A Positive Expected Value Does Not Cap Position Size. Compounding Arithmetic Does.

The gain required to undo a loss is always larger than the loss. That is not a psychological observation, it is a division problem. An account down 20% needs +25.00% on what remains. Down 50%, it needs +100.00%. The two figures separate because the loss is measured against the old balance while the recovery has to be earned on the smaller one that survived it.

Most written sizing rules stop one step short of that. They check whether the expected value of a trade is positive, and whether one loss at the chosen size is tolerable. Both checks can pass while the size that passed them still carries the account into a region the arithmetic above does not let it leave at any ordinary pace. The value that goes uncalculated is not the risk on the next trade. It is the compounded cost of a run of them, and the size of the gain that run then demands.

What follows is that arithmetic, computed rather than asserted, together with the one place in United States equity rules where the same arithmetic runs into a fixed dollar threshold instead of a soft one. No risk fraction is recommended anywhere below. The point is the shape of the curve, not a number to adopt from it.

The Recovery Cost Is a Division Problem

Take one balance and one loss. If an account falls by a fraction d of its peak, the equity that remains is 1 - d of the peak. Returning to the peak means growing that remainder by the missing amount, d, expressed as a fraction of what survived. So:

r = d / (1 - d)

Every value in the table below comes from that one expression. Nothing is estimated and nothing is sampled.

Drawdown d	Equity remaining	Gain required	Required gain as a multiple of the loss
5%	95%	+5.26%	1.05x
10%	90%	+11.11%	1.11x
20%	80%	+25.00%	1.25x
30%	70%	+42.86%	1.43x
40%	60%	+66.67%	1.67x
50%	50%	+100.00%	2.00x
60%	40%	+150.00%	2.50x
75%	25%	+300.00%	4.00x
90%	10%	+900.00%	10.00x

Two features of that table matter more than the individual entries. The first is that the penalty is mild for a long stretch and then stops being mild. Between a 5% and a 10% drawdown the required gain runs about 5% to 11%, close enough to the loss itself that treating them as interchangeable costs almost nothing. By 30% the required gain is already 1.43 times the loss. By 60% it is 2.50 times.

The second feature is that the curve has no kink in it. There is no threshold where the arithmetic changes character and warns anyone. It simply bends, continuously, and the bend is easiest to miss precisely in the range where a trading account spends most of its time.

The dashed line in that chart is worth a moment. It shows what a linear reading of the same loss would imply: lose 40%, make 40% back. That line is where intuition sits by default, and the vertical gap between it and the solid line is the part of the cost that never appears in a per-trade risk calculation.

Ten Losses in a Row Is an Ordinary Number

The recovery curve only becomes a sizing constraint once it is paired with a realistic estimate of how far down a normal losing run reaches. Assume a fixed-fractional scheme, meaning the risk is a constant percentage of current equity rather than of the starting balance, which is how most position-size formulas are written. Ten consecutive losses then leave (1 - f)^10 of the account.

Risk per trade	Equity after 10 straight losses	Drawdown	Gain required
1%	90.44%	9.56%	+10.57%
2%	81.71%	18.29%	+22.39%
5%	59.87%	40.13%	+67.02%
10%	34.87%	65.13%	+186.80%

Extend the run to twenty and the separation widens further: 81.79% of equity remains at 1%, 66.76% at 2%, 35.85% at 5%, and 12.16% at 10%. The required gains at twenty losses are +22.26%, +49.79%, +178.95% and +722.53% respectively.

Whether ten in a row is a fair thing to plan around depends on the win rate. Treating trades as independent draws with a fixed probability of winning, the chance that a run of ten consecutive losses appears somewhere inside a given number of trades can be computed exactly rather than guessed:

Win rate	Within 100 trades	Within 250	Within 500	Within 1,000
60%	0.6%	1.5%	3.0%	6.1%
55%	1.7%	4.4%	8.8%	17.0%
50%	4.4%	11.2%	21.5%	38.5%
45%	10.1%	24.4%	43.3%	68.2%
40%	20.5%	45.2%	70.5%	91.5%

A trend-following or breakout approach that wins 40% of the time and relies on large winners to carry the expectation has better than a two-in-three chance of hitting a ten-loss run inside 500 trades. At the same 40% win rate over 500 trades, a run of at least six consecutive losses is effectively certain and a run of at least eight arrives with 97.1% probability. These are not tail events being dragged in to make a point. They are the ordinary texture of a low win rate strategy.

The independence assumption is the weak part of the calculation, and it does not fail in a helpful direction. Correlated positions, a single volatility regime, and one repeated setup all push real losing runs longer than the independent case, not shorter.

Expected Value and Survival Are Two Different Calculations

The reason this stays uncalculated is that a different calculation already appears to have settled the question. Expected value per trade, in units of the amount risked, is p*b - q where p is the win rate, b the payoff on a winner and q the loss rate. Scale the risk fraction and the expectation scales with it, in exact proportion. The sign never moves.

Compound growth does not behave that way. The per-trade growth rate of the account is g = p*ln(1 + b*f) + q*ln(1 - f), and the loss term in that expression falls toward negative infinity as f approaches 1. That single asymmetry, the same one visible in the recovery table, changes the shape of the whole function.

A worked case makes the gap concrete. Take a system with a 40% win rate and winners worth 2.5 times the amount risked. Expected value per trade is 0.4 x 2.5 - 0.6 x 1 = +0.40, a positive number at every position size from the smallest to the largest. Now put the same three inputs into the growth expression. Median compound growth peaks at a risk fraction of 16.0% and crosses zero at approximately 33.1%. Above that fraction the account is expected to shrink over time while the expected value per trade sits, unchanged, at +0.40.

