Bounds and approximations for sums of dependent log-elliptical random variables

Valdez, Emiliano A.; Dhaene, Jan; Maj, Mateusz; Vanduffel, Steven

doi:10.1016/j.insmatheco.2008.11.007

Cited by 40 publications

(21 citation statements)

References 27 publications

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“…Valdez et al (2009) investigate to which extent the general results on convex bounds of Section 2 can be applied to sums of non-independent log-elliptical random variables which incorporate sums of log-normals as a special case. First, they show that unlike the log-normal case, for general sums of log-ellipticals the convex lower bound (9) does no longer result in closed-form approximations for the different risk measures.…”

Section: Further Developments Of the Theorymentioning

confidence: 99%

Advanced Mathematical Methods for Finance

Nunno¹,

Øksendal²

2011

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Over the last decade, it has been shown that the concept of comonotonicity is a helpful tool for solving several research and practical problems in the domain of finance and insurance. In this paper, we give an extensive bibliographic overview -without claiming to be complete -of the developments of the theory of comonotonicity and its applications, with an emphasis on the achievements over the last five years. These applications range from pricing and hedging of derivatives over risk management to life insurance. ComonotonicityOver the last two decades, researchers in economics, financial mathematics and actuarial science have introduced results related to the concept of comonotonicity in their respective fields of interest. In this paper, we give an overview of the relevant literature in these research fields, with the main emphasis on the development of the theory and its applications in finance and insurance over the last five years. Although it is our intention to give an extensive bibliographic overview, due to the high number of papers on applications of comonotonicity, it is impossible to present here an exhaustive overview of the recent literature. Further, we restrict this paper to a description of how and where comonotonicity comes in and refer to the relevant papers for a detailed mathematical description. In order to make this paper self-contained, we also provide a short overview of the basic definitions and initial main results of comonotonicity theory, hereby referring to part of the older literature on this topic.The concept of comonotonicity is closely related to the following well-known result, which is usually attributed to both Hoeffding (1940) and Fréchet (1951): For any n-dimensional random vector X ≡ (X 1 , X 2 , . . . , X n ) with multivariate cumulative distribution function (cdf) F X and marginal univariate cdf's F X 1 , F X 2 , . . . , F Xn and for any x ≡ (x 1 , x 2 , . . . , x n ) ∈ R n it holds that F X (x) ≤ min (F X 1 (x 1 ) , F X 2 (x 2 ) , . . . , F Xn (x n )) .(1)In the sequel, the notation R n (F X 1 , F X 2 , . . . , F Xn ) will be used to denote the class of all random vectors Y ≡ (Y 1 , Y 2 , . . . , Y n ) with marginals F Y i equal to the respective marginals F X i . The set R n (F X 1 , F X 2 , . . . , F Xn ) is called the Fréchet class related to the random vector X.The upper bound in (1) is reachable in the Fréchet class R n (F X 1 , F X 2 , . . . , F Xn ) in the sense that it is the cdf of an n-dimensional random vector with marginals given by F X i , i = 1, 2, . . . , n. * Department of Mathematics, ECARES, Université Libre de Bruxelles, CP 210, 1050 Brussels, Belgium † Faculty of Business and Economics, Katholieke Universiteit, 3000 Leuven, Belgium. ‡ Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 S9, 9000 Gent, Belgium 1 In order to prove the reachability property, consider a random variable U , uniformly distributed on the unit interval (0, 1). Then one has that F −1 X 1 (U ), F −1 X 2 (U ), . . . , F −1 Xn (U ) ∈ ...

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Section: Further Developments Of the Theorymentioning

confidence: 99%

Advanced Mathematical Methods for Finance

Nunno¹,

Øksendal²

2011

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“…Actuarial applications of elliptical distributions are considered in Landsman and Valdez (2003) and Valdez and Dhaene (2004), amongst others. In the remainder of the paper we will only consider rvs with a finite mean.…”

Section: Elliptical Distributionsmentioning

confidence: 99%

“…see Valdez and Dhaene (2004). In the next two examples we make use of (47) to derive expressions for the CTEs of lognormal and logLaplace distributions.…”

Section: Lognormal and Logelliptical Distributionsmentioning

confidence: 99%

Some results on the CTE-based capital allocation rule

Dhaene

Henrard²,

Landsman

et al. 2008

Insurance: Mathematics and Economics

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Tasche [Tasche, D., 1999. Risk contributions and performance measurement. Working paper, Technische Universität München] introduces a capital allocation principle where the capital allocated to each risk unit can be expressed in terms of its contribution to the conditional tail expectation (CTE) of the aggregate risk. Panjer [Panjer, H.H., 2002. Measurement of risk, solvency requirements and allocation of capital within financial conglomerates. Institute of Insurance and Pension Research, University of Waterloo, Research Report 01-15] derives a closed-form expression for this allocation rule in the multivariate normal case. Landsman and Valdez [Landsman, Z., Valdez, E., 2002. Tail conditional expectations for elliptical distributions. North American Actuarial J. 7 (4)] generalize Panjer's result to the class of multivariate elliptical distributions.In this paper we provide an alternative and simpler proof for the CTE-based allocation formula in the elliptical case. Furthermore, we derive accurate and easy computable closed-form approximations for this allocation formula for sums that involve normal and lognormal risks.

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“…Frangos and Karlis, 2004;Valdez et al, 2009;Cherubini et al, 2011;Furman and Landsman, 2010, among many others).…”

Section: Introductionmentioning

confidence: 98%