Smoothed Moving Average (SMMA)

I. What is SMMA

The Smoothed Moving Average (SMMA) is a special type of moving average that achieves an extremely high degree of smoothness by blending the previous period’s moving average value with the current price. It is also known as RMA (Running Moving Average) or Wilder’s Smoothing, because J. Welles Wilder extensively used this smoothing method in his classic 1978 book New Concepts in Technical Trading Systems.

Historical Background

J. Welles Wilder is one of the most influential figures in the field of technical analysis, having invented RSI, ATR, ADX, Parabolic SAR, and many other classic indicators. When computing these indicators, Wilder consistently used a special smoothing method — SMMA. In the 1970s, when computing power was limited, SMMA’s concise recursive formula was ideal for manual calculation and early spreadsheets.

Indicator Classification

• Type: Overlay indicator, plotted on the price chart
• Category: Moving Averages
• Default Parameters: Period $n = 20$, data source is closing price (Close)

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II. Mathematical Principles and Calculation

Core Formula

The recursive formula for SMMA is:

$SMMA_t = \frac{SMMA_{t-1} \times (n - 1) + P_t}{n}$

Equivalent form:

$SMMA_t = SMMA_{t-1} + \frac{P_t - SMMA_{t-1}}{n}$

The initial value uses the simple moving average of the first $n$ periods:

$SMMA_1 = SMA_n = \frac{1}{n}\sum_{i=1}^{n} P_i$

Mathematical Relationship with EMA

EMA’s formula is $EMA_t = \alpha \cdot P_t + (1 - \alpha) \cdot EMA_{t-1}$, where $\alpha = \frac{2}{n+1}$.

Rewriting SMMA’s formula in a similar form:

$SMMA_t = \frac{1}{n} \cdot P_t + \frac{n-1}{n} \cdot SMMA_{t-1}$

$= \frac{1}{n} \cdot P_t + \left(1 - \frac{1}{n}\right) \cdot SMMA_{t-1}$

Therefore SMMA’s effective smoothing factor is $\alpha_{SMMA} = \frac{1}{n}$.

Setting $\frac{1}{n} = \frac{2}{m+1}$ (where $m$ is the equivalent EMA period) and solving:

$m = 2n - 1$

Thus SMMA(14) = EMA(27) and SMMA(20) = EMA(39).

Weight Decay

SMMA’s historical weight decay rate is $(1 - 1/n)^k$. The weight of the price $k$ periods ago is:

$w_k = \frac{1}{n} \left(1 - \frac{1}{n}\right)^k$

Since $\alpha = 1/n$ is smaller than EMA’s $\alpha = 2/(n+1)$, SMMA’s weight decay is slower and historical data influence persists longer.

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III. Python Implementation

import numpy as np
import pandas as pd

def smma(close: pd.Series, period: int = 20) -> pd.Series:
    """
    Calculate the Smoothed Moving Average (SMMA / RMA / Wilder's Smoothing)

    Parameters
    ----------
    close : pd.Series
        Closing price series
    period : int
        Calculation period, default is 20

    Returns
    -------
    pd.Series
        SMMA value series
    """
    # Method 1: Using the SMMA(n) = EMA(2n-1) equivalence
    # return close.ewm(span=2*period-1, adjust=False).mean()

    # Method 2: Direct recursive implementation (more faithful to Wilder's original definition)
    result = pd.Series(np.nan, index=close.index)

    # Initial value is SMA of first 'period' data points
    result.iloc[period - 1] = close.iloc[:period].mean()

    # Recursive calculation
    for i in range(period, len(close)):
        result.iloc[i] = (result.iloc[i - 1] * (period - 1) + close.iloc[i]) / period

    return result


def smma_numpy(close: np.ndarray, period: int = 20) -> np.ndarray:
    """
    SMMA implementation using numpy (high-performance version)
    """
    n = len(close)
    result = np.full(n, np.nan)

    if n < period:
        return result

    result[period - 1] = np.mean(close[:period])

    alpha = 1.0 / period
    for i in range(period, n):
        result[i] = result[i - 1] + alpha * (close[i] - result[i - 1])

