The Size of the Next One

The claim, stated the way its believers state it

Press when you are trading well and cut back when you are not. It is not superstition and it does not require believing in streaks. The argument is about you rather than about the market: a run of losses is evidence that something in your reading has stopped working, or that conditions have changed, or simply that you are tired, and a smaller position while you find out is cheap insurance. A run of wins is the opposite signal. Every desk that has ever had a risk manager has some version of this rule written down.

It also has the property that keeps a habit alive: it produced the biggest number in this module. Nothing else on the sheet comes close to 43.535.

The column this lesson adds

Module 12’s sheet has the entries and the rule’s exits from lesson 86 and the excursions from lesson 87. This lesson adds the only column a trader chooses fresh every time: the size. The rule is one sentence. Start at one unit; after a trade that finishes positive, double; after one that does not, halve. The seven results, net of the 0.1230 round trip and in the order they happened, are 2.577, 1.877, 2.377, 1.377, minus 0.423, 1.177 and 0.877.

That totals 43.535 on an average position of 7.8571 units. The comparison everyone makes is against 9.839, the flat record at one unit, and it is the wrong comparison, because a rule that ends up carrying nearly eight units is not competing with a rule carrying one. It is competing with a flat rule carrying nearly eight. Seven and six sevenths units flat on results totalling 9.839 makes 77.306.

So the habit made 43.535 where its own exposure was worth 77.306. It returned 0.5631 of what a flat position of the same average size returned, and it did that on a record where six of seven trades won.

The shuffle, run on the order instead of the prices

Lesson 63 shuffled the bar-to-bar moves to find out how much of a backtest was the search. The same instrument works here, one level up. Take the seven results and put them in every possible order. There are 5,040 of them, which is few enough to run all of them rather than sample. For each order, run the sizing rule, note the total, note the average position it ended up carrying, and divide the total by what a flat position of that same average size would have made. A rule with no dependence on order returns exactly 1 in all 5,040. This one does not.

The spread from 0.2559 to 1.5696 is the whole content of the table. The same seven trades, the same rule, the same average exposure, and the outcome runs from giving up three quarters of what the exposure earned to adding half again on top — decided entirely by which result came first. A trader who ran this rule and reported 43.535 would be reporting a fact about the sequence, and the sequence is the one thing in trading nobody selects.

Two details in the table are worth more than the headline. The median is 0.8763, so the typical order loses money to flat sizing rather than making it: this is not a coin flip with high variance, it is a rule with a cost and high variance on top. And the actual order ranks 855th of 5,040, which means four fifths of the orders that could have happened would have made the habit look better than it looked. Almost nobody you trade against has run this shuffle on their own results, and running it needs a list of trade outcomes and eight lines of code, because unlike every other test in this course it needs no prices at all.

The fifth trade, and what it cost to arrive fifth

Trade 5 loses 0.423 a share. It is the only losing trade on the sheet and it is the smallest number on it in absolute terms: every winner is bigger, and the largest winner is six times its size.

It arrives fifth. By then the rule has doubled four times, so the position is sixteen units, and the smallest result on the sheet is carried at the largest position on the sheet. It contributes minus 6.768, which is more than the first two winners contributed put together at their own sizes, 6.331, and the loss itself is the smallest per-share number on the sheet.

Now move it. Put the same loss first, ahead of everything, and change nothing else about the seven results or the rule. It is carried at one unit, contributes minus 0.423, and the rule halves to a half unit and works its way up from there. Put it last, after six straight doublings, and it is carried at sixty-four units and contributes minus 27.072, which is sixty-four times the same loss.

The rule did not decide how much to risk on the loser. The calendar did.

Sizing by the last result is a bet on the order, and the order was never yours.

That is why the habit survives contact with evidence. It is not that it never works — it works in 1,741 of the 5,040 orders, which is often enough that everybody has a good story about it. It is that whether it worked for you is a fact about the sequence you happened to receive, and a sequence is exactly the kind of thing a person reads intention into. The trader who doubled into a run and kept it has a lesson learned. The trader who doubled into the loss has a lesson learned too, and it is the opposite lesson, drawn from the same rule applied to the same seven trades.

So take your own last thirty results, in cash, and shuffle them a thousand times against whatever sizing rule you actually use.

What this does not settle

That varying size is always wrong. It is not, and the opposite habit shows why the finding is about order rather than about direction. Halve after a win and double after a loss — the rule every risk manager in the world would refuse to sign — returns 1.4923 on the order that actually happened and beats flat sizing in 3,371 of the 5,040 orders. Nothing in that pair of numbers is a recommendation. It is the same measurement pointing the other way, and it is the strongest evidence on the page that both rules are reading the order rather than the market.

That doubling is a realistic rule. It is not, and no desk would run it: four wins take the position to sixteen units and a fifth would take it to thirty-two, which lesson 75 would have stopped at the second. The rule was chosen because it is the clean, extreme version of a habit almost everybody runs in a milder form, and the milder forms behave the same way — multiplying by one and a half instead of two moves the median ratio only from 0.8763 to 0.8809 and takes the winning share down to 1,282 orders, not up.

That the seven results are independent, which the shuffle assumes. They are not, quite: they come from one rule on one series, and lesson 85 measured that series turning rather than walking, so a winner is somewhat more likely to be followed by something a mean-reverting series produces. The shuffle destroys whatever dependence there was, which is the point when the question is how much of the outcome was order — but it also means the 5,040 figure describes a world slightly tidier than the one the trades came from.

That seven results can measure a sizing rule. They cannot, and this is the same objection lesson 19 makes to every small record: seven is not a sample. What seven results can do is exhaust their own permutations, which is why this page runs all 5,040 rather than a simulation, and why the finding is stated as a spread rather than as an estimate. The claim is that the order matters this much, not that the rule is worth this much.

And the concession that costs most: the sheet is still not a book. Every column added so far — the exit, the stop, the size — has been a decision about one position at a time, and lesson 71 already found that positions are not one at a time; two rules at the correlation this course measures carry 1.03 independent bets rather than two. The module has one column left, and it is the one that explains why the other three are so hard to give up. Lesson 89 puts the best price every one of these seven trades ever showed onto the sheet, and finds that 14.60 a share was on the screen against 10.70 in the account: 3.90 that was genuinely there and genuinely left, which is 26.7 per cent of everything that ever showed.