Triple Exponential Moving Average (TEMA)

I. What is TEMA

The Triple Exponential Moving Average (TEMA) is a further extension of DEMA that achieves even lower lag than DEMA through a mathematical combination of three layers of EMA. It was also proposed by Patrick Mulloy in the same 1994 article published in Technical Analysis of Stocks & Commodities magazine.

Historical Background

While designing DEMA, Mulloy found that although the two-layer EMA combination significantly reduced lag, it did not completely eliminate it. He extended the method to a three-layer EMA combination, producing TEMA. In theory, TEMA nearly eliminates the lag of traditional EMA, at the cost of increased sensitivity to market noise and potential overshoot during sharp price changes.

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

TEMA is calculated in four steps:

Step 1: Calculate the EMA of price

$EMA_1 = EMA(P, n)$

Step 2: Apply EMA to $EMA_1$

$EMA_2 = EMA(EMA_1, n)$

Step 3: Apply EMA to $EMA_2$

$EMA_3 = EMA(EMA_2, n)$

Step 4: Combine to get TEMA

$TEMA = 3 \cdot EMA_1 - 3 \cdot EMA_2 + EMA_3$

Mathematical Derivation

Assuming the lag operator introduced by EMA is $L$, and the original price is $P$:

• $EMA_1 \approx P - L$ (lag $L$)
• $EMA_2 \approx EMA_1 - L = P - 2L$ (lag $2L$)
• $EMA_3 \approx EMA_2 - L = P - 3L$ (lag $3L$)

Substituting into the TEMA formula:

$TEMA = 3(P - L) - 3(P - 2L) + (P - 3L)$ $= 3P - 3L - 3P + 6L + P - 3L$ $= P + 0 \cdot L$

That is, first-order lag is theoretically completely eliminated. However, in practice, higher-order lag terms and nonlinear effects mean TEMA retains a very small residual lag.

Equivalent Interpretation

The TEMA formula can be rewritten as:

$TEMA = EMA_1 + (EMA_1 - EMA_2) + (EMA_1 - 2 \cdot EMA_2 + EMA_3)$

Where:

• $EMA_1$: Base EMA
• $(EMA_1 - EMA_2)$: First-order correction term (already present in DEMA)
• $(EMA_1 - 2 \cdot EMA_2 + EMA_3)$: Second-order correction term (new in TEMA)

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III. Python Implementation

import numpy as np
import pandas as pd

def ema(close: pd.Series, period: int) -> pd.Series:
    """Calculate EMA"""
    return close.ewm(span=period, adjust=False).mean()


def tema(close: pd.Series, period: int = 20) -> pd.Series:
    """
    Calculate the Triple Exponential Moving Average (TEMA)

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

    Returns
    -------
    pd.Series
        TEMA value series
    """
    ema1 = ema(close, period)
    ema2 = ema(ema1, period)
    ema3 = ema(ema2, period)
    return 3 * ema1 - 3 * ema2 + ema3


def tema_numpy(close: np.ndarray, period: int = 20) -> np.ndarray:
    """
    Manual TEMA implementation using numpy
    """
    alpha = 2.0 / (period + 1)
    n = len(close)

    def _ema_array(src: np.ndarray, start_idx: int) -> np.ndarray:
        """Calculate EMA for src starting from start_idx"""
        result = np.full(n, np.nan)
        valid = src[~np.isnan(src)]
        if len(valid) < period:
            return result
        first_valid = np.where(~np.isnan(src))[0][0]
        init_end = first_valid + period
        if init_end > n:
            return result
        result[init_end - 1] = np.mean(src[first_valid:init_end])
        for i in range(init_end, n):
            if not np.isnan(src[i]):
                result[i] = alpha * src[i] + (1 - alpha) * result[i - 1]
        return result

    ema1 = _ema_array(close, 0)
    ema2 = _ema_array(ema1, period - 1)
    ema3 = _ema_array(ema2, 2 * (period - 1))

    return 3 * ema1 - 3 * ema2 + ema3


# ========== Usage Example ==========
if __name__ == "__main__":
    np.random.seed(42)
    dates = pd.date_range("2024-01-01", periods=150, freq="D")
    price = 100 + np.cumsum(np.random.randn(150) * 0.8)

    df = pd.DataFrame({
        "date": dates,
        "open":  price + np.random.randn(150) * 0.3,
        "high":  price + np.abs(np.random.randn(150) * 0.6),
        "low":   price - np.abs(np.random.randn(150) * 0.6),
        "close": price,
        "volume": np.random.randint(1000, 10000, size=150),
    })
    df.set_index("date", inplace=True)

    # Compare multiple moving averages
    df["SMA_20"]  = df["close"].rolling(20).mean()
    df["EMA_20"]  = ema(df["close"], 20)
    df["DEMA_20"] = 2 * ema(df["close"], 20) - ema(ema(df["close"], 20), 20)
    df["TEMA_20"] = tema(df["close"], 20)

    print("=== Four Moving Averages Comparison (last 10 rows) ===")
    print(df[["close", "SMA_20", "EMA_20", "DEMA_20", "TEMA_20"]].tail(10))

    # TEMA trend signal
    df["tema_slope"] = df["TEMA_20"].diff()
    df["trend"] = np.where(df["tema_slope"] > 0, "UP",
                  np.where(df["tema_slope"] < 0, "DOWN", "FLAT"))

    print("\n=== TEMA Slope Trend Signal ===")
    print(df[["close", "TEMA_20", "tema_slope", "trend"]].tail(10))

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IV. Problems the Indicator Solves

1. Extremely Low-Lag Trend Detection

TEMA nearly eliminates first-order lag, making it the lowest-lag indicator among all EMA-based moving averages. In trending markets, TEMA can provide the earliest directional signal.

2. Rapid Reversal Detection

Because TEMA closely tracks price movement, when a trend reverses, the change in TEMA’s direction (slope) is the fastest. Traders can watch for sign changes in TEMA’s slope to identify reversal points.

3. EMA Replacement for Improved Strategy Performance

In any strategy that uses EMA, TEMA can serve as a low-lag substitute:

• TEMA crossover strategies
• TEMA-based MACD variants
• TEMA as a trend filter

4. High-Frequency and Short-Term Trading

TEMA’s fast response makes it particularly suited for intraday and high-frequency strategies, where every moment of lag can translate to significant profit differences.

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V. Advantages, Disadvantages, and Use Cases

Advantages

Advantage	Description
Minimal lag	Lowest lag among all EMA-based moving averages
Trend sensitive	Fastest response to trend changes
Elegant formula	A concise linear combination based on EMA
Universal substitute	Can replace EMA in any indicator system

Disadvantages

Disadvantage	Description
Notable overshoot	May exceed actual price during sharp price changes
Extremely noise sensitive	Overreacts to short-term fluctuations; many false signals in ranging markets
Longest startup period	Requires initialization data for three layers of EMA
Less smooth curve	Compared to SMA and EMA, TEMA’s curve is more jagged

Use Cases

• Strong trending markets: The stronger the trend, the more pronounced TEMA’s low-lag advantage
• Short-term and intraday trading: Scenarios requiring extremely fast signal response
• Post-confirmation entry: Use TEMA for precise entry after confirming a trend with indicators like ADX

Moving Average Family Lag Ranking

$SMA > WMA \approx EMA > DEMA > TEMA$

TEMA has the lowest lag but the highest noise sensitivity. The choice of moving average depends on the trader’s preference for the trade-off between “lag” and “smoothness.”