Thesis (MComm (Statistics and Actuarial Science))--Stellenbosch University, 2008. / The goodness-of- t of a statistical model describes how well it ts a set of observations. Measures of goodness-of- t typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, for example to test for normality, to test whether two samples are drawn from identical distributions, or whether outcome frequencies follow a speci ed distribution. Goodness-of- t for copulas is a special case of the more general problem of testing multivariate models, but is complicated due to the di culty of specifying marginal distributions. In this thesis, the goodness-of- t test statistics for general distributions and the tests for copulas are investigated, but prior to that an understanding of copulas and their properties is developed. In fact copulas are useful tools for understanding relationships among multivariate variables, and are important tools for describing the dependence structure between random variables. Several univariate, bivariate and multivariate test statistics are investigated, the emphasis being on tests for normality. Among goodness-of- t tests for copulas, tests based on the probability integral transform, Rosenblatt's transformation, as well as some dimension reduction techniques are considered. Bootstrap procedures are also described. Simulation studies are conducted to rst compare the power of rejection of the null hypothesis of the Clayton copula by four di erent test statistics under the alternative of the Gumbel-Hougaard copula, and also to compare the power of rejection of the null hypothesis of the Gumbel-Hougaard copula under the alternative of the Clayton copula. An application of the described techniques is made to a practical data set.

Thesis (MComm (Statistics and Actuarial Science)--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Life insurance and pension funds offer a wide range of products that are invested in a mix of assets. These portfolios (II), underlying the products, are rebalanced back to predetermined fixed proportions on a regular basis. This is done by selling the better performing assets and buying the worse performing assets. Life insurance or pension fund contracts can offer the client a minimum payout guarantee on the contract by charging them an extra premium (a). This problem can be changed to that of the pricing of a put option with underlying . It forms a liability for the insurance firm, and therefore needs to be managed in terms of risks as well. This can be done by studying the option’s sensitivities. In this thesis the premium and sensitivities of this put option are calculated, using different Monte Carlo methods, in order to find the most efficient method. Using general Monte Carlo methods, a simplistic pricing method is found which is refined by applying mathematical techniques so that the computational time is reduced significantly. After considering Antithetic Variables, Control Variates and Latin Hypercube Sampling as variance reduction techniques, option prices as Control Variates prove to reduce the error of the refined method most efficiently. This is improved by considering different Quasi-Monte Carlo techniques, namely Halton, Faure, normal Sobol’ and other randomised Sobol’ sequences. Owen and Faure-Tezuke type randomised Sobol’ sequences improved the convergence of the estimator the most efficiently. Furthermore, the best methods between Pathwise Derivatives Estimates and Finite Difference Approximations for estimating sensitivities of this option are found. Therefore by using the refined pricing method with option prices as Control Variates together with Owen and Faure-Tezuke type randomised Sobol’ sequences as a Quasi-Monte Carlo method, more efficient methods to price this option (compared to simplistic Monte Carlo methods) are obtained. In addition, more efficient sensitivity estimators are obtained to help manage risks. / AFRIKAANSE OPSOMMING: Lewensversekering en pensioenfondse bied die mark ’n wye reeks produkte wat belê word in ’n mengsel van bates. Hierdie portefeuljes (II), onderliggend aan die produkte, word op ’n gereelde basis terug herbalanseer volgens voorafbepaalde vaste proporsies. Dit word gedoen deur bates wat beter opbrengste gehad het te verkoop, en bates met swakker opbrengste aan te koop. Lewensversekeringof pensioenfondskontrakte kan ’n kliënt ’n verdere minimum uitbetaling aan die einde van die kontrak waarborg deur ’n ekstra premie (a) op die kontrak te vra. Die probleem kan verander word na die prysing van ’n verkoopopsie met onderliggende bate . Hierdie vorm deel van die versekeringsmaatskappy se laste en moet dus ook bestuur word in terme van sy risiko’s. Dit kan gedoen word deur die opsie se sensitiwiteite te bestudeer. In hierdie tesis word die premie en sensitiwiteite van die verkoopopsie met behulp van verskillende Monte Carlo metodes bereken, om sodoende die effektiefste metode te vind. Deur die gebruik van algemene Monte Carlo metodes word ’n simplistiese prysingsmetode, wat verfyn is met behulp van wiskundige tegnieke wat die berekeningstyd wesenlik verminder, gevind. Nadat Antitetiese Veranderlikes, Kontrole Variate en Latynse Hiperkubus Steekproefneming as variansiereduksietegnieke oorweeg is, word gevind dat die verfynde metode se fout die effektiefste verminder met behulp van opsiepryse as Kontrole Variate. Dit word verbeter deur verskillende Quasi-Monte Carlo tegnieke, naamlik Halton, Faure, normale Sobol’ en ander verewekansigde Sobol’ reekse, te vergelyk. Die Owen en Faure-Tezuke tipe verewekansigde Sobol’ reeks verbeter die konvergensie van die beramer die effektiefste. Verder is die beste metode tussen Baanafhanklike Afgeleide Beramers en Eindige Differensie Benaderings om die sensitiwiteit vir die opsie te bepaal, ook gevind. Deur dus die verfynde prysingsmetode met opsiepryse as Kontrole Variate, saam met Owen en Faure-Tezuke tipe verewekansigde Sobol’ reekse as ’n Quasi-Monte Carlo metode te gebruik, word meer effektiewe metodes om die opsie te prys, gevind (in vergelyking met simplistiese Monte Carlo metodes). Verder is meer effektiewe sensitiwiteitsberamers as voorheen gevind wat gebruik kan word om risiko’s te help bestuur.

