Building the average directional index from range extension and true range

What each bar contributes

Each bar is assigned exactly one net directional movement: upward extension, downward extension, or none. The assignment is made by comparing the current high-low range with the prior bar.

Positive directional movement is the upward part of the current bar that extends beyond the prior bar, or zero when that upward extension is not the larger of the two sides. Negative directional movement is the downward part of the current bar that extends beyond the prior bar, recorded as a positive magnitude when that downward extension is the larger of the two sides.

When a bar extends both above and below the prior range, only the larger outside portion is kept as directional movement. An inside bar, or a bar whose outside portions are equal, produces zero directional movement.

True range as a positive scale

True range is defined as the largest of the current high minus low, the current high minus the prior close, and the current low minus the prior close. It is always recorded as a positive value.

In the same construction, true range is the largest of the current high-low span and the two gaps from the prior close to the current high or low. Directional movement is later read against this scale, not against a raw price change.

Directional indicators over a lookback

A directional indicator is built by dividing that bar’s directional movement by true range. Both the upward and downward indicators are nonnegative ratios of range extension to realized range.

Those daily ratios are then accumulated over a lookback, commonly 14 bars and sometimes 13 or 20, by summing true ranges and summing each side’s directional movement over the same window. The raw one-bar ratio is less jumpy after that accumulation.

The construction treats a trend as a series of ranges that keep extending in a consistent direction, then expresses that extension as a function of true range rather than as a raw price change.

From two sides to one filter

The directional index is the absolute gap between the two directional indicators, scaled by their sum, so one number expresses how one-sided the lookback has been. The average directional index is a smoothed score of how much successive true ranges extend one way rather than both ways, used to judge whether a market is trending enough for a directional rule.

The finished filter is meant to classify whether a market is trending before a trend-following rule is applied. Methods built for trends and methods built for ranges can fail when used in the other regime.

A further average of the directional index, the average directional rating, can be used as a slower confirmation step. It is sometimes used as a slower confirmation that a trend filter has already crossed a construction threshold. A construction change such as requiring that average to rise through a threshold instead of merely sitting above it is presented as something that must be backtested on the intended series.