We model Normal Inverse Gaussian distributed log-returns with the assumption of stochastic volatility. We consider different methods of parametrization of returns and following the paper of Lindberg, [21] we assume that the volatility is a linear function of the number of trades. In addition to the Lindberg’s paper, we suggest daily stock volumes and amounts as alternative measures of the volatility. As an application of the models, we perform Value-at-Risk and Expected Tail Loss predictions by the Lindberg’s volatility model and by our own suggested model. These applications are new and not described in the literature. For better understanding of our caluclations, programmes and simulations, basic informations and properties about the Normal Inverse Gaussian and Inverse Gaussian distributions are provided. Practical applications of the models are implemented on the Nasdaq-OMX, where we have calculated Value-at-Risk and Expected Tail Loss for the Ericsson B stock data during the period 1999 to 2004.

Riskvärdering har under 90-talet blivit ett allt mer medvetet begrepp. Ett populärt instrument vid riskvärdering är Value at Risk då denna modell skapar ett gemensamt riskmått för olika typer av portföljer och derivat. VaR mäter den maximala värdeförändringen för en portfölj där sannolikheten och tidshorisonten är förutbestämd. I uppsatsen har en konfidensnivå på 95 procent antagits vilket medför att de verkliga förlusterna ska överstiga VaR en gång av tjugo. Icke-linjära instrument, såsom optioner, är svåra att riskvärdera då dess pris förändras oproportionerligt gentemot dess underliggande. För att beräkna VaR kan flertalet modeller appliceras och dessa har olika egenskaper. Det är därför av intresse att ta reda på om Delta-Normal metoden, Monte Carlo simulering och Historisk simulering ger samma svar vid riskvärdering av optioner. Vidare syftar denna uppsats till att söka svar på om dessa tre VaR-modeller ger ett tillfredsställande resultat på 95 procentig konfidensnivå. För att få svar på dessa funderingar har vi i empiriavsnittet genomfört två hypotesprövningar. Den första slutsatsen som kan dras av undersökningen är att det inte går att skilja på det VaR som Delta-Normal metoden och Historisk simulering tagit fram. Vid ett hypotestest för proportioner blev resultatet att endast för Monte Carlo simuleringen kunde inte nollhypotesen förkastas. Detta innebär att det finns stöd för att de verkliga förlusterna överstiger Monte Carlo simuleringens beräknade VaR en gång av tjugo.

Value at Risk (VaR) approach has been extensively used by investment and commercial banks since its development by JP Morgan in 1990s. As time passes, it has become interesting to investigate whether VaR could be used also by other financial intermediaries like pension funds and insurance companies. The aim of this paper is to outline Value at Risk (VaR) methodology by giving more emphasis on parametric approach which is used for empirical section and to investigate the applicability and usefulness of VaR in pension funds. After providing theoretical framework for VaR approach, the paper continues with pension fund systems in general and especially highlights AP funds of Swedish National pension fund system by trying to show why VaR could be an invaluable risk management tool for these funds together with other traditional risk measures used. Based on this given theoretical frame, a practical application of VaR –parametric or covariance/variance method- is executed on 50 biggest investments in the fixed income and equity portfolios of three selected Swedish national pension funds – AP1, AP2 and AP3. Results of one day VaR (DEAR) estimations on 30/12/2005 for each fund have been presented and it is aimed to show the additional information that could be obtained by using VaR and which is not always apparent from other risk measures employed by funds. According to the two traditional risk measures which are active risk and Sharpe ratio; AP2 and AP3 lie in the same risk level for 2005 which can create a contradiction by considering their different returns. On the other hand, obtained DEAR estimates show their different risk exposures even with the 50 biggest investments employed. The results give a matching relationship between return of funds and DEAR estimates meaning that; the fund with the highest return has the highest DEAR value and the fund with the lowest return has the lowest DEAR value; which is consistent with the main rule- “higher risk, higher return”. Thus, we can conclude that VaR could be applied additionally to get a better picture about real risk exposures and also to get valuable information on expected possible loss together with other traditional risk measures used. Key words: Value at Risk, DEAR, Pension funds, Risk management, Swedish pension plan, AP1, AP2, AP3

In this paper the performance of classical approaches and GARCH family models are evaluated and compared in estimation one-step-ahead VaR. The classical VaR methodology includes historical simulation (HS), RiskMetrics, and unconditional approaches. The classical VaR methods, the four univariate and two multivariate GARCH models with the Student’s t and the normal error distributions have been applied to 5 stock indices and 4 portfolios to determine the best VaR method. We used four evaluation tests to assess the quality of VaR forecasts: -                     Violation ratio -                     Kupiec’s test -                     Christoffersen’s test -                     Joint test The results point out that GARCH-based models produce far more accurate forecasts for both individual and portfolio VaR. RiskMetrics gives reliable VaR predictions but it is still substantially inferior to GARCH models. The choice of an optimal GARCH model depends on the individual asset, and the best model can be different based on different empirical data.

