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

   Daníelsson, J. and J.-P. Zigrand (2006). On time-scaling of risk and the square-root-of-time rule. Journal of Banking and Finance 30, 2701–2713. www.RiskResearch.org.

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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\'tre 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.

@ARTICLE{DanielssonZigrand2006,
 author =  {J{\'o}n Dan{\'\i}elsson  and Jean--Pierre Zigrand},
 title =   {On time-scaling of risk and the square-root-of-time
                 rule},
 journal = "Journal of Banking and Finance",
 volume =  {30},
 pages =   {2701-2713},
 year =    2006,
 note =    {\tt www.RiskResearch.org},
 url =     {RiskResearch.org},
}