Written by
Etan AyenEtan Ayen
Edmund TaylorEdmund Taylor

This article makes no claim about what any particular charting product does. I have not surveyed them, and I will not assert from memory what a piece of software puts on a screen. What follows is about the mathematics of a statistic and about what happens to that mathematics when it is computed outside the domain it was defined on. Where a reader's own platform sits in relation to this is a question the reader can answer in about a minute, and is better answered that way than taken from me.

The argument is narrow on purpose. I am not going to tell you that an industry cannot do arithmetic. The arithmetic is fine. What I am going to show you is that one label is doing the work of three different quantities, that the three are not interchangeable, and that nothing on the chart tells you which one you have got.

PART I. WHAT THE NUMBER IS FOR

Start with the thing itself.

Formula plate 01: The volume-weighted average price, defined.
Fig. 01

Every price at which the security traded, each weighted by the quantity that traded there, divided by the total quantity. It is the average price paid, by everybody, across some window. There is nothing controversial in the formula and there never has been.

The controversy is entirely in the window.

1.1 It was built to answer one question

The volume-weighted average price entered institutional practice as a benchmark for execution quality, and it exists because of a specific problem. A pension fund tells a broker to buy two million shares. The broker cannot buy two million shares at one price; the order has to be worked across hours, and the price moves while it is worked. When the day ends, how does the fund know whether it was served well or badly?

It compares the average price it actually paid against the average price everybody paid. That is the whole idea. Beat the session's volume-weighted average and the execution added value; miss it and the execution cost money. The comparison is the product, and the number exists to make the comparison possible.

The measure has a documented origin. Berkowitz, Logue and Noser set it out in The Journal of Finance in March 1988, developing execution cost against the volume-weighted average price over the trading day, tested across more than fourteen thousand actual transactions. The session bound is not something practice bolted on afterwards and it is not a convention that drifted in. It is in the paper that established the measure, in the definition, at the beginning.

This is why it resets. A benchmark has to bound the thing it is measuring. The order was worked today, against today's liquidity, in today's conditions, so the yardstick is today. The session reset is not a default someone chose for convenience. It is the definition of what is being measured. Remove it and you have not extended the benchmark; you have stopped computing the benchmark and started computing something else.

Formula plate 02: The session anchor, and what the reset does.
Fig. 02

1.2 The conflation, in the open

The most widely read reference explanation of this indicator on the public internet describes it, in the space of a few lines, as a technical analysis indicator and as a trading benchmark, and tells the reader that the line "looks similar to a moving average line, but is smoother."

I cite that not as authority. It is not authority for anything; it is a secondary explainer, and this article takes no fact from one. I cite it as evidence, because it demonstrates that the two ideas have collapsed into each other in the place where most people go to learn the term. A benchmark and an indicator are different instruments. A benchmark measures something that already happened against a standard. An indicator is consulted about what to do next. The same number cannot quietly be both, and a comparison to a moving average is the exact confusion this article is about, appearing in the first paragraph a newcomer reads.

PART II. THE THREE OBJECTS

Take the label above the intraday session and it stops denoting one quantity. Depending on how it is computed, it becomes one of three, and they behave differently enough that conclusions drawn from one do not transfer to another.

2.1 Object one: session-anchored, above the session, is degenerate

Put a session-anchored volume-weighted average on a chart whose bars are one day long, and ask where the line comes from.

Formula plate 03: One session per bar, the degenerate case.
Fig. 03

The session is the bar. The cumulative sums in the numerator and the denominator both open and close inside a single candle, so the calculation yields exactly one value per bar and then restarts. There is no evolution within the window, because the window and the bar are the same object.

What you get is a series of daily average traded prices. That is a real quantity and a useful one; it is a smoother price series than the close, and there are good reasons to look at it. But it is not the intraday line. The intraday line is a running measure that develops across a session and tells you where the day's participation is concentrated relative to now. The daily series is a sequence of independent point estimates with nothing running through them.

Join those points with a line and the chart asserts a continuity the mathematics does not contain. Nobody wrote that assertion down; it is a property of drawing a line between two dots.

2.2 Object two: re-anchored to a longer window, and the denominator problem

The alternative is to let the sums run: start at some point in the past and accumulate from there. This is a coherent calculation, and it is not what most people assume they are looking at.

Formula plate 04: Marginal weight under a growing denominator.
Fig. 04

Consider what the newest observation contributes. Its influence on the result is its own volume divided by all volume accumulated so far. On the first day of the window that fraction is large. Two hundred days in, it is roughly one two-hundredth of the total, if daily volumes are comparable. Two thousand days in, it is negligible.

