Quant from Scratch by QuantWright

Introduction to Quant Finance · Lesson 24

Capstone: read a strategy claim

One invented sales page, five claims, and the tools of course A: what each number can show, and what it leaves out.

Know the basics? Read the lesson notes

A sales page, written for this lesson

This last lesson of course A teaches nothing new. The page below is invented: it was written for this lesson, and its trading system (also called a strategy), a fixed set of rules for when to buy and sell, belongs to no real product, firm or person. A computer simulated (made up) the system's full trade list: the result of each of its 400 trades, in order. A trade is one buy and the sale that ends it.

An invented sales page

Example System 17

  1. Wins 7 trades in 10.
  2. Backtested on 400 trades over two years.
  3. Made €2,000 on a €10,000 account.
  4. Lowest point: only €300 below the starting balance.
  5. Risk four times as much per trade and it would have made €8,000.

A backtest runs a system over past prices to see what it would have made. All five claims are accurate: each matches the trade list. But the page leaves out what each trade costs to make; the lesson assumes a cost and says so.

A guess first. Once a cost is taken off every trade, how did the €10,000 account end the two years?

Keep your guess in mind. The answer comes with claim 3.

Claim 1: the win rate

Claim 1 of the invented page"Wins 7 trades in 10."

The share of trades that win is the win rate: here 70%. What does this claim leave out, so that it can't yet say whether the system makes money?

How long a trade lasts doesn't decide what it makes. The sizes do: a win rate counts the wins and says nothing of how big they were.
A win rate counts the wins and says nothing of how big they were.
That is in the claim: 7 in 10 won, so 3 in 10 lost. The sizes are missing: a win rate counts the wins and says nothing of how big they were.

The trade list has the sizes: every win made €50 and every loss took €100. So ten typical trades, 7 wins and 3 losses, make 7 × €50 − 3 × €100 = +€50, which is +€5 a trade.

This average result of one trade is the expectancy (lesson A1). It is positive, but thin. One loss takes 2 wins to pay for, so the system must win 2 trades in every 3, or 66.7%, for its average trade to make nothing: its break-even win rate. Its 70% is 3.3 points above that (a point is one percentage point).

Claim 2: the number of trades

Claim 2 of the invented page"Backtested on 400 trades over two years."

Is 3.3 points more than luck? A backtest's trades are a sample: the trades that happened to come up, out of all the system could make. Luck moves a sample's win rate away from the system's true win rate, the one it would settle at over very many trades. Lesson A10 measured how far: for samples of n trades, about 95 in 100 land within 100 ÷ √n points of the true win rate. That range is the band: a rule of thumb when one trade doesn't change the next one's chances, the true win rate is about 30% to 70%, and there are a few dozen trades or more. (√400, the square root of 400, is 20: 20 × 20 = 400.)

Picture a system with the same €50 wins and €100 losses but no edge: its average trade makes nothing, so its true win rate is the break-even 66.7%. Over 400 trades, where does its band reach?

100 ÷ √400 = 100 ÷ 20 = 5 points. So 66.7 ± 5: from 61.7% to 71.7%.
That is 1 point either side, which takes 10,000 trades (√10,000 = 100). For 400 trades: 100 ÷ √400 = 100 ÷ 20 = 5 points, so from 61.7% to 71.7%.
That is 20 points either side, the band for 25 trades (√25 = 5). For 400 trades: 100 ÷ √400 = 100 ÷ 20 = 5 points, so from 61.7% to 71.7%.

The claim's 70% is inside that band. So this trade list can't tell Example System 17 from a system that makes nothing. In the lesson's simulation, 80 of 1,000 trade lists from such a system won 280 trades or more (280 is 70% of 400).

That is "can't tell", not "no edge": the system was built with a true win rate of 70%, a small edge before costs.

Claim 3: the result

Claim 3 of the invented page"Made €2,000 on a €10,000 account."

The sum is right: 400 trades × €5 = €2,000. But costs are not in it: the trade list holds each trade's result before costs.

