In the pursuit of efficient methods of dimension reduction for multi-factor correlation systems and for sparsely populated and partially observed matrices, the problem of matrix completion within a low-rank framework is of particular significance. This dissertation presents the methods of spectral completion and convex relaxation, which have been successfully applied to the particular problem of lowrank completion and recovery of valid correlation matrices. Numerical testing was performed on the classical exponential and noisy Toeplitz parametrisations and, in addition, to real datasets comprising of FX rates and stock price data. In almost all instances, the method of convex relaxation performed better than spectral methods and achieved the closest and best-fitted low-rank approximations to the true, optimal low-rank matrices (for some rank-n). Furthermore, a dependence was found to exist on which correlation pairs were used as inputs, with the accuracy of the approximations being, in general, directly proportional to the number of input correlations provided to the algorithms.

Though it is customary to use standard Gaussian term structure models for term structure modelling, this becomes theoretically implausible in cases when nominal interest rates are near zero: Gaussian models can have arbitrarily large negative rates, whereas arbitrage considerations dictate that rates should remain positive (or very slightly negative at most). Black (1995) suggests that interest rates include an optionality which restricts them to non-negative values. This introduces a non-linearity at the zero-lower bound that makes these so-called shadow-rate models a computational challenge. This dissertation analyses the shadow-rate approximations suggested by Krippner (2013) and Priebsch (2013) for the Vasicek and ˇ arbitrage-free Nelson-Siegel (AFNS) models. We also investigate and compare the accuracy of the iterated extended Kalman filter (IEKF) with that of the unscented Kalman filter (UKF). We find that Krippner’s approach approximates interest rates within reasonable bounds for both the 1-factor Vasicek and AFNS models. Prieb- ˇ sch’s first-cumulant method is more accurate than Krippner’s method for a 1-factor Vasicek model, while Priebsch’s second-cumulant method is deemed impractical ˇ because of the computational time it takes. In a multi-factor AFNS model, only Krippner’s framework is feasible. Moreover, the IEKF outperforms the UKF in terms of filtering with no significant difference in run-time.

This dissertation examines the properties and some of the uses of compositional data. It gives a brief history of the distinction between 'normal' data and compositions, as well as the various methods of analysing compositional data. It is mainly concerned with performing regression analysis including compositions. In order to model a composition it is necessary to understand the nature of compositions and how to use standard statistical tools with them. This dissertation describes the simplex and several functions which can be performed in it, as well as introducing several useful covariate structures for compositional samples after transformation. It also introduces the transformations between the simplex and unconstrained real space. The dissertation concludes with four examples of regression analyses involving compositions.

Includes abstract. / Includes bibliographical references. / This paper investigates the relationship between scale and efficiency through the application of Data Envelopment Analysis (DEA) to a set of South African private hospitals over the three year period from 2007 to 2009. As part of the investigation, this paper provides a description of the current research into scale and efficiency with a focus on definition and measurement. It also provides an introduction to DEA as a tool for measuring the relationship between hospital scale and efficiency. Based on the underlying set of private hospitals, this investigation found that scale efficiency improvements are likely to be possible.

Includes bibliographical references. / The comparative assessment of physician performance, also known as ‘physician profiling’ is frequently used by healthcare funders. It aims to identify and improve the resource efficiency and quality of physician care. South African private healthcare funders use a wide range of profiling techniques; however, currently the use of frontier analysis is absent. This study explores the use of the non - parametric frontier analysis technique called Data Envelopment Analysis (DEA) for the profiling of physicians in South Africa. This is investigated by following a DEA profiling approach to evaluate the performance of 403 general/ paediatric surgeons in providing in - hospital services in 2012. A 7 - input 1 - output VRS DEA model is used to determine the efficiency of the surgeons. The profiling results are then analysed to determine their usefulness. It results reveal that 58 surgeons are efficient, representing only 14.4% of surgeons profiled. Therefore, the DEA approach reveals a large potential for efficiency improvements. The average efficiency score of inefficient surgeons is found to be 0.68. This means that, on average, inefficient surgeons have to decrease resource utilisation by 32% to achieve efficiency. The DEA approach is also found to be proficient at identifying the physicians presenting the most severe levels of inefficiency. 37 surgeons are found to be significantly inefficient. The approach also allows for the identification of peers against which inefficient surgeons are able to directly compare their practices. These results are determined to be of significant potential use to South African private healthcare funders. It is, however, noted that the analysis and results obtained was solely of a statistical nature. Closer consideration of the clinical appropriateness of the results is essential. In any case, this study concludes that a DEA profiling approach can be considered a useful technique in the comparison of physician performance in South Africa.