Thesis (MComm)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The Southern Ocean is a complex system, where the relationship between CO2 concentrations and its drivers varies intra- and inter-annually. Due to the lack of readily available in situ data in the Southern Ocean, a model approach was required which could predict the CO2 concentration proxy variable, fCO2. This must be done using predictor variables available via remote measurements to ensure the usefulness of the model in the future. These predictor variables were sea surface temperature, log transformed chlorophyll-a concentration, mixed layer depth and at a later stage altimetry. Initial exploratory analysis indicated that a non-parametric approach to the model should be taken. A parametric multiple linear regression model was developed to use as a comparison to previous studies in the North Atlantic Ocean as well as to compare with the results of the non-parametric approach. A non-parametric kernel regression model was then used to predict fCO2 and nally a combination of the parametric and non-parametric regression models was developed, referred to as the mixed regression model. The results indicated, as expected from exploratory analyses, that the non-parametric approach produced more accurate estimates based on an independent test data set. These more accurate estimates, however, were coupled with zero estimates, caused by the curse of dimensionality. It was also found that the inclusion of salinity (not available remotely) improved the model and therefore altimetry was chosen to attempt to capture this e ect in the model. The mixed model displayed reduced errors as well as removing the zero estimates and hence reducing the variance of the error rates. The results indicated that the mixed model is the best approach to use to predict fCO2 in the Southern Ocean and that altimetry's inclusion did improve the prediction accuracy. / AFRIKAANSE OPSOMMING: Die Suidelike Oseaan is 'n komplekse sisteem waar die verhouding tussen CO2 konsentrasies en die drywers daarvoor intra- en interjaarliks varieer. 'n Tekort aan maklik verkrygbare in situ data van die Suidelike Oseaan het daartoe gelei dat 'n model benadering nodig was wat die CO2 konsentrasie plaasvervangerveranderlike, fCO2, kon voorspel. Dié moet gedoen word deur om gebruik te maak van voorspellende veranderlikes, beskikbaar deur middel van afgeleë metings, om die bruikbaarheid van die model in die toekoms te verseker. Hierdie voorspellende veranderlikes het ingesluit see-oppervlaktetemperatuur, log getransformeerde chloro l-a konsentrasie, gemengde laag diepte en op 'n latere stadium, hoogtemeting. 'n Aanvanklike, ondersoekende analise het aangedui dat 'n nie-parametriese benadering tot die data geneem moet word. 'n Parametriese meerfoudige lineêre regressie model is ontwikkel om met die vorige studies in die Noord-Atlantiese Oseaan asook met die resultate van die nieparametriese benadering te vergelyk. 'n Nie-parametriese kern regressie model is toe ingespan om die fCO2 te voorspel en uiteindelik is 'n kombinasie van die parametriese en nie-parametriese regressie modelle ontwikkel vir dieselfde doel, wat na verwys word as die gemengde regressie model. Die resultate het aangetoon, soos verwag uit die ondersoekende analise, dat die nie-parametriese benadering meer akkurate beramings lewer, gebaseer op 'n onafhanklike toets datastel. Dié meer akkurate beramings het egter met "nul"beramings gepaartgegaan wat veroorsaak word deur die vloek van dimensionaliteit. Daar is ook gevind dat die insluiting van soutgehalte (nie beskikbaar oor via sateliet nie) die model verbeter en juis daarom is hoogtemeting gekies om te poog om hierdie e ek in die model vas te vang. Die gemengde model het kleiner foute getoon asook die "nul"beramings verwyder en sodoende die variasie van die foutkoerse verminder. Die resultate het dus aangetoon dat dat die gemengde model die beste benadering is om te gebruik om die fCO2 in die Suidelike Oseaan te beraam en dat die insluiting van altimetry die akkuraatheid van hierdie beraming verbeter.

