Understanding the EMA Formula

Why the Exponential Moving Average Uses (Period+1) in Its Denominator

The Core Question

In exponential moving average calculations, the smoothing factor α uses the formula:

α = 2 / (N + 1)

Many traders and analysts wonder: why include the "+1"? Why not simply use 2 / N?

The answer lies in a fundamental principle of moving average mathematics: lag equivalence.

The Lag Equivalence Principle

Both simple moving averages (SMAs) and exponential moving averages (EMAs) inherently lag behind the actual price data because they're based on past observations. The key insight is that an EMA and SMA with the same period exhibit approximately the same lag.

Simple Moving Average (SMA)

Equally weights all prices within a window. A 10-period SMA includes exactly 10 periods of data with equal weight.

Exponential Moving Average (EMA)

Gives exponentially decreasing weight to older data. Uses infinite lookback, but 86% of weight from last N periods.

Both experience a lag of approximately (N–1)/2 periods, where N is the period. This is not coincidence—it's mathematically derived.

Example: A 10-period EMA and a 10-period SMA both lag by approximately (10−1)/2 = 4.5 periods. This means both respond to price changes at nearly the same rate.

The Mathematics: Average Age of Data

The underlying principle involves the average age of data in the calculation. In exponential smoothing, this is calculated as:

Average Age = 1 / α

When we use α = 2/(N+1), we get:

Average Age = (N + 1) / 2

This creates mathematical parity with the SMA. Here's why this matters:

  • 10-period EMA: Average data age = (10+1)/2 = 5.5 periods
  • 10-period SMA: Average data age ≈ 5.5 periods (the middle of its window)
  • Both indicators: Respond to price changes at approximately the same speed

The "+1" in the denominator is the adjustment that ensures the exponentially decaying weight structure produces the same effective average data age as a comparably-sized simple average.

Why Not Just Use Period?

If you used α = 2 / N instead, several problems would occur:

  • The EMA would lag less than an equivalent-period SMA, breaking comparability
  • The indicator would be overly reactive to the most recent price movement
  • You'd lose the predictable relationship between period and responsiveness
  • It would violate the mathematical equivalence that makes EMAs useful for comparison

The "+1" ensures that when traders say "20-period EMA," it behaves similarly to a "20-period SMA" in terms of lag and overall responsiveness.

The Role of the Numerator "2"

It's important to note that the "2" is a convention, not a mathematical requirement. It represents the smoothing factor—a higher value gives more weight to recent prices.

The "2" was chosen because it strikes a practical balance between:

Some applications use different smoothing factors (1.5, 3, or custom values), but the denominator adjusts accordingly to maintain lag equivalence with the intended period.

Historical Origins: Hunter (1986)

The specific formula α = 2 / (N + 1) was popularized by J.S. Hunter's 1986 work on exponentially weighted moving averages (EWMA) for quality control applications in manufacturing.

Hunter's formulation became the industry standard in technical analysis because it provided an intuitive and mathematically sound relationship between the EMA period and its actual lag characteristics. This made EMAs practical and predictable for traders worldwide.

Quick Summary

  • The (N+1) denominator ensures EMAs have the same lag as equivalent-period SMAs
  • Both indicators have an average data age of approximately (N+1)/2 periods
  • This creates predictable and comparable behavior across different moving averages
  • Without the "+1", EMAs would be overly reactive and unpredictable relative to SMAs
  • The "2" in the numerator is a convention for balancing responsiveness and stability
  • The formula has been the standard since Hunter's 1986 quality control work

Key References

  1. The EMA has the same lag as an SMA with equivalent period; derived from weighted sum analysis of both indicators' lag characteristics.
  2. The average age of exponentially smoothed data is calculated as 1/α, where α is the smoothing factor.
  3. Hunter (1986) established the EWMA formula for quality control; this became the standard in technical analysis.
  4. The smoothing factor can vary; "2" is conventional but practitioners may use different values depending on responsiveness requirements.