Weighted Moving Average

The Weighted Moving Average addresses a fundamental limitation of the Simple Moving Average: treating all data points equally. In reality, recent prices are often more relevant to current market conditions than older ones. WMA solves this by assigning linearly increasing weights to each data point, with the most recent price receiving the highest weight.

$$WMA = \frac{\sum_{i=1}^{n} (w_i \times P_i)}{\sum_{i=1}^{n} w_i}$$

Where for linear weighting: $w_i = i$, so the oldest point gets weight 1, and the most recent gets weight $n$.

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The Mathematics: Why Linear Weights?

Expanded Formula

For an $n$-period WMA, the formula expands to:

$$WMA_n = \frac{1 \cdot P_1 + 2 \cdot P_2 + 3 \cdot P_3 + \cdots + n \cdot P_n}{1 + 2 + 3 + \cdots + n}$$

The denominator is the triangular number formula:

$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$

For example, a 5-period WMA divides by $\frac{5 \times 6}{2} = 15$.

Why Linear (Not Arbitrary)?

Linear weights create a constant rate of decay in importance. Each step back in time reduces influence by the same absolute amount. This provides:

• Predictable behavior — You know exactly how much weight each period contributes
• Symmetry — The weight distribution forms a simple triangle
• Balance — More responsive than SMA, but not as jumpy as exponential weighting

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Weight Distribution Visualization

flowchart LR
    subgraph Window["5-Period WMA"]
        P1["Day 1<br/>Weight: 1<br/>(6.7%)"]
        P2["Day 2<br/>Weight: 2<br/>(13.3%)"]
        P3["Day 3<br/>Weight: 3<br/>(20%)"]
        P4["Day 4<br/>Weight: 4<br/>(26.7%)"]
        P5["Day 5<br/>Weight: 5<br/>(33.3%)"]
    end
    
    subgraph Calc["Calculation"]
        SUM["Σ(Price × Weight)"]
        DIV["÷ 15"]
        WMA["WMA Value"]
    end
    
    P1 --> SUM
    P2 --> SUM
    P3 --> SUM
    P4 --> SUM
    P5 --> SUM
    SUM --> DIV --> WMA

Weight Distribution Table

Position	Age	Weight	Percentage of Total
1	Oldest	1	6.7%
2		2	13.3%
3	Middle	3	20.0%
4		4	26.7%
5	Newest	5	33.3%
Total		15	100%

The most recent price contributes 5× more to the average than the oldest price.

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Worked Example

Given 5 daily closing prices: $100,$102, $101,$104, $103

Day	Price	Weight	Price × Weight
1	$100	1	$100
2	$102	2	$204
3	$101	3	$303
4	$104	4	$416
5	$103	5	$515
Sum		15	$1,538

$$WMA = \frac{1538}{15} = 102.53$$

Compare to SMA: $\frac{100 + 102 + 101 + 104 + 103}{5} = 102.00$

The WMA is slightly higher because it weights the recent higher prices more heavily.

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Moving Average Comparison

flowchart TD
    subgraph Types["Moving Average Types"]
        SMA["SMA<br/>Equal weights<br/>All periods = 1/n"]
        WMA["WMA<br/>Linear weights<br/>Recent = n, Oldest = 1"]
        EMA["EMA<br/>Exponential decay<br/>Recent heavily weighted"]
    end
    
    subgraph Response["Price Responsiveness"]
        SLOW["Slowest response<br/>Most lag"]
        MED["Balanced response<br/>Moderate lag"]
        FAST["Fastest response<br/>Least lag"]
    end
    
    SMA --> SLOW
    WMA --> MED
    EMA --> FAST

Aspect	SMA	WMA	EMA
Weight scheme	Equal (1/n each)	Linear (1, 2, ..., n)	Exponential decay
Responsiveness	Slowest	Moderate	Fastest
Smoothness	Highest	Moderate	Lowest
Memory	None beyond window	None beyond window	Infinite (decaying)
Best for	Long-term trends	Balanced analysis	Short-term signals

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Missing Data

window means window observed bars — not calendar slots. A closed-market slot (weekend, holiday) or a genuine data gap never fabricates a value and never poisons the average with a stale or zero substitute.

• Gaps are bridged, not broken: the window slides over the bars that actually have data. A 5-period WMA spanning a weekend weights the five most recent trading days — Friday and the following Monday are adjacent, exactly as a trader reads the chart.
• Missing slots stay null: a data gap or closed-market slot produces null at that exact position in the output; it is never carried forward and never zero-filled.
• Warm-up is null, not zero: the first window - 1 observed bars have no full window behind them and are null. Once real history accumulates, every value is a genuine window-bar weighted average — there are no fabricated early values to discount.
• Live execution returns exactly one datapoint — the current bar's WMA — which is null when the current bar's data, or enough trading history behind it, is missing.