Thesis (MComm)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: This thesis examines the effect of liquidity on excess stock returns on the Johannesburg Stock Exchange (JSE) over the period 2003 to 2011. It builds on the findings of previous studies that found size, value and momentum effects to be significant in explaining market anomalies by adding a further explanatory factor, namely liquidity. A standard CAPM, as well as a momentum-augmented Fama-French (1993: 3) model are employed to perform regression analyses to examine the effect of the four variables on excess stock returns. Results suggested that the log of the stock‘s market value best captured the size effect, the earnings yield best captured the value effect and the previous three month‘s returns best captured the momentum effect. Five liquidity proxies are used: the bid-ask spread first proposed by Amihud (1986: 223), turnover, the price impact measure of Amihud (2002: 31) and two zero return measures proposed by Lesmond et al. (1999: 1113). Despite prior studies having found liquidity to be an influential factor, this thesis found the opposite to be true. This finding remains robust, irrespective of the type of liquidity measure used. While size, value and momentum are found to be significant to a certain extent in explaining excess stock returns over the period, liquidity is not found to be significant. This is a surprising result, given that the JSE is seen as an emerging market, which is generally regarded as illiquid. This fact is exacerbated by the fact that the JSE is a highly concentrated and therefore skewed market that is dominated by only a handful of shares. Hence liquidity is expected to be of utmost importance. The result that liquidity is however not a priced factor on this market is therefore an important finding that requires further analysis to determine why this is the case. In addition, significant non-zero intercepts remained, indicating continued missing risk factors. / AFRIKAANSE OPSOMMING: In hierdie tesis word die effek van likiditeit op oormaat aandeel-opbrengste op die Johannesburg Effektebeurs (JEB) ondersoek gedurende die periode 2003 tot 2011. Dit bou voort op die bevindinge van vorige studies wat toon dat grootte, waarde en momentum beduidend is in die verklaring van mark onreëlmatighede deur 'n addisionele verklarende faktor, likiditeit, toe te voeg. 'n Standaard kapitaalbateprysingsmodel (KBPM) sowel as 'n momentum-aangepaste Fama-French (1993: 3) model word gebruik om deur middel van regressie analise die effek van die vier veranderlikes op oormaat aandeel-opbrengste te ondersoek. Die resultate toon dat die grootte effek die beste verteenwoordig word deur die logaritme van die aandeel se mark kapitalisasie, die verdienste-opbrengs verteenwoordig die waarde effek en die vorige drie-maande opbrengskoerse verteenwoordig die momentum effek die beste. Vyf likiditeitsveranderlikes is gebruik: bod-en-aanbod spreiding voorgestel deur Amihud (1986: 223), omset, die prys-impak maatstaf van Amihud (2002: 31) en twee nul-opbrengskoers maatstawwe voorgestel deur Lesmond et al. (1999: 1113). Afgesien van die feit dat vorige studies die effek van likiditeit beduidend vind, word die teenoorgestelde in hierdie tesis gevind. Hierdie bevinding bly robuus, ongeag van die likiditeitsveranderlike wat gebruik word. Terwyl bevind is dat grootte, waarde en momentum beduidend is tot 'n sekere mate in die verklaring van oormaat aandeel-opbrengste tydens die periode, is geen aanduiding dat likiditeit 'n addisionele beduidende verklarende faktor is gevind nie. Hierdie bevinding is onverwags, aangesien die JEB beskou word as 'n ontluikende mark, wat normaalweg illikied is. Hierdie feit word vererger deur dat die JEB hoogs gekonsentreerd is en dus 'n skewe mark is wat oorheers word deur slegs 'n hand vol aandele. Dus word verwag dat likiditeit 'n baie belangrike faktor behoort te wees. Die bevinding dat likiditeit nie 'n prysingsfaktor op hierdie mark is nie, is dus 'n belangrike bevinding en vereis verdere analise om vas te stel waarom dit die geval is. Addisioneel word beduidende nie-nul afsnitte verkry, wat aandui dat daar steeds risiko faktore is wat nog nie geïdentifiseer is nie.

