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The math
Three pieces of arithmetic: what it takes to recover a loss, what frequent trading costs before any decision is made, and what an ordinary run of losses does to an account. None of them predict anything, which is exactly why none of them can be wrong.
How these fit together
These three formulas are one formula seen from three angles, and the angle is compounding. A loss compounds against the smaller balance that survives it, which is why recovery costs more than the fall. A cost compounds on every round trip, which is why frequency is expensive independently of whether the trades were right. A losing streak compounds too, which is why position size and not chart-reading is what usually decides whether an account survives. Every indicator in this reference is an attempt to estimate an edge that may not exist; these three numbers are known in advance and are the same for everyone.
- G = L / (1 − L) — a 30% fall needs +42.9%; a 60% fall needs +150%
- D = 1 − (1 − c)N — 0.10% a round trip, 250 round trips, 22.1% of the account gone
- p* = 1 / (1 + R) — at 2:1 you can be wrong two times in three and break even
- 1 − (1 − r)k — ten straight losses at 2% is an 18.3% hole, needing +22.4% back
- Costs are certain, edges are estimates — which is the whole argument for trading less often
- 0.5⁷ = 1 in 128 — a seven-loss row is the texture of a fair coin, not a broken method
The math of getting back to even
1Every figure on this page is a percentage, and percentages are not symmetric. A loss and the gain that undoes it are never the same number — and the gap between them widens the deeper the hole goes.
G — the gain required to return to the starting value
In percentage terms: G% = 100 × L% ÷ (100 − L%)
Why it happens
Percentage returns multiply rather than add. Losing and then recovering means (1 − L) × (1 + G) = 1, and solving that for G gives the formula above. The reason the two numbers differ is the base they are measured against: the loss is taken on the full amount, while the recovery is earned on what survived it. Every percent you gain afterwards is worth fewer dollars than every percent you lost, because there is less money left to earn it on.
The same arithmetic explains a smaller oddity: a 10% gain followed by a 10% loss does not return you to where you started. It leaves 99% — 1.10 × 0.90 = 0.99 — and the order makes no difference, because multiplication does not care.
Worked example
- Start$10,000
- Fall 30%10,000 × (1 − 0.30) = $7,000
- Gap to fill$3,000 — on a $7,000 base
- Gain needed0.30 ÷ 0.70 = 0.4286 → +42.9%
- Check7,000 × 1.4286 = $10,000
The 30% you lost takes a 42.9% gain to undo — half again as much work as the damage.
Scroll to take a bigger loss
Scroll on, or move across the curve, to read any level.
−0.5%+0.5%The whole curve, in numbers
| Loss taken | Gain required | |
|---|---|---|
| −5% | +5.3% | |
| −10% | +11.1% | |
| −20% | +25.0% | |
| −25% | +33.3% | |
| −30% | +42.9% | |
| −40% | +66.7% | |
| −50% | +100% | |
| −60% | +150% | |
| −70% | +233% | |
| −80% | +400% | |
| −90% | +900% |
Up to about 20% the two numbers stay close. Past 50% they part company completely: half your money gone needs a double to repair, and 90% gone needs a tenfold return.
Why it matters
This is the arithmetic behind every rule about cutting losses. A position allowed to fall 60% does not need a good year to recover — it needs a 150% return, which is far rarer than a 60% decline. It is also why professional risk management is stated as a fraction of capital per trade rather than a feeling: keeping each loss small keeps the required recovery inside the range of ordinary outcomes. The Volatility and statistics page covers the sizing side of that, where Average True Range is used to make every loss the same size.
What it costs to trade often
2Commissions are zero on most retail platforms now, which is not the same as trading being free. Every round trip still pays the spread and whatever the price moves between the decision and the fill. That charge is small, it is certain, and it compounds against you in the one direction costs always run.
N — round trips over the period
D — the share of capital consumed, before any gain or loss on the trades themselves
c = spread ÷ 2 + slippage + fees + financing
Why it compounds
Each round trip is charged on whatever capital is left after the last one, so the costs multiply in exactly the way returns do — and because every term is a loss, there is no year in which they happen to cancel out. A tenth of one percent sounds like nothing next to a daily price range of two or three percent. Over 250 round trips, which is one a day, it is 1 − 0.999250 = 22.1% of the account, and it is charged whether the trades were right or wrong.
This is the mechanism behind the 11.4% figure on the base-rates page. The most active households in that study picked stocks about as well as the least active ones. What separated them by six and a half percentage points a year was N.
Worked example
- Spread$0.02 on a $100 stock = 0.02%
- Pay half of itentering, and half again exiting
- Slippage+0.08% on a market order
- c≈ 0.10% per round trip
- One a day1 − 0.999²⁵⁰ = 22.1%
To finish the year flat, that account has to produce a 28.4% gross return — because a 22.1% hole needs 28.4% to fill, by the formula above.
