The Session Cycle

A session is the interval between two closures, and its edges concentrate activity for three structural reasons, none of which is mood. That much is real and published. What is not real is almost every number people attach to it. Split this module’s sixty bars into three consecutive twenties and the efficiency ratio reads 0.021, then 0.067, then 0.333 — a sixteenfold spread on a series that has no clock, no sessions and one generating rule. The mean bar range across the same three thirds reads 1.46, 1.25 and 1.34, and barely moves. The quantity that swings wildly under a split is the one nobody quotes per session; the quantity people do quote is the one that holds still.

Prerequisites: Lesson 40, which put the discontinuity at the boundary this lesson is about, and lesson 19, whose sample arithmetic decides whether any split of your record by the hour has told you anything.

The same sixty bars, cut into three

Quantity, per twenty bars	Bars 1–20	Bars 21–40	Bars 41–60	All sixty
Efficiency ratio	0.021	0.067	0.333	0.082
Net travel	0.9	0.8	4.8	5.8
Path length	43.1	12.0	14.4	71.0
Mean bar range	1.46	1.25	1.34	1.35
Summed range	29.1	25.1	26.8	81.0
Share of the day’s range	36%	31%	33%	100%
Swing points, lesson 32’s rule	3	5	4	15

The efficiency ratio is the row that would fool you. Read across it and the day looks like three different markets: a first third that goes nowhere at 0.021, a middle that is barely better at 0.067, and a last third at 0.333 that is sixteen times the first and would be called a clean trending phase by any rule in lesson 36. Nothing in the closes changes at bar 20 or at bar 40. The three readings are one process cut in three places.

You can see how ordinary that spread is by taking every contiguous twenty-bar window instead of just three. There are forty-one of them, the ratio runs from 0.009 to 0.587, the median is 0.209, and the middle half of the readings falls between 0.088 and 0.399. Fourteen of the forty-one windows read at or above the 0.333 that made the last third look like a trend. A reading of 0.333 on twenty bars of this tape is not a finding. Twenty-seven of the forty-one windows read below it, so it sits at roughly the two-thirds mark of what twenty bars of this tape do.

The same table carries a scale error worth naming, because it is lesson 38’s error wearing a clock. The whole-sixty ratio is 0.082 and the median twenty-bar window is 0.209 — two and a half times higher — for the reason lesson 38 gave: path length grows with the window while net travel does not, so a shorter window always reads more efficient. Compare a session’s ratio against the day’s and the session wins before anybody looks at the market. A per-phase number can only be compared with another number computed over the same number of bars.

Now read the quiet rows. Mean bar range is 1.46, 1.25 and 1.34: a spread of sixteen per cent between the widest third and the narrowest, on a split that moved the ratio by a factor of sixteen. Summed range divides 36 / 31 / 33, which is a third each to within a rounding error. Those two rows are also the ones drawing on two different sources — the highs and lows of bars 1 to 20 were published outright, while those of bars 21 to 60 come from lesson 35’s generating rule — and they still come out flat. On a series with no session structure, the quantity that measures how far price travels comes out even across the day, exactly as it should. That is the shape a null produces, and it is the shape against which a real U has to be measured.

The swing row hides a mechanical loss. Fifteen points on the whole series, but three plus five plus four is twelve. Lesson 32’s rule needs two bars either side, so it cannot evaluate the first two and last two bars of whatever window it is given. One window of sixty leaves fifty-six positions evaluable; three windows of twenty leave forty-eight. The split destroyed eight positions, and with them three of the day’s structural points. Cutting your chart at a session boundary blinds the structure rule at exactly the moment you most wanted it to see — the first bars after the open.

How long before a difference between two hours is real

None of this says the session shape is imaginary. It says a split of one day, or one week, cannot show it to you, and lesson 19 already published the arithmetic for how much would. The question “do my morning trades beat my midday trades” is a comparison of two rates, and comparing two rates needs far more observations than measuring one. For a difference to be established at conventional confidence and power, each group needs roughly 7.84 times the summed variance of the two, divided by the square of the difference between them.

The difference you think you see	Trades needed per phase	Days, at two trades a day in that phase
60% against 40%	94	47
55% against 45%	388	194
50% against 40%	384	192
50% against 45%	1,560	780

Two things in that table are worth more than the numbers themselves. The first is that the middle two rows are nearly identical, 388 against 384, although one compares 55 with 45 and the other 50 with 40. What sets the requirement is the size of the gap, not where the pair sits. The second is the last row: a five-point difference in hit rate, which is the sort of difference people reorganise their whole day around, takes fifteen hundred and sixty trades in each phase to establish. At two trades a day in that phase, that is three years of mornings and three years of middays before you may say the mornings were better.

In lesson 19’s own units the numbers are identical, because at a fixed two-to-one payoff a win rate and an expectancy are the same statement: 50 per cent against 40 per cent is 0.50R against 0.20R and still wants 384 trades a side. And the cost does not stop at the sample. Restricting yourself to one phase is a filter, so lesson 39’s two break-even points apply without amendment — the hours you delete must remove a higher share of your losers than of your winners to raise expectancy, and that ratio must beat your profit factor before it raises money — while the trade rate you gave up lengthens every measurement you will ever make. Three costs, and the clock is only visible in one of them.