From lesson A16: every trade pays transaction costs, win or lose: a fee, and prices a little worse than the ones shown. What is left after costs is the net result.

Slide the cost of one trade up from €0. Find the cost that leaves nothing of the €2,000, then go on to €7.

The same 400 trades at the cost you set (simulated)

Grey: the account before costs. Orange: after costs. Dashed: its €10,000 start.

Average trade after costs€5 − €0 = +€5
Total over 400 trades+€2,000
Account ended at€12,000

At €5 a trade the cost equals the average trade, and nothing is left: 400 × (€5 − €5) = €0.

This lesson assumes a cost of €7 a trade. That makes the average trade €5 − €7 = −€2, and 400 × −€2 = −€800: the account ended at €9,200. So the answer to the opening guess was "down".

Claim 4: the lowest point

Claim 4 of the invented page"Lowest point: only €300 below the starting balance."

Play the 400 trades. Grey is the account before costs, as the invented page counts it; orange, after the €7 cost.

The account over the 400 trades (simulated)

Dashed: the €10,000 start. Circles, at the end: each line's peak and the low after it. First grey circle: the lowest value before costs.

Trade0 of 400
Before costs€10,000
After costs€10,000

Before costs the claim is accurate: the account's lowest value was €9,700.

But lesson A17 measures a fall from the peak, the highest value so far, and not from the start. Before costs the account climbed to €11,550 and then fell to €10,500. That is €1,050 down, and 1,050 ÷ 11,550 = 9.1% of the peak. A fall from a peak is a drawdown, and the deepest one is the maximum drawdown.

After costs the maximum drawdown ran from €10,217 down to €8,330: 1,887 ÷ 10,217 = 18.5%.

Claim 5: four times the risk

Claim 5 of the invented page"Risk four times as much per trade and it would have made €8,000."

A trade's risk is what it loses when it loses: here €100. At four times the risk every loss takes €400 and every win makes €200, so before costs the €2,000 becomes 4 × €2,000 = €8,000, as the claim says.

Before costs the deepest fall was €1,050. What would it have been at four times the risk?

The trades are the same, but every loss in that fall is four times as big: 4 × €1,050 = €4,200.
Every loss in that fall is four times as big, not twice: 4 × €1,050 = €4,200.
Every loss in that fall is four times as big, like every win: 4 × €1,050 = €4,200.

From lesson A18: risking more makes every result bigger, wins and losses alike. In euros, four times the risk gives four times the fall, which the claim leaves out. As a percent of the peak it grows less, because the peak grows too: the same fall, at four times the risk, is 25.9% of its peak (€4,200 from €16,200), against 9.1%.

Per euro risked nothing changes: +€5 on €100 risked is +€20 on €400, the same 5% of the risk. So for trades like these, which don't change each other's chances and keep the same win rate, win and loss, risking more can't add an edge. Nor can raising the risk after losses (lesson A19), for the same reason. A risk limit, a loss set in advance at which trading stops (lesson A20), caps what can be lost and, by stopping trading, changes which trades happen. It adds no edge to the ones taken.

Theory

The tools for the average trade

The tools from claims 1 to 3.

E = p × W − (1 − p) × L
p* = L ÷ (W + L)
average net total = n × (E − c)
band ≈ p ± 100 ÷ √n points
E
the expectancy (lesson A1): the average result of one trade, before costs
p
the win rate: a share in E and p* (0.7 for 70%), with 1 − p the loss rate; a percent in the band (66.7, not 0.667)
W, L
the average win and the average loss, as positive sizes
p*
the break-even win rate (lesson A1): the win rate at which E = 0
c
the cost of one trade, paid win or lose (lesson A16)
n
the number of trades
band
the range that holds about 95 sample win rates in 100, for samples of n trades from a system whose true win rate is p (lesson A10)
Worked example: Example System 17 (invented)

E = 0.7 × 50 − 0.3 × 100 = +€5. p* = 100 ÷ (50 + 100) = 66.7%. Average net total = 400 × (5 − 7) = −€800. Band for a true win rate of 66.7%: 100 ÷ √400 = 5 points, so 61.7% to 71.7%.

