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Measuring and comparing risks of different types

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  • Aigner, Maximilian
  • Chavez-Demoulin, Valérie
  • Guillou, Armelle

Abstract

Being able to compare risk measures in practice is crucial in many applications such as in finance, insurance or environmental science. The difficulty is that the variables of interest are not always of the same nature, nor of the same type or scale. Thus the usual risk measures are often misleading and to solve this issue we propose to use the Expected Proportional Shortfall (EPS) which is scale invariant and thus, which does not depend on the unit of measurement. To estimate the EPS, an estimator of the tail index γ is required. The main asymptotic properties of our EPS estimator are provided under very general assumptions in case of d-variate β-mixing processes with Pareto-type marginals. Then, we propose a test statistic based on the EPS estimators to compare different risks, whatever their nature/type/scale are. Since the performances of the test statistic are poor when a biased estimate of γ is used, we propose to perform our EPS estimation with an asymptotically unbiased estimator for γ. The efficiency of our test statistic is illustrated in a simulation experiment and validated on an environmental dataset.

Suggested Citation

  • Aigner, Maximilian & Chavez-Demoulin, Valérie & Guillou, Armelle, 2022. "Measuring and comparing risks of different types," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 1-21.
    • Handle: RePEc:eee:insuma:v:102:y:2022:i:c:p:1-21
    DOI: 10.1016/j.insmatheco.2021.11.001
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    References listed on IDEAS

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    1. Lin, Zheng Yan & Choi, Yong-Kab, 1999. "Some limit theorems for fractional Lévy Brownian fields," Stochastic Processes and their Applications, Elsevier, vol. 82(2), pages 229-244, August.
    2. Julien Worms & Rym Worms, 2015. "A Test for Comparing Tail Indices for Heavy-Tailed Distributions via Empirical Likelihood," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(15), pages 3289-3302, August.
    3. Lobato I. N., 2001. "Testing That a Dependent Process Is Uncorrelated," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1066-1076, September.
    4. Severini,Thomas A., 2005. "Elements of Distribution Theory," Cambridge Books, Cambridge University Press, number 9780521844727, January.
    5. Einmahl, J.H.J. & Gantner, M. & Sawitzki, G., 2010. "The Shorth Plot," Other publications TiSEM 0bb67ddc-0dd1-4c13-9916-5, Tilburg University, School of Economics and Management.
    6. Einmahl, J.H.J. & Gantner, M. & Sawitzki, G., 2008. "The Shorth Plot," Other publications TiSEM 10b5cfb5-c502-46dc-8e51-5, Tilburg University, School of Economics and Management.
    7. Xiaofeng Shao, 2010. "A self‐normalized approach to confidence interval construction in time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(3), pages 343-366, June.
    8. Nicholas M. Kiefer & Timothy J. Vogelsang & Helle Bunzel, 2000. "Simple Robust Testing of Regression Hypotheses," Econometrica, Econometric Society, vol. 68(3), pages 695-714, May.
    9. Yuri Goegebeur & Armelle Guillou, 2013. "Asymptotically Unbiased Estimation of the Coefficient of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(1), pages 174-189, March.
    10. Chavez-Demoulin, Valérie & Guillou, Armelle, 2018. "Extreme quantile estimation for β-mixing time series and applications," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 59-74.
    11. Hoga, Yannick, 2017. "Change Point Tests For The Tail Index Of Β-Mixing Random Variables," Econometric Theory, Cambridge University Press, vol. 33(4), pages 915-954, August.
    12. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    13. Einmahl, J.H.J. & Gantner, M. & Sawitzki, G., 2008. "The Shorth Plot," Other publications TiSEM 10b5cfb5-c502-46dc-8e51-5, Tilburg University, School of Economics and Management.
    14. Belzunce, Félix & Pinar, José F. & Ruiz, José M. & Sordo, Miguel A., 2012. "Comparison of risks based on the expected proportional shortfall," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 292-302.
    15. Yannick Hoga, 2018. "Detecting Tail Risk Differences in Multivariate Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(5), pages 665-689, September.
    16. Xiaofeng Shao, 2015. "Self-Normalization for Time Series: A Review of Recent Developments," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(512), pages 1797-1817, December.
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    More about this item

    Keywords

    Asymptotically unbiased estimator; β-mixing; Convergence in distribution; Expected Proportional Shortfall; Hypothesis testing;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General

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