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On time-scaling of risk and the square–root–of–time rule

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  • Jean-Pierre Zigrand

    ()

  • Jon Danielsson

    ()

Abstract

Many financial applications, such as risk analysis and derivatives pricing, depend on time scaling of risk.� A common method for this purpose, though only correct when returns are iid normal, is the square root of time rule where an estimated quantile of a return distribution is scaled to a lower frequency by the square-root of the time horizon. The aim of this paper is to examine time scaling of risk when returns follow a jump diffusion process. It is argued that a jump diffusion is well-suited for the modeling of systemic risk, which is the raison d'etre of the Basel capital adequacy proposals. We demonstrate that the square root of time rule leads to a systematic underestimation of risk, whereby the degree of underestimation worsens with the time horizon,the jump intensity and the confidence level.� As a result,even if the square root of time rule has widespread applications in the Basel Accords, it fails to address the objective of the Accords.

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File URL: http://www2.lse.ac.uk/fmg/workingPapers/discussionPapers/fmg_pdfs/dp439.pdf
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Bibliographic Info

Paper provided by Financial Markets Group in its series FMG Discussion Papers with number dp439.

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Date of creation: Mar 2003
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Handle: RePEc:fmg:fmgdps:dp439

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Web page: http://www2.lse.ac.uk/fmg/

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  1. Peter Carr & Liuren Wu, 2003. "What Type of Process Underlies Options? A Simple Robust Test," Journal of Finance, American Finance Association, vol. 58(6), pages 2581-2610, December.
  2. Muller, Ulrich A. & Dacorogna, Michel M. & Olsen, Richard B. & Pictet, Olivier V. & Schwarz, Matthias & Morgenegg, Claude, 1990. "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis," Journal of Banking & Finance, Elsevier, vol. 14(6), pages 1189-1208, December.
  3. Guidolin, Massimo & Timmermann, Allan, 2006. "Term structure of risk under alternative econometric specifications," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 285-308.
  4. Bates, David S., 2000. "Post-'87 crash fears in the S&P 500 futures option market," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 181-238.
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Cited by:
  1. Saadi, Samir & Rahman, Abdul, 2008. "Evidence of non-stationary bias in scaling by square root of time: Implications for Value-at-Risk," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 18(3), pages 272-289, July.
  2. Gaglianone, Wagner Piazza & Linton, Oliver & Lima, Luiz Renato Regis de Oliveira, 2008. "Evaluating Value-at-Risk models via Quantile regressions," Economics Working Papers (Ensaios Economicos da EPGE) 679, Graduate School of Economics, Getulio Vargas Foundation (Brazil).
  3. Santos, André A. P. & Nogales, F. Javier & Ruiz, Esther & Dijk, Dick van, 2012. "Optimal portfolios with minimum capital requirements," Open Access publications from Universidad Carlos III de Madrid info:hdl:10016/15745, Universidad Carlos III de Madrid.
  4. Amadeo Alentorn & Sheri Markose, 2006. "Removing Maturity Effects of Implied Risk Neutral Densities and Related Statistics," Economics Discussion Papers 609, University of Essex, Department of Economics.
  5. Wang, Jying-Nan & Yeh, Jin-Huei & Cheng, Nick Ying-Pin, 2011. "How accurate is the square-root-of-time rule in scaling tail risk: A global study," Journal of Banking & Finance, Elsevier, vol. 35(5), pages 1158-1169, May.
  6. Alfred Mbairadjim Moussa & Jules Sadefo Kamdem & Arnold F. Shapiro & Michel Terraza, 2012. "Capital asset pricing model with fuzzy returns and hypothesis testing," Working Papers 12-33, LAMETA, Universtiy of Montpellier, revised Sep 2012.
  7. Amy S. K. Wong, 2006. "Basel II and the Risk Management of Basket Options with Time-Varying Correlations," International Journal of Central Banking, International Journal of Central Banking, vol. 2(4), December.
  8. Pérignon, Christophe & Smith, Daniel R., 2010. "Diversification and Value-at-Risk," Journal of Banking & Finance, Elsevier, vol. 34(1), pages 55-66, January.
  9. 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.
  10. Guy Kaplanski, Haim Levy, 2010. "The Two-Parameter Long-Horizon Value-at-Risk," Frontiers in Finance and Economics, SKEMA Business School, vol. 7(1), pages 1-20, April.
  11. Nikolaus Rab & Richard Warnung, 2010. "Scaling portfolio volatility and calculating risk contributions in the presence of serial cross-correlations," Papers 1009.3638, arXiv.org, revised Nov 2011.
  12. Al Janabi, Mazin A. M., 2009. "Asset Market Liquidity Risk Management: A Generalized Theoretical Modeling Approach for Trading and Fund Management Portfolios," MPRA Paper 19498, University Library of Munich, Germany.

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