Understanding the use of the square root of time to scale volatility

Time scaling of volatility

Discover how to scale standard deviation to different time horizons.

We know that risk increases with time: The longer we hold a position, the greater the potential loss. Following is a simple approximation to help you scale volatility estimates to a longer (or shorter) time horizon. Note, however, that this is just an approximation.

Volatility (or standard deviation) may be roughly approximated by scaling by the square root of time, assuming independent price moves. Note that we use the number of trading days (5 for 1 week, 21 for 1 month), as opposed to actual days to scale volatility.
Weekly volatility = daily volatility * sqrt(5) = daily volatility * 2.24
1 month VaR = 1 day VaR * sqrt(21) = 1 day VaR * 4.58

Intuitively, we can picture the square root of time scaling rule as follows: Imagine Zeus flipping coins every day--if heads come up, the stock market goes up, if tails come up, the stock market falls. Assuming 1% daily volatility (or standard deviation) in the markets, what would you expect weekly volatility to be?

We could just guess and multiply 1% by 5 to get 5%. However, it is unlikely (1/32 chance) that we'll have 5 bad days in a row, because every day's coin flip is independent--it has no memory of the past. An accurate prediction of a 5-day risk would 1% * square root of 5 = 2.24% (but only if Zeus is not biased!). Interested in why we use square root of time scaling?

Assumption

Square root of time scaling assumes independent price moves and constant volatility:
If there is significant mean reversion, time scaling will overestimate volatility (mean reversion is a statistical tendency to revert to a long-term average).
If there is significant trending, time scaling will underestimate volatility (trending is a statistical tendency to keep moving in one direction).
If volatility changes over time, or there are jumps, time scaling will be inaccurate.

Why use square root of time scaling?

From statistics, we know that we can aggregate the volatility of an independent process using the square root sum of the squares rule:

Total Std Dev = square root (Std Dev_1^2 + Std Dev_2^2 + .... + Std Dev_n^2)
Example

As a USD-based institution, you hold a 10-year JPY government bond. You are exposed to both FX and interest rate risk. The following are daily volatilities (1.65 standard deviation), according to RiskMetrics data on June 3, 1998, for interest rate and FX risk:

Price volatility of bond (interest rate risk) 0.38%
Volatility of JPY/USD (FX risk) 1.03%

Square root sum of the squares rule

Using the square root sum of the squares rule, total estimated risk assuming independence between FX and interest rate moves is as follows:
Estimate of total volatility = square root (0.38%^2+1.03%^2) = 1.1%. This means that we expect to lose 1.1% or more once every 20 days.
Using the RiskManager software to calculate, we get an almost identical VaR estimate:

In this case, assuming independence evidently did not sacrifice much accuracy (actual correlation between JPY/USD and 10-year bonds was 0.0185, or almost 0).

Continuing the coin flipping example, because risk is independent each day, we can use the square root sum of the squares rule to aggregate risk every day.
5-day risk = square root (day1 risk^2 + day2 risk^2 + day3 risk^2 + day4 risk^2 + day5 risk^2).
Assuming risk stays constant every day, we can simplify the expression as square root (5*day1 risk^2) = day 1 risk * square root (5).