The Unit That Moved

Where the unit cancels, and where it does not

R is one line of arithmetic. Take the change from each close to the next, take the standard deviation of those changes, and divide everything afterwards by it. That is a good choice and this page is not arguing with it. The argument is with measuring it once and then spending it for twenty-two lessons.

Rule out the obvious objection before reading any further. Thirty observations and twenty-nine observations measure a standard deviation badly, and a ratio of two badly measured numbers is worse than either, so perhaps any two halves would look like this. Draw twenty thousand series of fifty-nine moves from a single constant variance and split each one the same way. The median ratio is 1.000, the fifth percentile is 0.730 and the ninety-fifth is 1.369. The measured ratio is 2.711, and not one draw in twenty thousand reached it. Whatever else is true, these are not two halves of one thing.

Lesson 63 left this on the table in as many words. Its second bound said that the resampling null destroys volatility clustering, that the clustering is real, and that a crossover rule can genuinely trade it. The 2.711 above is that clustering, measured on the only sixty closes this course has, and the reason it belongs on this page rather than that one is that clustering does not merely change what a rule can earn. It changes the length of the ruler the earnings are measured with.

Now separate the figures this breaks from the figures it does not, because the course does not divide evenly. A t-statistic is a mean over a standard error and both are in the same units, so a constant divisor leaves it exactly where it was: lesson 63’s 3.65 is 3.65 whichever unit it is written in. The same goes for every ratio of one R quantity to another. All of module 8’s evidence about whether the winning rule is real survives this page untouched.

What does not survive is every absolute distance. Lesson 65’s tenth of an R, lesson 67’s eight-R line, lesson 64’s 0.317 of an R a trade, and lesson 63’s own 0.0796 of an R a round trip are all distances rather than ratios, and a distance in a unit that moved by a factor of 2.711 is two different distances wearing one name.

Almost nobody has measured their own R twice. It is the standard deviation of one bar’s change, computed on the first half of your record and then on the second, and the ratio of the two answers is the whole of this page.

Module 11’s list of the degrees of freedom this course did not have gains its fourth entry, and it is the last one: the unit itself, measured once and then spent as though a market only has one speed.

The final column says the two windows are not merely different sizes, they are different kinds of series. At two bars the full window has a variance ratio of 0.3434, which is a long way under the value of one that a random walk gives. Split it and the first thirty moves give 0.2560 and the last twenty-nine give 0.9202. The first-order autocorrelation of the daily moves tells the same story at minus 0.6594 over the whole window, minus 0.7490 in the first half and minus 0.0943 in the second. All of the mean reversion this window contains lives in the half where the winning rule took two of its seven trades.

The seven trades, priced twice

Take lesson 63’s winner, the 2-and-5, and write out its seven trades with the bar each was entered and exited on. Then price each one twice: once in the single R the course has been using, and once in the R of the window that trade actually lived through. The trade that runs from bar 26 to bar 39 crosses the split, so the full-window figure is the only honest divisor for it and it appears unchanged.

The fourth column averages 0.910 of an R and the fifth averages 1.101, and the t goes from 3.65 to 3.68. So the verdict does not move at all, which is the first thing worth saying: the winner is exactly as real as module 8 said it was, and nothing on this page rescues it or kills it.

The second thing is the shape underneath. Four of the seven trades sit wholly in the quiet half and three of those four lasted a single bar. The two big multiples in the fifth column, 1.826 and 1.560, are not big trades. They are ordinary trades divided by a small number.

The third thing is the bill, and this one does move. Lesson 63 charged the seven round trips 0.557 of an R, which is seven times 0.0796. Charge each round trip in the R of its own window and the total is 0.852 of an R, a factor of 1.53, because four of the seven were paid in the half where the same 0.1230 in cents buys 0.1631 of an R rather than 0.0602.

One rule in this course is immune, and it is worth naming because it is the only one. Lesson 75 sizes a position by dividing the money you will risk by the distance to the stop, so if R halves, the same two per cent buys twice as many shares and the risk carried is unchanged. A hundred thousand dollars risking two per cent buys 978 shares at the first half’s dispersion and 2,652 at the second half’s. The rule corrects itself, but only if you re-measure the input.

R was never a unit. It was an exchange rate, and this course quoted it once.

So measure your own R on the first half of your record and on the second, and take the ratio.

What this does not settle

That thirty moves and twenty-nine moves measure a dispersion. They measure one badly, and on their own the two numbers would be worth very little. The reason to believe the gap is the simulation rather than the measurement: twenty thousand draws from a constant variance, split the same way, never once reached 2.711. That test is the whole of the evidence here, and a reader who does not accept it should not accept the page.

That the second half is a random walk. A variance ratio of 0.9202 on twenty-nine moves is not a measurement of anything to two figures, and the honest statement is narrower: the strong mean reversion the full window shows cannot be found in the second half, which is a different claim from saying the second half has none.

That this window is a market. Sixty closes with a first-order autocorrelation of minus 0.6594 is not what a real instrument is likely to hand you, and a page that pretended otherwise would be doing what the eighty-four before it were built to avoid. What generalises is the question rather than the answer: measure the unit twice, on whatever your record is.

That the split is the right one. It was chosen for a single reason, which is that lesson 74 had already cut this window in half and this page wanted to cut it in the same place. A different cut gives different numbers, and nothing here says where the break belongs or whether a break is even the right model for a thing that probably moves continuously.

And the concession that costs most, which is also the last one this course will make: everything above is one instrument, one window and one unit, and the reason the unit was never re-measured is that re-measuring it needs more data than sixty closes can give. That is the shape of the whole course in one sentence. Every page has been an arithmetic you can reproduce on a record small enough to hold, and a record that small can carry a method and cannot carry a market. What you take away is not a system. It is the habit of dividing, of asking what the number is a number of, and of measuring the answer twice.