Both statements describe the same system. Neither is wrong. They answer different questions: expectation asks what the average trade contributes, and growth asks what the sequence does to a balance that carries forward. Only the second one is aware that a loss removes capital the next trade will be sized from.

Those two fractions, 16.0% and 33.1%, are the output of assumed inputs and should be read as arithmetic rather than as guidance. They assume the win rate and the payoff are known exactly and stay fixed. In live trading neither holds. A win rate estimated from a limited sample carries a wide confidence interval, and the growth curve is steeply penalising on the high side of its peak and forgiving on the low side, which means the direction in which an estimation error hurts is not symmetric either.

Where the Arithmetic Meets a Fixed Dollar Threshold

Percentages are abstract until an account has a floor underneath it that is denominated in dollars. In United States equity markets there is one.

According to the SEC's investor education materials, a pattern day trader is defined under FINRA rules as any customer who executes four or more day trades within five business days, provided those day trades represent more than six percent of the customer's total trades in the margin account over that same five business day period. A customer classified this way must have at least $25,000 in the account and may trade only in a margin account. A broker-dealer may also apply the classification proactively where it has a reasonable basis to believe a customer will trade this way.

Only those figures are stated here, because those are the ones checked against a regulator's own published material. Other details about the rule that circulate widely, including buying-power multiples and the exact timing of when the equity must be present, are left out rather than repeated on secondhand authority.

What the threshold does to the arithmetic is turn a drawdown percentage into a count of trades. An account starting at $30,000 sits 16.7% above the $25,000 figure. Applying a fixed-fractional risk to that account, the number of consecutive losses that carries it below the threshold is a matter of solving for the first n where 30,000 x (1 - f)^n < 25,000.

At 1% per trade, that takes 19 straight losses, ending at $24,785. At 2%, ten losses, ending at $24,512. At 5%, four losses, ending at $24,435. Four consecutive losses is not a losing streak in any meaningful sense. On the table above, a 40% win rate produces a run of at least four losses many times over the course of a few hundred trades.

The consequence is not only the drawdown. An account below the threshold that has been classified as a pattern day trader loses the ability to keep placing the trades that were supposed to earn the drawdown back. The recovery percentage from the first table still applies, but the method of earning it has been restricted at exactly the moment it is needed. That is a compounding of two separate constraints, and neither of them appears in a per-trade expected value calculation.

What Would Invalidate This

Several conditions make the framing above the wrong lens.

• Fixed-dollar rather than fixed-fractional risk. All of the streak arithmetic assumes risk is a percentage of current equity. An account risking a constant dollar amount decays linearly, not geometrically, and the numbers in the streak tables do not apply to it. The recovery curve d/(1-d) still does, because that expression is a property of the balance, not of the sizing scheme.
• A short and terminating horizon. The gap between expectation and compound growth only opens up over a sequence. A trader placing a small, fixed number of trades and then withdrawing the balance is closer to the expected value case than to the growth case.
• Outside capital or regular deposits. If the balance is replenished from income on a schedule, the account is not a closed compounding system and the recovery burden is shared with the deposits.
• Non-independent, non-stationary trades. The streak probabilities assume independent draws from a fixed win rate. Real results cluster by regime. The correct conclusion from this is that the tables understate the risk of long runs, not that they overstate it, but a trader with a genuinely regime-conditional model may be able to compute something better than the independent case.
• Cash accounts and non-margin structures. The $25,000 figure applies to the pattern day trader classification in a margin account. It is not a general minimum for holding or trading securities.
• Asymmetric loss sizes. The streak tables assume a full-size loss on each of the ten trades. A method whose exits regularly cut the loss to a fraction of the planned risk will not follow those rows, though the effective risk fraction in that case is simply a smaller number, not a different curve.

Concrete Framework

1. Compute the recovery cost of the drawdown already taken. Take the current balance against the account's peak, get d, and calculate d/(1-d). This is a two-second division that produces the actual size of the hole rather than the remembered size of it.
2. Fix a drawdown ceiling before choosing a size, not after. Decide what value of d the account and the operator can still work from, and read the required gain off the table. The required gain is the number to react to, since it is the one that has to be earned.
3. Convert that ceiling into a losing-run count. Solve (1 - f)^n for the risk fraction under consideration and find the n that reaches the ceiling. If that count is a number the strategy produces routinely, the ceiling and the size are inconsistent with each other.
4. Estimate the strategy's realistic run length from its own win rate. Use the streak probability table as a floor, not a forecast. A 40% win rate strategy that has never seen eight losses in a row has probably not yet traded enough.
5. Check both calculations, not one. Expected value confirms the edge exists. Compound growth confirms the size does not consume it. A sizing rule that only ever produces the first number has not tested the constraint that binds.
6. Locate any hard threshold beneath the account. The $25,000 pattern day trader minimum is one example, but margin maintenance levels, platform minimums and withdrawal commitments work the same way. Measure the distance to the nearest one in consecutive losses rather than in percent.
7. Re-run the arithmetic after every equity change, not on a calendar. Fixed-fractional sizing recalculates the position automatically; the drawdown ceiling and the loss count to the nearest threshold do not. Those move with every closed trade and go stale quietly.
8. Record the assumed win rate and payoff alongside the chosen size. The whole calculation rests on those two inputs. When realised results drift away from them, the size that was derived from them is no longer the size that was intended, and the drift is only visible if the original assumption was written down.

None of this makes a losing run less likely. It changes what the run costs, which is the only part of the sequence that position sizing controls at all.