# Quelle:Rh/Schmidt 2003

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 Autor R. Schmidt Titel Dependencies of Extreme Events in Finance Ort Ulm Jahr 2003 Anmerkung Dissertation URL http://www.wisostat.uni-koeln.de/Institut/LSSchmid/Ehemalige/RSchmidt/SchmidtPhDThesis.pdf Literaturverz. nein Fußnoten nein Fragmente 6

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Comovements and extreme events between financial asset-returns have significantly increased during recent time periods in almost all international markets. Dependencies between financial asset-returns have significantly increased during recent time periods in almost all international markets.
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Extreme events refer, for example, to extraordinary claims to insurance companies, crashes of equity markets or extreme losses in credit portfolios due to borrower defaults. Hence, extreme events occur rarely, i.e., only few extreme observations are available. Nevertheless, probabilities and dependence structures have to be assigned to extreme events due to their economic impact. Extreme events refer, for example, to extraordinary claims to insurance companies, crashes of equity markets, or extreme losses in credit portfolios due to borrower defaults. Hence, extreme events occur rarely, ergo, only few extreme observations are available; but probabilities and dependence structures have to be assigned to extreme events due to their economic impact.
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Indeed, standard portfolio selection is usually based on this approach, the Markowitz mean-variance theory and the Sharpe-Lintner-Mossin capital asset pricing model (CAPM). Further, the dependence structure of the asset returns is solely described by the Pearson correlation of the returns. This suffices in case the asset return distributions are multivariate normal. However, if we leave the Gaussian or more generally the elliptical world, a mere consideration of the correlation matrix often explains the dependence structure in a quite unsatisfying way. Pitfalls like a non-existing correlation or zero correlation for dependent random variables may occur. Especially the dependence structure of extreme events is usually poorly or incorrectly described by this measure. [Seite 5: 25-27]

Standard portfolio selection is usually based on the Markowitz mean-variance theory of risk and return, from 1952 on, and the Sharpe-Lintner-Mossin capital asset pricing model (CAPM) of 1964-66.

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In particular, the dependence structure of the asset returns is solely described by the covariance matrix Σ. This suffices in case the asset-return distributions are multivariate normal. However, if we leave the Gaussian world, a sole consideration of the covariance matrix often explains the dependence structure in a quite unsatisfying way. Pitfalls like a non-existing covariance or zero covariance for dependent random variables may occur. Especially the dependence structure of extreme events is usually poorly or incorrectly described by the covariance matrix.

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Suppose X1, ... , Xn be a sequence of iid non degenerate random variables with distribution function F. The primary concern of extreme value theory relates to extreme values or maximal values as for example, large asset returns or large portfolio losses. Let X, X(1), X(2), ... , X(m), ${\scriptstyle m\in \mathbb{N} }$, be independent, identically distributed n-dimensional random vectors with distribution function F. The primary concern of EVT relates to extreme values or maximal values (in practice: large asset returns or large portfolio losses) of the above random sample [...]
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This class of distributions provide us with techniques to trade off the bias of having insufficient data in practice and meaningful extrapolations beyond the range of given data. Frèchet [sic!] distributions are primarily applied in finance because of their unbounded support on the positive halfline and their relationship to so called heavy-tailed distributions. [Seite 12: 37-39]

EVT provides techniques to trade off the bias of having insufficient data in practice and meaningful extrapolations beyond the range of given data.

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Fréchet distributions (ξ > 0) are primarily applied in practice because of their unbounded support on the positive halfline and their relationship to so-called heavy-tailed distributions.

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Summarizing the above results, univariate EVT provides probabilistic tools to model the limiting distributions of normalized maxima and excesses over high thresholds. Regarding the parameter estimation of extreme value distributions it suffices to apply parametric estimation methods instead of nonparametric estimation methods which are less robust for small sample sizes. Summarizing the above results, univariate EVT provides probabilistic tools to model the limiting distributions of normalized maxima and excesses over high thresholds. Regarding the parameter estimation of extreme value distributions it suffices to apply parametric estimation methods instead of nonparametric estimation methods which are less robust for small sample sizes.
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