There are different types of risk, examples include credit risk, liquidity risk and financial risk. In DeMarzo & Berk (2011, s. 293) is a study presented which is based on the yield of different types of financial assets between 1925 and 2009, the study show that a high risk gave substantially higher reward. With the study as a background, it is interesting to study practical risk management within participants of the financial markets of Sweden. In risk management there are several theories about whether risk can be calculated and analyzed with scientific methods in practice. To generate new empirical data a qualitative method was used in the form of interviews. The selection, which was strategic, was based on mailed questionnaire sent to participants of the Swedish fund market. Theory can be problematic to apply in practice, since reality is often simplified in theory, as discussed by Franklin (2004). Franklin’s thoughts are accompanied by Baird (2010) in a similar discussion. The main model of the study is Value at Risk, which is recovered from Hull (2011) but has its origin from the financial company JP Morgan. Other models that are applied in the study are Capital Asset Pricing Model, CAPM, and the Sharpe ratio. There are known critiques against these models, which are discussed in this study. In the study it is shown that all the participants applied the model Value at Risk. The report also indicates that standard deviation has a central role in risk management. All the respondents were well aware of the critique against Value at Risk. To manage the flaws of the model they also used stress tests as a complement. The analysis of the study indicates that practical and theoretical application in many aspects are similar, the most apparent one being Value at Risk. Even though there are some differences, CAPM was indicated to have no practical use for any of the participants. Two vital factors for whether a model can be applied practically are the model’s simplicity and the need for assumptions to correlate with reality. Having completed this study, the conclusion that the participants successfully applied theoretical risk management models in practice can be validated.

This thesis investigates robust techniques for mean-variance (MV) portfolio optimization problems under the estimation risk in mean return. We evaluate the performance of the optimal portfolios generated by the min-max robust MV portfolio optimization model. With an ellipsoidal uncertainty set based on the statistics of the sample mean estimates, minmax robust portfolios equal to the ones from the standard MV model based on the nominal mean estimates but with larger risk aversion parameters. With an interval uncertainty set for mean return, min-max robust portfolios can vary significantly with the initial data used to generate the uncertainty set. In addition, by focusing on the worst-case scenario in the mean return uncertainty set, min-max robust portfolios can be too conservative and unable to achieve a high return. Adjusting the conservatism level of min-max robust portfolios can only be achieved by excluding poor mean return scenarios from the uncertainty set, which runs counter to the principle of min-max robustness. We propose a CVaR robust MV portfolio optimization model in which the estimation risk is measured by the Conditional Value-at-Risk (CVaR). We show that, using CVaR to quantify the estimation risk in mean return, the conservatism level of CVaR robust portfolios can be more naturally adjusted by gradually including better mean return scenarios. Moreover, we compare min-max robust portfolios (with an interval uncertainty set for mean return) and CVaR robust portfolios in terms of actual frontier variation, portfolio efficiency, and portfolio diversification. Finally, a computational method based on a smoothing technique is implemented to solve the optimization problem in the CVaR robust model. We numerically show that, compared with the quadratic programming (QP) approach, the smoothing approach is more computationally efficient for computing CVaR robust portfolios.

The purpose of this thesis is to evaluate five different volatility forecasting models that are used to calculate financial market risk. The models are used on both daily exchange rates and high-frequency intraday data from four different series. The results show that time series models fitted to high-frequency intraday data together with a critical value taken from the empirical distribution displayed the best forecasts overall.

The purpose with this study is to compare four different models to VaR in terms of accuracy, namely Historical Simulation (HS), Simple Moving Average (SMA), Exponentially Weighted Moving Average (EWMA) and Exponentially Weighted Historical Simulation (EWHS). These VaR models will be applied to one underlying asset which is the Brent Blend Oil using these confidence levels 95 %, 99 % and 99, 9 %. Concerning the return of the asset the models under two different assumptions namely student t-distribution and normal distribution will be studied

This thesis compares insurance premium principles with current financial risk paradigms and uses distorted probabilities, a recent development in premium principle literature, to synthesize the current models for financial risk measures in banking and insurance. This work attempts to broaden the definition of value-at-risk beyond the percentile measures. Examples are used to show how the percentile measure fails to give consistent results, and how it can be manipulated. A new class of consistent risk measures is investigated.

This thesis investigates robust techniques for mean-variance (MV) portfolio optimization problems under the estimation risk in mean return. We evaluate the performance of the optimal portfolios generated by the min-max robust MV portfolio optimization model. With an ellipsoidal uncertainty set based on the statistics of the sample mean estimates, minmax robust portfolios equal to the ones from the standard MV model based on the nominal mean estimates but with larger risk aversion parameters. With an interval uncertainty set for mean return, min-max robust portfolios can vary significantly with the initial data used to generate the uncertainty set. In addition, by focusing on the worst-case scenario in the mean return uncertainty set, min-max robust portfolios can be too conservative and unable to achieve a high return. Adjusting the conservatism level of min-max robust portfolios can only be achieved by excluding poor mean return scenarios from the uncertainty set, which runs counter to the principle of min-max robustness. We propose a CVaR robust MV portfolio optimization model in which the estimation risk is measured by the Conditional Value-at-Risk (CVaR). We show that, using CVaR to quantify the estimation risk in mean return, the conservatism level of CVaR robust portfolios can be more naturally adjusted by gradually including better mean return scenarios. Moreover, we compare min-max robust portfolios (with an interval uncertainty set for mean return) and CVaR robust portfolios in terms of actual frontier variation, portfolio efficiency, and portfolio diversification. Finally, a computational method based on a smoothing technique is implemented to solve the optimization problem in the CVaR robust model. We numerically show that, compared with the quadratic programming (QP) approach, the smoothing approach is more computationally efficient for computing CVaR robust portfolios.