Quant from Scratch by QuantWright

Introduction to Quant Finance · Lesson 17

Drawdown

What you'd have sat through: how far an account fell from its high, how long it stayed down, and how likely it is that a deeper fall is still to come.

Know the basics? Read the lesson notes

Up 28%, and the ride there

A simulated account starts with €10,000 and follows a made-up trading rule. Each trade wins or loses 1% of the account. The rule wins 55 trades in 100, so it has an edge: an advantage that makes money on average. Play its first 250 trades.

One account, 250 trades (simulated)

The account after each trade. Dashed: its €10,000 start.

Trade0 of 250
Account€10,000
Highest so far€10,000

The account ended at €12,808, up 28.1%. The simulation ran 1,000 accounts on this same rule; they differ only by luck. This one is typical: it ended in the middle (the median account). Now look at the ride. At trade 157 the account stood at €13,801. Then it fell: 62 trades later it was down to €11,724.

Whoever trades this rule sits through that fall without knowing how it ends. This lesson measures it: how deep, and how long.

Peak, drawdown, trough

Two words first. The peak is the highest value the account has reached so far, its starting value included. The drawdown is how far the account is below that peak, as a percent of the peak. At a new peak the drawdown is 0%.

Here is a small made-up account at the end of each of ten months. Step through them.

Ten months of a made-up account

Top: the account, with its peak so far dashed. Bottom: the drawdown, hanging below 0%.

Month1 of 10
Account€1,000
Peak so far€1,000
Drawdown0%

The fall after month 3 reached its lowest point, its trough, in month 5: €960. That is €240 below the €1,200 peak, and 240 ÷ 1,200 = 20%: the drawdown's depth.

Its length is 4 months: months 4 to 7 all sat below €1,200, until month 8 set a new peak. The word drawdown is used both for the fall at one moment and for the whole spell below a peak.

A month below the peak is a month under water. Length counts one spell; time under water counts them all: 5 of the 10 months here (months 4 to 7, and month 9).

The maximum drawdown

The deepest drawdown in an account's whole history is its maximum drawdown: the number quoted for an account's worst stretch.

The ten months again: €1,000, €1,100, €1,200, €1,080, €960, €1,020, €1,140, €1,250, €1,125, €1,300. What is their maximum drawdown?

Peak €1,200, trough €960: 240 ÷ 1,200 = 20%. It built up over two months: €120 down, then €120 more.
That €125 fall is the biggest one-month fall in euros, and a drawdown of 10% (125 ÷ 1,250). But a drawdown can build over several months: €1,200 down to €960 is 240 ÷ 1,200 = 20%.
A drawdown is measured from the peak, wherever the account started. By month 3 the account had reached €1,200, so at €960 it was 240 ÷ 1,200 = 20% below its high. Against the start it looks like only 4%.

How often below its peak?

Back to the simulated account. Its rule has an edge, and it ended its 250 trades up 28.1%.

After how many of those 250 trades was the account under water, below its peak so far?

Keep your guess in mind. On the next screen you see all 250 trades.

Under water

Under the account's line is its underwater chart: after every trade, how far the account sat below its peak. Where it touches 0%, the account is at a new peak. Slide through the trades and find the deepest point.

The same account and its underwater chart (simulated)

Top: the account, with its peak so far dashed. Bottom: its drawdown after each trade.

Account€10,000
Peak so far€10,000
Drawdown now0%
Under water so far0 of 0 trades

The deepest point came at trade 219: €11,724, which is €2,077 below the €13,801 peak, and 2,077 ÷ 13,801 = 15.0%. The account was still below that peak when the 250 trades ended, 93 trades later, so this drawdown is at least 93 trades long.

Slide on to trade 250 for the full count: after 217 of its 250 trades, 87 in 100, the account was under water. Only 33 trades set a new peak.

Why so few, with an edge? A new peak needs a win that takes the account beyond its old high. This rule wins 55 trades in 100 and loses 45, so on average it ends only 10 wins ahead per 100 trades. Most wins just win back earlier losses: in the simulation this account won 138 of its 250 trades (55 in 100 would be 137.5), and only 33 of those wins set a new peak.

This is normal for this rule: of the 1,000 accounts, 935 were under water after at least 8 of every 10 trades. A rule with a stronger edge spends less of its time under water.

Is the worst behind it?

