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Indicator Formula

Technical Details — Hull Moving Average (HMA)

Section titled “Technical Details — Hull Moving Average (HMA)”

Overview

Section titled “Overview”

The Hull Moving Average (HMA) reduces lag by applying a WMA of period √n to the difference between a half-period WMA and a full-period WMA. This clever technique produces a responsive average that can even overshoot price during strong moves.

Mathematical Derivation

Step 1 — Calculate Half-Period WMA

Section titled “Step 1 — Calculate Half-Period WMA”

Purpose: Get a fast-responding weighted average.

$$WMA_{n/2}[t] = WMA(Close, \lfloor n/2 \rfloor)$$

Where:

  • $$n$$ = Full period
  • $$\lfloor n/2 \rfloor$$ = Half period (rounded down)

What This Measures: Recent price trend with faster response

Step 2 — Calculate Full-Period WMA

Section titled “Step 2 — Calculate Full-Period WMA”

Purpose: Get a smoother weighted average.

$$WMA_n[t] = WMA(Close, n)$$

Where:

  • $$n$$ = Full lookback period

What This Measures: Broader price trend with more smoothing

Step 3 — Compute Difference Series

Section titled “Step 3 — Compute Difference Series”

Purpose: Emphasize recent price action by doubling fast MA and subtracting slow MA.

$$Diff[t] = 2 \times WMA_{n/2}[t] - WMA_n[t]$$

What This Measures: Amplified recent trend component

Step 4 — Apply Final WMA

Section titled “Step 4 — Apply Final WMA”

Purpose: Smooth the difference series with √n period WMA.

$$HMA[t] = WMA(Diff, \lfloor \sqrt{n} \rfloor)$$

Where:

  • $$\lfloor \sqrt{n} \rfloor$$ = Square root of period (rounded down)

What This Measures: Final smoothed average with minimal lag

Compact Formula Summary

$$HMA[t] = WMA(2 \times WMA(Close, \lfloor n/2 \rfloor) - WMA(Close, n), \lfloor \sqrt{n} \rfloor)$$

Default Parameter: Period (n) = 14

Complete Calculation Example

HMA uses nested WMAs. With Period = 16:

  • Half-period WMA(8) captures recent trend
  • Full-period WMA(16) captures broader trend
  • Diff = 2 × WMA(8) - WMA(16) amplifies recent movement
  • Final WMA(4) (√16 = 4) smooths the result

The result tracks price closely with minimal lag.

Key Takeaways from the Example
  1. Lowest Lag: HMA’s double-weighted technique makes it the fastest-responding moving average
  2. Can Overshoot: During strong trends, HMA may temporarily move ahead of price — unique among MAs
  3. Turn Detection: HMA changes direction faster than other MAs, useful for detecting trend turns early
  4. Nested Computation: Requires more calculation steps than simple MAs, but period n is the only parameter