Many risk and derivatives-pricing applications require scaling risk from one time horizon to another. The common device is the square-root-of-time rule — scale an estimated return-distribution quantile to a lower frequency by multiplying by the square root of the horizon — which is correct only when returns are IID normal. Daníelsson and Zigrand examine time-scaling of quantiles when returns follow a jump diffusion, argued to be well suited to modeling the systemic risk that motivates the Basel capital-adequacy rules, and show that the square-root-of-time rule leads to a systematic underestimation of risk that worsens with the horizon, the jump intensity, and the confidence level.
"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."
A clean, consequential result about a shortcut everyone knows is only approximate but uses anyway. The paper's contribution is to pin down the direction and drivers of the error: it is not random slippage but a systematic under-statement that grows exactly where it is most dangerous — long horizons, fat-tailed jumps, extreme quantiles — which is the regime capital buffers are meant to cover. That the Basel framework leaned on the rule makes the finding pointed rather than academic. On the wiki it is a natural caution on the multi-step VaR material and a companion to the Alexander-Leigh warning that convenient risk conventions can fail precisely in the tails; the caveat, as always, is model-specific — the magnitude depends on the assumed jump process.