Lesson 2 of 25 · 7 min
How a realised volatility is actually computed
Five arithmetic steps on a short list of returns, and the last step hides a divisor most people never see.
Realised volatility sounds like a property of an asset. It is the output of five arithmetic steps applied to a particular list of returns, and changing the list changes the output. This is the one figure in the whole subject you can compute by hand, and doing it once removes most of the mystery from the rest of the track.
The five steps, in order:
- Collect the returns over your chosen window.
- Compute their mean.
- Subtract the mean from each return to get the deviations.
- Square each deviation and average the squares, dividing by one fewer than the number of observations.
- Take the square root. That is the volatility, expressed for whatever period the returns covered.
Take ten illustrative daily returns, in percent: 0.83, −1.47, 0.29, 2.16, −0.62, −3.08, 1.74, 0.05, −0.91, 1.38. They sum to 0.37, so the mean is 0.037 percent. The deviations are 0.793, −1.507, 0.253, 2.123, −0.657, −3.117, 1.703, 0.013, −0.947 and 1.343.
Square those and you get 0.6288, 2.2710, 0.0640, 4.5071, 0.4316, 9.7157, 2.9002, 0.0002, 0.8968 and 1.8036, which sum to 23.21921. Divide by nine rather than by ten and the average squared deviation is 2.57991. The square root of that is 1.6062 percent, and that is the daily volatility of this series.
daily volatility = square root of ( sum of squared deviations ÷ (n − 1) )
- daily volatility
- The answer, expressed for the same interval the returns covered. Here, one session.
- sum of squared deviations
- Each return minus the mean return, squared, then all of them added together.
- n
- The number of returns in the window. Ten in this example.
The divisor deserves a moment. Dividing by nine rather than ten corrects for the fact that the mean was estimated from the same data, and it matters most when the window is short. Dividing by ten here gives 1.5238 percent instead of 1.6062 percent — about five percent lower, from a choice no published figure discloses.
Key terms
- Realised volatility
- The volatility computed from returns that have already happened. Track 3 lesson 6 named it; this is the arithmetic that produces it.
- Deviation
- A single return minus the mean of the window it sits in. The raw material of every step after the second.
- Divisor correction
- Dividing by one fewer than the number of observations, because the mean was estimated from the same numbers. Its effect shrinks as the window lengthens.
Worked example
Ten returns, computed all the way through
Illustrative daily returns in percent: 0.83, −1.47, 0.29, 2.16, −0.62, −3.08, 1.74, 0.05, −0.91, 1.38.
- Sum of the returns
- 0.37
- Mean return
- 0.037%
- Sum of squared deviations
- 23.21921
- Average, dividing by nine
- 2.57991
- Daily volatility
- 1.6062%
- Same figure dividing by ten instead
- 1.5238%
0.37 ÷ 10.
The square root of 2.57991.
A five percent difference from the divisor alone.
Every number in this track descends from an arithmetic chain of exactly this shape. Nothing more sophisticated is happening underneath.
Check yourself
Four daily returns, in percent: 1.2, −0.8, 0.4, −1.6. What is the daily volatility, and what is it annualised at 252 sessions?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.