Value at Risk: the square-root-of-time rule

The rule in one line

If daily returns are independent and identically distributed with a mean of about zero, the standard deviation over T days is the one-day standard deviation times √T. Value at Risk (VaR) is a quantile of the return distribution, so under the same assumptions it scales the same way.

This is why a risk report can quote a one-day VaR and a ten-day VaR from a single calculation, and why regulators historically asked for a 10-day horizon without asking banks to estimate ten-day moves directly.

Worked example

Take a portfolio worth $10,000,000 with a daily return standard deviation of 1.2%. Use a 99% one-tailed confidence level, where the normal z-score is 2.3263.

Now scale it:

Notice that ten days of risk is about 3.2 times one day, not ten times. Risk grows with the square root of time because good and bad days partly offset each other when they are independent.

Changing the confidence level

The z-score is the only thing that changes. At 95% it is 1.645, at 99% it is 2.326, and at 99.9% it is 3.090. The same $10m portfolio has a one-day 95% VaR of about $197,382.

What the rule assumes

• Independent returns. Today's move says nothing about tomorrow's. If returns trend or mean-revert, the true multi-day risk is higher or lower than √T suggests.
• Constant volatility. Real volatility clusters: calm periods and stressed periods. Scaling a calm-period number understates risk going into a crisis.
• Zero mean. Over a few days the drift is negligible, but over longer horizons it is not.
• Normal (or at least stable) tails. Scaling the quantile of a fat-tailed distribution by √T is only an approximation.
• A fixed portfolio. The rule holds the positions constant. Options, whose exposure changes as the market moves, do not scale cleanly, and neither does a portfolio you would really rebalance or be unable to exit.

Where it shows up in exams and in practice

Market-risk capital under the Basel framework has long been built on a 10-day, 99% VaR, which is why the √10 scaling is so familiar. The Fundamental Review of the Trading Book moves the headline measure to expected shortfall at 97.5% and replaces a single 10-day horizon with liquidity horizons that differ by risk factor, partly because of the weaknesses above.

In an exam, expect the rule as a calculation (scale this one-day VaR) and as a conceptual question (which assumption fails here, and in which direction does the error go). A common trap is applying √T to a mean as well as to the volatility; the mean scales with T, not √T.

More guides

• Duration and convexity: how good is the approximation?A ten-year bond repriced at four yield shocks, exactly and with the duration and convexity estimates.
• Initial margin and the uncleared margin rules, explainedWho has to post it, how it is sized, and why the phase-in dates matter.
• Valuing an interest rate swap: a three-year worked exampleDiscount factors, the par swap rate and the value of a 4% swap, step by step.
• Expected shortfall versus VaR: what changes in the tailWhy regulators moved from 99% VaR to 97.5% expected shortfall, with normal and fat-tailed numbers.