This thesis challenges several concepts in finance. Firstly, it is the Markowitz's solution to the portfolio problem. It introduces a new method which de- noises the covariance matrix - the cornerstone of the portfolio management. Random Matrix Theory originates in particle physics and was recently intro- duced to finance as the intersection between economics and natural sciences has widened over the past couple of years. Often discussed Efficient Market Hypothesis is opposed by adopting the assumption, that financial returns are driven by Paretian distributions, in- stead of Gaussian ones, as conjured by Mandelbrot some 50 years ago. The portfolio selection is set in a framework, where Expected Shortfall replaces the standard deviation as the risk measure. Therefore, direct optimi- sation of the portfolio is implemented to be compared with the performance of the classical solution and its denoised counterpart. The results are evalu- ated in a controlled environment of Monte Carlo simulation as well as using empirical data from S&P 500 constituents. 1

This paper analyses the impact of banking mergers on systemic risk, with in particular if internationalization prior to acquisition increases systemic risk. By using the marginal expected shortfall methodology for an international sample of mergers, a significant increase in systemic risk is found as a result of mergers in the financial sector. Moreover, if a bank is operating internationally prior to acquisition, this increases systemic risk. Additionally, there is evidence of a too-big-to-fail motive for relatively smaller banks to use mergers to become systemically important. The results confirm that consolidation in the financial sector increases fragility of the financial system.

The 2008/9 financial crisis intensified the search for realistic return models, that capture real market movements. The assumed underlying statistical distribution of financial returns plays a crucial role in the evaluation of risk measures, and pricing of financial instruments. In this dissertation, we discuss an empirical study on the evaluation of the traditional portfolio risk measures, and option pricing under the hybrid Brownian motion model, developed by Shaw and Schofield. Under this model, we derive probability density functions that have a fat-tailed property, such that “25-sigma” or worse events are more probable. We then estimate Value-at-Risk (VaR) and Expected Shortfall (ES) using four equity stocks listed on the Johannesburg Stock Exchange, including the FTSE/JSE Top 40 index. We apply the historical method and Variance-Covariance method (VC) in the valuation of VaR. Under the VC method, we adopt the GARCH(1,1) model to deal with the volatility clustering phenomenon. We backtest the VaR results and discuss our findings for each probability density function. Furthermore, we apply the hybrid model to price European style options. We compare the pricing performance of the hybrid model to the classical Black-Scholes model. / Dissertation (MSc)--University of Pretoria, 2017. / National Research Fund (NRF), University of Pretoria Postgraduate bursary and the General Studentship bursary / Mathematics and Applied Mathematics / MSc / Unrestricted

In recent years, the question of whether Expected Shortfall is possible to backtest has been a hot topic after the findings of Gneiting in 2011 that Expected Shortfall lacks a mathematical property called elicitability. However, new research has indicated that backtesting of Expected Shortfall is in fact possible and that it does not have to be very difficult. The purpose of this thesis is to show that Expected Shortfall is in fact backtestable by providing six different examples of how a backtest could be designed without exploiting the property of elicitability. The different approaches are tested and their performances are compared against each other. The material can be seen as guidance on how to think in the initial steps of the implementation of an Expected Shortfall backtest in practice. / De senaste åren har frågan om huruvida det är möjligt att hitta backtester som validerar Expected Shortfall varit ett omdiskuterat ämne efter att Gneiting 2011 visade att Expected Shortfall saknade den matematiska egenskapen som kallas elicitabilitet. Ny forskning tyder på att det går att validera Expected Shortfall och att det inte behöver vara alltför svårt. Syftet med den här uppsatsen är att visa att det går att hitta metoder som backtestar Expected Shortfall. Vi gör det genom att visa utförandet av sex olika metoder som validerar Expected Shortfall utan att använda sig av elicitabilitet. De olika metoderna testas och deras egenskaper jämförs mot varandra. Materialet kan ses som en guide i hur man ska tänka i de första stegen i implementeringen av en metod för att backtesta Expected Shortfall.

Value-at-Risk (VaR) has long been the standard risk measure in financial risk management. However, VaR suffers from critical shortcomings as a risk measure when it comes to quantifying the most severe risks, which was made especially apparent during the financial crisis of 2007–2008. An alternative risk measure addressing the shortcomings of VaR known as expected shortfall (ES) is gaining popularity and is set to replace VaR as the standard measure of financial risk. This thesis introduces how extreme value theory can be applied in estimating ES using an unconditional peaks-over-threshold method. This includes giving an introduction to the theoretical foundations of the method. An application of this method is also performed on five different assets. These assets are chosen to serve as a proxy for the more broad asset classes of equity, fixed income, currencies, commodities and cryptocurrencies. In terms of ES, we find that cryptocurrencies is the riskiest asset and fixed income the safest.

In this work, we look into the practical applicability of three joint quantile and expected shortfall regression backtests. The strict, auxiliary, and intercept ESR backtests are applied to the historical log returns of the OMX Stockholm 30 market-weight price index. We estimate the conditional variance using GARCH models for various rolling window lengths and refitting frequencies. We are particularly interested in the rejection rates of the one-sided intercept ESR backtest as it is comparable to the current standard of backtests. The one-sided test is found to perform well when the conditional variance is estimated by either the GARCH(1,1), GJR-GARCH(1,1), or EGARCH(1,1) coupled with student’s t-innovation residuals and a rolling window size of 1000 days.

