Since the introduction univariate GARCH models number of available models have grown rapidly and has been extended to the multivariate area. This paper compares three different multivariate GARCH models and they are evaluated using out of sample Value at Risk of dif- ferent portfolios. Sector portfolios are used with different market capitalization. The models compared are the DCC,CCC and the GO-Garch model. The forecast horizon is 1-day, 5-day and 10-day ahead forecast of the estimated VaR limit. The DCC performs best with regards to both conditional anc unconditional violations of the VaR estimates.

Cette thèse étudie la gestion et la couverture du risque en s’appuyant sur la Value-at-Risk (VaR) et la Value-at-Risk Conditionnelle (CVaR), comme mesures de risque. La première partie propose un modèle d’évolution de prix que nous confrontons à des données réelles issues de la bourse de Paris (Euronext PARIS). Notre modèle prend en compte les probabilités d’occurrence des pertes extrêmes et les changements de régimes observés sur les données. Notre approche consiste à détecter les différentes périodes de chaque régime par la construction d’une chaîne de Markov cachée et à estimer la queue de distribution de chaque régime par des lois puissances. Nous montrons empiriquement que ces dernières sont plus adaptées que les lois normales et les lois stables. L’estimation de la VaR est validée par plusieurs backtests et comparée aux résultats d’autres modèles classiques sur une base de 56 actifs boursiers. Dans la deuxième partie, nous supposons que les prix boursiers sont modélisés par des exponentielles de processus de Lévy. Dans un premier temps, nous développons une méthode numérique pour le calcul de la VaR et la CVaR cumulatives. Ce problème est résolu en utilisant la formalisation de Rockafellar et Uryasev, que nous évaluons numériquement par inversion de Fourier. Dans un deuxième temps, nous nous intéressons à la minimisation du risque de couverture des options européennes, sous une contrainte budgétaire sur le capital initial. En mesurant ce risque par la CVaR, nous établissons une équivalence entre ce problème et un problème de type Neyman-Pearson, pour lequel nous proposons une approximation numérique s’appuyant sur la relaxation de la contrainte / This thesis studies the risk management and hedging, based on the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR) as risk measures. The first part offers a stocks return model that we test in real data from NSYE Euronext. Our model takes into account the probability of occurrence of extreme losses and the regime switching observed in the data. Our approach is to detect the different periods of each regime by constructing a hidden Markov chain and estimate the tail of each regime distribution by power laws. We empirically show that powers laws are more suitable than Gaussian law and stable laws. The estimated VaR is validated by several backtests and compared to other conventional models results on a basis of 56 stock market assets. In the second part, we assume that stock prices are modeled by exponentials of a Lévy process. First, we develop a numerical method to compute the cumulative VaR and CVaR. This problem is solved by using the formalization of Rockafellar and Uryasev, which we numerically evaluate by Fourier inversion techniques. Secondly, we are interested in minimizing the hedging risk of European options under a budget constraint on the initial capital. By measuring this risk by CVaR, we establish an equivalence between this problem and a problem of Neyman-Pearson type, for which we propose a numerical approximation based on the constraint relaxation

We specify a class of non-linear and non-Gaussian models for which we estimate and forecast the conditional distributions with daily frequency. We use these forecasts to calculate VaR measures for three different equity markets (US, GB and Japan). These forecasts are evaluated on the basis of different statistical performance measures as well as on the basis of their economic costs that go along with the forecasted capital requirements. The results indicate that different performance measures generate different rankings of the models even within one financial market. We also find that for the three markets the improvement in the forecast by non-linear models over linear ones is negligible, while non-gaussian models significantly dominate the gaussian models. / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

This project focuses on risk analysis of corporate bond portfolios. We separate the total risk of the portfolio into three parts, which are market risk, credit risk and liquidity risk. The market risk component is quantified by value-at-risk (VaR) determined by change in yield to maturity of the bond portfolio. For the credit risk component, we calculate default probabilities and losses in the event of default and then compute credit VaR. Next, we define a factor called basis which is the difference between the Credit Default Swap (CDS) spread and its corresponding corporate bond yield spread (z-spread or OAS). We quantify the liquidity risk by using the basis. In addition, we also introduce a Fama-French multi-factor model to analyze factor significance to the corporate bond portfolio.

This project focuses on risk analysis of corporate bond portfolios. We divide the total risk of the portfolio into three parts, which are market risk, credit risk and liquidity risk. The market risk component is quantified by value-at-risk (VaR) which is determined by change in yield to maturity of the bond portfolio. For the credit risk component, we calculate default probabilities and losses in the event of default and then compute credit VaR. Next, we define a factor called `basis' which is the difference between the Credit Default Swap (CDS) spread and its corresponding corporate bond yield spread (z-spread or OAS). We quantify the liquidity risk by using the basis. In addition we also introduce a Fama-French multi-factor model to analyze the factor significance to the corporate bond portfolio.

