http://repub.eur.nl/res/aut/37/ List of Publications en http://repub.eur.nl/static-eur/img/logo.png http://repub.eur.nl http://repub.eur.nl/res/pub/8747/ 2007-02-14T00:00:00Z The stability of the financial system at higher loss levels is either characterized by asymptotic dependence or asymptotic independence. If asymptotically independent, the dependency, when present, eventually dies out completely at the more extreme quantiles, as in case of the multivariate normal distribution. Given that financial service firms' equity returns depend linearly on the risk drivers, we show that the marginals' distributions maximum domain of attraction determines the type of systemic (in-)stability. A scale for the amount of dependency at high loss lovels is designed. This permits a characterization of systemic risk inherent to different financial network structures. The theory also suggests the functional form of the economically relevant limit copulas. http://repub.eur.nl/res/pub/12370/ 2006-02-24T00:00:00Z Suppose $X_1, X_2, \\ldots$ are independent subexponential random variables with partial sums $S_n$. We show that if the pairwise sums of the $X_i$’s are subexponential, then $S_n$ is subexponential and $(S_n > x) ∼ \\sum_{i}^n P(X_i > x)(x \\rightarrow \\infty)$. The result is applied to give conditions under which $P(\\sum_{1}^\\infty c_iX_i > x)$ as $x \\rightarrow \\infty$, where $c_1, c_2, \\ldots$ are constants such that $\\sum_{1}^\\infty c_iX_i$ is a.s. convergent. Asymptotic tail probabilities for bivariate linear combinations of subexponential random variables are given. These results are applied to explain the joint movements of the stocks of reinsurers. Portfolio investment and retrocession practices in the reinsurance industry expose different reinsurers to the same subexponential risks on both sides of their balance sheets. This implies that reinsurer’s equity returns can be asymptotically dependent, exposing the industry to systemic risk. http://repub.eur.nl/res/pub/1800/ 2004-11-05T00:00:00Z Asymptotic tail probabilities for bivariate linear combinations of subexponential random variables are given. These results are applied to explain the joint movements of the stocks of reinsurers. Portfolio investment and retrocession practices in the reinsurance industry, for reasons of diversification, exposes different reinsurers to the same risks on both sides of their balance sheets. Assuming, in line with the industry practice that the risk drivers follow subexponential distributions, we derive (under mild conditions) when the reinsurer's equity returns are asymptotically dependent, exposing the industry to systemic risk. http://repub.eur.nl/res/pub/6615/ 2004-09-12T00:00:00Z http://repub.eur.nl/res/pub/1055/ 2003-08-07T00:00:00Z We give a sufficient condition for i.i.d. random variables X1,X2 in order to have P{X1-X2>x} ~ P{|X1|>x} as x tends to infinity. A factorization property for subexponential distributions is used in the proof. In a subsequent paper the results will be applied to model fragility of financial markets. http://repub.eur.nl/res/pub/541/ 2002-11-11T00:00:00Z It has been known for a long time that for bootstrapping the probability distribution of the maximum of a sample consistently, the bootstrap sample size needs to be of smaller order than the original sample size. See Jun Shao and Dongsheng Tu (1995), Ex. 3.9,p. 123. We show that the same is true if we use the bootstrap for estimating an intermediate quantile. http://repub.eur.nl/res/pub/12390/ 2000-01-01T00:00:00Z http://repub.eur.nl/res/pub/7711/ 1999-10-14T00:00:00Z The paper characterizes first and second order tail behavior of convolutions of i.i.d. heavy tailed random variables with support on the real line. The result is applied to the problem of risk diversification in portfolio analysis and to the estimation of the parameter in a MA(1) model. http://repub.eur.nl/res/pub/1564/ 1999-03-30T00:00:00Z For samples of random variables with a regularly varying tail estimating the tail index has received much attention recently. For the proof of asymptotic normality of the tail index estimator second order regular variation is needed. In this paper we first supplement earlier results on convolution given by Geluk et al. (1997). Secondly we propose a simple estimator of the tail index for finite moving average time series. We also give a subsampling procedure in order to estimate the optimal sample fraction in the sense of minimal mean squared error. http://repub.eur.nl/res/pub/1547/ 1998-08-13T00:00:00Z We characterize second order regular variation of the tail sum of F together with a balance condition on the tails interms of the behaviour of the characteristic function near zero. http://repub.eur.nl/res/pub/7796/ 1997-08-15T00:00:00Z The theory of stable probability distributions and their domains of attraction is derived in a direct way (avoiding the usual route via infinitely divisible distributions) using Fourier transforms. Regularly varying functions play an important role in the exposition.