Leading and lagging indicators

What every indicator has in common 

An indicator is a function. Prices go in, a second series comes out, and that series is drawn beside the bars or in a pane beneath them. The input is always a set of numbers the chart already contains: closes, opens, highs, lows, and on some instruments the number of contracts traded. The output is arithmetic performed on those numbers. Nothing else is available to the calculation, because nothing else exists at the moment it runs.

That reads as obvious written down, and it is the sentence most descriptions of indicators quietly abandon a paragraph later. A value plotted at the right hand edge of a chart was computed from bars to the left of it. It cannot consult the bar that has not printed. Whatever an indicator is said to do, it does it with a window of history and nothing else, and every difference between one indicator and the next is a difference in what the arithmetic does to the same history.

Two things arithmetic can do to a price series 

The useful division is not between indicators that see ahead and indicators that do not, since none of them do. It is between the two operations available to a calculation that reads a list of past prices.

The first is smoothing. A smoother replaces each bar's price with a summary of the bars behind it, most simply the arithmetic mean of a stated number of closes, recalculated as each bar completes. The result varies less than the series it came from, because the extremes inside the window are averaged against the ordinary values beside them. A moving average is the plain case of this, and everything built on averages inherits its properties: an average of an average, a band drawn a fixed distance either side of one, a difference between two of different lengths.

The second is normalising. A normaliser does not summarise the level of price at all. It measures something about recent change, the size of the up moves against the size of the down moves, or where the latest close sits between the highest high and the lowest low of a window, and it expresses the answer on a fixed bounded scale. The output has no units and no relationship to the instrument's price. The same reading can be produced by an instrument quoted in thousands and by one quoted in fractions of a unit, which is what the scale is for: it makes a comparison possible where raw prices allow none. An indicator of this shape is conventionally called an oscillator, because a bounded series has nowhere to go except back and forth inside its bounds.

• A smoother returns a price. It is drawn on the same axis as the bars, it moves in the instrument's own units, and it says nothing about the speed at which price arrived where it is.
• A normaliser returns a position on a scale. It is drawn in a separate pane, it carries no units, and it says nothing about the level price is at.
• Both read the same completed bars from the same chart. Neither has an input the other lacks.

Where the two labels come from 

Lagging is the label conventionally attached to the first family, and the reason is mechanical rather than rhetorical. An average holds a fixed number of bars, so a change confined to the newest bar moves the average by a fraction of the distance it moved the price. The average reaches its own turning point after the series it summarises reached one, and the gap between the two widens as the window lengthens.

Leading is the label conventionally attached to the second family, and it is a far weaker claim than the word suggests. A measure of change can fall while the level it was computed from is still rising, because a smaller rise is still a rise. Nothing has been foreseen when that happens. A deceleration that had already occurred has been measured, and the measurement became available at the same moment as every other fact about that bar.

Neither label describes foresight 

Placed side by side, the labels shrink to their real size. One family arrives late by construction. The other arrives early relative to the level, also by construction, and pays for that with readings after which the price series does nothing at all. Neither drew on information from the future, because the future is not among the inputs. The phrase leading indicator, read literally, would name a calculation that consults a bar which has not printed. No such calculation exists, on any chart, on any platform, in any tradition.

The distinction that survives is narrow, and worth keeping anyway. A bounded normalised series reaches the edge of its scale at a different point in the same history than a price average changes direction. That is a difference in the timing of two arithmetic objects computed from one set of bars. It is not a difference in what either of them knows, because neither of them knows anything.

The trade off the labels are really about 

One setting is common to both families: the lookback, the number of completed bars the calculation reads. Nearly everything practitioners argue about follows from it, and the arithmetic is simple enough to state exactly.

In an average, every bar in the window carries an equal share of the result. A short window gives each bar a large share, so the output moves a long way when one bar is unusual. A long window gives each bar a small share, so the same unusual bar barely disturbs the output. The identical logic governs a normalised measure: a short lookback lets a single bar carry a reading to the edge of the scale, and a long lookback does not.

That is the whole trade off, and it has no solution. A shorter lookback produces an output that turns sooner and turns more often, so a larger share of its turns are followed by no change in the price series at all. A longer lookback produces fewer such turns and produces each one later. The two errors are traded against one another rather than reduced together, because they are the same quantity read from opposite ends.

A reading not followed by any change in the price series is conventionally called a false signal, and the phrase misleads in an instructive way. The calculation did not fail. It reported the window it was given, correctly, and that window happened to contain a fluctuation rather than the beginning of something. Traditions differ on which of the two errors is the greater nuisance, which is why platform defaults differ from one another, why the same indicator is published with different periods in different textbooks, and why a lookback stated in a book is a convention rather than a finding.

Why two indicators of the same family add little 

Charts commonly carry several indicators at once, and the argument for the practice is that agreement between separate measures counts for more than any one measure alone. The argument holds only where the measures are genuinely separate.

Two smoothers of similar length read the same closes over largely overlapping windows. Two bounded oscillators of similar length read the same closes over largely overlapping windows and map them onto scales of the same shape. In both cases the second calculation shares most of its inputs with the first, so its output moves with the first almost by construction. When they agree, nothing has been confirmed. One measurement has been performed twice and counted twice, and the impression of agreement is stronger than the single reading it was built from, which is why the redundancy is worth naming rather than merely wasteful.

Practitioners who take the objection seriously answer it by pairing measures that read different attributes of the same bars: one describing direction, another describing the size of recent bars, another describing the number of contracts traded. That reduces the overlap without removing it, since all of them still read one history. There is no agreed threshold for how different two measures must be before their agreement carries meaning, and no procedure in common use for testing the question on a given chart, which is worth holding onto before any arrangement of indicators is described as confirmation.

Where practitioners disagree 

Two disagreements sit underneath this whole part of the module and neither is settled. The first is whether leading is a category at all. One tradition holds that every indicator lags by construction, that a measure of change is simply a lagging measure of a different quantity, and that the leading label describes marketing rather than arithmetic. Another holds that the distinction earns its keep, because bounded measures behave differently around turning points and a vocabulary that ignores the difference loses something real. Both sides describe the same calculations correctly and disagree about what to call them.

The second is larger. One school argues that an indicator is a compression of the price series and can therefore contain nothing the price series does not, so reading the bars themselves is the more direct route to the same information. Another argues that compression is the entire point, that reading raw bars by eye is inconsistent from one person and one day to the next, and that a stated calculation at least returns the same number from the same data every time. The second argument concedes the first one's premise and answers a different question, which is why the exchange recurs in every generation of trading literature without resolving.