Includes bibliographical references. / On retirement individuals are expected to live on what they have saved over their working life-times if they are not to be dependent on the state or family. However, investments made over the working life-time may be inadequate to provide fully for retirement. Whilst it is obvious that one must save enough to retire on, it is not always clear how much is enough, or how best to save this. This can, and usually does, result in individuals retiring with less income than they need or expect (Masilela, 2011a), and in many cases, causes them to become dependent on their families or the state.

Includes abstract. / Includes bibliorahical references. / This dissertation presents an eclectic mix around a central theme of alpha, or value-add. It comprises four essays that are concerned with various theoretical and empirical aspects of alpha. The primary objective is to provide new perspectives in the major areas of modelling alpha; namely, performance measurement, opportunity forecasting and tradability.

This study aims to test the suitability of using Bayesian probabilistic models to predict bankruptcy of JSE-listed companies. A sample of 132 companies is considered with fourteen years of financial statement information and macroeconomic indicators used as predictor variables. Various permutations of Bayesian models are tested relating to different learning algorithms, intervals of discretisation and scoring metrics. In contrast to previous research, we explore a variety of evaluation measures and it is found that predictive accuracy for bankrupt firms does not exceed 70% in any model augmentation. On comparison to other popular models such as the Altman Z-score and the logit model, it is found that Bayesian networks produce marginally better predictive accuracy. Furthermore, a comparison to previous research on the same subject is carried and reasons for significantly different results are considered. Finally, the reasons for low predictive accuracies is considered with issues relating specifically to South Africa being discussed.

Includes abstract. / Includes bibliographical references. / There is little existing South African literature relating to hospital efficiency that allows for differences in case mix across hospitals. One of the primary motivations for this dissertation is to help fill this gap in the literature by examining the impact that adjusting for differences in case mix has on efficiency scores. Data Envelopment Analysis (DEA) is chosen as the efficiency measurement method because of its exibility and ease of handling multiple inputs and outputs. A number of DEA models are applied to a sample of South African private hospitals for the years 2008 to 2011 inclusive. Three different case-mix adjustment techniques are investigated and their ability to capture differences in case mix is assessed. The three techniques investigated are: a case-mix adjustment factor (constructed using Diagnosis-Related Groups (DRGs)) to adjust outputs; including the case-mix adjustment factor as an additional output; and disaggregating hospital admissions into broad categories which are used as outputs. A comparison of the unadjusted model with the case-mix adjusted model reveals that omitting the adjustment can have a considerable impact on efficiency scores. Whilst little difference is noted in average efficiency scores for the group of hospitals, 90% for the unadjusted model and 92% for the adjusted model in 2011, there are substantial differences between the adjusted and unadjusted efficiency scores of individual hospitals. On comparison of the three different techniques investigated, it is evident that if there is sufficient data to construct a case-mix adjustment factor, case-mix adjusted admissions should be used, rather than using the factor as an additional output variable. In the case where insufficient data is available, disaggregating admissions does capture some of the differences in case mix but a substantial amount of power is lost as a result of increasing the number of output variables.

Bibliography: pages 100-108. / The salient feature under examination in this thesis is the assumption that the error terms, ZD(t) and Zy(t), are normally distributed. This assumption is common to most of the stochastic asset models that are in widespread use within the actuarial profession. An example is the well known Wilkie model (Wilkie (1984, 1995)).