Risk management cannot create an edge
Position sizing cannot turn a losing strategy into a winning one. It changes how fast the account moves and how dispersed the outcomes are; it does not change the sign of the expected result. At zero edge with costs, lowering the risk per trade makes passing a funded-account exam less likely, not more.
The identity
Let e be the gross edge per unit of risk and c the cost per unit of risk. For any sizing rule that only uses past information,
The sum of risk is always positive, so the sign of the expected profit is the sign of e − c. A sizing rule can make the sum larger or smaller; it cannot flip the sign. I checked this with six sizing rules at two cost levels over 100,000 simulated paths: the largest deviation from the identity was |z| = 1.01, well inside noise.
With a real edge, sizing matters a great deal: it sets how much of that edge you collect and how rough the ride is. It sets magnitude, not sign. "Risk management is useless" is the wrong lesson.
Break-even win rates
For a reward-to-risk ratio of R, the win rate needed to break even is (1 + c)/(1 + R). Without costs:
| Reward : risk | Break-even win rate |
|---|---|
| 1 : 1 | 50.0% |
| 1 : 2 | 33.3% |
| 1 : 3 | 25.0% |
| 1 : 5 | 16.7% |
Costs raise every row: the numerator becomes 1 + c.
Smaller risk at zero edge
A funded-account exam is a race between a profit target and a loss limit. In a simplified version, +10% target, −10% limit, 1:2 trades, zero gross edge and a cost of 0.05 of the risk per trade:
| Risk per trade | Chance of passing |
|---|---|
| 2% fixed | 41.5% |
| 1% fixed | 36.6% |
| 0.5% fixed | 26.3% |
| 0.25% fixed | 11.9% |
| 1% → 0.5% → 0.25% ladder | 15.1% |
With zero cost, every row is 48–51%. Costs are what make small risk expensive: the smaller each bet, the more bets it takes to reach the barrier, and each one pays the cost.
The cost of 0.05 of the risk is one value from a scan, not a measured figure for any instrument; the real number depends on the market and the stop distance. This is a single barrier, not a real two-step exam with daily limits.
Small risk magnifies the sign
The same barrier with a gross edge of +0.10 per unit of risk: passing becomes 56.0% at 2% risk and 87.1% at 0.25%. At zero gross edge the same two rows are 43.8% and 11.9%.
So "lower your risk" helps a trader with a positive net edge and hurts one without. The advice cannot tell which of the two you are.
The slow version
With zero gross edge (1:2, win rate one in three) and a cost of 0.05 of the risk per trade, the average balance after 1,000 trades, as a fraction of the starting balance:
| Risk per trade | Average balance after 1,000 trades |
|---|---|
| 2% | 0.37 |
| 1% | 0.61 |
| 0.5% | 0.78 |
| 0.25% | 0.88 |
The dial changes the speed of the decline, not its direction. This is an average; a single account can be luckier for a while.
Repeated attempts
At zero edge, buying more attempts makes at least one pass close to certain. With a 36.6% chance per attempt, the chance of at least one pass is 89.7% within 5 attempts and 98.9% within 10. That assumes independent attempts; the same trades placed in several accounts are not independent.
A pass after several attempts is what zero edge predicts, not evidence against it.
The story that explains every loss
A widely taught breakdown of trading success gives technical analysis 10% and psychology, risk and account management the other 90%. As it is usually told, "success needs strategy, risk management and psychology" leaves no outcome that would show it wrong: every loss can be assigned afterwards to one of the three.
That does not make psychology irrelevant. It means the claim, in that form, cannot be tested, and the identity above sets a limit on what the risk part of it can do.
Method: closed-form results checked by Monte Carlo simulation with fixed seeds. Code and outputs are in a private research repository and available on request.