Calculation of the Value at Risk (VaR) measure, of a portfolio, can be done using Monte Carlo simulations of that portfolio's potential losses over a specified period of time. Regulators, such as the US Securities and Exchange Commission, and Exchanges, such as the New York Stock Exchange, establish regulatory capital requirements for firms. These regulations set the amount of capital that firms are required to have on hand to safeguard against market loses that can occur. VaR gives us this specific monetary value set by Regulators and Exchanges. The specific amount of capital on hand must satisfy that, for a given confidence level, a portfolio's loses over a certain period of time, will likely be no greater than the capital required a firm must have on hand. The scenario used will be one of a Risk Manager position in which this manager inherited a portfolio that was set up for a client beginning in April 1992. The portfolio will have to meet certain parameters. The initial portfolio is worth $61,543,328.00. The risk manager will be responsible for the calculation of the Value at Risk measure, at five percent, with a confidence level of 95% and 20 days out from each of the 24 business quarters, over a six year period, starting in 1992 and ending in 1996.

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.

In last few years Value-at-Risk (Var) is a very popular and frequently used risk measure. Risk measure VaR is used in most of the financial institutions. VaR is popular thanks to its simple interpretation and simple valuation. Valuation of VaR is a problem if we assume a few dependent risks. So VaR is estimated in a practice. In presented thesis we study theory of stochastic bounding. Using this theory we obtain bounds for VaR of sum a few dependent risks. In next part of presented thesis we show how we can generalize obtained bounds by theory of copulae. Then we show numerical algorithm, which we can use to evaluate bounds, when exact analytical evaluate isn't possible. In a final part of presented thesis we show our results on practical examples.

Stochastic solvency testing methods have existed for more than 20 years, yet there has been little research conducted in this area, particularly in Australia. This is for a number of reasons, the most pertinent of which being the lack of computing capabilities available in the past to implement more sophisticated techniques. However, recent advances in computing have made stochastic solvency testing possible in practice and have resulted in a trend towards this being done in advanced studies. ¶ The purpose of this thesis is to develop a realistic solvency testing model in a form that can be implemented by Australian Life Insurers, in anticipation that the Australian insurance regulator, APRA, will ultimately follow the world trend and require stochastic solvency testing to be carried out in Australia. The model is constructed from three interconnected stochastic sub-models used to describe the economic environment and the mortality and lapsation experience of the portfolio of policies under consideration. Australian economic and Life Insurance data is used to fit a number of possible sub-models, such as generalised linear models, over-dispersion models and asset models, and the ``best'' model is selected in each case. The selected models are a modified CAS/SOA economic sub-model; either a Poisson or negative binomial (NB1) distribution (depending on the policy type considered) as the mortality sub-model; and a normal-Poisson lapsation sub-model. ¶ Based on tests carried out using this model, it is demonstrated that, for portfolios of level and yearly-renewable term insurance business, the current deterministic solvency capital requirements provide little protection against insolvency. In fact, for the test portfolios of term insurance policies considered, the deterministic capital requirements have levels of sufficiency of less than 2% (on a Value at Risk basis) when compared to the change in capital distribution over a three year time horizon. This is of concern, as yearly-renewable term insurance comprises a significant volume of Life Insurance business in Australia, with there being over 426,000 yearly-renewable term insurance policies on the books of Australian Life Insurers in 1999 and more business expected since then. ¶ A sensitivity analysis shows that the results of the stochastic asset requirement calculations are sensitive to the choice of sub-model used to forecast economic variables and to the choice of formulae used to describe the mean mortality and lapsation rates. The implication of this is that, if APRA were to require Life Insurers to calculate their solvency capital requirements on a stochastic basis, some guidance would need to be provided regarding the components of the solvency testing model used. The model is not, however, sensitive to whether an allowance is made for mortality or lapsation rate over-dispersion, nor to whether dependency relationships between mortality rates, lapsation rates and the economy are allowed for. Thus, over-dispersion and dependency relationships between the sub-models can be ignored in a stochastic solvency testing model without significantly impacting the calculated solvency requirements.

We utilise several asset pricing models that allow for discontinuities in the returns and volatility time series in order to obtain estimates of Value-at-Risk (VaR). The first class of model that we use mixes a continuous diffusion process with discrete jumps at random points in time (Poisson Jump Diffusion Model). We also apply a purely discontinuous model that does not contain any continuous component at all in the underlying distribution (Variance Gamma Model). These models have been shown to have some success in capturing certain characteristics of return distributions, a few being leptokurtosis and skewness. Calibrating these models onto the returns of an index of Australian stocks (All Ordinaries Index), we then use the resulting parameters to obtain daily estimates of VaR. In order to obtain the VaR estimates for the Poisson Jump Diffusion Model and the Variance Gamma Model, we introduce the use of an innovation from option pricing techniques, which concentrates on the more tractable characteristic functions of the models. Having then obtained a series of VaR estimates, we then apply a variety of criteria to assess how each model performs and also evaluate these models against the traditional approaches to calculating VaR, such as that suggested by J.P. Morgan???s RiskMetrics. Our results show that whilst the Poisson Jump Diffusion model proved the most accurate at the 95% VaR level, neither the Poisson Jump Diffusion or Variance Gamma models were dominant in the other performance criteria examined. Overall, no model was clearly superior according to all the performance criteria analysed, and it seems that the extra computational time required to calibrate the Poisson Jump Diffusion and Variance Gamma models for the purposes of VaR estimation do not provide sufficient reward for the additional effort than that currently employed by Riskmetrics.