WMA: Weighted Moving Average for Recent Prices

Key Takeaways

• WMA weighted moving average uses linear weights 1, 2, ..., N, so the newest close pulls hardest on the line.
• The denominator is the triangular number N(N+1)/2, which keeps the weights normalized to one.
• The most common mistake is assuming a 20-period WMA reacts like a 20-period EMA; the WMA is faster on entry.
• WMAs are the building block of the Hull Moving Average and several adaptive smoothing methods.

What It Is

A weighted moving average computes the mean of the last N closes after multiplying each by a weight that grows with recency. The weights form an arithmetic sequence (1, 2, 3, ..., N), so the most recent bar gets weight N and the oldest gets weight 1.

The WMA sits between the SMA, which gives every price equal weight, and the EMA, which decays weights exponentially over an infinite history. Inside its window the WMA reacts faster than an SMA of the same length, and unlike an EMA it forgets old prices completely.

The Intuition

Equal weighting feels wrong on fast horizons. Yesterday's close usually carries more information about where price is heading than a print from N bars ago. The WMA codifies that intuition with a clean linear ramp.

Why linear and not exponential? Two reasons. First, the math stays inside a finite window, which makes the WMA easy to reset, slice across instruments, or rebuild from scratch on any bar. Second, the linear weighting is the natural choice when you want a smooth ramp of importance without the long tail of an EMA.

How It Works

The formula is:

WMA = (P1*1 + P2*2 + P3*3 + ... + PN*N) / (1 + 2 + 3 + ... + N)

The denominator simplifies to the triangular number N(N+1)/2. For a 20-period WMA the divisor is 20 * 21 / 2 = 210. The most recent price contributes N/210 of the average; the oldest contributes 1/210.

Note the indexing convention. Here P1 is the oldest price in the window and PN is the most recent, which matches how most charting packages document the formula. Some textbooks reverse the index, which gives the same result but can confuse a manual check.

Lag profile: a length-N WMA has an effective lag of roughly (N - 1) / 3 bars in steady-state. That is faster than an SMA of length N (lag (N - 1) / 2) and roughly matches a 2N / 3 EMA. So a 21-period WMA reacts more like a 14-period EMA than like a 21-period EMA.

Worked Example

Take five daily closes: 100, 102, 104, 103, 110. Weights are 1, 2, 3, 4, 5; the divisor is 1+2+3+4+5 = 15.

WMA = (100*1 + 102*2 + 104*3 + 103*4 + 110*5) / 15
    = (100 + 204 + 312 + 412 + 550) / 15
    = 1578 / 15
    = 105.2

Compare to the SMA of the same five prices, which is 103.8. The WMA sits 1.4 points higher because the newest close of 110 carries five times the weight of the oldest 100.

Now roll the window forward by one bar. New sequence: 102, 104, 103, 110, 108.

WMA = (102*1 + 104*2 + 103*3 + 110*4 + 108*5) / 15
    = (102 + 208 + 309 + 440 + 540) / 15
    = 1599 / 15
    = 106.6

The WMA climbed from 105.2 to 106.6 despite the most recent close falling from 110 to 108. The newer high weight on 108 plus the loss of the old 100 dominated the move.

Common Mistakes

1. Confusing window length with smoothing. A 20-period WMA is not as smooth as a 20-period EMA. If you want EMA-like smoothing, lengthen the WMA by roughly a factor of 1.5 to 2.

2. Forgetting the divisor. Some implementations omit the N(N+1)/2 normalization and produce values that are not on the price scale. Always check that the WMA prints near current price on a stable instrument.

3. Mixing index conventions. Some references put weight N on the oldest price instead of the newest. The math comes out the same only if you reverse the price sequence. Pick one convention and stay with it.

4. Building Hull-style chains without checking lengths. Hull's HMA uses two WMAs of length N/2 and N. Substituting other lengths breaks the lag-cancellation that makes HMA work.

5. Reading single crosses as signals. Like SMA and EMA, WMA crosses whipsaw in sideways markets. Treat them as alerts and pair with a trend filter or volatility regime.

Frequently Asked Questions

What is WMA weighted moving average in simple terms? The WMA weighted moving average is a price average where newer bars count more than older ones. It uses linear weights inside a fixed window.

How does WMA affect investment decisions? Active swing traders use WMAs in place of SMAs to get faster confirmation of new trends without the spike sensitivity of very short EMAs. The Hull Moving Average, popular in retail tools, is built from WMAs.

What is a real-world example of WMA use? WMAs are the math inside the Hull Moving Average, which charting platforms like TradingView and Fidelity display as a default option. They also appear in volume-weighted price calculations.

How can investors use WMAs effectively? Match the length to your horizon, expect faster reaction than a same-length SMA, and pair the line with a longer trend filter. Use it as a building block for HMA when you want minimal lag.

How is WMA different from EMA? WMA uses linear weights inside a finite window. EMA uses an exponential decay that never fully forgets older prices, which makes its tail heavier and its center of mass slightly further back.

Sources

1. Corporate Finance Institute. "Weighted Moving Average (WMA)." https://corporatefinanceinstitute.com/resources/career-map/sell-side/capital-markets/weighted-moving-average-wma/
2. Investopedia. "Weighted Average." https://www.investopedia.com/terms/w/weightedaverage.asp
3. Fidelity Learning Center. "Moving Averages." https://www.fidelity.com/learning-center/trading-investing/technical-analysis/technical-indicator-guide/ema
4. CME Group Education. "Understanding Moving Averages." https://www.cmegroup.com/education/courses/technical-analysis/understanding-moving-averages

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.