# 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?

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EWMA is an outdated model.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.

• If EWMA is outdated, what is today's standard in modeling & forecasting daily volatility? A different GARCH model? Not everyone uses HF stuff and realized volatility... Option implied volatility is only available for assets with a liquid derivatives market…
– Alex
Aug 17 '20 at 16:37
• as said EWMA 1996's weighting scheme tapers too fast and doesn't reflect financial autocorrelation decay. if you're confident enough to stop using an inaccurate estimator, switch to the EWMA 2006 at least. Besides these, we're here 15 years into the future and still articles avoid sharing results for how their estimators perform for horizons longer than 5 days ahead. Aug 17 '20 at 16:39
• Are you aware of any studies which show that EWMA 2006 predicts volatility significantly better than EWMA 1996 or other standard GARCH models?
– Alex
Aug 17 '20 at 16:42
• RiskMetrics 2006 didn't I say. if an independent study is what you want, I guess one evaluation from JP Morgan/Morgan Stanley (the quants who made/acquired EWMA and RiskMetrics) was good enough, meaning any further studies would be redundant. These were the same guys who invented Value-at-Risk (VaR) Aug 17 '20 at 16:45
• if you cut it off like you describe, then it won't sum to 1 but if you let it go all the way back, back and back, then, by it's definition, the weights will sum to 1.0. Also, as someone above said, it's the corresponding half life that matters rather than the value of $\lambda$. You could define the ewma in the opposite manner: $\lambda (1-\lambda)^{n}$ and then $\lambda^{*} = (1-\lambda)$ will give you the same weightings as the weightings that $\lambda$ gives with your expression. Aug 18 '20 at 0:22

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".

• -1, the question is not about estimation of lambda, but rather interpretation of weights for a given lambda. Mar 29 at 13:57