Holds when
  • one trade's result doesn't change the next one's chances, and p, W and L stay the same throughout;
  • every trade pays the same cost;
  • for the band: p is the true win rate, between about 30% and 70%, with at least a few dozen trades.
Breaks when

A backtest's p, W and L come from one sample: they are estimates, not the true values (lesson B5, Sampling error and confidence intervals).

New numbers. A second invented page says: "Wins 45% of its trades. Average win €120, average loss €80. 500 trades." At a cost of €6 a trade, what is its average net total, in euros?

E = 0.45 × 120 − 0.55 × 80 = 54 − 44 = +€10. After costs: 10 − 6 = €4 a trade, and 500 × 4 = €2,000. Its break-even win rate is 80 ÷ (120 + 80) = 40%, and 45% is 5 points above it.
E = 0.45 × 120 − 0.55 × 80 = 54 − 44 = +€10. After costs: 10 − 6 = €4 a trade, and 500 × 4 = €2,000. If you got 5,000, you left out the costs: 500 × €6 = €3,000. Its break-even win rate is 80 ÷ (120 + 80) = 40%, and 45% is 5 points above it.
Theory

The tools for the fall and the risk

The tools from claims 4 and 5.

DDt = (peakt − Vt) ÷ peakt
MDD = the largest DDt
fixed amount: Vt = Vt−1 + A × Rt
Vt
the account's value after trade t; V0 is its starting value
peakt
the highest value up to trade t, the starting value included
DDt
the drawdown after trade t (lesson A17), as a decimal: 0.20 is 20%
MDD
the maximum drawdown (lesson A17)
A
the fixed amount risked on every trade (lesson A18)
Rt
the result of trade t as a multiple of its risk (lesson A18): −1 for a loss of the whole risk, +0.5 for a win of half of it
Worked example: Example System 17 (invented)

Before costs, the largest DD = (11,550 − 10,500) ÷ 11,550 = 9.1%, the MDD. Risk: a win is R = +0.5 and a loss R = −1. With A = €100 a win adds 100 × 0.5 = €50. With A = €400 a loss takes 400 × 1 = €400, so the deepest fall is four times as big in euros (€4,200) but 25.9% of its higher peak.

Holds when
  • no money is added or taken out, and every trade's value is counted;
  • for the fixed amount: a loss is never bigger than the risk set for it.
Breaks when

Values far apart, such as year-ends, miss a deeper fall in between.

New numbers. A third invented page gives its starting balance and then five year-end values: €20,000, €25,000, €21,000, €24,000, €19,000, €23,000. It says: "Lowest point: only 5% below the starting balance." What is the maximum drawdown of the six values, in percent?

The peak is €25,000 and the lowest value after it €19,000: (25,000 − 19,000) ÷ 25,000 = 6,000 ÷ 25,000 = 24%. Against the €20,000 start that is 1,000 ÷ 20,000 = 5% down: the claim is accurate and leaves out the fall from the peak.
The peak is €25,000 and the lowest value after it €19,000: (25,000 − 19,000) ÷ 25,000 = 6,000 ÷ 25,000 = 24%. If you got 16, you stopped at €21,000; the fall went on to €19,000. If you got 5, you measured from the start: €19,000 is 1,000 ÷ 20,000 = 5% below it, so the claim is accurate and leaves out the fall from the peak.

What course A can't check

Every check so far took the trade list as given. Three things can make a trade list look better than its system is, and none of them shows in the list's own numbers. Course B, Backtesting Essentials, teaches each one, and course E, Testing Popular Ideas, uses them on well-known trading ideas. Turn them over.

  1. How many versions were tried.

    If many versions of a system are backtested and only the best is shown, part of its result is luck. Lesson B7, The best of many.

