The Drawdown You Should Expect

What a drawdown is made of

A drawdown has two parents. One is your edge, or the absence of it. The other is the spread of your trade outcomes, which exists whether or not the edge does. Every trader in a hole attributes it to the first parent. The arithmetic below says the second one does nearly all of the work, and it says so with an overlap you can measure rather than an argument you have to accept.

Here is the harness, in full, because everything on this page comes out of it. A trade’s outcome is drawn from a normal distribution whose standard deviation is one R and whose mean is d, the edge, in the same units. The worst drawdown of a run is the largest fall from any running peak to any later running total, with the peak starting at zero, so a run that never gets above where it started still has a drawdown. Each figure below is 20,000 independent runs of 156 trades from seed 20260903.

One R on this page is one standard deviation of a trade, and that is not lesson 18’s R, where a loser is exactly one R and the spread of outcomes is 1.49 of them. The translation is one division and it is worth doing now, because it puts a lesson you have already read into this table. Lesson 18’s system wins 45 per cent of the time at two to one, so it makes 0.35R a trade with a standard deviation of 1.4925R. Divide: its edge in this page’s units is 0.2345, which lands between the fourth and fifth rows below.

The harness is also checkable against something that is not a simulation. With no edge at all, continuous time gives a mean worst drawdown of the square root of πT/2, which over 156 trades is 15.65R. The simulation returns 14.53R, and the gap is not an error: a walk that only looks once a trade misses the extremes between looks. Refine it to 256 looks a trade and it returns 15.61R.

True edge, R a trade	Luckiest tenth	Median	Unluckiest tenth	Unluckiest hundredth
0.00, nothing there	7.56	13.26	23.33	33.95
0.05	6.35	10.65	18.59	27.81
0.10	5.51	8.82	14.96	22.57
0.20	4.29	6.52	10.49	15.71
0.30	3.50	5.17	8.03	11.66

Read it down the columns rather than across the rows, because the columns are where the lesson is. The luckiest tenth of dead systems reach 7.56R. The typical live one at a tenth of an R reaches 8.82R. Those two numbers are in the wrong order for the story everybody tells about drawdowns, and no amount of staring at your own equity curve will put them back, because the distributions they come from overlap over almost their whole length. A depth of eight R is comfortably inside the ordinary range of both.

So turn the depth into the only question worth asking about it. If you are D down, how much more likely is that under a dead system than under a live one? It is one exceedance count on each of two simulations, divided.

Six R: reached by 97.0 per cent of dead systems and 84.6 per cent of live ones, a ratio of 1.15.

Eight R: 87.2 per cent against 59.5 per cent, a ratio of 1.47.

Twelve R: 58.5 per cent against 22.6 per cent, a ratio of 2.59.

Twenty R: 18.2 per cent against 2.3 per cent, a ratio of 7.78.

A ratio of 1.47 means that being eight R down multiplies whatever odds you already held on the system being dead by 1.47, and by no more than that. It is evidence, faintly. It is not remotely enough to act on, and it is the strongest thing a reader has in the moment they usually act. The depth that would actually settle something is twenty R, at which the ratio finally reaches 7.78 — and twenty R is a depth that arrives to two live systems in a hundred and to eighteen dead ones, so waiting for it means most of your genuinely dead systems never trip it either.

Almost nobody who has ever switched a system off has computed that ratio at the depth they switched it off at. Two simulations and two exceedance counts, and an evening is enough for both.

The sheet this module has been building since lesson 62 gains a sixth column here, and it is the depth at which you will stop, written down before you start. Beside it write the fraction of live systems that depth discards, from the exceedance figures above, because a stopping rule without that fraction next to it is a number somebody picked. Lesson 70 is where the whole sheet gets spent at once.

The test that does answer the question

Depth fails as evidence because depth is bounded and evidence is not. A drawdown can only get so deep before you are out of money, but the number of trades you have taken keeps going up, and every one of them carries a little information about which system you are holding. So stop measuring the hole and start accumulating the trades.

Wald’s sequential test keeps one running number: the log of how much more likely your trades are under a live system than under a dead one. After each trade it adds a term. For an outcome x, testing a mean of zero against a mean of d:

L → L + d x − d²/2

Two lines are drawn before the first trade. Declare it alive when L rises to the log of (1 − β) over α; declare it dead when L falls to the log of β over (1 − α). At a 5 per cent chance of each error those are +2.9444 and −2.9444, and the symmetry is a consequence of choosing the two error rates equal rather than an assumption.

One substitution, in full. At an edge of a tenth of an R, every trade adds 0.10x − 0.005 to L. A live system’s trades average +0.10, so its average step is 0.10 times 0.10 minus 0.005, which is +0.005. A dead system’s trades average zero, so its average step is −0.005. Each walks toward its own line at the same speed, and each has 2.9444 to travel. Divide: 2.9444 by 0.005 is 589 trades. At forty trades a month, 14.7 months.

Edge you are testing for	Step per trade, d²/2	Trades to a verdict	Months at forty a month
0.05	0.00125	2,356	58.9
0.10	0.00500	589	14.7
0.20	0.02000	148	3.7
0.30	0.04500	66	1.6

The trade counts round up and the months divide the unrounded expectation by forty, which is why the last row reads 66 and 1.6 rather than 66 and 1.7.

