The South African fixed interest rate market has historically had very little liquidity beyond 15 - 20 years. Most financial institutions are currently prepared to quote and trade interest rate risk up to a maximum term of 30 years. Any trades beyond 30 years usually attract very onerous spreads and raise relevant questions regarding an appropriate level of mid-rates. However, there are many South African entities whose business involves taking on exposure to interest rates beyond 30 years, such as life insurance companies and pension funds. These entities have historically used very traditional approaches to hedging their interest rate exposures across the whole term structure and have typically done little to gain any further protection. We can generalise the problems faced by any entity exposed to long term interest rate risk in South Africa: 1. The inadequacy of traditional matching methods (i.e. immunisation and bucketing) to cope with the long term interest rate risks. 2. The non-observability of interest rate data beyond the maximum term in the yield curve. Associated with this is the inability to adequately quantify interest rate risk. 3. The lack of liquidity in long term interest rate markets. Associated with this is the inability to adequately hedge long term interest rate risk. We examine various traditional approaches to matching / hedging interest rate risk using information available at observable / tradable terms on the nominal yield curve. We then look at the reasons why these approaches are not suitable for hedging long term interest rate risk. Some modern methods to forecast and hedge long term interest rate risks are then examined and the possibility of their use in managing long term interest risk is explored. On the back of these investigations, we propose a number of possible yield curve extrapolation procedures and methodology for performing calibrations. Using some general theoretical hedging results, we perform a case study which analyses the performance of various theoretical hedges over a historical period from October 2001 to March 2007. The results indicate that extrapolation and hedging of the yield curve is able to significantly reduce Value-At-Risk of long term interest rate exposures. A second case study is then performed which analyses performance of the various theoretical hedges using out-of-sample simulated yield curve data. We find that there appears to be a significant benefit to the use of yield curve extrapolation techniques, specifically when used in conjunction with a hedging strategy. In some cases we find that the more simple extrapolation techniques actually increase risk (significantly) when used in conjunction with hedging. However, for some of the more advanced techniques, risk can be significantly reduced. For an entity looking to deal with long term interest rate risk, we find that the choice of extrapolation technique and hedging strategy go hand-in-hand. For this reason the cost of hedging and reduction in risk are strongly correlated. The results we obtain suggest that it is necessary to weigh the benefits against the cost of hedging. Further, this cost seems to increase with increasing reduction in risk. The research and results presented here are related to those in the paper Long Term Forecasting and Hedging of the South African Yield Curve presented by Thomas and Maré at the 2007 Actuarial Convention in South Africa. Copyright / Dissertation (MSc)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

In this research, subsonic rarefied flows over a flat-plate at constant pressure are investigated using the direct simulation Monte Carlo (DSMC) technique. An infinitely thin plate (either finite or semi-infinite) with zero angle of attack is considered. Flows with a Mach number of 0.102 and 0.4 and a Reynolds number varying between 0.063 and 246 are considered covering most of the transitional regime between the free-molecule and the continuum limits. A two-dimensional DSMC code of G.A. Bird is used to simulate these flows, and the code is modified to examine the effects of various inflow and outflow boundary conditions. It is observed that simulations of the subsonic rarefied flows are sensitive to the applied boundary conditions. Several extrapolation techniques are considered for the evaluation of the flow properties at the inflow and outflow boundaries. Among various alternatives, four techniques are considered in which the solutions are found to be relatively less sensitive. In addition to the commonly used extrapolation techniques, in which the flow properties are taken from the neighboring boundary cells of the domain, a newly developed extrapolation scheme, based on tracking streamlines, is applied to the outflow boundaries, and the best results are obtained using the new extrapolation technique together with the Neumann boundary conditions. With the new technique, the flow is not distorted even when the computational domain is small. Simulations are performed for various freestream conditions and computational domain configurations, and excellent agreement is obtained with the available experimental data.

Les fonctionnelles de la densité couramment utilisées ont rencontrées un succès spectaculaire dans la modélisation des systèmes physiques, chimiques, et biologiques. Toutefois, elles se sont avérées inadaptées pour décrire certaines situations, comme par exemple les forces de dispersion de London ou les phénomènes de corrélation forte. Dans le cadre de cette thèse, nous nous sommes intéressés à des développements récents de la formulation de l’énergie de corrélation exprimée à partir du théorème de fluctuation-dissipation et connexion adiabatique, visant à pallier ces problèmes. En particulier, différentes implémentations des méthodes au-delà de l’approximation de la phase aléatoire, qui permettent la prise en compte de la contribution d’échange dans le calcul de l’énergie de corrélation, ont été comparées. Ensuite, afin de réduire drastiquement la complexité numérique, une procédure d’orthogonalisation des vecteurs utilisées pour représenter la matrice diélectrique a été développée. Ces méthodes ont ensuite été appliquées au calcul de l’énergie de liaison de petits complexes moléculaires. La formulation de l’énergie de corrélation de la théorie de perturbation de Møller-Plesset dans le contexte matrice diélectrique est aussi présentée et testée. En parallèle, des calculs utilisant les méthodes semi-empiriques numériquement efficaces ont été conduits sur trois ensembles de molécules afin d’en tester les performances concernant les énergies de liaisons en les comparant aux valeurs de références disponibles dans la littérature / Commonly used density functionals have encountered a spectacular success in the modelling of physical, chemical or biological systems. However, they have proven to be unsuitable to describe some situations, such as London’s dispersion forces or strong correlation behaviour. In this thesis, we have been interested in recent developments in the formulation of the correlation energy from the adiabatic connection fluctuation dissipation theorem, to overcome these problems. In particular, different implementations of methods beyond the random phase approximation, which allow to take into account the exchange contribution in the computation of the correlation energy, have been compared. Then, in order to drastically decrease the numerical complexity, an orthogonalization procedure of the vectors used to represent the dielectric matrix has been developed. Then these approaches were applied to the calculation of the binding energy of small molecular complexes. The formulation of the correlation energy of the Møller-Plesset perturbation theory within the dielectric matrix context is also presented and tested. In parallel, calculations using numerically efficient semi-empirical methods were conducted over three molecular sets in order to test their performances regarding the binding energies by comparing them to reference values available in the literature