Quant from Scratch by QuantWright

Introduction to Quant Finance · Lesson 20

Rules as risk limits

Daily loss caps, drawdown caps and prop-firm rules: what a limit that stops trading does to an account, and what it can't do.

Know the basics? Read the lesson notes

A limit that stops trading

A risk limit, also called a cap, is set before trading starts: when losses reach a set line, trading stops. Reaching the line is a breach.

A simulated account starts with €10,000 and takes five trades a day. Its trading rule, the way it picks its trades, is made up. Every trade risks a fixed €100: a win adds €100, a loss takes €100. It wins 55 trades in 100, so 100 trades make 55 × €100 − 45 × €100 = €1,000: +€10 a trade on average, its edge.

Its risk limit is a maximum loss of €1,000: at €9,000 or below, the account stops for good. Play its 60 days.

One account with a maximum loss of €1,000 (simulated)

Orange: the account. Grey: trades it never took. Dark: the €9,000 line. Dashed: the €10,000 start.

Day0 of 60
Account€10,000
Above its line€1,000

On the third trade of day 8 the account reached €9,000 and stopped: ten more losses than wins.

The trades it never took would have carried it down to €8,700, then back up to €9,800 by day 60. The limit can't know what comes next.

Of 1,000 such accounts, the limit stopped 134.

A limit for one day

Swap the maximum loss for a daily loss limit of €300: once a day's loss, measured from the account's value when the day began, reaches €300, the account takes no more trades until the next day.

Does such a limit improve results? Test it on a trading rule with no edge: a coin flip that wins 50 trades in 100. 1,000 simulated coin-flip accounts traded for 60 days under it.

Of the trades they took, about how many in 100 won?

A losing run is only known afterwards. After three losses the next trade is still a coin flip. The accounts took 284,822 trades and won 141,993 of them: 49.9 in 100.
The accounts took 284,822 trades and won 141,993 of them: 49.9 in 100.
Every stopped day does end on a loss. But every other trade counts too: the accounts took 284,822 trades and won 141,993 of them, which is 49.9 in 100.

Static or trailing

A limit chooses which trades are taken, not how a trade ends. For trading rules like these, where a trade's chance never depends on earlier trades, that adds no edge: the coin-flip accounts won 49.9 trades in 100 with the daily limit and 49.9 without it. It caps a day's loss at €300; without it, five losses in a day lose €500.

The maximum loss so far was measured from the start: a static limit, whose line never moves. A trailing limit is measured from the account's peak, its highest value so far: its line sits €1,000 below the peak, follows the account up and never moves back down.

Another of the 1,000 accounts: switch its limit to trailing.

One account, two kinds of maximum loss (simulated)

Orange: the account. Grey: trades never taken. Dark: its line. Dotted: the static line.

The line€9,000, fixed
The accountended day 60 at €12,600

With the static limit this account ended day 60 at €12,600.

With the trailing limit it stopped on day 28. It had peaked at €12,000, so its line stood at 12,000 − 1,000 = €11,000. It fell back to that line while still €1,000 above its start.

The peak is never below the start, so the trailing line is never below the static one. It stops every account the static line stops, and more: in this simulation, at €100 a trade, 701 of the 1,000 within 60 days, against 134.

A made-up evaluation

A prop firm (here: a company that sells trading evaluations to individual traders, often called "funded trader" programmes; not the same as the proprietary trading firms that employ traders) sells an evaluation, often called a challenge. The trader pays a fee and trades an account the firm provides, often a simulated one, under fixed rules. Traders who meet them may be paid a share of the profit they make, on the firm's terms. Fees, refunds, rules and payouts differ between firms and change.

The rulesA made-up evaluation, invented for this lesson and not copied from any firm. Real evaluations use their own numbers and definitions, which can change, and some may look similar.
  • Profit target: reach €10,900, a gain of €900 (9%).
  • Maximum loss: €1,200 (12%), static, where the accounts so far had €1,000. At €8,800 or below the account is closed.
  • Daily loss limit: €500 (5%). A day that loses €500 or more closes the account for good. The earlier €300 limit only paused trading for the day.
  • Minimum trading days: 4. The account must trade on at least 4 days: the target can't count before day 4. Until then it keeps trading and can fall back.