Practice 1 of 3

"Wins 60% of its trades"

A new invented claim"Wins 60% of its trades." The page also says: 64 trades, and every win and every loss is the same size.

With wins and losses of the same size, the break-even win rate is 50%. Could a system with no edge, truly at 50%, show 60% over 64 trades? (√64 = 8.)

Ten points is a lot over thousands of trades, but not over 64. 100 ÷ √64 = 100 ÷ 8 = 12.5 points either side of 50%: from 37.5% to 62.5%. 60% is inside, so 64 trades can't tell. That is not the same as "no edge".
100 ÷ √64 = 100 ÷ 8 = 12.5 points either side of 50%: from 37.5% to 62.5%. 60% is inside, so 64 trades can't tell. That is not the same as "no edge".
More trades make the band narrower, so 60% gets harder for a system with no edge, not easier. Over 64 trades: 100 ÷ √64 = 100 ÷ 8 = 12.5 points either side of 50%, from 37.5% to 62.5%. 60% is inside, so 64 trades can't tell. That is not the same as "no edge".
Practice 2 of 3

"Risk €200 a trade and it makes €12,000"

A new invented claim"Made €3,000 with a deepest fall of only €600, risking €50 a trade. Risk €200 a trade and it makes €12,000."

With the same trades at €200 a trade, what would the deepest fall have been, in euros?

Hint: how many times €50 is €200? Risking more makes every result bigger, wins and losses alike.

€200 is 4 times €50. The gain: 4 × 3,000 = €12,000, as the claim says. The deepest fall: 4 × 600 = €2,400, which the claim leaves out.
€200 is 4 times €50. The gain: 4 × 3,000 = €12,000, as the claim says. The deepest fall: 4 × 600 = €2,400, which the claim leaves out. If you got 600, you kept the fall as it was: the losses grow with the risk too.
Practice 3 of 3

"Losses are capped"

A new invented claimYou check an invented page and find that its system's average trade is −€3 after costs. The page adds: "A safety limit stops trading once the account is €1,000 down. Losses are capped, so over time the system comes out ahead."

What can that limit do?

The limit caps what can be lost and, by stopping trading, changes which trades happen. Every trade taken before it stops still averages −€3, so "comes out ahead" doesn't follow: a limit adds no edge to the trades taken.
More trades at −€3 each add up to a bigger loss, until the limit stops them. A limit stops trading; it adds no edge to the trades taken.
It does change one thing: trading stops at €1,000 down, so the loss is capped. What it can't change is the average trade, which stays −€3.
✓

Lesson done

A claim can be accurate and still leave out the number that decides it. A win rate needs the sizes of the wins and losses. A result needs its trade count, its costs and its fall from the peak. A bigger risk multiplies the losses too. On the invented page, five accurate claims came with an average trade of −€2 after an assumed cost.

Key ideas

Trade list
the result of every trade a system made, in order
Look-ahead bias
a backtest using something not yet known when a trade was made (lesson B2)
Held-out data
prices kept aside while a system is adjusted, to check it on prices it never used (lesson B9)

Formulas

expectancy: E = p × W − (1 − p) × L  ·  break-even win rate: p* = L ÷ (W + L)

average net total = n × (E − c)  ·  band ≈ p ± 100 ÷ √n points

maximum drawdown: MDD = the largest (peakt − Vt) ÷ peakt  ·  fixed amount: Vt = Vt−1 + A × Rt

Conditions: trades that don't change each other's chances, with the same win rate, sizes and cost throughout; for the band, p the true win rate (about 30% to 70%) and a few dozen trades or more; no money added or taken out; a loss never bigger than its risk.

Lesson notes

A claim about a trading system can be accurate and still leave out the number that decides it. These notes ask five questions of one invented sales page, each with a tool from course A, then list what the tools cannot check.

The page was written for this lesson. Its trading system (also called a strategy), a fixed set of rules for when to buy and sell, is called Example System 17 and belongs to no real product, firm or person. A computer simulated its full trade list, the result of every trade in order: 400 trades, standing for two years, on a €10,000 account. Each of the page's five claims matches that list.

