📏 SMA — Simple Moving Average
The SMA is the most literal way to define a "trend": the unweighted average of the last \(N\) closing prices. Every EMA, Bollinger Band and Donchian midline in this catalogue builds on the same rectangular-window idea.
💡 Financial Meaning
Because every observation in the window counts equally, the SMA reacts to new data more slowly than an EMA of the same length, but it also has zero phase distortion relative to its window — it is not "biased" toward recent or old prices. Traders use SMA crossovers (e.g. the 50/200-day "golden cross", where a "day" is a trading session) as the textbook long-horizon trend signal.
🔢 Mathematical Formula
where \(C_t\) is the closing price at time \(t\). Equivalently, as a running update:
which shows the SMA is a finite-memory filter: the oldest sample is dropped exactly as the newest one enters.
⚙️ Parameters
| Parameter | Key | Default | Description |
|---|---|---|---|
| Period (\(N\)) | period |
20 | Lookback window in sessions. Higher → smoother, slower. |
🎛️ Signal Processing Equivalent — Rectangular FIR Filter
The SMA is a Finite Impulse Response (FIR) low-pass filter with a rectangular (boxcar) window of length \(N\). Its frequency response is a \(\operatorname{sinc}\) function, with the first null at \(\omega = 2\pi/N\) — frequencies above that are attenuated, but with significant sidelobes (ripple) that let some high-frequency noise leak through, unlike a well-tuned IIR design.
flowchart LR
subgraph Window["Sliding window of N=20 closes"]
C1["C(t-19)"] --> C2["..."] --> C3["C(t)"]
end
Window -->|"unweighted average"| SMA["📏 SMA(t)"]Group delay
A rectangular FIR filter of length \(N\) has a constant group delay of \((N-1)/2\) samples — exactly the "lag" traders complain about. This is the price paid for the SMA's perfectly flat, unbiased weighting.