The statistic becomes progressively deaf. Its responsiveness to new information decays toward zero as the window lengthens, and the observations nearest the anchor keep their weight permanently, because nothing ever leaves the sum.

That decay deserves one condition rather than an airy limit, because it is a property of the volume series and not an axiom. It holds whenever the partial sums outgrow the terms, which is true of any market whose volume is stationary or grows at most linearly. It is not unconditional: if volume grew geometrically at rate r, the newest observation would retain a permanent weight of (r minus 1) over r rather than fading. No traded instrument behaves that way for long, so the conclusion stands; but it stands on an assumption, and the assumption should be visible.

Now set that beside the thing it is so often compared to.

Formula plate 05: Fixed lookback against cumulative weight.
Fig. 05

A simple moving average of length N gives every observation in its window a weight of exactly one over N, and drops the oldest observation each time it takes a new one. Its responsiveness is constant and its memory is finite. Those are the two properties that make it behave the way a reader of charts expects a line to behave.

A cumulative volume-weighted average has neither property. Its memory is total, and its responsiveness declines on trend, roughly as one over n when volumes are comparable. On trend, not monotonically: a single heavy session late in the window will lift the newest observation's weight above the one before it. The direction is reliable; the step-by-step ordering is not, and saying otherwise would be a claim the arithmetic does not support.

There is a fairness problem with the comparison as drawn, and it is worth closing rather than leaving for a reader to spot. A simple moving average differs from a cumulative volume-weighted average in two ways, not one: bounded memory, and equal weights instead of volume weights. To isolate the property under discussion, compare against a rolling-window volume-weighted average instead. Same volume weighting, bounded window. Now only one thing varies, and the contrast is exactly the one this section is about.

Two lines that look alike on a screen, doing opposite things with time. This is why "smoother than a moving average" is such an expensive sentence: the smoothness is not a filtering choice, it is the visible symptom of an average that is progressively refusing to move.

2.3 Object three: computed from bars, and the proxy that is not the price

There is a third case, and it is the quietest of the three, because it is invisible even to someone who has thought carefully about the first two.

The definition in Fig. 01 requires every trade. Above the intraday timeframe, the data available is usually not every trade; it is a summary. Open, high, low, close, and total volume for the bar. The individual prints inside the bar are gone.

So the calculation substitutes a stand-in for the price at which that volume traded.

Formula plate 06: The OHLCV proxy, typical price.
Fig. 06

Typical price, the mean of high, low and close, is the usual choice. It is a reasonable stand-in and it is not the answer. It assumes, in effect, that volume was distributed within the bar in a way that makes that particular average representative. Nothing guarantees that, and the summary data cannot tell you whether it held.

Work an example, and let it be a coherent one, because a constructed bar has to obey its own open, high, low and close or it proves nothing. Take a bar with a high of 110, a low of 90, and a close of 94: heavy trade up near the highs, then a fade into the bell. Typical price is (110 + 90 + 94) over 3, which is exactly 98.0. Now suppose 80 percent of the volume traded around 106 during that upper phase, and the remaining 20 percent around 93 on the way down.

Formula plate 07: Proxy error, worked.
Fig. 07

The true volume-weighted average for that bar is 0.8 times 106 plus 0.2 times 93, which is 103.4. The proxy says 98.0. The error is 5.4 points, 27 percent of the bar's entire range, and it is invisible from the summary data because the summary data is what destroyed the information.

Now bound it, because "the error can be large" is the kind of sentence that should always be replaced by a number.

Formula plate 08: The error is bounded, and by how much.
Fig. 08

The true volume-weighted average is a convex combination of prices that all traded inside the bar, so it lies in the interval from L to H. The proxy is (H + L + C) over 3 with C itself in that interval, so it lies in the narrower band from (H + 2L)/3 to (2H + L)/3. Subtract the extremes and the largest possible discrepancy is two thirds of the bar's range, attained exactly when every share trades at one extreme and the bar closes at the other. The worked example above uses about 41 percent of that maximum, so it is a middling case rather than a contrived one.

Two things follow. First, the error is bounded, and the bound is the range, which the chart is already showing you. Second, nothing else in the summary data narrows it: within those two thirds, the true value can sit anywhere, and open, high, low, close and total volume cannot tell you where. Wide, one-sided bars therefore admit the biggest errors, and wide one-sided bars are exactly the ones a person opens the chart to understand. The estimate is loosest precisely when it is most consulted.