Thesis (MComm)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: In this study Portfolio Opportunity Distributions (PODs) is applied as an alternative performance evaluation method. Traditionally, Broad-Market Indices or peer group comparisons are used to perform performance evaluation. These methods however have various biases and other problems related to its use. These biases and problems include composition bias, classification bias, concentration, etc. R.J. Surz (1994) introduced PODs in order to eliminate some of these problems. Each fund has its own opportunity set based on its style mandate and constraints. The style mandate of the fund is determined by calculating the fund’s exposure to the nine Surz Style Indices through the use of Returns-Based Style Analysis (RBSA). The indices are created based on the style proposed by R.J. Surz (1994). Some adjustments were made to incorporate the unique nature of the South African equity market. The combination of the fund’s exposures to the indices best explains the return that the fund generated. In this paper the fund’s constraints are based on the regulation requirements imposed on the funds in South Africa by the Collective Investment Schemes Control Act No. 45 of 2002 (CISCA). Thousands of random portfolios are then generated based on the fund’s opportunity set. The return and risk of the simulated portfolios represent the possible investment outcomes that the manager could have achieved given its opportunity set. Together the return and risk of the simulated portfolios represent a range of possible outcomes against which the performance of the fund is compared. It is also possible to determine the skill of the manager since it can be concluded that a manager who consistently outperforms most of the simulated portfolios shows skill in selecting shares to be included in the portfolio and assigning the correct weights to these shares. The South African Rand depreciated quite a bit during the period under evaluation and therefore funds invested large portions of their assets in foreign investments. These investments mostly yielded very high or very low returns compared to the returns available in the domestic equity market which impacted the application of PODs. Although the PODs methodology shows great potential, it is impossible to conclude with certainty whether the PODs methodology is superior to the traditional methods based on the current data. / AFRIKAANSE OPSOMMING: In hierdie studie word Portefeulje Geleentheids Verdelings (“PODs”) bekendgestel as ‘n alternatiewe manier om die obrengste van bestuurders te evalueer. Gewoonlik word indekse en die vergelyking van die fonds met soortgelyke fondse gebruik om fondse te evalueer. Die metodes het egter verskeie probleme wat met die gebruik daarvan verband hou. Die probleme sluit onder andere in: die samestelling en klassifikasie van soortgelyke fondse, die konsentrasie in die mark, ens. R.J. Surz (1994) het dus Portefeulje Geleentheids Verdelings (“PODs”) bekendgestel in ‘n poging om sommige van die probeleme te elimineer. Elke fonds het sy eie unieke geleentheids versameling wat gebaseer is op die fonds se styl en enige beperkings wat op die fonds van toepassing is. Die fonds se styl word bepaal deur die fonds se blootstelling aan die nege Surz Styl Indekse te meet met behulp van opbrengs-gebaseerde styl analise (“RBSA”). Die indekse is geskep gebaseer op die metode wat deur R.J. Surz (1994) voorgestel is. Daar is egter aanpassings gemaak om die unieke aard van die Suid-Afrikaanse aandele mark in ag te neem. Die kombinasie van die fonds se blootstelling aan die indekse verduidelik waar die fonds se opbrengs vandaan kom. In die navorsingstuk is die beperkings wat van toepassing is op die fonds afkomstig uit die regulasie vereistes wat deur die “Collective Investment Schemes Control Act No. 45 of 2002 (CISCA)” in Suid-Afrika op fondse van toepassing is. Duisende ewekansige portefeuljes word dan gegenereer gebaseer op die fonds se unieke groep aandele waarin die fonds kan belê. Die opbrengs en risiko van die gesimuleerde portefeuljes verteenwoordig al die moontlike beleggings uitkomste wat die fonds bestuurder kon gegenereer het gegewe die fonds se unieke groep aandele waarin dit kon belê. Die opbrengs en risiko van al die gesimuleerde portefeuljes skep saam ‘n verdeling van moontlike beleggings uitkomste waarteen die opbrengs en risiko van die fonds vergelyk word. Hierdie proses maak dit moontlik om die fonds bestuurder se vermoë om beter as meeste van die gesimuleerde portefeuljes te presteer te bepaal. Die aanname kan gemaak word dat ‘n bestuurder wat konsekwent oor tyd beter as meeste van die gesimuleerde portefeuljes presteer oor die vermoë beskik om die regte aandele te kies om in die portefeulje in te sluit en ook die regte gewigte aan die aandele toe te ken. Die Suid-Afrikaanse Rand het heelwat gedepresieer tydens die evaluasie periode en daarom het fondse groot porsies van hul beleggings oorsee belê. Die beleggings het dus of heelwat groter of heelwat kleiner opbrengste gehad in vergelyking met die opbrengste beskikbaar in die plaaslike aandelemark en dit het die toepassing van PODs beïnvloed. PODs toon baie potential, maar dit is egter onmoontlik om met die huidige data stel vas te stel of dit ‘n beter metode is.