Minimum Data Requirement

The window parameter defines how many observed bars are needed. If your input has fewer calendar points than the window, the module raises an error; if it has enough calendar points but too few observed (non-null) bars, the warm-up prefix simply stays null until enough real observations accumulate.

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Parameter Selection

Window	Behavior	Typical Use
5-10	Very responsive, noisy	Scalping, intraday
10-20	Quick signals, some noise	Day trading, swing entries
20-50	Balanced trend following	Swing trading
50-100	Smooth, delayed signals	Position trading
100-200	Very smooth, significant lag	Long-term trend identification

Choosing Your Window

Consider:

1. Trading frequency — Shorter windows for more frequent trades
2. Asset volatility — Higher volatility may need longer windows to reduce noise
3. Signal clarity vs. timeliness — Longer = clearer but later signals

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Trading Applications

1. Trend Identification

Condition	Interpretation
Price > WMA	Bullish bias
Price < WMA	Bearish bias
WMA slope rising	Uptrend strengthening
WMA slope falling	Downtrend strengthening
WMA flattening	Trend exhaustion or consolidation

2. Dynamic Support/Resistance

In trending markets, the WMA often acts as a dynamic level:

• Uptrend: Price pulls back to WMA, then bounces (support)
• Downtrend: Price rallies to WMA, then reverses (resistance)

3. Crossover Strategies

Signal	Condition
Bullish	Short WMA crosses above Long WMA
Bearish	Short WMA crosses below Long WMA
Confirmation	Both WMAs sloping in same direction after cross

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Real-World Applications

1. Momentum Transition Detection

WMA's linear weighting makes it particularly useful for detecting early momentum shifts. When price action changes character, the WMA responds faster than an SMA but without the jitter of an EMA.

Application: Monitor the WMA slope (rate of change). A flattening WMA after a sustained trend often precedes a reversal before price itself shows clear reversal patterns.

Strategy concept:

1. Calculate WMA on daily closes (20-period)
2. Track the difference between consecutive WMA values (WMA slope)
3. When slope transitions from positive to near-zero after an uptrend, consider tightening stops
4. Combine with volume analysis for confirmation

2. Paired WMA for Entry Timing

Using two WMAs of different periods creates a system that balances responsiveness with noise reduction.

Classic pairing: 10-period WMA (fast) with 30-period WMA (slow)

Why WMA over SMA for this?: The linear weighting ensures both averages respond proportionally to recent price changes, reducing the likelihood of false crossovers during consolidation.

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Advantages

• Recent price emphasis — Newer data influences the average more
• Predictable weighting — Linear weights are intuitive and transparent
• Balanced responsiveness — Faster than SMA, smoother than EMA
• Universal applicability — Works on any asset class and timeframe

Limitations

• Still lags — All moving averages lag price; WMA just lags less than SMA
• Arbitrary weight scheme — Linear weights aren't derived from market theory
• Window dependency — Signals vary significantly with window choice
• No infinite memory — Unlike EMA, data beyond the window has zero influence

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Further Reading

• Weighted Moving Average - Investopedia — Calculation and basic interpretation
• Weighted Moving Average - Fidelity — Practical trading applications
• Triangular Numbers - Wikipedia — Mathematical foundation of WMA denominator

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Indicator Assessment

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quadrantChart
    title WMA Indicator Assessment
    x-axis Low Complexity --> High Complexity
    y-axis Low Reliability --> High Reliability
    quadrant-1 High Value
    quadrant-2 Specialized
    quadrant-3 Limited Use
    quadrant-4 Accessible
    Stocks: [0.35, 0.70]
    Crypto: [0.35, 0.65]

Dimension	Stocks	Crypto	Notes
Popularity	6/10	5/10	Less popular than SMA and EMA; niche usage among traders seeking balanced responsiveness
Reliability	7/10	6.5/10	Consistent across asset classes; linear weighting provides predictable behavior
Complexity	3.5/10	3.5/10	Simple concept—linear weights—but slightly more complex than SMA
Versatility	7/10	7/10	Works on any timeframe and asset; can be paired with other MAs for crossover strategies
Recent Interest	5/10	5/10	Steady but modest interest; EMA often preferred for its infinite memory