1,000 accounts with the 55-in-100 edge take it, each trading five times a day at €100 a trade until it reaches the target or breaches a limit; the simulation stops at day 250. How many reach the target first?

Keep your guess in mind for the next screen.

1,000 accounts, four evaluation rules

Each square is one account, coloured by what happened first. Change one of the evaluation's rules at a time.

1,000 accounts with an edge in the made-up evaluation (simulated)

€100 a trade.

ShowingAll four rules
Target first735 of 1,000
Daily limit first235
Maximum loss first30

Under all four rules, 735 of the 1,000 accounts reached the target first. 235 breached the daily limit first and 30 the maximum loss.

In this simulation, at €100 a trade, the daily limit closed the most accounts: five losses in one day lose €500. Without it, 927 reached the target.

Without the minimum days: 736, barely a change. With the maximum loss trailing, €1,200 below the peak: 715. The trading rule won 55 in 100 throughout.

These counts belong to this invented evaluation, this simulated trading rule and this trade size, not to any real evaluation, whose rules, costs and traders' edges differ.

The same evaluation with no edge

Now 1,000 coin-flip accounts, with no edge, take the same evaluation at €100 a trade. How many reach the target first?

Keep your guess in mind for the next screen.

Luck, and the risk per trade

Look at the coin flip, then at a third made-up trading rule that wins 45 trades in 100: the edge is against it. Try each at €50 and €200 a trade.

1,000 accounts in the made-up evaluation, all four rules (simulated)

ShowingEdge, €100 a trade
Target first735 of 1,000
Daily limit first235
Maximum loss first30
Still trading at day 2500

With no edge, 419 of the 1,000 accounts reached the target first at €100 a trade, by luck alone: about 400 was the closest guess. Like every count here, that holds for this invented evaluation, this coin and this trade size, not for any real evaluation.

The risk per trade moved the counts, and which way depends on the edge. Accounts that reached the target first, at €50, €100 and €200:

  • edge for the trading rule: 994, 735, 439
  • no edge: 547, 419, 258
  • edge against: 36, 114, 141

With the edge we gave the trading rule, the smallest risk reached the target most often; with the edge against, the largest did. A smaller trade needs more trades to cover €900, so the edge, for or against, acts over more trades.

With only the target and the maximum loss, size barely moved the no-edge count: 547, 577 and 529. Here the daily limit closes more accounts as the trade grows: five losses in a day reach it at €100, three at €200, and none can at €50 (five trades lose €250 at most).

This counts only how often the target was reached first, not what an account keeps afterwards. And the simulation knows each edge because we set it: the edge of a real trading rule is not known, only estimated (lesson A10). So this lesson names no size.

Theory

Where each line sits

A drawdown is how far an account has fallen from its peak (lesson A17). Some firms' rule lists call the two kinds of maximum loss static drawdown and trailing drawdown, though only the trailing one is measured from the peak.

static line = V0 − M
trailing line = peak − M
daily line = B − D
V0
the account's starting value
M
the maximum loss, in money
peak
the account's highest value so far, its starting value included; it never moves down
B
the account's value when the day began
D
the daily loss limit, in money
line
breached when the account, valued after every finished trade, is at or below it
Worked example: your trailing account

M = 1,000. At the start the peak was 10,000, so both lines sat at 10,000 − 1,000 = €9,000. At its peak of 12,000 the trailing line stood at 12,000 − 1,000 = €11,000. In the evaluation, a day that begins at €10,300 has its daily line at 10,300 − 500 = €9,800.

Holds when
  • no money is added or taken out;
  • the trailing line keeps following the peak (some firms stop it rising once it reaches the starting value).
Breaks when

A firm may measure another way: on equity (trades still running counted) or on balance (finished trades only), all day or only at the day's end. The same trades can then breach under one definition and not another.

An account starts at €50,000 with a trailing maximum loss of €3,000. It climbs to €54,200, falls to €52,000 and climbs back to €53,500. Where is its line now?

