M.Comm. / This study analyzes the effect of the subprime crisis on portfolio allocation from the perspective of dependence structure. Empirical evidence has proved that the multivariate normal distribution is inadequate to model portfolio asset return distribution - firstly because the empirical marginal distributions of asset returns are skewed and fat tailed; and secondly because it does not consider the possibility of extreme joint co-movement of asset returns (Fama and French, 1993; Richardson and Smith, 1993; Géczy, 1998; Longin and Solnik, 2001; Mashal and Zeevi, 2002). This study employs Archimedean copulas to capture both the dependence structure and the asymmetry of asset returns in the tails of the empirical distributions.

Copulas have been known in the statistical literature for many years, and have become useful tools in modeling dependence structure of multivariate random variables, overcoming some of the drawbacks of the commonly-used correlation measures. Goodness-of-fit tests for copulas play a very important role in evaluating the suitability of a potential input copula model. In recent years, many approaches have been proposed for constructing goodness-of-fit tests for copula families. Among them, the so-called “blanket tests" do not require an arbitrary data categorization or any strategic choice of weight function, smoothing parameter, kernel, and so on. As preliminaries, some background and related results of copulas are firstly presented. Three goodness-of-fit test statistics belonging to the blanket test classification are then introduced. Since the asymptotic distributions of the test statistics are very complicated, parametric bootstrap procedures are employed to approximate critical values of the test statistics under the null hypotheses. To assess the performance of the three test statistics in the low dependence cases, simulation studies are carried out for three bivariate copula families, namely the Gumbel-Hougaard copula family, the Ali-Mikhail-Haq copula family, and the Farlie-Gumbel-Morgenstern copula family. Specifically the effect of low dependence on the empirical sizes and powers of the three blanket tests under various combinations of null and alternative copula families are examined. Furthermore, to check the performance of the three tests for higher dimensional copulas, the simulation studies are extended to some three-dimensional copulas. Finally the three goodness-of-fit tests are applied to two real data sets. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Assignment (MComm)--University of Stellenbosch, 2008. / Bibliography. Also available via the Internet.

Las redes sociales virtuales como Facebook y Twitter están muy difundidas en nuestras vidas cotidianas y generan un montón de datos de intercambios. Planteamos un modelo estocástico para Twitter que nos permite estudiar la dinámica de la red y el comportamiento de los usuarios sobre su saturación. Para estudiar este modelo estocástico se utiliza la herramienta estadística de cópulas que analiza la dependencia de variables aleatorias. Este trabajo de tesis proponemos una generalización del estimador por núcleos de cópulas para serie de tiempos presentado por Fermanian y Scaillet en 2002. Dicha generalización se extiende a procesos estocásticos de difusión. A partir de éste estimador, se puede analizar la probabilidad de saturación de un usuario de Twitter y otras medidas vinculadas con esta saturación. / Virtual social networks like Facebook and Twitter are very spread in daily life. Using it generates an incredible amount of exchange information. In this work we propose an stochastic model for Twitter that allows the study of network dynamics and users behavior specially concern with saturation. To study this model we use a statistical tool named as copulas that analices dependence between random variables. In this thesis we propose a generalization of the non-parametric copula estimator presented by Fermanian and Scaillet in 2002. This generalization reaches continuos process as diffusion. From this estimator we can analize profile saturation probability and other measures related with saturation.

Copulas are real functions representing the dependence structure of the distribution of a random vector, and measures of concordance associate with every copula a numerical value in order to allow for the comparison of different degrees of dependence. We first introduce and study a group of transformations mapping the collection of all copulas of fixed but arbitrary dimension into itself. These transformations may be used to construct new copulas from a given one or to prove that certain real functions on the unit cube are indeed copulas. It turns out that certain transformations of a symmetric copula may be asymmetric, and vice versa. Applying this group, we then propose a concise definition of a measure of concordance for copulas. This definition, in which the properties of a measure of concordance are defined in terms of two particular subgroups of the group, provides an easy access to the investigation of invariance properties of a measure of concordance. In particular, it turns out that for copulas which are invariant under a certain subgroup the value of every measure of concordance is equal to zero. We also show that the collections of all transformations which preserve symmetry or the concordance order or the value of every measure of concordance each form a subgroup and that these three subgroups are identical. Finally, we discuss a class of measures of concordance in which every element is defined as the expectation with respect to the probability measure induced by a fixed copula having an invariance property with respect to two subgroups of the group. This class is rich and includes the well-known examples Spearman's rho and Gini's gamma.

