The Day Every Stop Hits

The maximum, and how often it is the outcome

Portfolio heat is the sum of the stops: four positions risking two per cent each is eight per cent of heat, and that is the right object rather than a crude one. A standard deviation is a summary of a distribution and nobody loses a summary. Eight per cent is a number you can actually be handed, in full, on one afternoon, and it is handed to you exactly when every stop hits.

Take the standard deviation first anyway, because lesson 71 owes it to you and because seeing it settles what it is good for. The variance of a sum of N positions, each with the same variance and an average pairwise correlation of r, is the single-position variance multiplied by N times one plus N minus one times r. One plus three times 0.9464 is 3.8392. Four times that is 15.3568. Its square root is 3.9188, and at two per cent a position that is 7.8376 per cent. So the day’s standard deviation sits between the four per cent the independent story gives and the eight per cent that arrives when the four move as one, and it sits 98 per cent of the way to the eight. The distinction the textbooks draw between the summary and the maximum survives the algebra and does not survive the measurement.

Now the number nobody prints. To ask how often every position goes against you at once, you need a model, so here is one small enough to state completely. Give each position a standard normal outcome for the day and write it as the square root of r times a common shock Z, plus the square root of one minus r times its own shock. Every pair is then correlated at exactly r. All N go against you when every one of the private shocks falls below minus the square root of r, times Z, over the square root of one minus r. That is a single integral in a single variable, and the answers below come from evaluating it on a grid of 4,001 points from minus twelve to twelve.

Read the first column down and then the fifth. Independent, the all-against day is the thing you build a career without seeing: one in 256 at eight positions, which is once a year. At 0.9464 it is one in 2.7, which is eight times a month. That is a factor of ninety-four, and it is bought with no change to any position size, any stop, or any rule.

The second table is the one to keep, because it puts the two faces beside each other on a book that grows the way a real one does, at two per cent a position.

The second column doubles from four positions to eight. The fourth column moves from one day in 2.5 to one day in 2.7. Adding four positions doubled what the bad day costs and bought a 10 per cent longer wait for it, which is the worst trade in this course: you have paid in size and been paid in almost nothing. The reflex that says a fifth position spreads the risk is reading the second column and ignoring the fourth, and the fourth is the one that decides whether you are still trading in a month.

Almost nobody has multiplied their own risk per trade by their own position count and then asked how often that product arrives in full. It is one multiplication and one lookup, and the lookup is the only part that needs the correlation lesson 71 told you how to measure.

The card module 9 started in lesson 71 gains a second column, and it is the heat, in per cent of the account, beside the frequency of the day on which all of it arrives. Lesson 75 is where the whole card gets spent at once.

Six days to a career’s worst drawdown

Take lesson 69’s account: 50,000 dollars, two per cent a trade, so one R is 1,000 dollars. Put four positions on it, each with its stop where lesson 21 puts it, and give it the daily-loss switch lesson 69 used, three per cent of the account.

Heat: four positions at 1,000 dollars of risk each is 4,000 dollars, which is 4R and eight per cent of the account.

The daily-loss switch: three per cent is 1,500 dollars, which is 1.5R.

The all-against day, at lesson 71’s measured 0.9464: 0.4046, or one day in 2.5.

Those three lines do not fit together, and the way they fail to fit is the point. The switch is set at 1.5R. The book can hand you 4R in an afternoon. So the switch is not a limit on the day at all: it is a limit on what you may do after the day has already cost you more than it was set to allow, which is lesson 69’s finding arriving from the other direction. And it is beaten not occasionally but 23.85 per cent of the time.

Now put the frequency against a depth. Lesson 18 put the median worst drawdown of the system it has been carrying at 9R, and that is a number for an entire career: the deepest hole a working system digs, half the time. Four positions hand you 4R on the all-against day, so 2.25 of those days is 9R. At one day in 2.5, the 2.25 arrive in 5.6 trading days.

Six days. If the four were independent it would be thirty-six, and if you had one position it would not be a question this arithmetic could answer at all.

Heat is a maximum, and correlation is the schedule.

That is stated at its worst, and it should be read as an ingredient rather than a path: the days in between contain winners, and the actual drawdown is shallower than the sum of the bad days alone. What survives the correction is the comparison. The ingredient for a career-worst drawdown accumulates six times faster than the independent story says, and it does so on a book that any checklist would call diversified.

Which makes the cap computable rather than conventional. If a daily-loss limit is going to mean anything, the full heat has to fit inside it, so the number of positions is the limit divided by the risk per trade. A three per cent limit at two per cent a trade permits 1.5 positions, which is to say one. At one per cent a trade it permits three. At half a per cent it permits six, which is where the standard advice about position size and the standard advice about diversification finally stop contradicting each other.

So divide your daily-loss limit by your risk per trade tonight, and never carry more positions than the answer.

What this does not settle

That the equicorrelated normal is your book. It has one parameter standing in for a whole matrix, and lesson 71’s own worked example showed why that hurts: four rules whose average correlation is 0.7695 contained one pair at 0.9648, and a book with structure like that behaves worse than its average suggests. The normal also has thin tails, and every honest measurement of market returns finds fatter ones, so the all-against frequencies in the first table are floors rather than estimates.

That every position going against you means every stop hits. It does not. A position can move against you all day and finish well short of its stop, so the first table counts days on which the whole book is red rather than days on which the whole heat is paid. The frequency of the full eight per cent is lower than 23.85 per cent, and the frequency of an unpleasant day is exactly that. The table is the right shape and the wrong label if you read it as the heat arriving in full every time.

That 0.9464 is your correlation. It is lesson 71’s measurement on one rule family on one instrument, which is the situation this course has been building, and it is not a market constant. Four unrelated instruments would give a lower number and a longer wait. A crowded trade in a falling market would give a higher one, because correlations rise in the tail, and the day the frequency matters most is the day the number is worst.

That the daily-loss limit is the control. Lesson 69 established that a loss limit is a postcondition: it reads loss, and loss exists only after exposure does. Everything on this page sharpens that finding rather than fixing it. The cap derived here has to be enforced where lesson 69 put it, before the order is sent, by asking the broker what is already on rather than by watching a profit-and-loss figure fall.

That heat and drawdown are the same measurement. Heat is one day’s maximum and a drawdown is a path, and the nine days in the worked example are an accumulation of the bad days with the good ones left out. A book that hands you 4R on Monday and takes 1R back on Tuesday has not moved 5R. The comparison between nine days and thirty-six survives that objection because both sides ignore the same winners; the absolute nine does not.

And the concession that costs most: this page priced a book in which every position is the same size, which is the assumption that let two per cent times four be written as eight. Nothing here says the four should be the same size, and they should not be. Lesson 73 asks how much goes in each and finds that the weights minimising this book’s variance are 0.746, 0.078, minus 0.672 and 0.848: a 9.72 per cent reduction in the standard deviation of the day, bought by shorting two thirds of one of your own rules and carrying 2.35 dollars of position for every dollar of book.