1. A win rate: how big are the wins and the losses?

The first claim is "Wins 7 trades in 10". The win rate is the share of trades that win: 280 of the 400, or 70%. It says nothing of how big the wins and losses were. In the trade list every win made €50 and every loss took €100.

The average result of one trade is the expectancy (lesson A1): win rate × average win − loss rate × average loss, here 0.7 × €50 − 0.3 × €100 = €35 − €30 = +€5. The break-even win rate, at which the average trade makes nothing, is average loss ÷ (average win + average loss): 100 ÷ 150 = 66.7%. The 70% is 3.3 points above it (a point is one percentage point).

2. A trade count: what could luck do?

The second claim is "Backtested on 400 trades over two years". A backtest runs a system over past prices to see what it would have made. Its trades are a sample: the trades that happened to come up, out of all the system could make. Luck moves a sample's win rate away from the system's true win rate, the one it would settle at over very many trades. For samples of n trades, about 95 in 100 land within 100 ÷ √n points of the true win rate (lesson A10). That range is the band: a rule of thumb for trades that don't change each other's chances, true win rates from about 30% to 70%, and at least a few dozen trades.

A system with no edge is one whose average trade makes nothing. With the same €50 wins and €100 losses its true win rate is the break-even 66.7%, and over 400 trades its band reaches 100 ÷ √400 = 100 ÷ 20 = 5 points either side: from 61.7% to 71.7%. The claimed 70% is inside, so the trade list can't tell this system from one that makes nothing: in a simulation of 1,000 trade lists from such a system, 80 won 280 trades or more.

That is "can't tell", not "no edge": the invented system was built with a true win rate of 70%, a small edge before costs.

3. A result: are the costs in it?

The third claim is "Made €2,000 on a €10,000 account". The sum is right, 400 × €5 = €2,000, but costs are not in it: the trade list holds each trade's result before costs.

Transaction costs are paid on every trade, win or lose: a fee, and prices a little worse than the ones shown. What is left after them is the net result (lesson A16). A cost of €5 a trade, equal to the average trade, leaves nothing. Assume a cost of €7 a trade: then the average trade after costs is 5 − 7 = −€2 and the net total is 400 × −€2 = −€800: the account ended at €9,200.

4. A lowest point: measured from where?

The fourth claim is "Lowest point: only €300 below the starting balance". Before costs that is accurate: the account's lowest value was €9,700. But a drawdown is a fall from the peak, the highest value so far, and not from the start (lesson A17). Before costs the account reached €11,550 and later fell to €10,500: 1,050 ÷ 11,550 = 9.1%. That was its maximum drawdown, the deepest fall from a peak. After costs the maximum drawdown ran from €10,217 to €8,330: 1,887 ÷ 10,217 = 18.5%.

5. A bigger risk: do the losses grow too?

The fifth claim is "Risk four times as much per trade and it would have made €8,000". A trade's risk is what it loses when it loses: here €100. At four times the risk a win makes €200 and a loss takes €400: risking more makes every result bigger, wins and losses alike (lesson A18). Before costs the €2,000 becomes €8,000, as claimed, and the deepest fall of €1,050 becomes 4 × 1,050 = €4,200, which the claim leaves out. As a percent of the peak it grows less, because the peak grows too: the same fall, at four times the risk, is 25.9% of its peak (€4,200 from €16,200), against 9.1%.

Per euro risked nothing changes: +€5 on €100 risked is +€20 on €400, the same 5% of the risk. So for trades like these, which don't change each other's chances and keep the same win rate, win and loss, risking more can't add an edge. Nor can raising the risk after losses (lesson A19), for the same reason. A risk limit, a loss set in advance at which trading stops (lesson A20), caps what can be lost and, by stopping trading, changes which trades happen. It adds no edge to the ones taken.

What these tools cannot check

All five checks take the trade list as given. Three things can make a trade list look better than its system is, and none of them shows in the list's own numbers.