Drag by cost and frequency
| Cost per round trip | 10 | 50 | 100 | 250 | 500 |
|---|---|---|---|---|---|
| 0.02% | 0.2% | 1.0% | 2.0% | 4.9% | 9.5% |
| 0.05% | 0.5% | 2.5% | 4.9% | 12% | 22% |
| 0.10% | 1.0% | 4.9% | 9.5% | 22% | 39% |
| 0.25% | 2.5% | 12% | 22% | 47% | 71% |
| 0.50% | 4.9% | 22% | 39% | 71% | 92% |
Read across a row to see what frequency costs at a fixed spread; read down a column to see what a wider spread costs at a fixed frequency. Options, penny stocks and thin small caps live in the bottom rows, not the top ones.
Why it matters
This is the only number on the site that is known in advance. A strategy’s edge is an estimate with error bars around it; its cost is arithmetic. That asymmetry is why the first question worth asking about any method is not whether it wins more often than it loses, but whether its average win clears c by enough to be worth the variance — and why a method that trades ten times less often needs a far smaller edge to end up ahead.
The arithmetic of a losing streak
3Two numbers decide whether a method can work at all, and neither of them is the win rate on its own. The first is how much a winner pays relative to a loser. The second is how deep an ordinary run of bad luck goes before it is over.
p* — the win rate at which the method breaks exactly even
Drawdown after k straight losses at r risk each: 1 − (1 − r)k
Why the win rate alone tells you nothing
Expected value per trade is p × W − (1 − p) × L. Setting that to zero and writing R = W / L gives the formula above, and it says something most trading talk gets backwards: a method that wins 40% of the time is profitable at 2:1 and ruinous at 1:2, while a method that wins 70% of the time is ruinous at 1:3. The win rate is one of two inputs and is meaningless quoted alone.
The second half is about survival rather than expectation. Losing streaks are not rare events; they are what a fair coin looks like up close. At an even win rate, a run of seven losses has a probability of 0.5⁷, which is one in 128 — so a trader taking a position a day should expect to live through one about twice a year. The method does not have to be broken for that to happen, and the account still has to be intact afterwards.
How often a fair coin does this
- 3 losses in a row — 0.53 = 1 in 8, so roughly about 31 times in 250 trades
- 5 losses in a row — 0.55 = 1 in 32, so roughly about 8 times in 250 trades
- 7 losses in a row — 0.57 = 1 in 128, so roughly about 2 times in 250 trades
- 10 losses in a row — 0.510 = 1 in 1024, so roughly once every 4 years in 250 trades
None of these are signs that anything has gone wrong. They are the expected texture of a 50% win rate, and any position size that cannot absorb the seven-loss row is sized for a market that does not exist.
What a streak costs, and what it takes back
| Risked per trade | 5 in a row | 10 in a row | 15 in a row | 20 in a row |
|---|---|---|---|---|
| 1% | −4.9%needs +5.2% | −9.6%needs +10.6% | −14.0%needs +16.3% | −18.2%needs +22.3% |
| 2% | −9.6%needs +10.6% | −18.3%needs +22.4% | −26.1%needs +35.4% | −33.2%needs +49.8% |
| 5% | −22.6%needs +29.2% | −40.1%needs +67.0% | −53.7%needs +116% | −64.2%needs +179% |
The second line in each cell is the break-even gain from the first panel, applied to the hole the streak dug. At 5% a trade, twenty consecutive losses is a 64% drawdown needing a 179% return — which is why the professional convention lands between 0.5% and 2%, and why it is a fraction of capital rather than a dollar figure.
The win rate you need, by reward-to-risk
| Reward : risk | Break-even win rate | |
|---|---|---|
| 0.5 : 1 | 66.7% | |
| 1 : 1 | 50.0% | |
| 1.5 : 1 | 40.0% | |
| 2 : 1 | 33.3% | |
| 3 : 1 | 25.0% | |
| 5 : 1 | 16.7% |
Everything above 1:1 needs a majority of winners; everything below it does not. This is the arithmetic that makes a 30%-win-rate trend method and an 80%-win-rate mean-reversion method equally sensible as descriptions — and equally unproven as predictions.
Why it matters
These two formulas are the part of trading that is genuinely solved. Which way price goes next is not known and may not be knowable; what a given bet size does to an account over a run of losses is fully determined and takes thirty seconds to work out. Almost every account that ends at zero does so through the second problem rather than the first — not because the reading of the chart was wrong, but because the position was sized as though a seven-loss row could not happen. Average true range, on the Volatility and statistics page, is the usual tool for turning a chosen r into a number of shares.