How Mean and Volatility Scale Over Short Time Intervals

Summary

The document explains why the mean return may be treated as negligible relative to dispersion over a very short time interval. Its answer states that expected return accumulates in proportion to elapsed time, while standard deviation grows with the square root of elapsed time. For a sufficiently small interval, the mean therefore shrinks faster than the scale of random variation, making its contribution small in many short-horizon variance calculations.

The response gives a brief scaling argument rather than a derivation from a particular return process or an empirical test. It does not establish whether a one-day interval is small enough for a given application, and that depends on the asset, model, and units used. A second answer simply notes that a daily average return is tiny; this is an intuition, not a general justification. The discussion is best read as a time-scale approximation, not a rule that the mean can always be omitted.

Key ideas

• Expected return is described as scaling linearly with elapsed time.
• Standard deviation is described as scaling with the square root of elapsed time.
• At sufficiently short intervals, the mean can be small relative to random variation.
• Whether a daily step justifies neglecting the mean depends on the setting and model assumptions.
• The response offers a scaling intuition, not a derivation or empirical validation.

Tags

# Why can we neglect the mean in the variance when the time step is very small?


# Why can we neglect the mean in the variance when the time step is very small?












Can anyone tell me why we can neglect the mean in the variance when the time step is very small? See the following picture:

Usually, we choose a time step of one day. Is it small enough?

## Answer by fni (score 2)

https://quant.stackexchange.com/a/32585

The average return scales linearly with the time period, i.e. $R_N = N R_1$, while the standard deviation scales with the square root, i.e. $\sigma_N = \sqrt{N}\sigma_1$. As the period becomes really small, $\sqrt{N}$ becomes much bigger than $N$.

## Answer by Randor (score 0)

https://quant.stackexchange.com/a/32572

it is because "R-bar" over 1 day is a very tiny number , virtually zero

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.