@article { ISI:000345588000005,
	title = {The effect of aggregation on extremes from asymptotically independent light-tailed risks},
	journal = {EXTREMES},
	volume = {17},
	number = {4},
	year = {2014},
	month = {DEC},
	pages = {615-631},
	publisher = {SPRINGER},
	type = {Article},
	address = {233 SPRING ST, NEW YORK, NY 10013 USA},
	abstract = {Portfolio risk diversification is a well established concept in finance. While aggregation of several risky assets generally reduces the overall investment risk due to diversification, the effectiveness of diversification depends on the stochastic behaviour of the returns of the assets comprising the portfolio. The paper proposes a new approach to quantifying the effect of portfolio tail diversification by looking at the asymptotic behaviour of the ratio of the largest loss on the portfolio to the sum of the largest losses on the individual investments held on the stand-alone basis. It is assumed that independent and identically distributed random vectors from the underlying distribution, describing the behaviour of returns on the assets in the portfolio, can be scaled to converge onto a deterministic limit set. This property is satisfied by a number of distributions commonly used in finance. Several analytical examples are given to illustrate the proposed asymptotic diversification index as a measure of the effect of risk aggregation on extremes as well as to quantify the impact of dimension on the diversification and as a tool in optimal portfolio selection.},
	keywords = {Asymptotic independence, Light-tailed distributions, Limit set, Risk diversification, Tail risk, Weak tail dependence coefficient},
	issn = {1386-1999},
	doi = {10.1007/s10687-014-0192-y},
	author = {Nolde, Natalia}
}
@article { ISI:000327908200006,
	title = {Geometric interpretation of the residual dependence coefficient},
	journal = {JOURNAL OF MULTIVARIATE ANALYSIS},
	volume = {123},
	year = {2014},
	month = {JAN},
	pages = {85-95},
	publisher = {ELSEVIER INC},
	type = {Article},
	address = {525 B STREET, STE 1900, SAN DIEGO, CA 92101-4495 USA},
	abstract = {The residual dependence coefficient was originally introduced by Ledford and Tawn (1996) [25] as a measure of residual dependence between extreme values in the presence of asymptotic independence. We present a geometric interpretation of this coefficient with the additional assumptions that the random samples from a given distribution can be scaled to converge onto a limit set and that the marginal distributions have Weibull-type tails. This result leads to simple and intuitive computations of the residual dependence coefficient for a variety of distributions. (C) 2013 Elsevier Inc. All rights reserved.},
	keywords = {Asymptotic independence, Geometric approach, Limit set, Multivariate density, Residual dependence coefficient, Sample clouds},
	issn = {0047-259X},
	doi = {10.1016/j.jmva.2013.08.018},
	author = {Nolde, Natalia}
}
@article { ISI:000312614700014,
	title = {Sensitivity of the limit shape of sample clouds from meta densities},
	journal = {BERNOULLI},
	volume = {18},
	number = {4},
	year = {2012},
	month = {NOV},
	pages = {1386-1404},
	publisher = {INT STATISTICAL INST},
	type = {Article},
	address = {428 PRINSES BEATRIXLAAN, 2270 AZ VOORBURG, NETHERLANDS},
	abstract = {The paper focuses on a class of light-tailed multivariate probability distributions. These are obtained via a transformation of the margins from a heavy-tailed original distribution. This class was introduced in Balkema et al. (J. Multivariate Anal. 101 (2010) 1738-1754). As shown there, for the light-tailed meta distribution the sample clouds, properly scaled, converge onto a deterministic set. The shape of the limit set gives a good description of the relation between extreme observations in different directions. This paper investigates how sensitive the limit shape is to changes in the underlying heavy-tailed distribution. Copulas fit in well with multivariate extremes. By Galambos{\textquoteright}s theorem, existence of directional derivatives in the upper endpoint of the copula is necessary and sufficient for convergence of the multivariate extremes provided the marginal maxima converge. The copula of the max-stable limit distribution does not depend on the margins. So margins seem to play a subsidiary role in multivariate extremes. The theory and examples presented in this paper cast a different light on the significance of margins. For light-tailed meta distributions, the asymptotic behaviour is very sensitive to perturbations of the underlying heavy-tailed original distribution, it may change drastically even when the asymptotic behaviour of the heavy-tailed density is not affected.},
	keywords = {extremes, Limit set, limit shape, meta distribution, regular partition, sensitivity},
	issn = {1350-7265},
	doi = {10.3150/11-BEJ370},
	author = {Balkema, Guus and Embrechts, Paul and Nolde, Natalia}
}