The peak so far is €54,200, and the line never moves down: 54,200 − 3,000 = €51,200.
The peak so far is €54,200, and the line never moves down: 54,200 − 3,000 = €51,200. If you got 50,500, you measured from where the account is now. If you got 47,000, that is the static line.
Theory

No edge: which line comes first?

Take a trading rule with no edge and only two ways to stop: a target above the start, or a static maximum loss below it. Which comes first is the classic gambler's ruin problem. Write P for the share of accounts that reach the target first; the rest, 1 − P, reach the maximum loss first. No edge means each trade makes nothing on average, and stopping at a line set in advance can't change that. So all the accounts together still average zero when they stop: what the winners gain equals what the losers lose: P × T = (1 − P) × M. Move P × M to the left, P × (T + M) = M, and divide by T + M:

P = M ÷ (T + M)
P
the share of accounts that reach the target first, as a decimal (0.75 is 75 in 100), for a trading rule with no edge and no other way to stop
T
the gain that reaches the target, in money
M
the loss that reaches the maximum loss, in money
Worked example: the evaluation with two rules

Keep only the target and the maximum loss: T = 900, M = 1,200. P = 1,200 ÷ (900 + 1,200) = 1,200 ÷ 2,100 = 0.571: about 571 accounts in 1,000. Run that way, the simulation counted 577 coin-flip accounts at the target: 6 more, an ordinary chance difference between runs.

Holds when
  • the trading rule has no edge and no costs;
  • every trade risks the same amount, one that divides the gain and the loss, so accounts stop exactly on a line;
  • the target and a static maximum loss are the only ways to stop.
Breaks when

The trading rule has an edge, pays trading costs (lesson A16, Costs), or faces more limits.

A trading rule with no edge risks a fixed amount per trade. Its account stops at a gain of €500 or a loss of €2,000. About how many accounts in 100 stop at the gain?

P = 2,000 ÷ (500 + 2,000) = 2,000 ÷ 2,500 = 0.80: 80 in 100. With these made-up numbers, four accounts in five stop at the gain, with no edge.
P = 2,000 ÷ (500 + 2,000) = 2,000 ÷ 2,500 = 0.80: 80 in 100. Check: 80 × €500 = €40,000 gained, 20 × €2,000 = €40,000 lost. If you got 20, you put the gain on top: that is the share that stops at the loss.

What this means for your backtest

A backtest runs a trading rule over past prices to see what it would have made. Two points.

  1. A limit is checked on the path (the account's value day by day), not on the summary.

    A final gain and a maximum drawdown can't show whether a daily limit was breached on the way.

Practice 1 of 3

"I have €2,600 of room"

The situationA friend's made-up account started at €25,000 with a trailing maximum loss of €2,000. It has been as high as €27,400 and now stands at €25,600. "My line is at €23,000, so I have €2,600 of room," they say.

How many euros is the account above its line?

Hint: trailing line = peak − M.

The line follows the peak: 27,400 − 2,000 = €25,400. The account is 25,600 − 25,400 = €200 above it. €23,000 would be a static line.
The line follows the peak: 27,400 − 2,000 = €25,400. The account is 25,600 − 25,400 = €200 above it. €23,000 would be a static line: 25,000 − 2,000, which gives the friend's €2,600.
Practice 2 of 3

"78 accounts in 100"

The situationA made-up seller writes: "I ran my trading rule on 100 accounts, with the same risk on every trade. Each stopped at a gain of €500 or a loss of €1,500. 78 stopped at the gain. That is proof of an edge."

With no edge at all, about how many accounts in 100 would stop at the gain?

Hint: P = M ÷ (T + M).

P = 1,500 ÷ (500 + 1,500) = 1,500 ÷ 2,000 = 0.75: 75 in 100. The seller's 78 is 3 more. Each account is one try, and lesson A10's margin of error for 100 tries is about 100 ÷ √100 = 10 points either side: no proof of an edge.
P = 1,500 ÷ (500 + 1,500) = 1,500 ÷ 2,000 = 0.75: 75 in 100. The seller's 78 is 3 more. Each account is one try, and lesson A10's margin of error for 100 tries is about 100 ÷ √100 = 10 points either side: no proof of an edge. If you got 25, you put the gain on top.
Practice 3 of 3

Two numbers, two limits

The situationYour own research: a backtest on a €10,000 account ended up 12%, with a maximum drawdown of 7% measured after every trade.