Due to turbulence in the financial market throughout history, stress testing has become a growing part of the risk analysis performed by clearing houses. Events connected to previous crises have increased the demand for prudent risk exposure, and in this thesis we investigate regulators view on how CCPs should construct risk scenarios to meet best practice for stress testing their members’ composite portfolios. A method based on multivariate t-distributions and copula-transformations applied to historical time series data, is proposed for constructing an independent scenario generator which should be used as a compliment to other, more knowledge-based methods. The method was implemented in Matlab to test the theory in practice, and experiments were setup for pure stock portfolios as well as for derivative based portfolios. Backtests were then carried out to validate the underlying theory on historical data spanning 25 years in total. Results show that the method proposed in this thesis indeed has the potential to be a useful approach for creating stress scenarios. Its ability to render specific levels of plausibility seems to show a sufficient level of consistency with real life data, and further research is thereby justified.

In this thesis we are interested in the calculation of economic capital for the to- tal loss which is the sum of partial dependent losses, whose dependence structure is described by Archimedean and hierarchical Archimedean copulas. Firstly, the concept of economic capital and the ways of its aggregation are introduced. Then the basic definitions and properties of copulas are listed, as well as the depen- dence measures. After that we work with definition and properties of Archimedean copulas and their simulation. We also mention the most popular families of Ar- chimedes copulas. Next, hierarchical Archimedean copulas are defined, as well as the algorithm for their sampling. Finally, we present methods for estimating the parameters of copulas and the recursive algorithm for estimating the hierarchical Archimedean copula structure. In the last chapter we perform simulation studies of selected models using hierarchical Archimedes copulas. 1

Copulas are multivariate probability distributions, as well as functions which link marginal distributions to their joint distribution. These functions have been used extensively in finance and more recently in other disciplines, for example hydrology and genetics. This study has two components, (a) the development of copula-based mathematical tools for use in all industries, and (b) the application of distorted copulas in structured finance. In the first part of this study, copulabased conditional expectation formulae are described and are applied to small data sets from medicine and hydrology. In the second part of this study we develop a method of improving the estimation of default risk in the context of collateralized debt obligations. Credit risk is a particularly important application of copulas, and given the current global financial crisis, there is great motivation to improve the way these functions are applied. We compose distortion functions with copula functions in order to obtain greater flexibility and accuracy in existing pricing algorithms. We also describe an n-dimensional dynamic copula, which takes into account temporal and spatial changes. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical sciences, 2009

