Title: | Multivariate Regular Variation |
Author: | Bernardo, Alexandre |
Contributor: | University of Helsinki, Faculty of Science |
Publisher: | Helsingin yliopisto |
Date: | 2020 |
Language: | eng |
URI: |
http://urn.fi/URN:NBN:fi:hulib-202009304164
http://hdl.handle.net/10138/319783 |
Thesis level: | master's thesis |
Degree program: |
Matematiikan ja tilastotieteen maisteriohjelma
Master's Programme in Mathematics and Statistics Magisterprogrammet i matematik och statistik |
Specialisation: |
Sovellettu matematiikka
Applied Mathematics Tillämpad matematik |
Discipline: | none |
Abstract: | In insurance and reinsurance, heavy-tail analysis is used to model insurance claim sizes and frequencies in order to quantify the risk to the insurance company and to set appropriate premium rates. One of the reasons for this application comes from the fact that excess claims covered by reinsurance companies are very large, and so a natural field for heavy-tail analysis. In finance, the multivariate returns process often exhibits heavy-tail marginal distributions with little or no correlation between the components of the random vector (even though it is a highly correlated process when taking the square or the absolute values of the returns). The fact that vectors which are considered independent by conventional standards may still exhibit dependence of large realizations leads to the use of techniques from classical extreme-value theory, that contains heavy-tail analysis, in estimating an extreme quantile of the profit-and-loss density called value-at-risk (VaR). The need of the industry to understand the dependence between random vectors for very large values, as exemplified above, makes the concept of multivariate regular variation a current topic of great interest. This thesis discusses multivariate regular variation, showing that, by having multiple equivalent characterizations and and by being quite easy to handle, it is an excellent tool to address the real-world issues raised previously. The thesis is structured as follows. At first, some mathematical background is covered: the notions of regular variation of a tail distribution in one dimension is introduced, as well as different concepts of convergence of probability measures, namely vague convergence and $\mathbb{M}^*$-convergence. The preference in using the latter over the former is briefly discussed. The thesis then proceeds to the main definition of this work, that of multivariate regular variation, which involves a limit measure and a scaling function. It is shown that multivariate regular variation can be expressed in polar coordinates, by replacing the limit measure with a product of a one-dimensional measure with a tail index and a spectral measure. Looking for a second source of regular variation leads to the concept of hidden regular variation, to which a new hidden limit measure is associated. Estimation of the tail index, the spectral measure and the support of the limit measure are next considered. Some examples of risk vectors are next analyzed, such as risk vectors with independent components and risk vectors with repeated components. The support estimator presented earlier is then computed in some examples with simulated data to display its efficiency. However, when the estimator is computed with real-life data (the value of stocks for different companies), it does not seem to suit the sample in an adequate way. The conclusion is drawn that, although the mathematical background for the theory is quite solid, more research needs to be done when applying it to real-life data, namely having a reliable way to check whether the data stems from a multivariate regular distribution, as well as identifying the support of the limit measure. |
Subject: |
multivariate regular variation
heavy-tailed distribution risk limit measure hidden regular variation insurance estimation |
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