Updating EWMA Volatility Forecasts with Squared Returns
Summary
The discussion explains a one day ahead exponentially weighted moving average volatility update. At the end of a day, the prior forecast for that day's variance is combined with the newly observed squared return, using a decay parameter to weight the old forecast and the latest observation. The resulting value is the next day's variance forecast; it is a recursively constructed series, not a variance that must be separately supplied at every step.
Variance is generally the expected squared deviation from the mean. For daily asset returns, the mean is often treated as negligible, making squared returns a practical approximation to the variance input in this update. That approximation may be less appropriate at weekly, monthly, or annual frequencies, where the mean can matter more. The answers explain the formula conceptually but do not assess forecast accuracy, select a decay parameter, or address other modeling choices, so they should not be read as validation of a particular volatility model.
Key ideas
- EWMA updates combine the previous variance forecast with the latest squared return.
- The updated value becomes the forecast for the following period.
- Variance measures squared deviations from the mean, while squared returns omit mean adjustment.
- For daily returns, treating the mean as negligible can make squared returns a useful approximation.
- The approximation may be weaker at longer sampling intervals.
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Full text
# What is the difference between squared returns and variance?
# What is the difference between squared returns and variance?
I am trying to calculate 1-day ahead volatility forecasts using the exponentially weighted moving average, however I am unsure on how to read the formula provided within Risk-Metrics Technical Documentation for one day ahead forecasts. That formula is
$σ_{1,t+1|t}^2=λ σ_{1,t|t-1}^2+(1-λ) r_{1,t}^2$
(This is equation 5.3 on page 81 of this document)
Can someone please explain the difference between the variance and the squared returns? Both of these components are required for the calculation, however I was using squared returns as my variance for the series. Thanks
## Answer by Alex C (score 2, accepted)
https://quant.stackexchange.com/a/38915
This equation shows how you update your forecast. (In a word: recursively).
At the close of business on day t, when the day's return $r_{1,t}$ becomes available, you take a weighted average of:
- The forecast you had made yesterday for today, $σ_{1,t|t-1}^2$. (Hopefully you wrote that number down yesterday and you still have the piece of paper on which you wrote it down, otherwise you are in trouble).
- and, the square return for today
the number you thus compute is your forecast $σ_{1,t+|t}^2$ for tomorrow.
(So you are computing this time series of $\sigma^2$ values, it is not a variance taken from somewhere else and used in the calculation).
The justification for this method is what @phdstudent said, namely that the expected $r$ is negligible and so is being left out of the calculation.
## Answer by phdstudent (score 9)
https://quant.stackexchange.com/a/38847
Usually the formula for the sample variance of a stock is given by:
\begin{equation} Var(R_{i}) = E (R_t - E(R_t))^2 \end{equation}
If you are using daily data to compute the variance then the second term: $E(R_t) \approx 0$, therefore you can drop it from the computation. Which yields:
\begin{equation} Var(R_{i}) \approx E (R_t)^2 \end{equation}
With weekly, monthly, annual data this is no longer a good approximation.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.