The first is how many versions were tried: if only the best of many backtested versions is shown, part of its result is luck (lesson B7, The best of many). The second is whether the backtest saw the future: using something not yet known when a trade was made is look-ahead bias (lesson B2). The third is whether any data was held out: a system adjusted until it fits past prices has only shown that it fits them, and a fair check uses held-out data, prices kept aside and never used for the adjusting (lesson B9, In-sample, out-of-sample and walk-forward). Course B, Backtesting Essentials, teaches these checks, and course E, Testing Popular Ideas, uses them on well-known trading ideas.

Formula: the tools for the average trade

E = p × W − (1 − p) × L
p* = L ÷ (W + L)
average net total = n × (E − c)
band ≈ p ± 100 ÷ √n points
E
the expectancy (lesson A1): the average result of one trade, before costs
p
the win rate: a share in E and p* (0.7 for 70%), with 1 − p the loss rate; a percent in the band (66.7, not 0.667)
W, L
the average win and the average loss, as positive sizes
p*
the break-even win rate (lesson A1): the win rate at which E = 0
c
the cost of one trade, paid win or lose (lesson A16)
n
the number of trades
band
the range that holds about 95 sample win rates in 100, for samples of n trades from a system whose true win rate is p (lesson A10)
Holds whenBreaks when

A backtest's p, W and L come from one sample: they are estimates, not the true values (lesson B5, Sampling error and confidence intervals).

Formula: the tools for the fall and the risk

DDt = (peakt − Vt) ÷ peakt
MDD = the largest DDt
fixed amount: Vt = Vt−1 + A × Rt
Vt
the account's value after trade t; V0 is its starting value
peakt
the highest value up to trade t, the starting value included
DDt
the drawdown after trade t (lesson A17), as a decimal: 0.20 is 20%
MDD
the maximum drawdown (lesson A17)
A
the fixed amount risked on every trade (lesson A18)
Rt
the result of trade t as a multiple of its risk (lesson A18): −1 for a loss of the whole risk, +0.5 for a win of half of it
Worked example: Example System 17 (invented)

Risk: a win is R = +0.5 and a loss R = −1. With A = €100 a win adds 100 × 0.5 = €50. With A = €400 a loss takes 400 × 1 = €400, so the deepest fall is four times as big in euros (€4,200) but 25.9% of its higher peak.

Holds whenBreaks when

Values far apart, such as year-ends, miss a deeper fall in between.

One more worked example

Another invented claim, start to finish

An invented page says: "Wins 55% of its trades. 100 trades. Lowest point: only 10% below the starting balance." Its wins and losses are €80 each, a trade costs €10, and after costs its account stood at €5,000, €6,000, €5,400, €4,500 and €4,800. Take these five as all the values this toy account had; the first, €5,000, is its starting balance.

Sizes: E = 0.55 × 80 − 0.45 × 80 = 44 − 36 = +€8. Break-even: 80 ÷ (80 + 80) = 50%.

Luck: at a true 50%, the band over 100 trades is 100 ÷ √100 = 10 points either side, from 40% to 60%. The claimed 55% is inside, so 100 trades can't tell.

Costs: average net total = 100 × (8 − 10) = −€200, against +€800 before costs. The account's last value agrees: 5,000 − 200 = €4,800.

Fall: from the €6,000 peak to €4,500 is 1,500 ÷ 6,000 = 25%. Against the €5,000 start it is 500 ÷ 5,000 = 10%, as claimed.

Common mistakes

Check yourself

  1. A system wins 80% of its trades. Its average win is €20 and its average loss €100. What is its expectancy, and what is its break-even win rate?
    Answer

    E = 0.8 × 20 − 0.2 × 100 = 16 − 20 = −€4 a trade. Break-even: 100 ÷ (20 + 100) = 83.3%. Its 80% is below that.

  2. A system with no edge has wins and losses of the same size, so its true win rate is 50%. Where does its band reach over 100 trades?
    Answer

    100 ÷ √100 = 100 ÷ 10 = 10 points either side: from 40% to 60%.

