Risk research: On time-scaling of risk and the square-root-of-time rule

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

Danielsson, J. and J.-P. Zigrand (2006). On time-scaling of risk and the square-root-of-time rule. 30, 2701-2713.

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 dtre 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,
    title={On time-scaling of risk and the square-root-of-time rule},
    author={J{\'o}n Dan{\'i}elsson  and Jean--Pierre Zigrand },
  	url= {https://ssrn.com/abstract=567123},
  	journal=JBF,
    year=2006,
	volume=30,
	issue=10,
	pages={2701-2713},
	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 dtre 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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