    return result


# ========== Usage Example ==========
if __name__ == "__main__":
    np.random.seed(42)
    dates = pd.date_range("2024-01-01", periods=120, freq="D")
    price = 100 + np.cumsum(np.random.randn(120) * 0.5)

    df = pd.DataFrame({
        "date": dates,
        "open":  price + np.random.randn(120) * 0.2,
        "high":  price + np.abs(np.random.randn(120) * 0.5),
        "low":   price - np.abs(np.random.randn(120) * 0.5),
        "close": price,
        "volume": np.random.randint(1000, 10000, size=120),
    })
    df.set_index("date", inplace=True)

    # SMMA compared with EMA/SMA
    df["SMA_14"]  = df["close"].rolling(14).mean()
    df["EMA_14"]  = df["close"].ewm(span=14, adjust=False).mean()
    df["SMMA_14"] = smma(df["close"], period=14)
    df["EMA_27"]  = df["close"].ewm(span=27, adjust=False).mean()  # Equivalence check

    print("=== SMMA(14) vs EMA(14) vs EMA(27) Comparison ===")
    print(df[["close", "SMA_14", "EMA_14", "SMMA_14", "EMA_27"]].tail(10))
    print("\nNote: SMMA(14) and EMA(27) values are very close (except for initialization differences)")

    # SMMA application in RSI calculation
    delta = df["close"].diff()
    gain = delta.where(delta > 0, 0.0)
    loss = (-delta).where(delta < 0, 0.0)

    avg_gain = smma(gain, period=14)
    avg_loss = smma(loss, period=14)

    rs = avg_gain / avg_loss
    rsi = 100 - (100 / (1 + rs))

    df["RSI_14"] = rsi
    print("\n=== RSI(14) Calculated Using SMMA ===")
    print(df[["close", "RSI_14"]].tail(10))

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IV. Problems the Indicator Solves

1. Core Smoothing Engine for Wilder’s Indicators

SMMA’s most important application is not as a standalone moving average, but as the internal smoothing component of the following classic indicators:

Indicator	Role of SMMA
RSI	Smoothing average gain and average loss
ATR	Smoothing the True Range
ADX	Smoothing the directional movement index
Parabolic SAR	Internal smoothing step in the calculation

2. Ultra-Smooth Long-Term Trend Line

Because SMMA decays more slowly than EMA, it produces a smoother curve, making it suitable as a long-term trend reference line. SMMA(20) has smoothness comparable to EMA(39) or SMA(39).

3. Reducing Trading Noise

In ranging markets, SMMA’s ultra-smooth characteristics produce fewer crossover signals than EMA, helping to reduce overtrading.

4. Dynamic Support / Resistance

SMMA’s extremely high smoothness makes it an excellent dynamic support/resistance level. Since the curve changes slowly, price interactions with SMMA are more regular.

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V. Advantages, Disadvantages, and Use Cases

Advantages

Advantage	Description
Extremely high smoothness	Smoother than same-period EMA; strong noise filtering capability
Standard component	The standard smoothing method for classic indicators like RSI, ATR, and ADX
Efficient recursion	$O(1)$ recursive update; very computationally efficient
Long track record	Validated through decades of market use

Disadvantages

Disadvantage	Description
High lag	SMMA(n) lag is close to EMA(2n-1); much slower than same-period EMA
Sluggish response	Responds very slowly to short-term price changes
Confusion risk	The distinction from EMA is often overlooked, leading to calculation errors
Rarely used independently	Seldom used as a standalone trading indicator

Use Cases

• Internal indicator component: Must be used when calculating RSI, ATR, and ADX
• Long-term trend assessment: As an ultra-smooth long-term trend reference line
• Low-frequency trading: Suitable for investors who prefer infrequent trading

SMMA vs EMA Comparison

Feature	SMMA(14)	EMA(14)	EMA(27)
Smoothing factor $\alpha$	0.0714	0.1333	0.0714
Smoothness	High	Medium	High
Lag	High	Medium	High
Equivalence	= EMA(27)	-	= SMMA(14)