M.Com. (International Accounting) / Post-employment benefits under IAS 19 include defined benefit plans (DBP’s) and defined contribution plans. The accounting for defined contribution plans is fairly straightforward, since accrual accounting is applied and the employer entity’s legal or constructive obligation is limited to the amount the employer entity agrees to contribute to the defined contribution plan. In contrast, the accounting for DBP’s is complicated and provides a promise/guarantee of future benefits and the investment and actuarial risk of the plan lies with the employer entity. The literature review indicated that accounting and presentation of DBP’s is complicated because of the long-term nature of the liability/asset that is raised for the plan. There are many uncertainties involved in estimating the liability as this involves looking into the future and making estimates and assumptions about the future. The literature also indicated factors such as the market performance of assets, and inaccurate or unrealistic assumptions and decisions that delay making payments to DBP’s affects the funding status. Actuaries and accountants differ somewhat in the roles they play in determining the amount for DBP’s, with accountants choosing the accrued benefit method. The problem with DBP’s is that they are of a long-term nature and require estimates and assumptions to be made in calculating the DBP liability/asset. The long-term nature affects the adequacy of the liability/asset recognised for DBP’s and the related disclosure in the financial statements of large listed companies. The objective of the minor dissertation is to perform a content analysis on the presentation and disclosure of DBPs in the financial statements of a sample of Johannesburg Stock Exchange listed companies in South Africa. The research approach applied includes a broad assessment of the current status of DBP’s and defined contribution plans operated by the top 40 Johannesburg Stock Exchange (JSE) listed companies, followed by a quantitative and qualitative assessment on the disclosures provided by these companies’ financial statements.

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, 2006. / In recent times many theories have surfaced posing challenging threats to the Efficient Market Hypothesis. We are entering an exciting era of financial economics fueled by the urge to have a better understanding of the intricate workings of financial markets. Many studies are emerging that investigate the relationship between stock market predictability and efficiency. This paper studies the existence of calendar-based patterns in equity returns, price momentum and earnings momentum in the South African securities market. These phenomena are commonly referred to in the literature as security market imperfections, financial market puzzles and market anomalies. We provide evidence that suggests that they do exist in the South African context, which is consistent with findings in various international markets. A vast number of papers on the subject exist in the international arena. However, very few empirical studies on the South African market can be found in the public domain. We aim to contribute to the literature by investigating the South African case.

Thesis (MComm (Statistics and Actuarial Science))--University of Stellenbosch, 2007. / The use of options on various stock markets over the world has introduced a unique opportunity for investors to hedge, speculate, create synthetic financial instruments and reduce funding and other costs in their trading strategies. The power of options lies in their versatility. They enable an investor to adapt or adjust her position according to any situation that arises. Another benefit of using options is that they provide leverage. Since options cost less than stock, they provide a high-leverage approach to trading that can significantly limit the overall risk of a trade, or provide additional income. This versatility and leverage, however, come at a price. Options are complex securities and can be extremely risky. In this document several aspects of trading and valuing some exotic options are investigated. The aim is to give insight into their uses and the risks involved in their trading. Two volatility-dependent derivatives, namely compound and chooser options; two path-dependent derivatives, namely barrier and Asian options; and lastly binary options, are discussed in detail. The purpose of this study is to provide a reference that contains both the mathematical derivations and detail in valuating these exotic options, as well as an overview of their applicability and use for students and other interested parties.

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.