  3. A trade list shows +€9 a trade before costs over 300 trades. Each trade costs €4. What is the average net total?
    Answer

    300 × (9 − 4) = 300 × 5 = €1,500.

  4. An account stood at €8,000, €10,000, €9,000, €7,500 and €9,500, with no value skipped. What is its maximum drawdown, and how far below its start did it get?
    Answer

    The peak is €10,000 and the lowest value after it €7,500: 2,500 ÷ 10,000 = 25%. Against the €8,000 start, €7,500 is 500 ÷ 8,000 = 6.25% down.

  5. Name the three things about a trade list that these tools cannot check.
    Answer

    How many versions were tried before this one was shown; whether the backtest used something not yet known when a trade was made (look-ahead bias); and whether any data was held out, kept aside and never used while the system was adjusted.

Try it yourself

Run the script. Every number in this lesson comes from one Python script with a fixed random seed (below). Change the win chance, the win and loss sizes or the cost, and see which of the five claims still match the trade list.

By hand, in a spreadsheet. Type a trade list down column A, starting in A2: 1 for a win, 0 for a loss. Put 10000 in B1 and in C1. In B2 put =B1+IF(A2=1,50,-100)-7, the account after costs. In C2 put =MAX(C1,B2), its peak so far. In D2 put =(C2-B2)/C2, its drawdown. Copy the three down. =MAX(D:D) is the maximum drawdown. Then change the 7.

The code behind every number (Python, seed 2026111017)
Loading…

Going further: the other 999 trade lists, and the next 400 trades

The simulation drew 1,000 trade lists from the same invented system and used the one with the middle number of wins. Before costs, 918 of the 1,000 accounts ended above their €10,000 start. After the €7 cost, 299 did: an average trade of −€2 means a loss on average, yet luck still put about 3 lists in 10 ahead after 400 trades.

So a trade list that ended with a gain after costs would not have shown that the system has an edge after costs. One list is one run of luck, whichever way it points.

Lesson A17 has one more tool for a worst stretch. If a system behaves the same after its backtest as during it, every stretch as long as the backtest is equally likely to hold the deepest drawdown. So trading on for k times the backtest's length gives a chance of about k ÷ (k + 1) that the deepest is still to come. For 6 more years after a 2-year backtest, k = 3 and the chance is about 3 ÷ 4 = 75%.

For the invented system's next 400 trades, k = 1 and the formula gives 50%. In the lesson's simulation, after costs, 517 of the 1,000 trade lists met a deeper fall in euros in their next 400 trades.

For the preciseSeed 2026111017, one numpy Generator, three draws of 1,000 lists × 400 trades: the system (a win with chance 0.70), a system at its break-even win rate (chance 2/3), and the system's next 400 trades. The list used is row 15: its 280 wins are the middle number of wins among the 1,000 lists (the 500th when sorted), and it is the lowest row with that many (36 lists share it). The middle 950 of the 1,000 break-even lists showed win rates from 62.3% to 71.0% (the rule of thumb says 61.7% to 71.7%), and the middle 950 lists of the system itself from 65.8% to 74.5%. With the €7 cost the break-even win rate is (100 + 7) ÷ 150 = 71.3% (lesson A16's formula), above the system's true 70%. Before costs the list's account was below its peak after 336 of its 400 trades, and after costs after 394. At twice the risk the deepest fall before costs was €2,100. At four times the risk, with four times the cost, the account would have fallen from €10,868 to €3,320, a maximum drawdown of 69.5%; it never reached €0. Counted as a percent of the peak, 573 of the 1,000 lists met a deeper drawdown in their next 400 trades, more than the 517 counted in euros: after costs the accounts shrink, so the same fall in euros is a bigger share of a lower peak. The list used here met a shallower one in its next 400 trades (11.6%).

Honest notes

Where this comes from

This lesson adds no new idea. Each tool's sources are listed in its own lesson; the main ones are: