The diploma thesis "Value at Risk: Historical simulation, variance covariance method and Monte Carlo" aims to value the risk which real bond portfolio bears. The thesis is decomposed into two major chapters: Theoretical and Practical chapters. The first one speaks about VaR and conditional VaR theory including their advantages and disadvantages. Moreover, there are described three basic methods to calculate VaR and CVaR with adjustments to each method in order to increase the reliability of results. The last chapter brings results of VaR and CVaR computation. Many graphs, tables and images are added to the result section in order to make the outputs more visible and well-arranged.

Nos últimos anos, o value-at-risk tem se tornado uma ferramenta amplamente utilizada nas principais instituições financeiras, inclusive no Brasil. Dentre suas vantagens, destaca-se a possibilidade de se resumir em um único número os riscos de mercado incorridos e incorporar neste valor tanto a exposição da instituição quanto a volatilidade do mercado. O objetivo principal deste estudo é verificar a eficácia dos modelos mais conhecidos de value-at-risk - RiskMetrics(TM) e Simulação Histórica - na mensuração dos riscos de mercado de carteiras de renda fixa compostas por instrumentos pré-fixados em reais. No âmbito da alocação de capital para atendimento aos órgãos de regulamentação, o estudo estende-se também ao modelo adotado pelo Banco Central do Brasil. No decorrer do estudo, discute-se ainda as vantagens e desvantagens apresentadas, bem como o impacto que as peculiaridades do mercado brasileiro exercem sobre as hipóteses assumidas em cada um dos modelos. / Value-at-Risk (VaR) has become the primary tool for the systematic measuring and monitoring of market risk in most financial institutions. VaR is a statistical measure that comprises not only the exposure but also the market volatility in a single number. The main purpose of this work is to evaluate the performance of the well-known value-at-risk models - RiskMetrics(TM) and Historical Simulation - in the Brazilian fixed-income market. In the scope of capital allocation related to banking regulation, this study also extends briefly to the model adopted by the Brazilian Central Bank. Additionally, the underlying assumptions of these models are analyzed in the Brazilian financial market context. Also, this study discusses the advantages and disadvantages presented by the RiskMetrics and the Historical Simulation models.

The goal of the paper is to compare different approach in calculation of capital requirement of market risk for options on stock's basket and describe their impact on selected instrument. The first part of the paper describes possible approaches for the capital requirement calculation, namely Standardized approach and Internal model approach, and the theoretical base for option pricing. An instrument with the embedded option on equities was chosen to show the impact. Although the instrument is valued using Monte Carlo simulation, one chapter is devoted to Black-Scholes model as the base model for option pricing. Powered by TCPDF (www.tcpdf.org)

>Magister Scientiae - MSc / Financial companies such as investment and commercial banks as well as insurance companies, mutual and pension funds hold assets in the form of financial instruments in portfolios. Nowadays, financial instruments have proliferated so much that there are so many forms of them namely: derivatives, common stock, corporate and government bonds, foreign exchange and contracts. With so many financial instruments, companies can have very large and diversified portfolios for which they must quantify the risk. With high profile calamities that have rocked the financial world lately, the need for better risk management has never been so in demand as before. Value-at-Risk (VaR) is the latest addition in the investor's toolkit as far as measurements of risk is concerned. This new measure of risk complements well the existing risk measures that exist.Unfortunately, VaR is not unanimous and it has attracted a lot of critics over the years. This research thesis is threefold: to introduce the reader to the VaR concept; to discuss the different methods that exist to calculate VaR; and, finally, to simulate the VaR of a portfolio of government bonds. The first part of this research is to introduce the reader to the general idea of risk forms and its management, the role that the existing risk measures have played so far and the coming up of the new technique, which is VaR. The pros and cons that accompany a new technique are discussed as well as the history of VaR. The second part is about the different methods that exist to compute the VaR of a portfolio. Usually, VaR methodologies fall into three categories namely: Parametric; Historical; and Monte Carlo. In this research, the advantages and disadvantages of these three methods are discussed together with a step-wise method on how to proceed to calculate the VaR of a portfolio using any of the three methods. The practical side of this thesis deals about the VaR simulation of a portfolio of financial instruments. The chosen financial instruments are four South African government bonds with different characteristics. VaR for this particular portfolio will then be simulated by the three main methods. Eleven different simulations are run and they are compared against a Control Simulation (Benchmark Portfolio) to see how factors influencing VaR measure cope under different conditions. The main idea here was to check how VaR measures can change under different portfolio characteristics and to interpret these changes. Moreover, the VaR estimates under the three different methods will be compared

Market risk is the risk of capital loss due to unexpected changes in market prices. One risk measure used to estimate market risk is Value at Risk (VaR). The common historical simulation methodology of VaR forecasting usually does not capture the time-varying volatilities associated with financial data. Therefore, dynamic factor models (DFM) are employed to improve VaR forecasting. The paper’s main focus is to use different volatility model specifications in the DFM to evaluate which is the most appropriate for VaR forecasting. The volatility models considered are the Constant Conditional Correlation (CCC-) GARCH, the Dynamic Conditional Correlation (DCC-) GARCH, and the corrected Dynamic Conditional Correlation (cDCC-) GARCH. The method is applied to an empirical dataset consisting of Swedish large-cap stocks between 2017-2021 where two different portfolios are used, the equally- and the value-weighted portfolio. The data purposefully includes the COVID-19 pandemic such that the models can be compared during less- and more volatile periods. The method is further evaluated in a simulation study where randomized portfolio weights are used. It is found that the VaR forecasts produced by the three different model specifications are similar throughout the entire sample. Therefore the most restricted volatility model (CCC-GARCH) is recommended.

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.

In this thesis the performance of the quantile based CAV iaR models is evaluated and compared with GARCH models for predicting the Value at Risk. This is done by one step ahead out of sample prediction. The one step ahead out of sample prediction is done for the 500 observations at the end of the sample. To calculate the predictions a rolling forecast is used. This means that the sample that is used to do the one step ahead predictions is equally sized for all 500 predictions. Then tests are performed to evaluate the predictive power of the forecasts. The tests that are used to evaluate the predictions are: the dynamic quantile test, the Kupiec test and the Christoffersens test. The data that is used in the analysis are two stock indexes and one exchange rate index. What is concluded from the thesis is that the models perform good in general for the Stockholmsb ̈orsen data. For the First north data the 1% V aR produced too high risk predictions so the exceedance rate became too low. For the 5% V aR the predictions were more accurate. For the exchange rate data the predictions from the models were generally good as well.

This paper explores the relationship between Chinese rare earth metals (REMs) and the industries in the U.S and Europe that heavily rely on them. The study uses the EGARCH(1,1)-ARMA(1,0) process for conditional volatility and incorporates it into VAR(8) framework for forecast error variance decomposition to evaluate the static and dynamic volatility spillovers using daily data from the 2nd of January 2018 to the 3rd of March 2023. The liaison of risk management is also consolidated through the incorporation of Value at Risk and Event Study. Our findings indicate that the volatility interconnectedness between the Chinese REMs market and computer and electronics, electric vehicle, and wind energy industries exhibits relatively low volatility spillover to and from each other. Value at Risk measures suggests complexity in assessing the potential short-term losses for REM equity, leading to difficulties in risk management. Establishing and utilizing a derivatives market could be beneficial for future notice. However, the study also highlights that severe geopolitical risk or conflict could enable extreme levels of financial risk due to the global supply dominance of the Chinese quasi-monopolistic construct and the elements' overall criticality in the sustainable energy transition. The study also highlights the infeasibility of Western nations decoupling themselves from the Chinese REM supply. Various factors such as the pace of advancement in sourcing alternatives, technological advancements, and recycling technology are the main drivers of ineligibility. The forecasted global demand for REMs is also expected to increase significantly, primarily driven by the renewable and sustainable energy transition worldwide, further straining the possibility of independence. Therefore, the pace of advancement of these factors must collectively supersede that of the forecasted demand to mitigate the risk. Keywords: Rare Earth Metals, Interconnectedness, Conditional Volatility, Risk Management, Value at Risk, Event Study.

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.

This research presents a bi-level stochastic network interdiction model on an attack graph to enable a risk-averse resource constrained cyber network defender to optimally deploy security countermeasures to protect against attackers having an uncertain budget. This risk-averse conditional-value-at-risk model minimizes a weighted sum of the expected maximum loss over all scenarios and the expected maximum loss from the most damaging attack scenarios. We develop an exact algorithm to solve our model as well as several acceleration techniques to improve the computational efficiency. Computational experiments demonstrate that the application of all the acceleration techniques reduces the average computation time of the basic algorithm by 71% for 100-node graphs. Using metrics called mean-risk value of stochastic solution and value of risk-aversion, numerical results suggest that our stochastic risk-averse model significantly outperforms deterministic and risk-neutral models when 1) the distribution of attacker budget is heavy-right-tailed and 2) the defender is highly risk-averse.