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Asymmetric Power Distribution: Theory and Applications to Risk Measurement

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  • Ivana Komunjer

Abstract

Theoretical literature in finance has shown that quantifying the risk of financial time series amounts to measuring their expected shortfall, also known as tail Value at Risk. Unfortunately, little empirical work has been devoted to the problem of modeling and inference of such risk measures and, in particular, to their estimation. In this paper, we construct a parametric estimator for the expected shortfall based on a new family of densities, which we call the Asymmetric Power Distribution (APD). The APD family extends the Generalized Power Distribution to cases where the data exhibits asymmetry. We provide a detailed description of the properties of an APD random variable, such as its quantiles, moments and moment related parameters. Moreover, we discuss the problem of simulation of such random variables and provide maximum likelihood estimates of the APD density parameters. The study of asymptotic properties of the latter falls outside the standard framework due to the non-differentiability of the APD log-likelihood. An empirical application to six daily financial market series reveals that returns tend to be asymmetric, with innovations which cannot be modeled by either Laplace (double-exponential) or Gaussian distribution, even if we allow the latter to be asymmetric. Under a more general assumption that the return innovations are APD, we are able to compute expected shortfalls and corresponding confidence intervals and thus compare the riskiness of the series examined

Suggested Citation

  • Ivana Komunjer, 2004. "Asymmetric Power Distribution: Theory and Applications to Risk Measurement," Econometric Society 2004 Latin American Meetings 44, Econometric Society.
  • Handle: RePEc:ecm:latm04:44
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    References listed on IDEAS

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    Cited by:

    1. Haas Markus, 2010. "Skew-Normal Mixture and Markov-Switching GARCH Processes," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 14(4), pages 1-56, September.
    2. Jin-Ray Lu & Chiang-Chang Hwang & Yi-Chun Chen & Chu-Ting Wen, 2013. "Including More Information Content to Enhance the Value at Risk Estimation for Real Estate Investment Trusts," International Journal of Financial Research, International Journal of Financial Research, Sciedu Press, vol. 4(3), pages 25-34, July.
    3. Li, Xiao-Ming & Rose, Lawrence C., 2009. "The tail risk of emerging stock markets," Emerging Markets Review, Elsevier, vol. 10(4), pages 242-256, December.
    4. Yong Bao & Aman Ullah, 2009. "Expectation of Quadratic Forms in Normal and Nonnormal Variables with Econometric Applications," Working Papers 200907, University of California at Riverside, Department of Economics, revised Jun 2009.
    5. Douch, Mohamed & Farooq, Omar & Bouaddi, Mohammed, 2015. "Stock price synchronicity and tails of return distribution," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 37(C), pages 1-11.
    6. Rombouts Jeroen V. K. & Bouaddi Mohammed, 2009. "Mixed Exponential Power Asymmetric Conditional Heteroskedasticity," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 13(3), pages 1-32, May.
    7. Yu, Dalei & Bai, Peng & Ding, Chang, 2015. "Adjusted quasi-maximum likelihood estimator for mixed regressive, spatial autoregressive model and its small sample bias," Computational Statistics & Data Analysis, Elsevier, vol. 87(C), pages 116-135.
    8. Mendoza-Velázquez, Alfonso & Galvanovskis, Evalds, 2009. "Introducing the GED-Copula with an application to Financial Contagion in Latin America," MPRA Paper 46669, University Library of Munich, Germany, revised 01 Feb 2010.
    9. Zhu, Dongming & Zinde-Walsh, Victoria, 2009. "Properties and estimation of asymmetric exponential power distribution," Journal of Econometrics, Elsevier, vol. 148(1), pages 86-99, January.
    10. Huber, Peter & Oberhofer, Harald & Pfaffermayr, Michael, 2017. "Who creates jobs? Econometric modeling and evidence for Austrian firm level data," European Economic Review, Elsevier, vol. 91(C), pages 57-71.
    11. Vijverberg, Chu-Ping C. & Vijverberg, Wim P.M. & Taşpınar, Süleyman, 2016. "Linking Tukey’s legacy to financial risk measurement," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 595-615.
    12. Fischer, Matthias, 2012. "A skew and leptokurtic distribution with polynomial tails and characterizing functions in closed form," FAU Discussion Papers in Economics 03/2012, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    13. repec:eee:ecmode:v:68:y:2018:i:c:p:611-621 is not listed on IDEAS
    14. Bera Anil K. & Galvao Antonio F. & Montes-Rojas Gabriel V. & Park Sung Y., 2016. "Asymmetric Laplace Regression: Maximum Likelihood, Maximum Entropy and Quantile Regression," Journal of Econometric Methods, De Gruyter, vol. 5(1), pages 79-101, January.
    15. J. Miguel Marin & Genaro Sucarrat, 2015. "Financial density selection," The European Journal of Finance, Taylor & Francis Journals, vol. 21(13-14), pages 1195-1213, November.
    16. Nieto, Maria Rosa & Ruiz, Esther, 2016. "Frontiers in VaR forecasting and backtesting," International Journal of Forecasting, Elsevier, vol. 32(2), pages 475-501.
    17. Yi-Ting Chen, 2016. "Testing for Granger Causality in Moments," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 78(2), pages 265-288, April.
    18. Komunjer, Ivana & Vuong, Quang, 2010. "Efficient estimation in dynamic conditional quantile models," Journal of Econometrics, Elsevier, vol. 157(2), pages 272-285, August.
    19. Kaiping Wang, 2014. "Modeling Stock Index Returns using Semi-Parametric Approach with Multiplicative Adjustment," Journal for Economic Forecasting, Institute for Economic Forecasting, vol. 0(4), pages 65-75, December.
    20. Vijverberg, Wim P. & Hasebe, Takuya, 2015. "GTL Regression: A Linear Model with Skewed and Thick-Tailed Disturbances," IZA Discussion Papers 8898, Institute for the Study of Labor (IZA).
    21. Mendoza, Alfonso. & Galvanovskis, Evalds., 2014. "La cópula GED bivariada. Una aplicación en entornos de crisis," El Trimestre Económico, Fondo de Cultura Económica, vol. 0(323), pages .721-746, julio-sep.
    22. Zhu, Dongming & Galbraith, John W., 2011. "Modeling and forecasting expected shortfall with the generalized asymmetric Student-t and asymmetric exponential power distributions," Journal of Empirical Finance, Elsevier, vol. 18(4), pages 765-778, September.

    More about this item

    Keywords

    expected shortfall; value-at-risk; generalized power distribution;

    JEL classification:

    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)

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