Within Central Counterparty Clearing, the Clearing House collects Initial Margin from its Clearing Members. The Initial Margin can be calculated in many ways, one of which is by applying the commonly used risk measure Value-at-Risk. However, Value-at-Risk has one major flaw, namely its inability to encapsulate Tail Risk. Due to this, there has for long been a desire to replace Value-at-Risk with Expected Shortfall, another risk measure that has shown to be much better suited to encapsulate Tail Risk. That said, Value-at-Risk is still used over Expected Shortfall, something which is mainly due to the fact that there is no consensus regarding how one should backtest Expected Shortfall. The goal of this thesis is to evaluate some of the most commonly proposed methods for backtesting Expected Shortfall. In doing this, several non-parametric backtests of Expected Shortfall are investigated using simulated data as well as market data from different types of securities. Moreover, this thesis aims to shed some light on the differences between Value-at-Risk and Expected Shortfall, highlighting why a change of risk measure is not as straightforward as one might believe. From the investigations of the thesis, several backtests are found to be sufficient for backtesting the Initial Margin with Expected Shortfall as the risk measure, the so called Minimally Biased Relative backtest showing the overall best performance of the looked at backtests. Further, the thesis visualizes how Value-at-Risk and Expected Shortfall are two risk measures that are inherently different in a real-world setting, emphasizing how one should be careful making conversions between the two based upon parametric assumptions.

This thesis aims to analyze some of the available methods that aid in risk estimation based on econometric models, as well as to propose some new ones. Some of the questions that are expected to be answered include which distribution to choose to obtain better risk estimates for series with abnormal behaviours, how to determine whether the distribution in parametric conditional models is a Student’s t, and how to assess whether an asset’s risk helps predict the risk of another asset. In Chapter 1, we estimate several cryptocurrencies’ Expected Shortfall using different error distributions and GARCH-type models for conditional variance. ur goal is to examine which distributions perform better and to check which component of the specification plays a more crucial role in estimating Expected Shortfall. The performance of the estimations is conducted using a backtesting technique with a rolling-window approach. Results show that, in the case of Bitcoin, it is important to use a distribution with at least two parameters that control its shape and an extension of the GARCH model, whether it be the NGARCH or the CGARCH model. On the other hand, other smaller cryptocurrencies yield good enough risk predictions with the Student’s t distribution and a GARCH model. The fact that the main measures of financial risk are focused on the tail of the distribution of returns highlights the importance of the choice of an appropriate distribution model. Chapter 2 develops a procedure for consistently testing the specification of a Student’s t distribution for the innovations of a dynamic model. This contributes to the existing literature by providing a test for Student’s t distributions in conditional mean and variance models with a parameter-free test statistic and, thus, a known asymptotic distribution, avoiding the use of more computationally costly resampling techniques such as bootstrapping. The specific expressions needed for the computation of the test statistic are obtained by adapting the generic test of Bai (2003), which is based on the Khmaladze (1988) transformation of the model residuals. Finally, in Chapter 3, the concept of Granger causality in Expected Shortfall (ES) is introduced, along with a testing procedure to detect this type of predictive relationship between return series. Granger causality in Expected Shortfall is here defined as the predictive ability of tail values of a series over future tail values of another series on average. This definition may help in analyzing whether past values of an asset in extreme risk affect future extreme risk values of another asset. The main contribution of this chapter is a test for detecting this type of causality, based on the test for Granger causality in VaR by Hong et al. (2009). An empirical application on financial institutions from different industries (banking, insurance, and diversified financials) is presented to analyze the risk spillovers in the US financial market. The contribution of this thesis to the field of financial econometrics focuses on the market risk of financial assets, both in its modeling through the metric known as Expected Shortfall suggested in the Basel III Accords and in its utility beyond capital requirements. The results highlight the importance of a good specification of the chosen distribution model for risk estimation - especially in high-risk assets such as cryptocurrencies - and a test is proposed to verify if the conditional distribution in parametric models used for risk predictions is or is not a Student’s t distribution. Finally, a Granger causality test in Expected Shortfall is proposed, which allows for studying risk propagation in tails of return distributions. The proposed test can be used to investigate interconnections within and between markets as a complement when evaluating systemic risk. Other potential applications include improving Expected Shortfall forecasts by including causing variables as regressors in estimations, studying the inclusion of certain asset pairs in the same portfolio based on how they interact in the riskiest situations, or constructing networks of extreme risk propagation. / Esta tesis doctoral ha sido financiada mediante una ayuda FPU por el Ministerio de Educación, Cultura y Deporte (FPU17/06227).

Neste trabalho são definidos, discutidos e estimados o Valor em Risco e o Expected Shortfall. Estas são medidas de Risco Financeiro de Mercado muito utilizadas por empresas e investidores para o gerenciamento do risco, aos quais podem estar expostos. O objetivo foi apresentar e utilizar vários métodos e modelos para a estimação dessas medidas e estabelecer qual o modelo mais adequado dentro de determinados cenários. / In this work Value at Risk and Expected Shortfall are defined, discussed and estimated . These are measures heavily used in Financial Market Risk, in particular by companies and investors to manage risk, which they may be exposed. The aim is to present and use several methods and models for estimating those measures and to establish which model is most appropriate in certain scenarios.

Neste trabalho são definidos, discutidos e estimados o Valor em Risco e o Expected Shortfall. Estas são medidas de Risco Financeiro de Mercado muito utilizadas por empresas e investidores para o gerenciamento do risco, aos quais podem estar expostos. O objetivo foi apresentar e utilizar vários métodos e modelos para a estimação dessas medidas e estabelecer qual o modelo mais adequado dentro de determinados cenários. / In this work Value at Risk and Expected Shortfall are defined, discussed and estimated . These are measures heavily used in Financial Market Risk, in particular by companies and investors to manage risk, which they may be exposed. The aim is to present and use several methods and models for estimating those measures and to establish which model is most appropriate in certain scenarios.