Did it stay inside a static maximum loss of €1,000 and a daily loss limit of €500?

The account was never more than 7% below a peak, and a peak is never below the start. So it never fell below 93% of €10,000: €9,300, above the €9,000 line. The daily limit needs the path: the deepest fall was 7% of a peak of €10,000 or more, so at least €700, perhaps in one day.
The ending says nothing about the days on the way. The maximum loss did hold: the account was never more than 7% below a peak, and a peak is never below the start, so it never fell below €9,300. But its deepest fall was 7% of a peak of €10,000 or more, so at least €700, and it could have come in one day.
One of them is known. The account was never more than 7% below a peak, and a peak is never below the start, so it never fell below €9,300: the static maximum loss held. The daily limit does need the path: the deepest fall was 7% of a peak of €10,000 or more, so at least €700, and it could have come in one day.
✓

Lesson done

A risk limit sets how much a day or an account can lose. In this lesson's simulation it did not change the share of trades won. In its made-up evaluation at €100 a trade, 735 of 1,000 accounts with a 55-in-100 edge reached the target first; so did 419 of 1,000 with no edge.

Key ideas

Risk limit (cap)
a line set before trading starts: when losses reach it, trading stops; reaching it is a breach
Maximum loss
a risk limit on the whole account: at its line the account stops for good
Static limit
a maximum loss measured from the starting value; its line never moves
Trailing limit
a maximum loss measured from the peak; its line follows the account up, never down
Daily loss limit
a risk limit on one day's loss, measured from the value when the day began
Evaluation (challenge)
a paid test sold by a prop firm (a company selling trading evaluations to individual traders): an account the firm provides, often simulated, traded under fixed rules
Gambler's ruin problem
which an account reaches first, a set gain or a set loss

Formulas

static line = V0 − M  ·  trailing line = peak − M  ·  daily line = B − D

P = M ÷ (T + M)

Lines: M and D in money, the account valued after every finished trade. P: the share of accounts that reach a gain T before a loss M, for a trading rule with no edge or costs, the same risk on every trade and no other way to stop.

Lesson notes

A risk limit, also called a cap, is set before trading starts: when the account's value reaches a line set in advance, trading stops. It sets how much a day or an account can lose. In the lesson's simulation, where a trade's chance never depends on the trades before it, a limit did not change the share of trades won. These notes show what limits do to made-up accounts, not which limits to set or how much to risk.

Two pairs are easy to blur. A trading rule is how an account picks its trades; an evaluation's rules are the target and limits a firm sets. And a breach can pause trading for the day, or close the account for good: the notes say which each time.

A limit stops trading

Reaching a limit's line is a breach. A maximum loss is a limit on the whole account: at or below its line, the account stops for good.

The lesson's simulated accounts start with €10,000 and take five trades a day. Their trading rule, the way they pick their trades, is made up. Every trade risks a fixed €100: a win adds €100, a loss takes €100. It wins 55 trades in 100, so 100 trades make 55 × €100 − 45 × €100 = €1,000: +€10 a trade on average, its edge.

With a maximum loss of €1,000 the line is at €9,000. One account reached it on the third trade of day 8, ten more losses than wins, and stopped. Of 1,000 such accounts, the limit stopped 134 within 60 days. A limit can't know what comes next: the trades that first account never took would have carried it down to €8,700 and back up to €9,800 by day 60.

A limit for one day, and what it can't do

A daily loss limit works on one day: the loss is measured from the account's value when the day began. The lesson tried one of €300: once a day's loss reaches €300, the account takes no more trades until the next day. It was tested on a trading rule with no edge, a coin flip that wins 50 trades in 100, still risking €100 on every trade.

Over 60 days, 1,000 coin-flip accounts under that limit won 141,993 of the 284,822 trades they took: 49.9 in 100, and 49.9 in 100 without the limit.

