One series, three results
Use four invented prices: 100, 102, 104 and 110. The simple average of the last three is about 105.33. A linearly weighted average assigns weights 1, 2 and 3 to 102, 104 and 110; the weighted sum is 640, the weight total is six and the result is about 106.67. The larger weight on the latest jump raises the average but adds no knowledge of a fifth price.
For an exponential average with coefficient 0.5 and initial value 100, each new estimate is half the new price plus half the previous estimate. The sequence is 100, 101, 102.5 and 106.25. Initialisation and coefficient are explicit; another start on a short series gives another result.
What the window means
A short window reacts faster and smooths less; a long window retains more history. In an SMA, an old observation leaves the window at once; in an EMA, its influence decays. Equal period counts therefore do not imply identical memory. Choosing the window from the most convenient historical picture is outcome tuning, not neutral description.
Hold the new price constant while an unusual old value leaves the window. The line may move materially without new economic information in today’s quote. This tests the filter’s construction and prevents assigning independent market meaning to every bend.
Why the line lags
After a sharp level change, earlier values remain in the average. Lag is not a software error; it follows from smoothing past data. More weight on new observations reduces memory and increases sensitivity to noise. One filter cannot be both completely resistant to random jumps and instantly responsive to every move without additional assumptions.
Two averages of one series are not independent witnesses. A crossing is a change in the relationship between two functions of past prices. A future-performance test needs a stated observation time, next available price, costs and control group. Several agreeing indicators do not multiply independent information.
Reproduce before interpreting
Ask the chart provider for the window, price type, adjustment rule, smoothing coefficient and initial value. Recalculate initial points on an artificial series, then test a constant price, one jump and steady growth. This checks implementation, not trading validity.
Do not transfer familiar settings between assets and frequencies without redefining the task: twenty daily observations and twenty minute observations represent different horizons. For recursive averages, request the history before the starting date. A correctly normalised average with weights summing to one should shift by the same constant when that constant is added to every input.
Sources and limits
[1] Fidelity Exponential Moving Average and Simple Moving Average materials: definitions and weighting, without signal validation.
[2] Foundations of Technical Analysis, NBER working paper 7613: operationalisation of chart patterns, not proof of profit.
The initial CME materials were not obtained in full. The listed complete primary materials support only the named definitions and arithmetic; trading interpretations and predictive promises were not validated.