Risk and position size
Drawdown and the arithmetic of recovery
Losses and gains are measured in percent but they do not cancel in percent, because they are applied to different balances. This asymmetry is the most important piece of arithmetic in trading and it takes one line to prove.
The one-line proof
Gain 10% and then lose 10%. Most people expect to be back where they started.
1,000 x 1.10 = 1,100 1,100 x 0.90 = 990 net -1.0%
The 10% gain was 100, applied to 1,000. The 10% loss was 110, applied to 1,100. The percentages were equal and the amounts were not, because a percentage is always a percentage of whatever the balance is at the time. Reverse the order and you get 990 again: 1,000 x 0.90 x 1.10 = 990. Multiplication does not care about the order, which is exactly why the asymmetry is not about luck.
The general form follows directly. After losing a fraction L you hold (1 - L) of what you had, so the gain g needed to get back satisfies (1 - L) x (1 + g) = 1:
g = L / (1 - L) L = 0.50 -> g = 0.50 / 0.50 = 1.00 = 100% L = 0.20 -> g = 0.20 / 0.80 = 0.25 = 25%
The table, and where it turns on you
Every row below is L / (1 - L). Check any of them on a calculator - the point of the table is that the right-hand column is not a rhetorical device.
down gain needed to get back the money left 5% 5.26% 95.0 10% 11.11% 90.0 20% 25.00% 80.0 30% 42.86% 70.0 40% 66.67% 60.0 50% 100.00% 50.0 60% 150.00% 40.0 70% 233.33% 30.0 80% 400.00% 20.0 90% 900.00% 10.0 per 100 of starting capital
Read down the second column and notice where it stops being linear. From 5% to 20% the required gain is close to the loss - a little worse, nothing dramatic. Around 30% the two columns separate. Past 50% the required gain is larger than everything you have left, and past 70% it is a different activity: 233% is not a recovery, it is a new career, started with 30% of the capital.
How a drawdown is actually built: consecutive losses
Nobody loses 50% in one trade if they size positions at all. Drawdowns are built out of runs, and runs of losses are completely ordinary - any sequence of independent outcomes produces them, and they arrive bunched rather than evenly spread.
Here is what ten consecutive full-risk losses do, at two different fractions of the account. The arithmetic is compounding, because each loss is a percentage of what is left after the previous one.
risking 2% of the account each time 0.98 ^ 10 = 0.8171 drawdown = 18.29% gain to recover = 0.1829 / 0.8171 = 22.39% risking 5% of the account each time 0.95 ^ 10 = 0.5987 drawdown = 40.13% gain to recover = 0.4013 / 0.5987 = 67.02%
The fraction more than doubled and the hole got nearly four times harder to climb out of, measured by the gain required. That is the asymmetry again, arriving through the back door: the cost of a larger fraction is not linear, and it is paid in the one scenario you cannot rule out - an ordinary run of losses.
Extend either column and the shape gets clearer. Twenty losses in a row at 2% leaves 0.98 ^ 20 = 0.6676, a 33.24% drawdown needing 49.79% to recover. At 5%, twenty losses leaves 0.95 ^ 20 = 0.3585, a 64.15% drawdown needing 178.94%. Check: 0.6415 / 0.3585 = 1.7894.
What this implies, stated carefully
This section is where a risk lesson usually tells you a number. This one does not, because the arithmetic above does not contain one. What it contains is a way to ask the question properly, and the question has three parts.
- How long a run of losses do you want to still be trading after? Pick the length honestly - longer than feels likely - and compute what it leaves. That is the computation, and its inputs are yours.
- What drawdown would make you stop? Not "would be bad" - would actually make you close the account or change everything. If a run of losses reaches that depth, the plan ended there regardless of what the arithmetic said afterwards.
- What drawdown would make you double up to get it back? This is the one people will not answer, and it is the one that empties accounts. The table above is the reason: at 50% down, the required gain genuinely is 100%, so the temptation to take more risk is a rational response to an impossible position. The place to fix it is before, in the fraction.
There is one more term the table leaves out, and it is unforgiving: a position liquidated on leverage can skip the rows entirely. Instead of walking down the table one loss at a time you arrive at the bottom of it in a single move. That is the next lesson.
The terms, defined
- Drawdown
- The fall from a high-water mark in the account, as a percentage of that high. Measured peak to trough, not from where you started.
- Recovery gain
- L / (1 - L). The gain on the reduced balance that returns you to the previous high, and it is always larger than the loss was.
- High-water mark
- The highest balance the account has reached. Drawdown is measured against it, which is why a drawdown can begin on a profitable day.
- Compounding
- Applying each result to the balance left by the previous one. It is what makes ten 5% losses a 40.13% hole rather than a 50% one, and what makes the recovery harder than the loss.
- Run of losses
- Consecutive losing trades. Ordinary in any sequence of uncertain outcomes, and the mechanism by which almost every real drawdown is built.
What this means in practice
- Memorise three rows: 20% needs 25%, 50% needs 100%, 70% needs 233%. They end most arguments about risk by themselves.
- Measure drawdown from the account high, not from your starting deposit.
- Compute what a long run of full-risk losses does to your account before you need the answer, using whatever fraction you have chosen.
- Treat the depth of a drawdown as the thing to control. Frequency is discomfort; depth is survival.
- Decide in advance what drawdown stops you, and write it down somewhere you will read it at the time.
- Never increase size to recover a drawdown. The table explains why the urge is logical and why acting on it is terminal.
Nothing in this lesson is financial advice or an instruction to trade. See the risk disclaimer. We never recommend a leverage setting - that decision is yours.