PART III. THE HONEST COUNTER-ARGUMENT

An argument that does not state its opposition properly is not an argument, so here is the opposition, at full strength.

Anchored volume-weighted averages are a deliberate, established, and entirely defensible technique. Fix the anchor at an event: a listing, an earnings gap, a capitulation low. Then accumulate from there. The resulting number answers a real question, namely what has the average participant paid since that event, and that question has genuine meaning for anyone thinking about who is holding what and at what cost.

Everything in section 2.2 still applies to it. The denominator still grows, the responsiveness still decays, the anchor still holds permanent weight. But those are not defects when the anchor is deliberate, because the question being asked is a cumulative one. A practitioner who anchors on purpose knows exactly which of the three objects is on the screen, knows why, and reads it accordingly. That practitioner is not making an error and this article does not say otherwise.

So the claim has to be stated precisely, because the precise version is the only one that survives:

The label does not distinguish the three cases, and an unlabelled statistic cannot be interpreted. A reader looking at a line marked VWAP on a chart above the intraday timeframe cannot tell from the label whether it is a per-bar average drawn as though continuous, a cumulative mean of unknown anchor and declining sensitivity, or an estimate built on a proxy whose error, though bounded by two thirds of each bar's range, is pinned no more tightly than that by anything in view. Those three have different memories, different responsiveness, and different error properties. Interpretation requires knowing which one you have, and the name does not say.

That is not a failure of arithmetic. It is a failure of specification, and it is the more serious of the two, because arithmetic errors get caught.

PART IV. WHAT IT ALL COMES TO

A measurement is a number plus the conditions under which it means something. Drop the conditions and you keep the number, which is worse than having nothing, because a number is persuasive and a number without its conditions is persuasive in exactly the same way.

The volume-weighted average price came with its conditions attached. It measures execution against the session in which execution happened. Everything difficult about it above that timeframe follows from carrying the name across a boundary and leaving the definition behind.

The practical form of this is short. If a volume-weighted line is on your chart, three questions settle what you are looking at, and they take a minute: where is the anchor, what price is being weighted, and how much does today move it. A tool that cannot answer all three has not given you a measurement. It has given you a line.

That distinction is the whole of the discipline, in this and in every other technical field I have worked in. Anyone can produce a number. Producing a number and being able to say precisely what it is a measurement of is a different job, and it is the one that carries liability.

DISCLAIMERS

  1. This article is not investment advice and must not be used as a basis for trading decisions. It concerns the mathematical definition and domain of a statistic, nothing more.
  2. No charting product, vendor, or platform was surveyed for this article, and none is named. No claim is made or implied about the default behaviour of any specific software. Readers wishing to know how their own tools compute and anchor this statistic should consult those tools' documentation directly.
  3. The worked example in section 2.3 is constructed, not observed. It is arithmetic chosen to make the mechanism visible, and it is labelled as such in the text. It is also internally consistent: the volume distribution described is compatible with the high, low and close it is given. An earlier draft of this article carried an example that was not, placing the closing print at 100 while asserting that the final block of volume traded near 92. That version is corrected here, and the correction is recorded rather than quietly made.
  4. Two claims in this article carry conditions, and both are stated where they are used. The decay of the newest observation's weight requires that a volume series' partial sums outgrow its terms, which is true of real markets and not true in general. The decline in responsiveness is on trend rather than monotonic; a single heavy session can raise one step above the last.
  5. The two-thirds bound in section 2.3 is derived in the text, not asserted, and it is tight: the extremal case that attains it is described. It was also checked numerically against several hundred thousand randomly generated bars, none of which exceeded it.
  6. The account of the statistic's origin as an execution benchmark reflects standard institutional practice as described in the transaction cost measurement literature. Where this article reasons rather than cites, it says so in the sentence.

FAIR USE NOTICE: Under Section 107 of the Copyright Act of 1976, allowance is made for fair use for purposes such as criticism, comment, news reporting, teaching, scholarship and research.

Source Citations

Ordered by authority. This article draws its mathematics from first principles; the citations below establish the statistic's role as an execution benchmark and the standard against which the conflation described in Part I is measured.

I. Standards and academic literature

II. Secondary reference, cited as evidence and not as authority

Note on sources not cited

No survey of charting platform behaviour was conducted, so none is cited, and no claim in this article depends on one. Readers will find a great deal written on this statistic in secondary trading literature; I have used none of it, because the mathematics here is elementary and derives from the definition in Fig. 01 without assistance.

Figures set by the author. Formula plates are original work.

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