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Value at Risk: Methodsintermediate

Time Scaling of VaR (Square-Root-of-Time)

To convert a one-period VaR to a longer horizon, the standard shortcut multiplies by the square root of time: VaRJ=VaR1J\mathrm{VaR}_J = \mathrm{VaR}_1\sqrt{J}. The rule comes from volatility growing with J\sqrt{J} when returns are independent and identically distributed (i.i.d.) with zero mean. This holds only under those assumptions. Under GARCH or volatility clustering, serial correlation, or fat tails, variance does not grow linearly with time, and square-root scaling is wrong, typically understating true multi-day risk.

Why it matters

Variance adds up over independent periods, so over JJ days the variance is JJ times the one-day variance and the standard deviation is J\sqrt{J} times as large. VaR, a multiple of standard deviation, inherits the same J\sqrt{J} factor. The whole argument rests on each day being an independent draw from the same distribution. The moment returns trend, cluster, or pile up in the tails, those independent variances no longer simply add and the tidy square-root rule breaks.

Formulas

Square-root-of-time scaling
VaRJ=VaR1 J\mathrm{VaR}_J = \mathrm{VaR}_1\,\sqrt{J}
Valid for i.i.d., zero-mean returns. JJ is the number of one-period units (e.g. trading days). A 10-day VaR is 10≈3.16\sqrt{10} \approx 3.16 times the one-day VaR.
Why: variance is additive under i.i.d.
σJ=σ1 J    ⇐    σJ2=J σ12\sigma_J = \sigma_1\,\sqrt{J} \;\;\Leftarrow\;\; \sigma_J^2 = J\,\sigma_1^2
Independence makes period variances add, so σJ2=Jσ12\sigma_J^2 = J\sigma_1^2. Serial correlation or time-varying volatility breaks the additivity and the J\sqrt{J} rule with it.

Worked examples

Scenario

A one-day 99% VaR is US$5M. Scale it to 10-day and 250-day (one-year) horizons using the square-root rule, and tabulate the results.

Solution

For 10 days, multiply by 10≈3.162\sqrt{10} \approx 3.162: US$5M times 3.162 is about US$15.8M. For 250 days, multiply by 250≈15.81\sqrt{250} \approx 15.81: US$5M times 15.81 is about US$79.1M. The Basel-style 10-day market-risk horizon is a common use of this rule. Each result assumes i.i.d. returns; if volatility clusters, the true 10-day loss is usually larger than US$15.8M.

Common mistakes

  • ✗Square-root scaling always works. It is valid only for i.i.d., zero-mean returns; under GARCH volatility clustering, serial correlation, or fat tails it is biased, usually downward.
  • ✗VaR scales linearly with the horizon. VaR scales with J\sqrt{J}, not JJ; a 4-day VaR is twice, not four times, the one-day VaR under the i.i.d. assumption.
  • ✗Square-root scaling also rescales the mean by J\sqrt{J}. The drift scales with JJ, not J\sqrt{J}; only the volatility term carries the square-root factor.
  • ✗A longer horizon always means proportionally more risk. Risk grows sublinearly under the square-root rule, and the rule itself may fail entirely once returns are autocorrelated.

Revision bullets

  • •Scale across horizons with VaRJ=VaR1J\mathrm{VaR}_J = \mathrm{VaR}_1\sqrt{J}
  • •Rests on i.i.d., zero-mean returns (variance additivity)
  • •Breaks under GARCH, serial correlation, or fat tails
  • •When it fails it usually understates multi-day risk
  • •Drift scales with JJ; only volatility scales with J\sqrt{J}

Quick check

A one-day 99% VaR is US$5M. Under square-root scaling, the 10-day 99% VaR is closest to

Square-root-of-time scaling of VaR can be seriously wrong when returns

Connected topics

VaR DefinitionParametric VaRFat TailsGARCH VaR

Sources

  1. Jorion (2007), Ch. 5
    Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007. ISBN 978-0-07-146495-6.
    States the square-root-of-time rule and the i.i.d. conditions under which it holds.
  2. Hull (2018), Ch. 12
    Hull, J. C. Risk Management and Financial Institutions. 5th ed. Wiley, 2018. ISBN 978-1-119-44811-2.
    Derives the time-scaling of VaR and the 10-day regulatory horizon convention.
How to cite this page
Dr. Phil's Quant Lab. (2026). Time Scaling of VaR (Square-Root-of-Time). Derivatives Atlas. https://phucnguyenvan.com/concept/frm-var-time-scaling
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