A limit chooses which trades are taken, not how a trade ends. For trading rules like these, where a trade's chance never depends on earlier trades, that adds no edge. It caps a day's loss at €300; without it, five losses in a day lose €500.

Static and trailing

A maximum loss measured from the start is a static limit: its line never moves. A trailing limit is measured from the account's peak, its highest value so far: its line sits a set amount below the peak, follows the account up and never moves back down. It is a limit on drawdown, how far an account has fallen from its peak (lesson A17).

A second simulated account, with a €1,000 maximum loss, shows the difference. With the static limit it traded all 60 days and ended at €12,600. With the trailing limit it stopped on day 28: it had peaked at €12,000, so its line stood at 12,000 − 1,000 = €11,000, and it fell back to that line while still €1,000 above its start.

The peak is never below the start, so the trailing line is never below the static one. The trailing limit stops every account the static limit stops, and more: in the lesson's simulation, at €100 a trade, 701 of the 1,000 accounts within 60 days, against 134.

A made-up evaluation

A prop firm (a company selling trading evaluations to individual traders) sells an evaluation, often called a challenge. The trader pays a fee and trades an account the firm provides, often a simulated one, under fixed rules. Traders who meet them may be paid a share of the profit they make, on the firm's terms. Fees, refunds, rules and payouts differ between firms and change.

The lesson's evaluation is made up, invented for the lesson and not copied from any firm. Real evaluations use their own numbers and definitions, which can change, and some may look similar. Its four rules: a profit target of €900 (reach €10,900); a static maximum loss of €1,200 (line at €8,800; the accounts above had €1,000); a daily loss limit of €500, where a day that loses €500 or more closes the account for good (the €300 limit above only paused trading for the day); and a minimum of 4 trading days (the account must trade on at least 4 days: the target can't count before day 4, and until then it keeps trading and can fall back).

Each account trades five times a day until it reaches the target or breaches a limit; the simulation stops at day 250. At €100 a trade, 735 of 1,000 accounts with the edge reached the target first, 235 breached the daily limit first and 30 the maximum loss. With one of the evaluation's rules changed at a time, 927 reached the target without the daily limit (at €100 a trade, five losses in one day lose €500), 736 without the minimum days, barely a change, and 715 with the maximum loss trailing, €1,200 below the peak. The trading rule won 55 in 100 throughout.

These counts belong to this invented evaluation, this simulated trading rule and this trade size. They do not carry over to any real evaluation, whose rules, costs and traders' edges differ.

No edge, and the risk per trade

The same evaluation was run on 1,000 coin-flip accounts. In this simulation, at €100 a trade, 419 of them reached the target first, by luck alone, against 735 with the edge. So in this made-up evaluation, reaching the target was weak evidence of an edge: an account at the target could be either kind. That shows a target and an edge are different things, not how often either turns up in a real evaluation.

The risk per trade moved the counts, and which way depends on the edge. A third made-up trading rule wins 45 trades in 100: the edge is against it. The counts below belong to this invented evaluation, these simulated trading rules and these trade sizes, not to any real evaluation. Accounts that reached the target first at €50, €100 and €200 a trade:

With the edge the lesson gave the trading rule, the smallest risk reached the target most often; with the edge against, the largest did. A smaller trade needs more trades to cover €900, so the edge, for or against, acts over more trades.

With only the target and the maximum loss, size barely moved the no-edge count: 547, 577 and 529. In the full evaluation the daily limit closes more accounts as the trade grows: five losses in a day reach it at €100, three at €200, and none can at €50 (five trades lose €250 at most).

This counts only how often the target was reached first, not what an account keeps afterwards. And the simulation knows each edge because the lesson set it: the edge of a real trading rule is not known, only estimated (lesson A10). So the lesson names no size.

A backtest runs a trading rule over past prices to see what it would have made. A limit is checked on the path (the account's value day by day), not on the summary: a final gain and a maximum drawdown (the deepest drawdown) can't show whether a daily limit was breached on the way.

