[go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/a/eee/jbfina/v93y2018icp92-104.html
   My bibliography  Save this article

Value at risk and expected shortfall based on Gram-Charlier-like expansions

Author

Listed:
  • Zoia, Maria Grazia
  • Biffi, Paola
  • Nicolussi, Federica
Abstract
This paper offers a new approach to modeling the distribution of a portfolio composed of either asset returns or insurance losses. To capture the leptokurtosis, which is inherent in most financial series, data are modeled by using Gram-Charlier (GC) expansions. Since we are interested in operating with several series simultaneously, the distribution of the sum of GC random variables is derived. This latter turns out to be a tail-sensitive density, suitable for modeling the distribution of a portfolio return-losses and, accordingly, can be conveniently adopted for computing risk measures such as the value at risk and the expected shortfall as well as some performance measures based on its partial moments. The closed form expressions of these risk measures are derived for cases when the density of a portfolio is the sum of GC expansions, either with the same or different kurtosis. An empirical application of this approach to a portfolio of financial asset indexes provides evidence of the comparative effectiveness of this technique in computing risk measures, both in and out of the sample period.

Suggested Citation

  • Zoia, Maria Grazia & Biffi, Paola & Nicolussi, Federica, 2018. "Value at risk and expected shortfall based on Gram-Charlier-like expansions," Journal of Banking & Finance, Elsevier, vol. 93(C), pages 92-104.
    • Handle: RePEc:eee:jbfina:v:93:y:2018:i:c:p:92-104
    DOI: 10.1016/j.jbankfin.2018.06.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378426618301195
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jbankfin.2018.06.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Jondeau, Eric & Rockinger, Michael, 2001. "Gram-Charlier densities," Journal of Economic Dynamics and Control, Elsevier, vol. 25(10), pages 1457-1483, October.
    2. Mills,Terence C. & Markellos,Raphael N., 2008. "The Econometric Modelling of Financial Time Series," Cambridge Books, Cambridge University Press, number 9780521710091, September.
    3. Lakshman Alles & Louis Murray, 2010. "Non-Normality and Risk in Developing Asian Markets," Review of Pacific Basin Financial Markets and Policies (RPBFMP), World Scientific Publishing Co. Pte. Ltd., vol. 13(04), pages 583-605.
    4. Peter F. Christoffersen & Francis X. Diebold & Til Schuermann, 1998. "Horizon problems and extreme events in financial risk management," Economic Policy Review, Federal Reserve Bank of New York, vol. 4(Oct), pages 109-118.
    5. Farinelli, Simone & Tibiletti, Luisa, 2008. "Sharpe thinking in asset ranking with one-sided measures," European Journal of Operational Research, Elsevier, vol. 185(3), pages 1542-1547, March.
    6. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    7. C. W. Granger & E. Maasoumi & J. Racine, 2004. "A Dependence Metric for Possibly Nonlinear Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(5), pages 649-669, September.
    8. Gallant, Ronald & Tauchen, George, 1989. "Seminonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications," Econometrica, Econometric Society, vol. 57(5), pages 1091-1120, September.
    9. Abu Bakar, S.A. & Hamzah, N.A. & Maghsoudi, M. & Nadarajah, S., 2015. "Modeling loss data using composite models," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 146-154.
    10. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    11. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    12. Fishburn, Peter C, 1977. "Mean-Risk Analysis with Risk Associated with Below-Target Returns," American Economic Review, American Economic Association, vol. 67(2), pages 116-126, March.
    13. Paul H. Kupiec, 1995. "Techniques for verifying the accuracy of risk measurement models," Finance and Economics Discussion Series 95-24, Board of Governors of the Federal Reserve System (U.S.).
    14. McNeil, Alexander J. & Frey, Rudiger, 2000. "Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach," Journal of Empirical Finance, Elsevier, vol. 7(3-4), pages 271-300, November.
    15. León, Angel & Moreno, Manuel, 2017. "One-sided performance measures under Gram-Charlier distributions," Journal of Banking & Finance, Elsevier, vol. 74(C), pages 38-50.
    16. José Curto & José Pinto & Gonçalo Tavares, 2009. "Modeling stock markets’ volatility using GARCH models with Normal, Student’s t and stable Paretian distributions," Statistical Papers, Springer, vol. 50(2), pages 311-321, March.
    17. Hayfield, Tristen & Racine, Jeffrey S., 2008. "Nonparametric Econometrics: The np Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 27(i05).
    18. Mittnik, Stefan & Paolella, Marc S. & Rachev, Svetlozar T., 2000. "Diagnosing and treating the fat tails in financial returns data," Journal of Empirical Finance, Elsevier, vol. 7(3-4), pages 389-416, November.
    19. Lee, Simon C.K. & Lin, X. Sheldon, 2012. "Modeling Dependent Risks with Multivariate Erlang Mixtures," ASTIN Bulletin, Cambridge University Press, vol. 42(1), pages 153-180, May.
    20. Kevin Dowd & David Blake, 2006. "After VaR: The Theory, Estimation, and Insurance Applications of Quantile‐Based Risk Measures," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 73(2), pages 193-229, June.
    21. Maasoumi, Esfandiar & Racine, Jeff, 2002. "Entropy and predictability of stock market returns," Journal of Econometrics, Elsevier, vol. 107(1-2), pages 291-312, March.
    22. Mills,Terence C. & Markellos,Raphael N., 2008. "The Econometric Modelling of Financial Time Series," Cambridge Books, Cambridge University Press, number 9780521883818.
    23. 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.
    24. Luca Bagnato & Valerio Potì & Maria Zoia, 2015. "The role of orthogonal polynomials in adjusting hyperpolic secant and logistic distributions to analyse financial asset returns," Statistical Papers, Springer, vol. 56(4), pages 1205-1234, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Wang, Tianyi & Liang, Fang & Huang, Zhuo & Yan, Hong, 2022. "Do realized higher moments have information content? - VaR forecasting based on the realized GARCH-RSRK model," Economic Modelling, Elsevier, vol. 109(C).
    2. Kwangmin Jung & Donggyu Kim & Seunghyeon Yu, 2022. "Next generation models for portfolio risk management: An approach using financial big data," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 89(3), pages 765-787, September.
    3. Gong, Xiao-Li & Xiong, Xiong, 2021. "Multi-objective portfolio optimization under tempered stable Lévy distribution with Copula dependence," Finance Research Letters, Elsevier, vol. 38(C).
    4. Piero Quatto & Gianmarco Vacca & Maria Grazia Zoia, 2021. "Modeling Portfolios with Leptokurtic and Dependent Risk Factors," Papers 2106.04218, arXiv.org.
    5. Md Akhtaruzzaman & Ramzi Benkraiem & Sabri Boubaker & Constantin Zopounidis, 2022. "COVID‐19 crisis and risk spillovers to developing economies: Evidence from Africa," Journal of International Development, John Wiley & Sons, Ltd., vol. 34(4), pages 898-918, May.
    6. M. D. Braga & C. R. Nava & M. G. Zoia, 2023. "Kurtosis-based risk parity: methodology and portfolio effects," Quantitative Finance, Taylor & Francis Journals, vol. 23(3), pages 453-469, March.
    7. Inés Jiménez & Andrés Mora-Valencia & Javier Perote, 2022. "Dynamic selection of Gram–Charlier expansions with risk targets: an application to cryptocurrencies," Risk Management, Palgrave Macmillan, vol. 24(1), pages 81-99, March.
    8. León, Ángel & Ñíguez, Trino-Manuel, 2020. "Modeling asset returns under time-varying semi-nonparametric distributions," Journal of Banking & Finance, Elsevier, vol. 118(C).
    9. Inés Jiménez & Andrés Mora-Valencia & Trino-Manuel Ñíguez & Javier Perote, 2020. "Portfolio Risk Assessment under Dynamic (Equi)Correlation and Semi-Nonparametric Estimation: An Application to Cryptocurrencies," Mathematics, MDPI, vol. 8(12), pages 1-24, November.
    10. Enrique Molina‐Muñoz & Andrés Mora‐Valencia & Javier Perote, 2021. "Backtesting expected shortfall for world stock index ETFs with extreme value theory and Gram–Charlier mixtures," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(3), pages 4163-4189, July.
    11. Lassance, Nathan & Vrins, Frédéric, 2021. "Portfolio selection with parsimonious higher comoments estimation," Journal of Banking & Finance, Elsevier, vol. 126(C).
    12. León, Ángel & Ñíguez, Trino-Manuel, 2021. "The transformed Gram Charlier distribution: Parametric properties and financial risk applications," Journal of Empirical Finance, Elsevier, vol. 63(C), pages 323-349.
    13. Quatto, Piero & Vacca, Gianmarco & Zoia, Maria Grazia, 2021. "A new copula for modeling portfolios with skewed, leptokurtic and high-order dependent risk factors," The North American Journal of Economics and Finance, Elsevier, vol. 58(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Luca Bagnato & Valerio Potì & Maria Zoia, 2015. "The role of orthogonal polynomials in adjusting hyperpolic secant and logistic distributions to analyse financial asset returns," Statistical Papers, Springer, vol. 56(4), pages 1205-1234, November.
    2. James Ming Chen, 2018. "On Exactitude in Financial Regulation: Value-at-Risk, Expected Shortfall, and Expectiles," Risks, MDPI, vol. 6(2), pages 1-28, June.
    3. Marie Kratz & Yen H Lok & Alexander J Mcneil, 2016. "Multinomial var backtests: A simple implicit approach to backtesting expected shortfall," Working Papers hal-01424279, HAL.
    4. Deepak K. Jadhav & Ramanathan Thekke Variyam, 2023. "Modified Expected Shortfall: a Coherent Risk Measure for Elliptical Family of Distributions," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 234-256, May.
    5. Brenda Castillo-Brais & Ángel León & Juan Mora, 2022. "Estimating Value-at-Risk and Expected Shortfall: Do Polynomial Expansions Outperform Parametric Densities?," Mathematics, MDPI, vol. 10(22), pages 1-17, November.
    6. León, Ángel & Ñíguez, Trino-Manuel, 2021. "The transformed Gram Charlier distribution: Parametric properties and financial risk applications," Journal of Empirical Finance, Elsevier, vol. 63(C), pages 323-349.
    7. Quatto, Piero & Vacca, Gianmarco & Zoia, Maria Grazia, 2021. "A new copula for modeling portfolios with skewed, leptokurtic and high-order dependent risk factors," The North American Journal of Economics and Finance, Elsevier, vol. 58(C).
    8. Kratz, Marie & Lok, Y-H & McNeil, Alexander J., 2016. "Multinomial VaR Backtests: A simple implicit approach to backtesting expected shortfall," ESSEC Working Papers WP1617, ESSEC Research Center, ESSEC Business School.
    9. León, Ángel & Ñíguez, Trino-Manuel, 2020. "Modeling asset returns under time-varying semi-nonparametric distributions," Journal of Banking & Finance, Elsevier, vol. 118(C).
    10. Kratz, Marie & Lok, Yen H. & McNeil, Alexander J., 2018. "Multinomial VaR backtests: A simple implicit approach to backtesting expected shortfall," Journal of Banking & Finance, Elsevier, vol. 88(C), pages 393-407.
    11. Marco Rocco, 2011. "Extreme value theory for finance: a survey," Questioni di Economia e Finanza (Occasional Papers) 99, Bank of Italy, Economic Research and International Relations Area.
    12. Lazar, Emese & Zhang, Ning, 2019. "Model risk of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 105(C), pages 74-93.
    13. Steven Kou & Xianhua Peng, 2016. "On the Measurement of Economic Tail Risk," Operations Research, INFORMS, vol. 64(5), pages 1056-1072, October.
    14. William B. Haskell & Wenjie Huang & Huifu Xu, 2018. "Preference Elicitation and Robust Optimization with Multi-Attribute Quasi-Concave Choice Functions," Papers 1805.06632, arXiv.org.
    15. Massimiliano Amarante, 2016. "A representation of risk measures," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 39(1), pages 95-103, April.
    16. Bakshi, Gurdip & Panayotov, George, 2010. "First-passage probability, jump models, and intra-horizon risk," Journal of Financial Economics, Elsevier, vol. 95(1), pages 20-40, January.
    17. Brandtner, Mario, 2018. "Expected Shortfall, spectral risk measures, and the aggravating effect of background risk, or: risk vulnerability and the problem of subadditivity," Journal of Banking & Finance, Elsevier, vol. 89(C), pages 138-149.
    18. Marcelo Brutti Righi & Paulo Sergio Ceretta, 2015. "Shortfall Deviation Risk: An alternative to risk measurement," Papers 1501.02007, arXiv.org, revised May 2016.
    19. Gabriele Canna & Francesca Centrone & Emanuela Rosazza Gianin, 2021. "Capital Allocation Rules and the No-Undercut Property," Mathematics, MDPI, vol. 9(2), pages 1-13, January.
    20. Enrique Molina‐Muñoz & Andrés Mora‐Valencia & Javier Perote, 2021. "Backtesting expected shortfall for world stock index ETFs with extreme value theory and Gram–Charlier mixtures," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(3), pages 4163-4189, July.

    More about this item

    Keywords

    Gram-Charlier expansions; Value at risk; Expected shortfall; Heavy tailed distributions;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • G1 - Financial Economics - - General Financial Markets

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jbfina:v:93:y:2018:i:c:p:92-104. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jbf .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.