Two-sided coherent risk measures and their application in realistic portfolio optimization

Chen, Zhiping; Wang, Yi

doi:10.1016/j.jbankfin.2008.07.004

Cited by 56 publications

(38 citation statements)

References 28 publications

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“…For each resample we draw the same number of uniform (0.1) random numbers as sample size and after sorting them in ascending order, we find the relevant quantile corresponding to the coverage level. This quantile is then used as the bootstrap coverage level, corresponding to which we obtain the VaR and ES as given by equations (10) and (14). Since, for a given bootstrap coverage level, this VaR equation needs to besolved numerically, the corresponding ES equation takes a long time to find a converging value.…”

Section: Estimation Of Risk Measures:methodology and Performancementioning

confidence: 99%

Do coherent risk measures identify assets risk profiles similarly? Evidence from international futures markets

Mozumder

Kabir

Dempsey

2017

Investment Management and Financial Innovations

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The authors consider Lévy processes with conditional distributions belonging to a generalized hyperbolic family and compare and contrast full density-based Lévy-expected shortfall (ES) risk measures and Lévy-spectral risk measures (SRM) with those of a traditional tail-based unconditional extreme value (EV) approach. Using the futures data of leading markets the authors find that ES and SRM often differ in recognizing the risk profiles of different assets. While EV (extreme value) is often found to be more consistent than Lévy models, Lévy measures often perform better than EV measures when compared with empirical values. This becomes increasingly apparent as investors become more risk averse.

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Section: Estimation Of Risk Measures:methodology and Performancementioning

confidence: 99%

Do coherent risk measures identify assets risk profiles similarly? Evidence from international futures markets

Mozumder

Kabir

Dempsey

2017

Investment Management and Financial Innovations

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“…Nevertheless, the copulae modelling approach produces in the great majority of cases multivariate distributions where a closed-form of the joint density is unknown. Salem and Mount (1974), Madan and Seneta (1990), Aït-Sahalia and Lo (2000), Scaillet (2004), and Chen and Wang (2008) were among the first papers to explore models with parametric non-normal distributions, following the second approach. Salem and Mount (1974) analysed the gamma distribution, while Madan and Seneta (1990) the variance-gamma distribution.…”

Section: Numerical Application: Quadratic Forms In Financementioning

confidence: 99%

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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“…Our study advances this stream because we present in a more complete fashion the SD component characteristics as a generalized deviation measure. Fischer (2003) and Chen and Wang (2008) consider combining mean and semideviations to different powers to form a coherent risk measure. However, SDR is defined for the tails, unlike the measures proposed by these authors.…”

Section: Introductionmentioning

confidence: 99%

Shortfall deviation risk: an alternative for risk measurement

Righi¹,

Ceretta²

2016

JOR

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We present the Shortfall Deviation Risk (SDR), a risk measure that represents the expected loss that occurs with certain probability penalized by the dispersion of results that are worse than such an expectation. SDR combines Expected Shortfall (ES) and Shortfall Deviation (SD), which we also introduce, contemplating two fundamental pillars of the risk concept -the probability of adverse events and the variability of an expectation -and considers extreme results. We demonstrate that SD is a generalized deviation measure, whereas SDR is a coherent risk measure. We achieve the dual representation of SDR, and we discuss issues such as its representation by a weighted ES, acceptance sets, convexity, continuity and the relationship with stochastic dominance. Illustrations with real and simulated data allow us to conclude that SDR offers greater protection in risk measurement compared with VaR and ES, especially in times of significant turbulence in riskier scenarios.

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Two-sided coherent risk measures and their application in realistic portfolio optimization

Cited by 56 publications

References 28 publications

Do coherent risk measures identify assets risk profiles similarly? Evidence from international futures markets

Do coherent risk measures identify assets risk profiles similarly? Evidence from international futures markets

Multivariate elliptical truncated moments

Shortfall deviation risk: an alternative for risk measurement

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