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Decomposing Portfolio Value-at-Risk: A General Analysis

Author

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  • Winfried G. Hallerbach

    (Erasmus University Rotterdam)

Abstract

An intensive and still growing body of research focuses on estimating a portfolio’s Value-at-Risk.Depending on both the degree of non-linearity of the instruments comprised in the portfolio and thewillingness to make restrictive assumptions on the underlying statistical distributions, a variety of analyticalmethods and simulation-based methods are available. Aside from the total portfolio’s VaR, there is agrowing need for information about (i) the marginal contribution of the individual portfolio components tothe diversified portfolio VaR, (ii) the proportion of the diversified portfolio VaR that can be attributed toeach of the individual components consituting the portfolio, and (iii) the incremental effect on VaR ofadding a new instrument to the existing portfolio. Expressions for these marginal, component and incremental VaR metricshave been derived by Garman [1996a, 1997a] under the assumption that returns are drawnfrom a multivariate normal distribution. For many portfolios, however, the assumption of normally distributedreturns is too stringent. Whenever these deviations from normality are expected to cause seriousdistortions in VaR calculations, one has to resort to either alternative distribution specifications orhistorical and Monte Carlo simulation methods. Although these approaches to overall VaR estimation have receivedsubstantial interest in the literature, there exist to the best of our knowledge no procedures for estimatingmarginal VaR, component VaR and incremental VaR in either a non-normal analytical setting or a MonteCarlo / historical simulation context.This paper tries to fill this gap by investigating these VaR concepts in a general distribution-freesetting. We derive a general expression for the marginal contribution of an instrument to the diversifiedportfolio VaR ? whether this instrument is already included in the portfolio or not. We show how in a mostgeneral way, the total portfolio VaR can be decomposed in partial VaRs that can be attributed to theindividual instruments comprised in the portfolio. These component VaRs have the appealing property thatthey aggregate linearly into the diversified portfolio VaR. We not only show how the standard results undernormality can be generalized to non-normal analytical VaR approaches but also present an explicitprocedure for estimating marginal VaRs in a simulation framework. Given the marginal VaR estimate,component VaR and incremental VaR readily follow. The proposed estimation approach pairs intuitiveappeal with computational efficiency. We evaluate various alternative estimation methods in an applicationexample and conclude that the proposed approach displays an astounding accuracy and a promisingoutperformance.

Suggested Citation

  • Winfried G. Hallerbach, 1999. "Decomposing Portfolio Value-at-Risk: A General Analysis," Tinbergen Institute Discussion Papers 99-034/2, Tinbergen Institute.
  • Handle: RePEc:tin:wpaper:19990034
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    Cited by:

    1. Tasche, Dirk, 2002. "Expected shortfall and beyond," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1519-1533, July.
    2. Marouane Airouss & Mohamed Tahiri & Amale Lahlou & Abdelhak Hassouni, 2018. "Advanced Expected Tail Loss Measurement and Quantification for the Moroccan All Shares Index Portfolio," Mathematics, MDPI, vol. 6(3), pages 1-19, March.
    3. Fermanian, Jean-David & Scaillet, Olivier, 2005. "Sensitivity analysis of VaR and Expected Shortfall for portfolios under netting agreements," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 927-958, April.
    4. Cheng-Der Fuh & Inchi Hu & Ya-Hui Hsu & Ren-Her Wang, 2011. "Efficient Simulation of Value at Risk with Heavy-Tailed Risk Factors," Operations Research, INFORMS, vol. 59(6), pages 1395-1406, December.
    5. Marius ACATRINEI, 2015. "Individual contributions to portfolio risk: risk decomposition for the BET-FI index," Computational Methods in Social Sciences (CMSS), "Nicolae Titulescu" University of Bucharest, Faculty of Economic Sciences, vol. 3(1), pages 75-80, June.
    6. Janecskó, Balázs, 2004. "A Bázel II. belső minősítésen alapuló módszerének közgazdasági-matematikai háttere és a granularitási korrekció elmélete [The economic and mathematical background to the Basel II internal ratings-b," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(3), pages 218-234.
    7. Monica Billio & Lorenzo Frattarolo & Loriana Pelizzon, 2016. "Hedge Fund Tail Risk: An investigation in stressed markets, extended version with appendix," Working Papers 2016:01, Department of Economics, University of Venice "Ca' Foscari".
    8. Marián Rimarčík, 2005. "Porovnanie prístupov na výpočet hodnoty v riziku menových portfólií [Comparison of approaches for value-at-risk estimation of foreign exchange portfolios]," Politická ekonomie, Prague University of Economics and Business, vol. 2005(3), pages 323-336.
    9. Dirk Tasche, 2002. "Expected Shortfall and Beyond," Papers cond-mat/0203558, arXiv.org, revised Oct 2002.
    10. Alexandre Kurth & Dirk Tasche, 2002. "Credit Risk Contributions to Value-at-Risk and Expected Shortfall," Papers cond-mat/0207750, arXiv.org, revised Nov 2002.
    11. Takaaki Koike & Mihoko Minami, 2017. "Estimation of Risk Contributions with MCMC," Papers 1702.03098, arXiv.org, revised Jan 2019.
    12. Serge Darolles & Christian Gouriéroux & Emmanuelle Jay, 2012. "Robust Portfolio Allocation with Systematic Risk Contribution Restrictions," Working Papers 2012-35, Center for Research in Economics and Statistics.
    13. Janecskó, Balázs, 2002. "Portfóliószemléletű hitelkockázat szimulációs meghatározása [Simulated determination of credit risk in portfolio terms]," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(7), pages 664-676.
    14. Simon Keel & David Ardia, 2011. "Generalized marginal risk," Journal of Asset Management, Palgrave Macmillan, vol. 12(2), pages 123-131, June.
    15. Yu Takata, 2018. "Application of Granularity Adjustment Approximation Method to Incremental Value-at-Risk in Concentrated Portfolios," Economics Bulletin, AccessEcon, vol. 38(4), pages 2320-2330.
    16. repec:dau:papers:123456789/4688 is not listed on IDEAS
    17. Takashi Kato, 2011. "Theoretical Sensitivity Analysis for Quantitative Operational Risk Management," Papers 1104.0359, arXiv.org, revised May 2017.
    18. Kaplanski, Guy & Kroll, Yoram, 2002. "VaR Risk Measures versus Traditional Risk Measures: an Analysis and Survey," MPRA Paper 80070, University Library of Munich, Germany.
    19. Akif Ince & Ilaria Peri & Silvana Pesenti, 2021. "Risk contributions of lambda quantiles," Papers 2106.14824, arXiv.org, revised Nov 2022.
    20. Takashi Kato, 2017. "Theoretical Sensitivity Analysis For Quantitative Operational Risk Management," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(05), pages 1-23, August.
    21. 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.
    22. Shrey Jain & Siddhartha P. Chakrabarty, 2020. "Does Marginal VaR Lead to Improved Performance of Managed Portfolios: A Study of S&P BSE 100 and S&P BSE 200," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 27(2), pages 291-323, June.

    More about this item

    Keywords

    Value-at-Risk; marginal VaR; component VaR; incremental VaR; non-normality; non-linearity; estimation; simulation;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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