A backtest runs a trading rule over past prices to see what it would have made. Treat each account's first 250 trades as its backtest, with its own maximum drawdown.

Then every account trades on for 750 more trades. The rule doesn't change: the same edge, the same 1% a trade. In how many of the 1,000 accounts do those later trades bring a drawdown deeper than the backtest's maximum?

Keep your guess in mind. On the next screen the accounts trade on.

More time, more chances

Each square is one of the 1,000 accounts. From the end of its backtest on, the peak starts fresh: the account's value at trade 250 is its first peak, as for someone who starts trading then. The square turns orange once the account has met a drawdown deeper than its backtest's maximum. Slide the trades forward to 750.

1,000 accounts after their backtests (simulated)

Orange: a deeper drawdown than the backtest's has come. The squares are sorted, so the orange ones sit together.

Compared with the backtest–
Met a deeper drawdown0 of 1,000

After 250 more trades, as many as the backtest had, 500 of the 1,000 accounts had met a deeper drawdown. After 750 more, 794 had.

Here is why. The 250 trades after the backtest follow the same rule as the backtest's 250, and luck in one stretch doesn't carry into the other: the stretches are independent. So each is equally likely to hold the deeper drawdown: 1 chance in 2.

Now take 750 trades: three stretches of 250, four with the backtest. They follow the same rule and are independent, so each of the four is equally likely to hold the deepest drawdown of all. So the chance that the backtest holds it is 1 in 4, and the chance that the worst comes later is 3 in 4: 750 accounts in 1,000. The count is a little higher, 794, because within the 750 trades a drawdown can also run from one stretch of 250 into the next.

Theory

Drawdown and maximum drawdown

The drawdown at any moment is the fall from the highest value so far, as a share of that value. The maximum drawdown is the largest one in the account's history.

peakt = the highest of V0, V1, …, Vt
DDt = (peakt − Vt) ÷ peakt
MDD = the largest of DD1, DD2, …, DDn
time under water = (how many DDt are above 0) ÷ n
Vt
the account's value at step t: after month t, or after trade t
V0
the account's starting value, before the first step, when there is one: the €10,000 before trade 1
n
how many steps there are
peakt
the peak at step t: the highest value so far, the starting value and Vt included
DDt
the drawdown at step t, as a decimal: 0.20 is 20%. It is 0 at a new peak
MDD
the maximum drawdown over all n steps
Worked example: your ten months

The ten month-end values have no starting value before them, so the peak begins at V1 = 1,000. Month 5: the highest value so far is 1,200 and V5 = 960, so DD5 = (1,200 − 960) ÷ 1,200 = 240 ÷ 1,200 = 0.20. All ten drawdowns: 0, 0, 0, 10%, 20%, 15%, 5%, 0, 10%, 0. The largest is the MDD: 20%. Five of the ten are above 0, so the time under water is 5 ÷ 10 = 50%.

Holds when
  • each value is the whole account at that moment;
  • no money is added or taken out along the way: a deposit would hide a fall.
Breaks when

The values are far apart: month-end values miss a fall that came and went inside a month. And it is one number from one account's run of luck: other luck gives another (lesson B11, Monte Carlo).

New numbers. An account at the end of six weeks: €500, €800, €720, €600, €840, €798. What is its maximum drawdown, in percent?

Peaks so far: 500, 800, 800, 800, 840, 840. Drawdowns: 0, 0, 80 ÷ 800 = 10%, 200 ÷ 800 = 25%, 0, 42 ÷ 840 = 5%. The largest is 25%.
Peaks so far: 500, 800, 800, 800, 840, 840. Drawdowns: 0, 0, 80 ÷ 800 = 10%, 200 ÷ 800 = 25%, 0, 42 ÷ 840 = 5%. The largest is 25%. If you got 16.7, you measured from €720 to €600; the peak was €800.
Theory

Is the worst still to come?

A value more extreme than every one before it is called a record. A maximum drawdown is a record, and more trading gives more chances to break it. If the rule behaves the same after the backtest as during it:

chance the deepest drawdown is still to come ≈ k ÷ (k + 1)
k
how many times as long as the backtest the trading after it lasts: 750 trades after a 250-trade backtest is k = 3. At k = 1 the chance is 1 ÷ 2: a coin flip
k + 1
the number of equal stretches, the backtest included; each is equally likely to hold the deepest drawdown
≈
about equal: after the backtest, drawdowns that run from one stretch into the next push the chance a little higher
Worked example: your 1,000 accounts

250 trades after a 250-trade backtest: k = 1, so 1 ÷ (1 + 1) = 1 ÷ 2 = 50%. Measured: 500 of 1,000. 750 trades after: k = 3, so 3 ÷ (3 + 1) = 3 ÷ 4 = 75%. Measured: 794 of 1,000, a little more.

Holds when
  • the rule behaves the same in every stretch: the same edge, the same size of trade;
  • one stretch's luck doesn't carry into the next (the stretches are independent);
  • the stretches hold about the same number of trades.
Breaks when

The edge fades or the market changes after the backtest. Then the formula no longer applies. If the edge fades, a deeper drawdown is more likely than it says (lesson C3, Regimes and decay).

A backtest covers 2 years. The rule then trades for 8 more years without changing. About what is the chance that its deepest drawdown is still to come?

20% is the chance that the backtest holds the deepest: 1 stretch in 5. The 8 years are k = 8 ÷ 2 = 4 times the backtest, so the answer is 4 ÷ 5 = 80%.
50% would be right for 2 more years (k = 1). The 8 years are k = 8 ÷ 2 = 4 times the backtest: 4 ÷ (4 + 1) = 4 ÷ 5 = 80%.
k = 8 ÷ 2 = 4, so 4 ÷ (4 + 1) = 4 ÷ 5 = 80%.

What this means for your backtest

Three things follow from this lesson. Turn them over one at a time.

  1. Read a maximum drawdown with its length and its climb back.

    The simulated account's worst was 15.0% deep and at least 93 trades long. Getting from €11,724 back to €13,801 takes a gain of 2,077 ÷ 11,724 = 17.7%: more than the fall, because a gain is measured on what's left (lesson A14).

Practice 1 of 3

"It never dipped below its start"

The situationA course seller shows a made-up account at the end of five months: €10,000, €15,000, €12,000, €10,500, €13,500. "It never dipped below its starting balance," the page says.

What is the account's maximum drawdown, in percent?

Hint: find the peak so far at each month. Then (peak − value) ÷ peak, and take the largest.

The peak is €15,000 and the deepest point after it €10,500: (15,000 − 10,500) ÷ 15,000 = 4,500 ÷ 15,000 = 30%. The seller's sentence is true, and it hides this: someone who joined at €15,000 was down 30%.
The peak is €15,000 and the deepest point after it €10,500: (15,000 − 10,500) ÷ 15,000 = 4,500 ÷ 15,000 = 30%. If you got 20, you stopped at €12,000; the fall went on to €10,500. The seller's sentence is true, and it hides this: someone who joined at €15,000 was down 30%.
Practice 2 of 3

Below its peak after 264 of 300 trades

The situationYour own research: a backtest of 300 trades ended well up. Its underwater chart shows the account below its peak after 264 of the 300 trades, and you wonder whether the rule is broken.

What does that time under water tell you?

New peaks are rare even with an edge. 264 ÷ 300 = 88%, and the lesson's account, whose rule has an edge, was under water after 87 trades in 100.
264 ÷ 300 = 88%. The lesson's account, whose rule has an edge, was under water after 87 trades in 100, and 935 of the 1,000 accounts after at least 80 in 100.
Both happen together. The lesson's account ended up 28.1% and was under water after 217 of its 250 trades: a few new peaks are enough to end higher.
Practice 3 of 3

"The worst I'll see is about 9%"

The situationA friend's backtest covers one year and shows a maximum drawdown of 9%. They plan to trade the rule for the next five years. "The worst I'll see is about 9%," they say.

If the rule keeps behaving as in the backtest, about what is the chance that those five years bring a drawdown deeper than 9%?

Five years are k = 5 times the backtest: 5 ÷ (5 + 1) = 5 ÷ 6, about 83%.
50% would be right for one more year (k = 1). Five years are k = 5 times the backtest: 5 ÷ (5 + 1) = 5 ÷ 6, about 83%.
1 in 6 is the chance that the backtest's year holds the deepest drawdown of all six. The chance that it comes later is 5 ÷ 6, about 83%.
And that is the kind case, where the rule keeps its edge. If the edge fades, a deeper drawdown is more likely still.
✓

Lesson done

A drawdown is how far an account is below its peak, as a percent of the peak. The maximum drawdown is the deepest one. In the lesson's simulation, an account whose rule has an edge spent most of its time under water, because new peaks are rare. And if a rule behaves the same after its backtest as during it, the longer it trades on, the more likely a deeper drawdown: about k ÷ (k + 1) for k times as long.

Key ideas

Peak
the highest value an account has reached so far, its starting value included
Drawdown
how far an account is below its peak, as a percent of the peak; 0% at a new peak
Trough
the lowest point of a drawdown, before the account climbs back
Maximum drawdown
the deepest drawdown in an account's history: the largest fall from a peak to a later trough
Length of a drawdown
how many values in a row (months, trades) sat below its peak
Under water
below the peak so far; time under water is the share of an account's history spent there
Underwater chart
an account's drawdown drawn at every step, hanging below 0%
Record
a value more extreme than every one before it, such as a new deepest drawdown

Formulas

DDt = (peakt − Vt) ÷ peakt  ·  MDD = the largest DDt  ·  time under water = values below the peak ÷ n

chance the deepest drawdown is still to come ≈ k ÷ (k + 1)

Lesson notes

A drawdown is how far an account is below its highest value so far. In the lesson's simulation, an account whose rule has an edge was under water, below its peak, after 87 trades in 100; a rule with a stronger edge spends less time there. A backtest runs a trading rule over past prices to see what it would have made. If the rule behaves the same after its backtest as during it, the longer it trades on, the more likely a drawdown deeper than the backtest's deepest.

Peak, drawdown, trough

The peak is the highest value an account has reached so far; its starting value counts. The drawdown is how far the account is below that peak, as a percent of the peak. At a new peak it is 0%.

Take a made-up account at the end of ten months: €1,000, 1,100, 1,200, 1,080, 960, 1,020, 1,140, 1,250, 1,125 and 1,300. By month 3 the peak is €1,200. In month 5 it reaches its trough, its lowest point: €960, €240 below the peak, and 240 ÷ 1,200 = 20%. That is the drawdown's depth.

Its length is how many values in a row sat below the peak: 4 months here, months 4 to 7, until month 8 set a new peak at €1,250. The word drawdown is used both for the fall at one moment and for the whole spell below a peak; the deepest keeps its own name, the maximum drawdown.

A month below the peak is a month under water. Length counts one spell; time under water counts them all: 5 of the 10 months here (months 4 to 7, and month 9).

The maximum drawdown

The deepest drawdown in an account's whole history is its maximum drawdown: 20% for the ten months. It is measured from the peak, wherever the account started. Against the €1,000 start, €960 looks like a fall of only 4%. It is also deeper than the 10% drawdown left by the biggest one-month fall in euros, €125 (from €1,250 to €1,125), because a drawdown can build over several months.

Under water most of the time

The lesson's simulation runs a made-up rule on 1,000 accounts of €10,000, which differ only by luck. Each trade wins or loses 1% of the account, and the rule wins 55 trades in 100, so it has an edge: an advantage that makes money on average. The account that ended in the middle of the 1,000 (the median account) finished its first 250 trades at €12,808, up 28.1%.

Yet after 217 of those 250 trades, 87 in 100, it was under water, and it set a new peak on only 33 trades. Its maximum drawdown ran from €13,801 at trade 157 down to €11,724 at trade 219: 2,077 ÷ 13,801 = 15.0%. The account was still below that peak when the 250 trades ended, 93 trades later, so this drawdown is at least 93 trades long. Getting from €11,724 back to €13,801 takes a gain of 2,077 ÷ 11,724 = 17.7%: more than the fall, because a gain is measured on what is left (lesson A14).

A new peak needs a win that takes the account beyond its old high. A rule that wins 55 trades in 100 and loses 45 ends only 10 wins ahead per 100 trades, on average. Most wins just win back earlier losses: in the simulation this account won 138 of its 250 trades (55 in 100 would be 137.5), and only 33 of those wins set a new peak. That is normal for this rule: of the 1,000 accounts, 935 were under water after at least 8 of every 10 trades.

So a stretch below the peak (a stretch is a run of trades in a row) can't tell you, on its own, whether a rule still works.

Is the worst still to come?

Treat each account's first 250 trades as its backtest, and let every account trade on with the same rule. From the end of the backtest on, the peak starts fresh: the account's value at trade 250 is its first peak, as for someone who starts trading then.

After 250 more trades, 500 of the 1,000 accounts had met a drawdown deeper than their backtest's maximum. After 750 more trades, 794 had.

The reason: the stretches follow the same rule and are independent (luck in one doesn't carry into the next), so each is equally likely to hold the deepest drawdown. With 250 more trades there are two stretches: 1 chance in 2, and the count was 500. With 750 more there are four: the chance that the worst comes after the backtest is 3 in 4, or 750 accounts in 1,000. That count, 794, came out a little higher, because within the 750 trades a drawdown can also run from one stretch of 250 into the next.

All of this assumes the rule behaves the same after the backtest as during it. If it doesn't, these chances no longer apply; if the edge fades, a deeper drawdown is more likely than they say.

Formula: drawdown and maximum drawdown

peakt = the highest of V0, V1, …, Vt
DDt = (peakt − Vt) ÷ peakt
MDD = the largest of DD1, DD2, …, DDn
time under water = (how many DDt are above 0) ÷ n
Vt
the account's value at step t: after month t, or after trade t
V0
the account's starting value, before the first step, when there is one: the €10,000 before trade 1
n
how many steps there are
peakt
the peak at step t: the highest value so far, the starting value and Vt included
DDt
the drawdown at step t, as a decimal: 0.20 is 20%. It is 0 at a new peak
MDD
the maximum drawdown over all n steps
Worked example: the ten months

The ten month-end values have no starting value before them, so the peak begins at V1. Values: 1,000, 1,100, 1,200, 1,080, 960, 1,020, 1,140, 1,250, 1,125, 1,300. Peaks so far: 1,000, 1,100, 1,200, 1,200, 1,200, 1,200, 1,200, 1,250, 1,250, 1,300. Month 5: DD = (1,200 − 960) ÷ 1,200 = 240 ÷ 1,200 = 0.20. All ten drawdowns: 0, 0, 0, 10%, 20%, 15%, 5%, 0, 10%, 0. The largest is the MDD: 20%. Five of the ten are above 0, so the time under water is 5 ÷ 10 = 50%.

Holds whenBreaks when

The values are far apart: month-end values miss a fall that came and went inside a month. And it is one number from one account's run of luck: other luck gives another (lesson B11, Monte Carlo).

Formula: the chance that the worst is still to come

A value more extreme than every one before it is called a record. A maximum drawdown is a record, and more trading gives more chances to break it. The formula holds if the rule behaves the same after the backtest as during it.

chance the deepest drawdown is still to come ≈ k ÷ (k + 1)
k
how many times as long as the backtest the trading after it lasts: 750 trades after a 250-trade backtest is k = 3. At k = 1 the chance is 1 ÷ 2: a coin flip
k + 1
the number of equal stretches, the backtest included; each is equally likely to hold the deepest drawdown
≈
about equal: after the backtest, drawdowns that run from one stretch into the next push the chance a little higher
Worked example: the 1,000 accounts

250 trades after a 250-trade backtest: k = 1, so 1 ÷ (1 + 1) = 1 ÷ 2 = 50%. Measured: 500 of 1,000. 750 trades after: k = 3, so 3 ÷ (3 + 1) = 3 ÷ 4 = 75%. Measured: 794 of 1,000, a little more.

Holds whenBreaks when

The edge fades or the market changes after the backtest. Then the formula no longer applies. If the edge fades, a deeper drawdown is more likely than it says (lesson C3, Regimes and decay).

One more worked example

Made-up round numbers

An account at the end of eight months: €4,000, 5,000, 4,500, 3,000, 4,800, 5,200, 4,940 and 5,300.

Peaks so far: 4,000, 5,000, 5,000, 5,000, 5,000, 5,200, 5,200, 5,300.

Drawdowns: 0, 0, 500 ÷ 5,000 = 10%, 2,000 ÷ 5,000 = 40%, 200 ÷ 5,000 = 4%, 0, 260 ÷ 5,200 = 5%, 0.

Maximum drawdown: 40%, from the €5,000 peak to the €3,000 trough. Its length: 3 months (months 3 to 5).

Time under water: 4 of the 8 months, so 4 ÷ 8 = 50%.

The climb back from the trough: 2,000 ÷ 3,000, about 67%.

If these eight months were a backtest and the rule traded 16 more months, behaving the same: k = 16 ÷ 8 = 2, so the chance of a drawdown deeper than 40% is about 2 ÷ 3, or 67%.

Common mistakes

Check yourself

  1. An account's values: €2,000, €2,400, €1,800, €2,100. What is its maximum drawdown?
    Answer

    The peak is €2,400 and the trough €1,800: 600 ÷ 2,400 = 25%.

  2. For the same four values, what is the time under water?
    Answer

    €1,800 and €2,100 are both below the €2,400 peak: 2 of the 4 values, so 50%.

  3. A backtest covers 1 year. The rule then trades 5 more years and keeps behaving the same. About what is the chance of a drawdown deeper than the backtest's maximum?
    Answer

    k = 5, so 5 ÷ (5 + 1) = 5 ÷ 6, about 83%.

  4. In the lesson's simulation, why did an account whose rule has an edge spend most of its time under water?
    Answer

    A new peak needs a win that takes the account beyond its old high. With 55 wins and 45 losses in 100, the account ends only about 10 wins ahead per 100 trades, so most wins just win back earlier losses.

  5. (from lesson A14) An account falls 20% from its peak. What gain gets it back to the peak?
    Answer

    25%. The gain is measured on what is left: €100 falls to €80, and €80 needs €20 more, which is 20 ÷ 80 = 25%.

Try it yourself

Run the script. Every number in this lesson comes from one Python script with a fixed random seed (below). Change the chance of a win, the size of a trade or the length of the backtest and see what moves.

By hand, in a spreadsheet. Type an account's values down column A. In B1 put =A1, in B2 put =MAX(B1,A2) and copy it down: the peak so far. In C1 put =(B1-A1)/B1 and copy it down: the drawdown at each step. =MAX(C:C) is the maximum drawdown, and =COUNTIF(C:C,">0")/COUNT(C:C) is the time under water. A chart of column C, turned upside down, is the underwater chart.

The code behind every number (Python, seed 2026111013)
Loading…

Going further: with no edge, the worst drawdown keeps growing

The script also ran the 1,000 accounts on a fair coin: 50 wins in 100, so no edge. Such an account is the random walk of lesson A2, and its typical distance from the start grows with the square root of the number of trades. Its worst drawdown grows the same way. Over 250 trades the maximum drawdown averaged 17.5%. Over 1,000 trades it averaged 32.8%: 1.88 times as deep (worked from the unrounded averages) for 4 times as many trades, close to √4 = 2.

With the 55-in-100 edge the same two averages were 11.1% and 16.4%: 1.48 times as deep. An edge slows the growth of the worst drawdown. It doesn't stop it.

Time under water tells the same story. Over the 1,000 trades the no-edge accounts were under water after 97.6% of their trades on average, and the accounts with the edge after 89.6%.

For the preciseSeed 2026111013, one numpy Generator. First draw: 1,000 accounts × 1,000 trades (the same rule, each account with its own luck), a win (× 1.01) with chance 0.55, else a loss (× 0.99), each account from €10,000. Second draw: the same on a fair coin. The account you played is "the middle account", row 29: its 138 wins in the first 250 trades are the middle number of wins among the 1,000 (the 500th when sorted), and it is the lowest row with that many (48 accounts share it). At its peak it had 95 wins and 62 losses, 33 ahead: one new peak for each. Drawdowns are compared after rounding to 9 decimals; ties (2 accounts at 250 trades, 0 at 750) are not counted as deeper. Counts of accounts with a deeper drawdown after the backtest: 500 within 250 trades, 705 within 500, 794 within 750. Taking the three 250-trade stretches after the backtest apart, each with its own peaks, the deepest of them beat the backtest in 764 accounts, against 750 from k ÷ (k + 1); the other 30 of the 794 come from drawdowns that cross a stretch edge. Backtest maximum drawdowns ran from 4.0% to 31.1%; half were above 10.5%. Over the 750 trades after, half of the accounts met a drawdown above 14.3%. For a walk with no drift, the expected maximum drawdown grows with the square root of time; with a steady upward drift it grows with the logarithm of time, much more slowly (Magdon-Ismail and others, below).

Honest notes

Where this comes from