Asymptotics of sums of lognormal random variables with Gaussian copula

Asmussen, Søren; Rojas-Nandayapa, Leonardo

doi:10.1016/j.spl.2008.03.035

Cited by 95 publications

(89 citation statements)

References 7 publications

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“…There has been a particular interest in understanding the tail behavior of the sum of asymptotically independent random variables with heavy tails (see Albrecher et al, 2006, Ko and Tang, 2008, Asmussen and Rojas-Nandayapa, 2008, Geluk and Tang, 2009, and Mitra and Resnick, 2009 3.1. Fréchet case.…”

Section: Main Results Under Asymptotic Independencementioning

confidence: 99%

Asymptotics for risk capital allocations based on Conditional Tail Expectation

Asimit

Furman

Tang

et al. 2011

Insurance: Mathematics and Economics

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Abstract. An investigation of the limiting behavior of a risk capital allocation rule based on the Conditional Tail Expectation (CTE) risk measure is carried out. More specifically, with the help of general notions of Extreme Value Theory (EVT), the aforementioned risk capital allocation is shown to be asymptotically proportional to the corresponding Value-atRisk (VaR) risk measure. The existing methodology acquired for VaR can therefore be applied to a somewhat less well-studied CTE. In the context of interest, the EVT approach is seemingly well-motivated by modern regulations, which openly strive for the excessive prudence in determining risk capitals.

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Section: Main Results Under Asymptotic Independencementioning

confidence: 99%

Asymptotics for risk capital allocations based on Conditional Tail Expectation

Asimit

Furman

Tang

et al. 2011

Insurance: Mathematics and Economics

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“…For simplicity, assume that each X i has mean µ and variance σ 2 > 0. This example is extensively considered in Asmussen and Rojas-Nandayapa (2008). We have already considered the case in which ρ = −1 in Example 3.3, so here we consider ρ ∈ (−1, 1).…”

Section: Examplesmentioning

confidence: 99%

“…Recent results can be found in Albrecher et al (2006), Klüppelberg and Resnick (2008), Wang and Tang (2006), Asmussen and Rojas-Nandayapa (2008), Alink et al (2004), Embrechts and Puccetti (2006), and Ko and Tang (2008). Approximating this probability helps us evaluate risk measures for investment portfolios as well as estimating credit risk.…”

Section: Introductionmentioning

confidence: 99%

Aggregation of rapidly varying risks and asymptotic independence

Mitra¹,

Resnick²

2009

Adv. Appl. Probab.

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We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks (X, Y ) such that P(X +Y > x) ∼ (constant) P(X > x). With the further assumption of nonnegativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions of X and Y are subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored, leading to an approximate solution of an optimization problem which is applied to portfolio design.

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“…In general, we propose that this approach be used to solve the SLN problem in closed-form: find a distribution with lognormal tails, and fit them to the tails of the SLN distribution [4], [27]. So far, we were the only ones to take this approach, and only with the MPLN distribution, but perhaps other (better) distribution functions exist that are amenable to this approach.…”

Section: Tail Properties On Lognormal Papermentioning

confidence: 99%

“…The slope in the +∞ tail of the SLN cdf on lognormal paper is known to be 1/max i σ i [4], [27]. We equate this slope with (10): 1 max…”

Section: A Upper Tail Slopementioning

confidence: 99%

Fitting the Modified-Power-Lognormal to the Sum of Independent Lognormals Distribution

Szyszkowicz

Yanıkömeroğlu

2009

GLOBECOM 2009 - 2009 IEEE Global Telecommunications Conference

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Abstract-We propose a new method for calculating a tight approximation to the distribution of the sum of independent lognormal random variables. We make use of a three-parameter modified-power-lognormal distribution function as the approximating distribution. We use theoretical results from our previous work on the tails of the distribution of the sum of lognormals to match the slope of the modified-power-lognormal function at both tails. This would not have been possible with many of the recently-proposed distribution functions, which do not behave properly in the tails. We then also use moment-matching to find the best curve match. Our method is mostly closed-form, requiring only one simple numerical integral.We compare our method with those in literature in terms of complexity and accuracy. We conclude that our method is more accurate than the simple (closed-form) methods, and much simpler to understand and implement than the more accurate methods which rely heavily on numerical integration.Index Terms-sum of lognormals, interference analysis

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Asymptotics of sums of lognormal random variables with Gaussian copula

Cited by 95 publications

References 7 publications

Asymptotics for risk capital allocations based on Conditional Tail Expectation

Asymptotics for risk capital allocations based on Conditional Tail Expectation

Aggregation of rapidly varying risks and asymptotic independence

Fitting the Modified-Power-Lognormal to the Sum of Independent Lognormals Distribution

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