Analytical VaR and Expected Shortfall for Quadratic Portfolios

Yueh, Meng-Lan; Wong, Mark

doi:10.3905/jod.2010.17.3.033

Cited by 12 publications

(6 citation statements)

References 10 publications

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“…In the case of elliptical distributions, Kamdem (2008) calculated the VaR and the expected shortfall of a quadratic portfolio for a mixture of elliptical distributions by an integral equation, 3 and Yueh and Wong (2010) provided analytic expressions for VaR and the expected shortfall when the risk factors are normally distributed by means of a Fourier transform. Our results improve on Kamdem (2008) and Yueh and Wong (2010), as we provide an analytic expression which is faster to calculate.…”

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confidence: 62%

Multivariate elliptical truncated moments

Arismendi

Broda

2017

Journal of Multivariate Analysis

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In this study, we derived analytic expressions for the elliptical truncated moment generating function (MGF), the zeroth-, first-, and second-order moments of quadratic forms of the multivariate normal, Student's t, and generalised hyperbolic distributions. The resulting formulae were tested in a numerical application to calculate an analytic expression of the expected shortfall of quadratic portfolios with the benefit that moment based sensitivity measures can be derived from the analytic expression. The convergence rate of the analytic expression is fast -one iteration -for small closed integration domains, and slower for open integration domains when compared to the Monte Carlo integration method. The analytic formulae provide a theoretical framework for calculations in robust estimation, robust regression, outlier detection, design of experiments, and stochastic extensions of deterministic elliptical curves results. Keywords: Multivariate truncated moments, Quadratic forms, Elliptical Truncation, Tail moments, Parametric distributions, Elliptical functionsThe first results on truncated moments were concerned with the linear truncated multivariate normal (MVN) distribution, and were provided by Tallis (1961). Tallis (1963) extended the results of linear truncations to the case of elliptical and radial truncation, and Tallis (1965) built on previous results to calculate the moments of a normal distribution with a plane truncation. Mantegna and Stanley (1994) used a truncated Lévy distribution to create a distribution where the sums have slow convergence towards the normal, providing first-and second-order moments. Masoom and Nadarajah (2007) calculated the truncated moments of a generalised Pareto distribution. Arismendi (2013) generalised the results of Tallis (1961) for higher-order moments, and for other elliptical distributions such as the Student's t and the lognormal distributions, and for a finite mixture of multivariate normal distributions. 1In this study, we derived analytical formulae for the calculation of the elliptical truncated moments of the multivariate normal distribution. We calculated an analytical expansion of the elliptical truncated moment generating function (MGF), and then derived this expression for the calculation of the elliptical truncated moments. Previous results on elliptical and radial truncated moments on multivarite normal distributions were provided by Tallis (1963). In this research we used the results of Ruben (1962) to derive the analytical expressions. We then applied the multivariate normal results to derive the multivariate Student's t (MST) and the multivariate generalised hyperbolic (MGH) elliptical truncated moments. Our results can be considered an extension of Ruben's (1962) results for the MST and MGH cases. The importance of elliptical truncated moments' expansions are evident in applications such as the design of experiments (Thompson, 1976;Cameron and Thompson, 1986), robust estimation (Cuesta-Albertos et al., 2008), outlier detection $ The author wish...

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confidence: 62%

Multivariate elliptical truncated moments

Arismendi

Broda

2017

Journal of Multivariate Analysis

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“…Duffie and Pan [2001] derived an analytical VaR formula based on a transform inversion of a quadratic approximation of portfolio values. Yueh and Wong [2010] harnessed Fourier transform techniques to compute the analytical VaR and ES for quadratic portfolios exposed to multivariate, normally distributed risk factors.…”

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confidence: 99%

Value at Risk and Expected Shortfall Improved Calculation Based on the Power Transformation Method

Leccadito

Toscano

Tunaru

2014

JOD

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“…The main focus of their study is on the choice of parameters to perform numerical integration of the characteristic function with a given error tolerance. Yueh and Wong (2010) use Fourier transform method to compute VaR and ES. Assuming that the risk factors are multivariate Gaussian, they derive closed-form formulas for these risk measures.…”

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confidence: 99%

Risk measures computation by Fourier inversion

Nguyen¹,

Nguyen

2017

JRF

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Purpose The purpose of this paper is to present the method for efficient computation of risk measures using Fourier transform technique. Another objective is to demonstrate that this technique enables an efficient computation of risk measures beyond value-at-risk and expected shortfall. Finally, this paper highlights the importance of validating assumptions behind the risk model and describes its application in the affine model framework. Design/methodology/approach The method proposed is based on Fourier transform methods for computing risk measures. The authors obtain the loss distribution by fitting a cubic spline through the points where Fourier inversion of the characteristic function is applied. From the loss distribution, the authors calculate value-at-risk and expected shortfall. As for the calculation of the entropic value-at-risk, it involves the moment generating function which is closely related to the characteristic function. The expectile risk measure is calculated based on call and put option prices which are available in a semi-closed form by Fourier inversion of the characteristic function. We also consider mean loss, standard deviation and semivariance which are calculated in a similar manner. Findings The study offers practical insights into the efficient computation of risk measures as well as validation of the risk models. It also provides a detailed description of algorithms to compute each of the risk measures considered. While the main focus of the paper is on portfolio-level risk metrics, all algorithms are also applicable to single instruments. Practical implications The algorithms presented in this paper require little computational effort which makes them very suitable for real-world applications. In addition, the mathematical setup adopted in this paper provides a natural framework for risk model validation which makes the approach presented in this paper particularly appealing in practice. Originality/value This is the first study to consider the computation of entropic value-at-risk, semivariance as well as expectile risk measure using Fourier transform method.

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Analytical VaR and Expected Shortfall for Quadratic Portfolios

Cited by 12 publications

References 10 publications

Multivariate elliptical truncated moments

Multivariate elliptical truncated moments

Value at Risk and Expected Shortfall Improved Calculation Based on the Power Transformation Method

Risk measures computation by Fourier inversion

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