On Time-Scaling of Risk and the Square-Root-Of-Time Rule

25 Pages Posted: 23 Jul 2004

See all articles by Jon Danielsson

Jon Danielsson

London School of Economics - Systemic Risk Centre

Jean-Pierre Zigrand

London School of Economics - Department of Finance, Systemic Risk Centre, and Financial Markets Group

Date Written: November 3, 2005

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.

Keywords: Square-root-of-time rule, time-scaling of risk, value-at-risk, systemic risk, risk regulation, jump diffusions

JEL Classification: G18, G20, D81

Suggested Citation

Danielsson, Jon and Zigrand, Jean-Pierre, On Time-Scaling of Risk and the Square-Root-Of-Time Rule (November 3, 2005). EFA 2004 Maastricht Meetings Paper No. 5339. Available at SSRN: https://ssrn.com/abstract=567123 or http://dx.doi.org/10.2139/ssrn.567123

Jon Danielsson

London School of Economics - Systemic Risk Centre ( email )

Houghton Street
London WC2A 2AE
United Kingdom
+44.207.955.6056 (Phone)

HOME PAGE: http://www.riskreasearch.org

Jean-Pierre Zigrand (Contact Author)

London School of Economics - Department of Finance, Systemic Risk Centre, and Financial Markets Group ( email )

Houghton Street
London WC2A 2AE
United Kingdom
+44 20 7955 6201 (Phone)
+44 20 7955 7420 (Fax)

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