So the honest version of the session claim is narrow and still useful. The mechanism says volume and travel concentrate at the edges, and you can verify that on your own instrument in an afternoon because volume is countable and needs no win rate. The mechanism says nothing about direction, so a per-session claim about direction is unsupported before you even test it. And a split of your own record by the hour will find the hours where you should not have been trading long before it can prove that any hour is better than another, because money already lost is not a hypothesis. A phase in which you paid more in costs than you collected in edge has already charged you, and you can stop paying it today without a test of any kind. The test is for the other claim — that one surviving hour is genuinely better than another — and that is the claim the table prices.

What this does not settle

That the three thirds of this series are anything like three sessions. They are three arbitrary cuts of a continuous tape, made to show what a cut does on its own. A real session boundary has an overnight interval, an auction at each end and a different population of participants on each side of it. The null is what the table supplies, and a null is a floor you must clear rather than a description of the building.

That the U-shape is measured here. It is not. The published finding is a U in volume and volatility across the session, established on real tapes over decades and cited below; this lesson reproduces neither its magnitude nor its shape, because a series with no clock cannot. What this lesson supplies is the mechanism that predicts a U, and the arithmetic that says how much data it takes to see one in your own record.

That the efficiency ratio is a bad measurement. It is a fine measurement of what it measures. The failure in the table is not the ratio’s; it is the comparison’s. Twenty-bar readings from one tape scatter from 0.009 to 0.587, and any statistic read off twenty observations will scatter like that. The lesson is about the width of that scatter, which is a property of the sample size and not of the statistic.

That the sample table settles what you should do tomorrow. It settles what you may claim, which is different. You can act on a hunch about the hours at any time; the arithmetic only tells you when you are entitled to call it established. Acting before then is a decision under uncertainty of the kind priced earlier in this course, where a run of eight losses at a 45 per cent win rate turns up in more than half of all two-hundred-trade samples, and it is not the same mistake as believing you have proof.

That a phase which lost money can be deleted without a test. The paragraph above says money already lost is not a hypothesis, and that is true of the money and false of the inference. A phase that came out negative over forty trades has cost you exactly what it cost you, and it may still have a positive expectancy: forty trades put a measured 45 per cent win rate anywhere between 30 and 60, which is wide enough to hold a good hour and a bad one at once. Deleting the phase stops a real loss and may also delete a real edge, and nothing in the record can yet say which. The cost is established and the cause is not, so the deletion is a decision under uncertainty like any other rather than the one free lunch on the page.

Problems

1. Count the volume, not the wins. Take twenty days of the instrument you trade, split each day into three equal blocks of time, and sum the volume in each block. Twenty days is nowhere near enough for a win rate and is plenty for this, because volume is counted rather than inferred and the differences the mechanism predicts are large. If your three blocks come out flat you are either trading an instrument without a session structure or your blocks are cut in the wrong places. If they come out as a U, you have measured on your own tape the one part of this lesson that is a market fact.
2. Compute your own null before you compute your split. Take the quantity you want to compare across hours — the efficiency ratio, the mean bar range, whatever it is — and compute it on every contiguous window of the length your split would use, ignoring the clock entirely. Write down the median and the middle half of that distribution. Only then compute it per phase. If your phase readings sit inside the middle half of the no-clock distribution, you have found nothing, and you now know that before you rearrange your morning.
3. Price the hour you are thinking of deleting. Split your own record into the phase you suspect and everything else, and count four numbers in each: winners, losers, total won, total lost. Two of the answers are available immediately and need no statistics. Did the suspect phase lose money outright? Did it lose more per trade than it won? Those are costs you have already paid, and you may act on them today. The third question — whether its hit rate is genuinely below the rest of the day — is the one the table above prices, and you will almost certainly find you cannot yet answer it. How to Collect a Base Rate is how the count is kept honest.

Sources. Robert A. Wood, Thomas H. McInish and J. Keith Ord, “An Investigation of Transactions Data for NYSE Stocks” (The Journal of Finance, 1985), for the founding measurement of the intraday pattern — returns and volume are elevated at the beginning and end of the trading day and quiet in between, measured on transaction data rather than asserted. Anat R. Admati and Paul Pfleiderer, “A Theory of Intraday Patterns: Volume and Price Variability” (The Review of Financial Studies, 1988), for the third mechanism above and for why it does not dissolve once everyone knows it: discretionary traders cluster where other discretionary traders are, which makes the clustering its own cause. Lawrence Harris, “A Transaction Data Study of Weekly and Intradaily Patterns in Stock Returns” (Journal of Financial Economics, 1986), for how carefully the pattern has to be measured before it can be believed — the paper spends most of its length on what the data does to the estimate rather than on the estimate.

The two busiest minutes of the session are the two in which the market is not trading continuously at all. At the open and at the close the book does not match order against order as they arrive; it collects them and clears them all at one price, which is a different mechanism with different arithmetic and a different set of things that can go wrong. Lesson 42 is about those two auctions: how a single clearing price is computed from a book of competing orders, why the price it produces can sit outside everything that traded around it, and what an order placed into one is actually agreeing to.

Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.