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.

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.