EWMA Volatility Weights, Truncation, and Decay Limitations
Summary
The document addresses whether finite-sample exponentially weighted moving average (EWMA) weights must sum to one when estimating volatility. With the usual exponential weighting rule and a decay factor of 0.94 over 22 observations, the questioner finds that the included weights sum to about 0.74, raising concern that the estimate will be too low. The accepted answer says finite EWMA weights need not sum to one and cites a RiskMetrics example with a similar window and a sum of 0.71.
The response advises against renormalizing weights merely to force their sum to one, but does not give a detailed comparison of the resulting variance estimates or establish that normalization is harmless in every implementation. It also cautions that exponential weights decay too quickly to capture long memory and autocorrelation decay in returns, describing EWMA and a later hyperbolic variant as limited models. Another answer notes that historical data can be used to estimate the decay parameter, without explaining that procedure.
Key ideas
- Finite-window EWMA weights can sum to less than one because older observations are omitted.
- The document cites a RiskMetrics example whose weights over a similar window also sum to less than one.
- The accepted answer questions whether renormalizing the weights improves the estimate but gives no full analysis of that choice.
- The response argues that exponential decay may taper too quickly to represent long memory and autocorrelation decay.
Tags
Full text
# Do the weights of the exponentially weighted moving average (EWMA) have to sum to 1?
# Do the weights of the exponentially weighted moving average (EWMA) have to sum to 1?
I am currently trying to calculate a volatility by using the EWMA model because it is said to yield better results than just using an equal weighted calculation approach. However I am a bit confused when it comes to using or choosing the lambda term.
According to various sources, in finance (especially risk management) a lambda of 0.94 is very common. Now lets imagine I work with a lookback period of n = 22. Now calculating the weights according to $ (1 - \lambda) (\lambda)^{w_n} $, where $\lambda$ = 0.94 and n is between 0 and 21, I get:
```
n EWMA weight Equal weight
0 0.06 0.045
1 0.056 0.045
..
21 0.016 0.045
sum 0.74 1
```
Now taking the sum of the EWMA weights, I get a value of 0.74. Now if I would be using this lambda (and its weights), wouldn't I get a significantly undervalued volatility considering that the sum of my weights is "only" 0.74? Can a lambda of 0.94 only be used with much larger n's, where the weights sum to almost 1?
Thanks in advance
## Answer by develarist (score 2, accepted)
https://quant.stackexchange.com/a/57422
The weights generated by EWMA do not have to sum to 1. Page 81 of the RiskMetrics 1996 document where EWMA was introduced shows an example with 22 observations, similar to yours, that uses the same value for lambda, and their weight series sums to 0.71.
Instead of worrying if this could underestimate the resulting volatility, it would be better to ask, what could really go wrong by changing the last or first weight in such a way that all the weights do sum to 1. would anyone really care?
EWMA is an outdated model. same goes for hyperbolic EWMA which succeeded it in RiskMetrics 2006, which recognized that the exponential weighting scheme does not properly reflect long memory and autocorrelation decay in financial returns because EWMA 96's weighting scheme tapers too fast.
## Answer by user54453 (score -4)
https://quant.stackexchange.com/a/63061
In fact, you can use the historical data to estimate the lambda, see the paper at "https://www.sciencedirect.com/science/article/abs/pii/S0304407617301926".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.