Formula: where each line sits

A drawdown is how far an account has fallen from its peak (lesson A17). Some firms' rule lists call the two kinds of maximum loss static drawdown and trailing drawdown, though only the trailing one is measured from the peak.

static line = V0 − M
trailing line = peak − M
daily line = B − D
V0
the account's starting value
M
the maximum loss, in money
peak
the account's highest value so far, its starting value included; it never moves down
B
the account's value when the day began
D
the daily loss limit, in money
line
breached when the account, valued after every finished trade, is at or below it
Worked example: the second account

M = 1,000. At the start the peak was 10,000, so both lines sat at 10,000 − 1,000 = €9,000. At its peak of 12,000 the trailing line stood at 12,000 − 1,000 = €11,000. In the evaluation, a day that begins at €10,300 has its daily line at 10,300 − 500 = €9,800.

Holds whenBreaks when

A firm may measure another way: on equity (trades still running counted) or on balance (finished trades only), all day or only at the day's end. The same trades can then breach under one definition and not another.

Formula: no edge, which line comes first

Take a trading rule with no edge and only two ways to stop: a target above the start, or a static maximum loss below it. Which comes first is the classic gambler's ruin problem. Write P for the share of accounts that reach the target first; the rest, 1 − P, reach the maximum loss first. No edge means each trade makes nothing on average, and stopping at a line set in advance can't change that. So all the accounts together still average zero when they stop: what the winners gain equals what the losers lose: P × T = (1 − P) × M. Move P × M to the left, P × (T + M) = M, and divide by T + M:

P = M ÷ (T + M)
P
the share of accounts that reach the target first, as a decimal (0.75 is 75 in 100), for a trading rule with no edge and no other way to stop
T
the gain that reaches the target, in money
M
the loss that reaches the maximum loss, in money
Worked example: the evaluation with two rules

Keep only the target and the maximum loss: T = 900, M = 1,200. P = 1,200 ÷ (900 + 1,200) = 1,200 ÷ 2,100 = 0.571: about 571 accounts in 1,000. Run that way, the simulation counted 577 coin-flip accounts at the target, at €100 a trade in the invented evaluation: 6 more, an ordinary chance difference between runs.

Holds whenBreaks when

The trading rule has an edge, pays trading costs (lesson A16, Costs), or faces more limits (419 of 1,000 coin-flip accounts at €100 a trade in this simulation of the invented evaluation).

One more worked example

Made-up round numbers

An account starts at €20,000 and risks €300 on every trade. It has a trailing maximum loss of €1,500 and a daily loss limit of €600.

Day 1: three wins in a row, so the account ends at 20,000 + 3 × 300 = €20,900, a new peak. The trailing line moves up to 20,900 − 1,500 = €19,400.

Day 2 begins at €20,900, and its first two trades lose: €20,600, then €20,300. The day's loss is 20,900 − 20,300 = €600, which reaches the limit: a breach of the daily limit, with the account still €900 above its trailing line. If the limit is the trader's own, as with the lesson's €300 limit, trading pauses until the next day. In the lesson's evaluation the same breach would close the account for good.

Separately: a trading rule with no edge and no costs, the same €300 at risk on every trade, stops at a gain of €1,200 or a loss of €1,800, with no other rule. P = 1,800 ÷ (1,200 + 1,800) = 1,800 ÷ 3,000 = 0.60: 60 accounts in 100 stop at the gain. Check: 60 × €1,200 = €72,000 gained and 40 × €1,800 = €72,000 lost.

Common mistakes

Check yourself

  1. An account starts at €30,000 with a maximum loss of €2,500. Its highest value so far is €33,000. Where is the line if the limit is static, and where if it is trailing?
    Answer

    Static: 30,000 − 2,500 = €27,500. Trailing: 33,000 − 2,500 = €30,500.

  2. A day begins with the account at €12,400, and the daily loss limit is €500. At what value is the limit breached?
    Answer

    A day's loss is measured from the value when the day began: at 12,400 − 500 = €11,900 or below.

  3. Why does a trailing limit stop every account that a static limit of the same size stops?
    Answer

    The peak is never below the start, so the trailing line is never below the static line. An account at or below the static line is at or below the trailing line too.

