-based and PAC-Bayesian interval estimators. The thesis discusses a number of approaches to obtaining such estimators. The rst type of training sample interval estimator to receive attention is estimators based on classical covering number arguments. A number of these estimators were generalized in various directions. Typical generalizations included: extension of results from misclassi cation loss to other loss functions; extending results to allow arbitrary ghost sample size; extending results to allow arbitrary scale in the relevant covering numbers; and extending results to allow arbitrary choice of in the use of symmetrization lemmas. These extensions were applied to covering number-based estimators for various measures of deviation, as well as for the special cases of misclassi - cation loss estimators, realizable case estimators, and margin bounds. Extended results were also provided for strati cation by (algorithm- and datadependent) complexity of the decision class. In order to facilitate application of these covering number-based bounds,

Thesis (MComm (Statistics and Actuarial Science))--University of Stellenbosch, 2006. / An overview of modern and historical interest rate model theory is given with the specific aim of derivative pricing. A variety of stochastic interest rate models are discussed within a South African market context. The various models are compared with respect to characteristics such as mean reversion, positivity of interest rates, the volatility structures they can represent, the yield curve shapes they can represent and weather analytical bond and derivative prices can be found. The distribution of the interest rates implied by some of these models is also found under various measures. The calibration of these models also receives attention with respect to instruments available in the South African market. Problems associated with the calibration of the modern models are also discussed.

Thesis (MComm (Statistics and Actuarial Science))--University of Stellenbosch, 2007. / The aim of this study is to assess the welfare of South Africa in terms of poverty and inequality. This is done using the Income and Expenditure Survey (IES) of 2000, released by Statistics South Africa, and reviewing the distribution of income in the country. A brief literature review of similar studies is given along with a broad de nition of poverty and inequality. A detailed description of the dataset used is given together with aspects of concern surrounding the dataset. An analysis of poverty and income inequality is made using datasets containing the continuous income variable, as well as a created grouped income variable. Results from these datasets are compared and conclusions made on the use of continuous or grouped income variables. Covariate analysis is also applied in the form of biplots. A brief overview of biplots is given and it is then used to obtain a graphical description of the data and identify any patterns. Lastly, the conclusions made in this study are put forward and some future research is mentioned.

Thesis (MComm)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Accurate estimation of Value at Risk (VaR) and Expected Shortfall (ES) is critical in the management of extreme market risks. These risks occur with small probability, but the financial impacts could be large. Traditional models to estimate VaR and ES are investigated. Following usual practice, 99% 10 day VaR and ES measures are calculated. A comprehensive theoretical background is first provided and then the models are applied to the Africa Financials Index from 29/01/1996 to 30/04/2013. The models considered include independent, identically distributed (i.i.d.) models and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) stochastic volatility models. Extreme Value Theory (EVT) models that focus especially on extreme market returns are also investigated. For this, the Peaks Over Threshold (POT) approach to EVT is followed. For the calculation of VaR, various scaling methods from one day to ten days are considered and their performance evaluated. The GARCH models fail to converge during periods of extreme returns. During these periods, EVT forecast results may be used. As a novel approach, this study considers the augmentation of the GARCH models with EVT forecasts. The two-step procedure of pre-filtering with a GARCH model and then applying EVT, as suggested by McNeil (1999), is also investigated. This study identifies some of the practical issues in model fitting. It is shown that no single forecasting model is universally optimal and the choice will depend on the nature of the data. For this data series, the best approach was to augment the GARCH stochastic volatility models with EVT forecasts during periods where the first do not converge. Model performance is judged by the actual number of VaR and ES violations compared to the expected number. The expected number is taken as the number of return observations over the entire sample period, multiplied by 0.01 for 99% VaR and ES calculations. / AFRIKAANSE OPSOMMING: Akkurate beraming van Waarde op Risiko (Value at Risk) en Verwagte Tekort (Expected Shortfall) is krities vir die bestuur van ekstreme mark risiko’s. Hierdie risiko’s kom met klein waarskynlikheid voor, maar die finansiële impakte is potensieel groot. Tradisionele modelle om Waarde op Risiko en Verwagte Tekort te beraam, word ondersoek. In ooreenstemming met die algemene praktyk, word 99% 10 dag maatstawwe bereken. ‘n Omvattende teoretiese agtergrond word eers gegee en daarna word die modelle toegepas op die Africa Financials Index vanaf 29/01/1996 tot 30/04/2013. Die modelle wat oorweeg word sluit onafhanklike, identies verdeelde modelle en Veralgemeende Auto-regressiewe Voorwaardelike Heteroskedastiese (GARCH) stogastiese volatiliteitsmodelle in. Ekstreemwaarde Teorie modelle, wat spesifiek op ekstreme mark opbrengste fokus, word ook ondersoek. In hierdie verband word die Peaks Over Threshold (POT) benadering tot Ekstreemwaarde Teorie gevolg. Vir die berekening van Waarde op Risiko word verskillende skaleringsmetodes van een dag na tien dae oorweeg en die prestasie van elk word ge-evalueer. Die GARCH modelle konvergeer nie gedurende tydperke van ekstreme opbrengste nie. Gedurende hierdie tydperke, kan Ekstreemwaarde Teorie modelle gebruik word. As ‘n nuwe benadering oorweeg hierdie studie die aanvulling van die GARCH modelle met Ekstreemwaarde Teorie vooruitskattings. Die sogenaamde twee-stap prosedure wat voor-af filtrering met ‘n GARCH model behels, gevolg deur die toepassing van Ekstreemwaarde Teorie (soos voorgestel deur McNeil, 1999), word ook ondersoek. Hierdie studie identifiseer sommige van die praktiese probleme in model passing. Daar word gewys dat geen enkele vooruistkattingsmodel universeel optimaal is nie en die keuse van die model hang af van die aard van die data. Die beste benadering vir die data reeks wat in hierdie studie gebruik word, was om die GARCH stogastiese volatiliteitsmodelle met Ekstreemwaarde Teorie vooruitskattings aan te vul waar die voorafgenoemde nie konvergeer nie. Die prestasie van die modelle word beoordeel deur die werklike aantal Waarde op Risiko en Verwagte Tekort oortredings met die verwagte aantal te vergelyk. Die verwagte aantal word geneem as die aantal obrengste waargeneem oor die hele steekproefperiode, vermenigvuldig met 0.01 vir die 99% Waarde op Risiko en Verwagte Tekort berekeninge.