Value at Risk Bounds for Portfolios of Non-Normal Returns

Luciano, Elisa; Marena, Marina

doi:10.2139/ssrn.274608

Cited by 5 publications

(3 citation statements)

References 10 publications

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“…Tasche, 2002). As a consequence, finding worst case scenarios for given marginal distributions of V, W in (4.1) is easy in case of ES (take the comonotonic scenario) and non-trivial in case of VaR (see Embrechts et al, 2003;Luciano and Marena, 2003).…”

Section: Defining a Diversification Measurementioning

confidence: 99%

Measuring Sectoral Diversification in an Asymptotic Multi-Factor Framework

Tasche

2006

SSRN Journal

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We investigate a multi-factor extension of the asymptotic single risk factor (ASRF) model that underlies the capital charges of the "Basel II Accord". In this extended model, it is still possible to derive closed-form solutions for the risk contributions to Value-at-Risk and Expected Shortfall. As an application of the risk contribution formulae we introduce a new concept for a diversification measure. The use of this new measure is illustrated by an example calculated with a two-factor model. The results with this model indicate that, thanks to dependence on not fully correlated systematic sectors, there can be a substantial reduction of risk contributions by sectoral diversification effects. *

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Section: Defining a Diversification Measurementioning

confidence: 99%

Measuring Sectoral Diversification in an Asymptotic Multi-Factor Framework

Tasche

2006

SSRN Journal

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“…Let X-{X x ,..., X n ) be a portfolio of multivariate losses, where the marginal losses X t have distributions F ( {x), i = 1,..., n. Suppose one is interested in the maximum VaR and CVaR for the aggregate loss S(X) = Xi + ... + X n whenever XeD(F u ..., F n ) belongs to the set of all multivariate losses with given marginals Fj(x). The evaluation of the maximum VaR of a portfolio with fixed margins is related to Kolmogorov's problem treated among others in Makarov(1981), Ruschendorf(1982), Frank et al(1987), Denuit et al(1999), Durrleman et al(2000), Luciano and Marena(2001) The result (5.1) yields a simple recipe for the calculation of the maximum CVaR for the aggregate loss of portfolios given incomplete information about the marginal losses, say X, eD l m :-Z>4((-°°, °°); lA, -,/4)> ™ e {2,4}, i = 1,..., n, for which it is known that the corresponding stop-loss ordered extremal random variables XJ™l msa have absolutely continuous distributions (Section 3 for m = 2, Section 4, proof of Theorem 4.2 for m = 4).…”

Section: The Maximum Cvar For Portfolios Under Incomplete Informationmentioning

confidence: 99%

Analytical Bounds for two Value-at-Risk Functionals

Hürlimann¹

2002

ASTIN Bull.

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Based on the notions of value-at-risk and conditional value-at-risk, we consider two functionals, abbreviated VaR and CVaR, which represent the economic risk capital required to operate a risky business over some time period when only a small probability of loss is tolerated. These functionals are consistent with the risk preferences of profit-seeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stop-loss order). This result is used to bound the VaR and CVaR functionals by determining their maximal values over the set of all loss and profit functions with fixed first few moments. The evaluation of CVaR for the aggregate loss of portfolios is also discussed. The results of VaR and CVaR calculations are illustrated and compared at some typical situations of general interest.

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“…Compare this to the situation when VaR α is used as risk measure. Then there is no easy general upper bound for the risk of X + Y , and finding the joint distribution of X and Y which yields the maximum value for VaR α (X + Y ) is a non-trivial task (Embrechts et al, 2001;Luciano and Marena, 2001).…”

Section: B)mentioning

confidence: 99%

Expected Shortfall and Beyond

Tasche

2002

Statistical Data Analysis Based on the L1-Norm and Related Methods

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Financial institutions have to allocate so-called economic capital in order to guarantee solvency to their clients and counterparties. Mathematically speaking, any methodology of allocating capital is a risk measure, i.e. a function mapping random variables to the real numbers. Nowadays value-at-risk, which is defined as a fixed level quantile of the random variable under consideration, is the most popular risk measure. Unfortunately, it fails to reward diversification, as it is not subadditive.In the search for a suitable alternative to value-at-risk, Expected Shortfall (or conditional value-at-risk or tail value-at-risk ) has been characterized as the smallest coherent and law invariant risk measure to dominate value-at-risk. We discuss these and some other properties of Expected Shortfall as well as its generalization to a class of coherent risk measures which can incorporate higher moment effects. Moreover, we suggest a general method on how to attribute Expected Shortfall risk contributions to portfolio components. JEL classification D81, C13.

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Value at Risk Bounds for Portfolios of Non-Normal Returns

Cited by 5 publications

References 10 publications

Measuring Sectoral Diversification in an Asymptotic Multi-Factor Framework

Measuring Sectoral Diversification in an Asymptotic Multi-Factor Framework

Analytical Bounds for two Value-at-Risk Functionals

Expected Shortfall and Beyond

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