  4. A trading rule with no edge stops at a gain of €1,000 or a loss of €1,000, with no other rule. What share of accounts stops at the gain?
    Answer

    P = 1,000 ÷ (1,000 + 1,000) = 0.50: 50 in 100.

  5. In the lesson's simulation, 1,000 coin-flip accounts traded under a €300 daily loss limit. What did the limit change, and what did it leave alone?
    Answer

    It changed how many trades were taken and the most a day could lose: €300, against €500 without it. It left the share of trades won alone: 49.9 in 100 with the limit and without.

Try it yourself

Run the script. Every number in this lesson comes from one Python script with a fixed random seed (below). Change the target, the limits, the trades per day or the chance of a win and see what moves.

By hand, in a spreadsheet. Type a list of results down column A, starting in A2: 100 for a win, -100 for a loss. Put 10000 in B1. In B2 put =B1+A2, the account. In C2 put =MAX(B$1:B2)-1000, the trailing line. In D2 put =IF(B2<=C2,"breach",""). Copy all three down. For a static line, put 9000 in every row of column C.

The code behind every number (Python, seed 2026111016)
Loading…

Going further: a rule that loses on average

The lesson's third trading rule wins 45 trades in 100 and so loses on average. In the made-up evaluation, under all four rules, 36 of 1,000 accounts reached the target first at €50 a trade, 114 at €100 and 141 at €200.

For this trading rule the smallest risk per trade reached the target least often. A small risk needs many trades to cover the same distance, and for a losing trading rule every extra trade is a step down on average. With the edge it was the other way round: more trades gave the edge more time to show. That is a count of targets reached first, not of what an account keeps afterwards.

So the count at each risk per trade depends on the edge, and the order can flip with it. The lesson knows its three edges because it set them. For a real trading rule the edge can only be estimated, which is why these counts are not a guide to how much to risk.

For the preciseSeed 2026111016, one numpy Generator, three draws of 1,000 accounts × 1,250 trades: a win with chance 0.55, then 0.50, then 0.45. The account played first is row 8, the lowest row that the static line stops within 60 days; the second is row 0, the lowest row that the trailing line stops and the static line never does (567 accounts are like that, and 299 are stopped by neither). Of the 701 accounts the trailing line stopped, 437 were above their start when it did. Of the 134 accounts the static line stopped, 120 would have ended day 60 above €9,000 had they traded on. With the €300 daily limit the accounts with the edge won 54.9 trades in 100, and 54.9 without it; they ended at €12,847 on average with the limit and €12,947 without, because the limit skipped 10,960 trades of a trading rule that makes money on average. Had the 134 accounts the static line stopped traded on to day 60, 86 would have ended above their start and 48 at or below it. At €50 a trade the worst possible day loses €250, so the daily limit can't be breached; at €200, three losses in a day lose €600. In the evaluation the checks after each trade run in this order: maximum loss, daily limit, target. With no edge at €50 a trade, 412 accounts breached the maximum loss first and 41 were still trading when the simulation stopped at day 250 (2 with the trailing maximum loss); with the edge at €50, 1 was. Every run at €100 or €200, and every run of the third trading rule, was finished by then. At €200 with the edge, 321 accounts stood at or above the target before day 4 and had to keep trading; 259 of them were at the target again from day 4 on and 62 breached a limit first. Without the minimum days, 501 reached the target at €200. With only the target and the maximum loss, the textbook formula for a trading rule with an edge, (1 − rb) ÷ (1 − ra+b) with r = 45 ÷ 55, a steps up to the target and b steps down to the maximum loss (each rounded up to a whole step), gives 992, 924 and 787 accounts in 1,000 at €50, €100 and €200; the simulation counted 994, 927 and 773. For the coin flip under those two rules the counts were 547 (with 41 accounts still trading), 577 and 529. The formula gives 571 at €50 and €100. At €200 the trade does not divide the €900 target: an account passes it only at €11,000, five steps up against six down, and 6 ÷ 11 gives 545. Each count is one run of 1,000 accounts, so it carries luck of its own.

Honest notes

Where this comes from