Now draw the two lines somewhere a trader can actually see them, which is on the running total in R. L is a straight line in that total: after n trades L is 0.10 times the total, less 0.005n. Set L to each boundary and solve, and the two lines are the total equal to n times d over two, minus and plus 2.9444 divided by d.

At 156 trades, the horizon lesson 65 fixed, that puts the dead line at −21.6R and the alive line at +37.2R, while a live edge of a tenth of an R expects to have made +15.6R. Both of those are worth a second read. The test will not call a system dead until it is 21.6R down, which is two and a half times the median drawdown a live system produces and deeper than the unluckiest tenth of them. And it will not call it alive at +15.6R, which is exactly what a live one was supposed to deliver. At the point where a real edge has done precisely what it promised, this test still has nothing to say about it.

One more line, and it is an identity rather than a coincidence. At 588.89 trades — the unrounded expectation, 14.7 months in — the dead line sits at exactly zero. The algebra is n times d over two against 2.9444 over d, and at n equal to twice 2.9444 divided by d squared those two quantities are the same number, so they cancel. Until month fifteen, a system that has made you nothing at all has not yet failed.

A drawdown measures your variance, not your edge.

Wald’s boundaries are an approximation, and an approximation can be checked rather than believed. Feed the test trades from a true edge of a tenth of an R, 20,000 runs, seed 20260903: it returns 19,092 alive and 908 dead, a false-dead rate of 4.5 per cent against the 5 per cent it was built for. Feed it trades from nothing at all: 19,035 dead and 965 alive, 4.8 per cent false-alive. The median verdict arrives at 435 trades and the mean at 545, both under Wald’s 589, because the expectation is pulled up by the minority of runs that wander for a long time.

Set that against lesson 19’s fixed-sample formula, which wanted 7.85 divided by the square of the edge-to-noise ratio. At a tenth of an R that is 785 trades, and what it bought was a four-in-five chance of noticing a real edge. Wald’s two lines buy nineteen in twenty in both directions and reach a median verdict in 435. Divide those: the sequential test is doing strictly better work on 55 per cent of the sample, and the reason is that it stops as soon as the answer is clear instead of waiting for a date somebody set in advance.

So draw both lines on your own equity curve tonight, before the drawdown, and stop reading the depth.

What this does not settle

That trades are independent and normally distributed. Neither is true of a real record, and the two failures point in opposite directions. Skew flatters the table: run lesson 18’s actual outcomes, +2R and −1R at 45 per cent, and the median worst drawdown over 200 trades is 9.00R, while the normal harness at the same mean and the same standard deviation gives 9.68R, an overstatement of 7.6 per cent, because a distribution whose winners are large and whose losers are small climbs out of holes faster than a symmetric one. Clustering points the other way and is probably larger: losses that arrive together dig deeper than losses that arrive scattered, and nothing on this page models that at all.

That the second decimal means anything. Two things put it in doubt, and they compound. One R is the standard deviation of your own trade outcomes, estimated from the same short record you are worried about, and on 156 observations its own standard error is about 5.7 per cent; every depth in the first table scales with it, so the 8.82 is 8.82 give or take half an R before anything else is counted. The simulation adds its own wobble: re-run that median at four other seeds and it lands on 8.84, 8.83, 8.75 and 8.84. Two decimals are printed because a table needs one format for every cell.

That the test tests the right hypothesis. Wald’s ratio compares two specific means, zero and a tenth of an R, and a real system is usually neither. Feed it trades from a true edge of 0.05 — half of what it is testing for, and still a perfectly real edge — and it splits almost exactly down the middle, calling 49.91 per cent of them alive and 50.09 per cent dead on a median of 672 trades. The test is not malfunctioning there. It was never asked about that case, and a reader who runs it at a tenth of an R has quietly agreed to treat a halved edge as a coin toss.

That waiting is available, and this is the concession that costs most. Everything above says the honest answer takes 589 trades and 14.7 months, and that the dead line at the 156-trade horizon sits at 21.6R down. A trader who is 21.6R down at two per cent a trade has lost about a third of the account, and the arithmetic on this page tells them to keep going. It cannot tell them how, because whether you can hold is a question about position size and about the money, not about the evidence. Lesson 20 is where the answer to the problem this lesson diagnoses actually lives, and all this page can do is say that the diagnosis and the treatment are different subjects.

And the one that undercuts the table itself: it is indexed by the true edge, which is the single quantity you never have. You cannot look up your own row. Read forwards, the table is unusable. Read backwards it says only that no depth you could currently be at rules any row out. That is a thinner result than five columns of figures make it look, and it is the honest version of what a drawdown tells you.

Which leaves the case this page has kept setting aside, because it is neither of the two hypotheses the test compares. Lesson 68 takes an edge that does not die but decays. Feed the test a tenth of an R fading linearly to nothing over exactly the 589 trades it needs, and it declares the system dead 65.46 per cent of the time and alive 34.53 per cent, on a median of 627 trades — so by the time the verdict lands, whichever way it lands, the thing it was testing has already gone.