Title: Modelling dependence between hydrological and meteorological variables measured on several stations Author: Bc. Marie Turčičová Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Daniela Jarušková CSc., Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mathematics Abstract: The aim of the thesis is to explore the dependence of daily discharge averages of the Opava river on high daily precipitation values in its basin. Three methods are presented that can be used for analyzing the dependence between high values of random variables. Their application on the studied data is also given. First it is the tail-dependence coefficient that measures the dependence between high values of two continuous random variables. The model for the high quantiles of the discharge at a given precipitation value was first determined non-parametrically by quantile regression and then parametrically through the peaks-over-threshold (POT) method. Keywords: extremal dependence, tail-dependence coefficient, quantile regression, peaks over threshold method

This thesis provides an analysis of systemic risk in the US banking sector. We use conditional value at risk (∆CoVaR), marginal expected shortfall (MES) and cross-quantilogram (CQ) to statistically measure tail-dependence in return series of individual institutions and the system as a whole. Wavelet multireso- lution analysis is used to study systemic risk in the time-frequency domain. De- composition of returns on different scales allows us to isolate cycles of 2-8 days, 8-32 days and 32-64 days and analyze co-movement patterns which would oth- erwise stay hidden. Empirical results demonstrate that filtering out short-term noise from the return series improves the forecast power of ∆CoVaR. Eventu- ally, we investigate the connection between statistical measures of systemic risk and fundamental characteristics of institutions (size, leverage, market to book ratio) and conclude that size is the most robust determinant of systemic risk.

本篇論文主要是想探討在2008年全球金融危機發生後，美國與亞洲國家股票市場之間的相關性是否發生明顯的改變。藉由2005年至2012年美國、新加坡、台灣、日本和泰國的股票市場資料，來觀察各國股票市場的相關性是否產生不對稱的現象，首先檢定美國對其他四個國家有無產生蔓延效果，並藉由不同期間的資料來檢定蔓延效果以看出各國之間是否在極端的情況下產生尾端相關性，最後，再使用不同的關聯結構函數配適出最適合資料的模型。 / The main idea of this paper is to show whether or not that stock market between U.S and Asian country has been obviously changed after 2008 financial crisis. For the sake of observing if there is or not occurred inconsistence phenomenon in each country’s stock market, we use the information from U.S、Singapore、Taiwan、Japan and Thailand since 2005 to 2012. First, look in that if U.S has contagion affects to other four countries and, checkup the contagion effects through the information from different period to find the tail dependence in extreme situation. Finally, to dispose a model which is the most suitable for the information by using different Copula functions.

It is of great importance to find an analytical copula that will represent the empirical lower tail dependence. In this study, the pairwise empirical copula are estimated using data of the S&P 500 stocks during the period 2007-2010.Different optimization methods and measures of dependence have been used to fit Gaussian, t and Clayton copula to the empirical copulas, in order to represent the empirical lower tail dependence. These different measures of dependence and optimization methods with their restrictions, point at different analytical copulas being optimal. In this study the t copula with 5 degrees of freedom is giving the most fulfilling result, when it comes to representing lower tail dependence. The t copula with 5 degrees of freedom gives the best representation of empirical lower tail dependence, whether one uses the 'Empirical maximum likelihood estimator', or 'Equal Ƭ' as an approach.

La thèse est composée de trois parties. La première présente un certain nombre de mesures de dépendance extrême. Une application sur les actions et les obligations de 49 pays montre que la théorie des valeurs extrêmes multivariées conduit aux résultats différents de ceux issus du coefficient de corrélation, mais relativement proches de ceux obtenus du rho de Spearman conditionnel multivarié. Cette partie évalue aussi le risque de pertes importantes simultanées. La deuxième partie examine les déterminants des co-mouvements extrêmes entre 5 pays core et 49 pays non core. Les mécanismes de transmission des chocs varient de la période moins récente à la période récente, des pays développés aux pays émergents, des chocs normaux aux chocs extrêmes. La troisième partie étudie le rôle de valeur refuge de l’or sur la période 1986-2012. Les gains positifs extrêmes de l'or peuvent être liés aux pertes extrêmes du S&P. Cependant, ce lien n'est pas toujours valable, il évolue dans le temps et serait conditionné par d'autres facteurs. / The thesis consists of three parts. The first part introduces a number of measures of extreme dependency. An application on stock and bond markets of 49 countries shows the multivariate extreme value theory leads to results which are different from those from the correlation coefficient, but relatively close to those obtained from multivariate conditional Spearman's rho. This part also assesses the risk of simultaneous losses. The second part examines the determinants of extreme co-movements between 5 core countries and 49 non-core countries. Transmission mechanisms of shocks vary from less recent to recent period, from developed to emerging markets, from normal to extreme shocks. The third part examines the role of safe haven of gold over the period 1986-2012. Extreme positive gains of gold can be linked to extreme losses of S&P. However, this relationship is not always valid, it evolves over time and could be determined by other factors.

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In 2005 Attilio Meucci presented his article Beyond Black-Litterman: Views on Non-Normal Markets which introduces the copula opinion pooling approach using generic non-normal market assumptions. Copulas and opinion pooling are used to express views on the market which provides a posterior market distribution that smoothly blends an arbitrarily distributed market prior distribution with arbitrarily chosen views. This thesis explains how to use this method in practice and investigates its performance in different investment situations. The method is tested on three portfolios, each showing some different feature. The conclusions that can be drawn are e.g. that the method can be used in many different investment situations in many different ways, implementation and calculations can be made within a few seconds for a large data set and the method could be useful for portfolio managers using mathematical methods. The presented examples together with the method generate reasonable results.

Quantification of risks is one of the pillars of the contemporary insurance industry. Natural catastrophes and their modelling represents one of the most important areas of non-life insurance in the Czech Republic. One of the key inputs of catastrophe models is a spatial dependence structure in the portfolio of an insurance company. Copulas represents a more general view on dependence structures and broaden the classical approach, which is implicitly using the dependence structure of a multivariate normal distribution. The goal of this work, with respect to absence of comprehensive monographs in the Czech Republic, is to provide a theoretical basis for use of copulas. It focuses on general properties of copulas and specifics of two most commonly used families of copulas -- Archimedean and elliptical. The other goal is to quantify difference between the given copula and the classical approach, which uses dependency structure of a multivariate normal distribution, in modelled flood losses in the Czech Republic. Results are largely dependent on scale of losses in individual areas. If the areas have approximately a "tower" structure (i.e., one area significantly outweighs others), the effect of a change in the dependency structure compared to the classical approach is between 5-10% (up and down depending on a copula) at 99.5 percentile of original losses (a return period of once in 200 years). In case that all areas are approximately similarly distributed the difference, owing to the dependency structure, can be up to 30%, which means rather an important difference when buying the most common form of reinsurance -- an excess of loss treaty. The classical approach has an indisputable advantage in its simplicity with which data can be generated. In spite of having a simple form, it is not so simple to generate Archimedean copulas for a growing number of dimensions. For a higher number of dimensions the complexity of data generation greatly increases. For above mentioned reasons it is worth considering whether conditions of 2 similarly distributed variables and not too high dimensionality are fulfilled, before general forms of dependence are applied.