Un nombre importants de défauts de prêts immobiliers ainsi que l'eff ondrement du march é immobilier ont entraî n é la faillite de plusieurs banques d'investissement aux Etats Unies. Ces facteurs ont aussi d éclench é la derni ère crise fi nanci ère. Ces év énements ont donné lieu à un pan de travaux cherchant à expliquer les facteurs d éterminant les d éfauts simultan és des prêts immobiliers. Cette th èse apporte des preuves suppl émentaires montrant l'importance de la d épendance des d éfauts lors de la gestion des portefeuilles des prêts immobiliers. Cette th èse comporte trois chapitres identifi ant les facteurs cl és d éterminants la d épendance des d éfauts de prêts immobiliers et des prix immobiliers. Nous montrons que mesure plus pr écise du risque de cr édit est possible en tenant compte des facteurs mentionn ées ci-dessous. Dans le chapitre 1, nous analysons la variation de la dépendance de 13 indices de prix r égionaux. Nous estimons une chaîne de Markov cach ée multivari ée avec deux r égimes équid epéndants. Nous mod élisons la probabilit é de transition en utilisant la croissance du taux d'int érêt, et du rapport prêt-valeur. Nos r ésultats montrent que la d épendance r égionale moyenne des prix immobiliers varie dans le temps. De plus cette d épendance est li ée an changement du taux d'int érêt et au rapport prêt-valeur. En consid érant un sous- échantillon de r égions m étropolitaines, nous montrons aussi qu'une baisse du rapport prêt valeur est associ ée à une plus forte probabilit é d'être dans un r égime de forte d épendance d écrite par une copule en arborescence canonique. Dans le Chapitre 2, nous utilisons une vraisemblance composite de copule (composite likelihood copula), et une fonction Mat érn, nous analysons la d épendance de d éfaut par paires d'un ensemble de prêts immobiliers titris és, a haut risques (subprime mortgages), provenant de la r égion de Los Angeles, entre 2000 et 2011. Nos r ésultats montrent que la d épendance des d éfauts est aff ect ée par la distance g éographique entre les prêts, la moyenne et la di fférences dyadiques de variables telle que le rapport prêt-valeur, le cr édit scoring FICO et le revenu au niveau l'arrondissement. De plus nous identifi ons un eff et de contagion o u un indice de changement des prix immobiliers r égionaux n égatifs et un haut taux de d éfauts augmente la d épendance des d éfauts. En fin notre mod èle donne une bonne estimation de la Value at Risk du nombre de d éfauts dans un bloc de prêts titris és. Dans le chapitre 3, nous analysons l'éfficacit e d'un portefeuille de prêts immobiliers titris és à haut risque (subprime). Nous estimons l'ésp erance et la variance des rendements en utilisant des probabilit e de d éfauts obtenu a partir d'un mod èle de d épendance de d éfaut par paires. Nous analysons les 13 plus larges bloc de prêts immobiliers, titris és entre 2001 et 2005. Nos r ésultats montrent que la diversi cation des blocs de prêts n' étaient pas optimale. De plus, nous montrons qu'il est possible de d'avantage diminuer le risque associ é bloc de prêts en tenant compte des risque non g éographique. / The high number of mortgage defaults along with the collective collapse in regional house prices have led to bankruptcies of Wall Street investment banks and triggered the last financial crisis. This phenomenon have led to a growing body of research seeking to understand how such mortgage defaults tend to occur together. This thesis adds to the body of evidence that dependence between mortgages as well as house prices needs to be seriously taken into consideration in managing the risk of mortgage pools. This thesis consists of three chapters that focus on identifying the factors affecting the dependence between house prices and mortgage defaults. We show how less risky mortgage portfolios can be constructed if we consider the factors mentionned below. In Chapter 1, we analyze time variations in the dependence of 13 regional house price indices. We estimate a multivariate hidden Markov copula model, with two equidependent regimes, and we allow the Markov transition probabilities to vary with changes in interest rates and leverage, measured by the Loan to value ratio (LTV). Our results provide evidence of time-variation in the average dependence in regional house prices. Besides they shows that house price dependence is strongly related to leverage and changes in interest rates. In addition, using a reduced set of Southwestern metropolitan statistical areas (MSAs), we further show that a decrease in leverage is associated with a higher probability of being in an asymmetric high dependence regime, described by a canonical vine copula. In Chapter 2, using a composite likelihood copula approach and a Mat'ern function, we analyze the pairwise dependence of defaults within a set of securitized subprime mortgages originated in Los Angeles between 2000 and 2011. Our results show that default dependence is affected by geographic proximity, as well as dyadic averages and differences in a number of mortgage-specific and local economic variables, such as FICO credit scores, Loan to Value (LTV) and zip code level income. In addition, we find evidence of a contagion effect, whereby negative local house price index returns and high lagged default rates increase default dependence. Our pairwise dependence model also delivers good estimates of Value at Risk for the number of defaults in a pool of mortgages. In Chapter 3, we analyze the mean variance efficiency of pools of securitized subprime mortgages. We estimate the means and variances of the returns from the default probabilities derived from a multinomial logit and a copula-based pairwise default dependence model. We examine the 13 largest mortgage pools that were securitized between 2001 and 2005. Our results first show that the mortgage portfolios were not optimally diversified. Secondly considering non-geographic risk factors leads to less risky optimal portfolios.

Copulas are real functions representing the dependence structure of the distribution of a random vector, and measures of concordance associate with every copula a numerical value in order to allow for the comparison of different degrees of dependence. We first introduce and study a group of transformations mapping the collection of all copulas of fixed but arbitrary dimension into itself. These transformations may be used to construct new copulas from a given one or to prove that certain real functions on the unit cube are indeed copulas. It turns out that certain transformations of a symmetric copula may be asymmetric, and vice versa. Applying this group, we then propose a concise definition of a measure of concordance for copulas. This definition, in which the properties of a measure of concordance are defined in terms of two particular subgroups of the group, provides an easy access to the investigation of invariance properties of a measure of concordance. In particular, it turns out that for copulas which are invariant under a certain subgroup the value of every measure of concordance is equal to zero. We also show that the collections of all transformations which preserve symmetry or the concordance order or the value of every measure of concordance each form a subgroup and that these three subgroups are identical. Finally, we discuss a class of measures of concordance in which every element is defined as the expectation with respect to the probability measure induced by a fixed copula having an invariance property with respect to two subgroups of the group. This class is rich and includes the well-known examples Spearman's rho and Gini's gamma.