The aim of this thesis is to improve risk measurement estimation by incorporating extra information in the form of constraint into completely non-parametric smoothing techniques. A similar approach has been applied in empirical likelihood analysis. The method of constraints incorporates bootstrap resampling techniques, in particular, biased bootstrap. This thesis brings together formal estimation methods, empirical information use, and computationally intensive methods. In this thesis, the constraint approach is applied to non-parametric smoothing estimators to improve the estimation or modelling of risk measures. We consider estimation of Value-at-Risk, of intraday volatility for market risk, and of recovery rate densities for credit risk management. Firstly, we study Value-at-Risk (VaR) and Expected Shortfall (ES) estimation. VaR and ES estimation are strongly related to quantile estimation. Hence, tail estimation is of interest in its own right. We employ constrained and unconstrained kernel density estimators to estimate tail distributions, and we estimate quantiles from the fitted tail distribution. The constrained kernel density estimator is an application of the biased bootstrap technique proposed by Hall & Presnell (1998). The estimator that we use for the constrained kernel estimator is the Harrell-Davis (H-D) quantile estimator. We calibrate the performance of the constrained and unconstrained kernel density estimators by estimating tail densities based on samples from Normal and Student-t distributions. We find a significant improvement in fitting heavy tail distributions using the constrained kernel estimator, when used in conjunction with the H-D quantile estimator. We also present an empirical study demonstrating VaR and ES calculation. A credit event in financial markets is defined as the event that a party fails to pay an obligation to another, and credit risk is defined as the measure of uncertainty of such events. Recovery rate, in the credit risk context, is the rate of recuperation when a credit event occurs. It is defined as Recovery rate = 1 - LGD, where LGD is the rate of loss given default. From this point of view, the recovery rate is a key element both for credit risk management and for pricing credit derivatives. Only the credit risk management is considered in this thesis. To avoid strong assumptions about the form of the recovery rate density in current approaches, we propose a non-parametric technique incorporating a mode constraint, with the adjusted Beta kernel employed to estimate the recovery density function. An encouraging result for the constrained Beta kernel estimator is illustrated by a large number of simulations, as genuine data are very confidential and difficult to obtain. Modelling high frequency data is a popular topic in contemporary finance. The intraday volatility patterns of standard indices and market-traded assets have been well documented in the literature. They show that the volatility patterns reflect the different characteristics of different stock markets, such as double U-shaped volatility pattern reported in the Hang Seng Index (HSI). We aim to capture this intraday volatility pattern using a non-parametric regression model. In particular, we propose a constrained function approximation technique to formally test the structure of the pattern and to approximate the location of the anti-mode of the U-shape. We illustrate this methodology on the HSI as an empirical example.

This work investigated a numerical approach to the search of a maximum heat transfer rate density (the overall heat transfer dissipated per unit of volume) from a two-dimensional laminar multiscale array of cylinders in cross-flow under an applied fixed pressure drop and subject to the constraint of fixed volume. It was furthermore assumed that the flow field was steady state and incompressible. The configuration had two degrees of freedom in the stationary state, that is, the spacing between the cylinders and the diameter of the smaller cylinders. The angular velocity of the cylinders was in the range 0 ≤ ϖ, ≤ 0.1. Two cylinders of different diameters were used, in the first case, the cylinders were aligned along a plane which lay on their centrelines. In the second case, the cylinder leading edge was aligned along the plane that received the incoming fluid at the same time. The diameter of the smaller cylinder was fixed at the optimal diameter obtained when the cylinders were stationary. Tests were conducted for co-rotating and counterrotating cylinders. The results were also compared with results obtained in the open literature and the trend was found to be the same. Results showed that the heat transfer from a rotating array of cylinders was enhanced in certain cases and this was observed for both directions of rotation from an array which was aligned on the centreline. For rotating cylinders with the same leading edge, there is heat transfer suppression and hence the effect of rotation on the maximum heat transfer rate density is insignificant. This research is important in further understanding of heat transfer from rotating cylinders, which can be applied to applications ranging from contact cylinder dryers in the chemical processes industry and rotating cylinder electrodes to devices used for roller hearth furnaces. / Dissertation (MEng)--University of Pretoria, 2011. / Mechanical and Aeronautical Engineering / unrestricted

Antarctic mass balance is mainly controlled by surface mass balance (SMB, i.e. the net effect of precipitations at the surface of the ice sheet) and ice discharge at its margins, mostly through ice shelves. These floating ice bodies made from ice flowing from the continent to the ocean are buttressed by ice rises (elevation of the sea floor on which ice shelf re-grounds) such as the Derwael Ice Rise (DIR) in Dronning Maud Land (DML). In addition to this role important to consider in the future contribution of Antarctica to sea level rise, ice rises are also “climate dipsticks” helping to reconstruct the climate of the past centuries to millennia at high resolution. Due to their coastal location, they witness the changes happening there more rapidly than inland. Furthermore, their internal stratigraphy forms arches that allow to assess their stability, to date their own formation and therefore, in some cases, to constrain the past extension of the ice sheet at the scale of several millennia. As part of the IceCon project :Constraining ice mass changes in Antarctica, this thesis aimed to drill a 120 m ice core (named IC12 for the IceCon project, 2012) at the divide of the DIR and perform physico-chemical analyses to study its density and its internal annual layering with the aim of reconstructing SMB of the last two centuries. We also recorded a virtual image of the borehole using an optical televiewer (OPTV) to assess the ability of this instrument to reconstruct a density profile and measure vertical strain rates when the logging is repeated in the same borehole after a sufficient period of time (here, 2 years).The results show a general increase in snow accumulation rates (SMB) of 30-40% during the 20th century, particularly marked during the last 20-50 years. SMB variability is governed to a large extent by atmospheric circulation and to a lesser extent by variations in sea ice cover. The vertical velocity profile measured from repeat borehole OPTV was applied to refine SMB correction and the results fall in the error range of the corrections made using a model previously developed to study the DIR’s stability. This thesis also contributed to characterizing the spatial variability of SMB across the DIR by dating internal reflection horizons (IRHs), former surfaces of the DIR buried under subsequent snow layers and detected using radio-echo-sounding, and by measuring the density profile of IC12. SMB is found to be 2.5 times higher on the upwind slope than on the downwind slope due to the orographic effect. This pattern is regularly observed on ice rises in DML and stresses the importance of adopting a sufficient spatial resolution (5 km) in climate models.Finally, the technical developments allowing to rapidly reconstruct a density profile from the OPTV image of a borehole contributed to improving our knowledge of two features of Antarctic ice shelves, namely melt ponds, influencing surface mass balance and subglacial channels, influencing basal mass balance. Specifically, the results show that density is 5 % higher in surface trenches associated with subglacial channels, and that ice below melt ponds can reach the density of bubble-free ice due to melting and refreezing processes, with implications on ice shelf viscosity. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished