Quant from Scratch by QuantWright

Introduction to Quant Finance · Lesson 6

Fat tails

A bell curve says a day 5 standard deviations out comes once in thousands of years. Simulated days with fat tails and the same standard deviation brought one every few years.

Know the basics? Read the lesson notes

How rare is a 5-standard-deviation day?

A day's return is what a market gained or lost that day, in percent. The standard deviation is how far a typical day lands from the average day: the width of the bell in lesson A5 (The bell curve). That bell, the normal distribution, says how often a day lands far out: about 68 days in 100 within one standard deviation of the average, 95 within two and 99.7 within three.

Picture days drawn from a bell curve with an average of 0% and a standard deviation of 1%. A 5-standard-deviation day moves 5% or more, up or down. A year holds 252 trading days. For scale, the bell gives a 3-standard-deviation day either way about once in 370 days, and a 4 about once in 63 years. How often should a 5 come?

Once a year would be 1 day in 252. This far out the bell is far stingier: about once in 6,922 years.
About once in 63 years is the bell's figure for 4 standard deviations. At 5 it is about once in 6,922 years.
About once in 6,922 years.

Lesson A5's rule, carried further out: the bell puts about 0.57 days in a million at 5 standard deviations or more, either way. That is 1 day in about 1,744,278, and 1,744,278 ÷ 252 ≈ 6,922 years. Ten years hold 2,520 trading days, so the bell expects 2,520 × 0.57 ÷ 1,000,000 ≈ 0.0014 such days in ten years.

For the preciseThe chance is computed from the bell's formula, not simulated: 2 × (1 − Φ(5)) = 0.000000573, where Φ is the bell's cumulative chance. In a spreadsheet: =2*(1-NORM.S.DIST(5,TRUE)). At 4 standard deviations it is 63.3 days in a million, about once in 63 years.

Ten years of fat-tailed days

Now days with the same 1% standard deviation, drawn from a different curve: one with fat tails, meaning more days far from the average than a bell curve has. Studies of real daily returns have reported such tails since the 1960s (sources at the end); every day in this lesson is simulated. Play ten simulated years: 2,520 days, each a fresh draw that ignores the days before it. Dashed lines mark +5% and −5%.

Ten years of fat-tailed days (simulated)

Each line is one day's return. Colour: 5% or more either way. Dashed: ±5%.

Day0 of 2,520
5-SD days so far0
Biggest so far–

This series held 9 days of 5% or more, and its biggest day moved +11.94%: about 12 standard deviations. One series is one draw, so count many. The computer drew 1,000 ten-year series from the same curve. Together they held 2,190 days of 5 standard deviations or more: 2.19 per ten years on average, and 891 of the 1,000 series had at least one. So the series you played, with 9, was a wild draw.

1,000 ten-year series of bell-curve days held 2, against the 1,000 × 0.0014 ≈ 1.44 the bell predicts. In this simulation, the fat-tailed series' 2,190 such days were about 1,500 times the 1.44 the bell predicts.

Thicker and thinner tails

The curve here is Student's t distribution: a family of hill shapes like the bell, set by one number, its degrees of freedom, written ν ("nu"). A small ν gives fat tails; as ν grows, the curve closes in on the bell. Every setting below is scaled to the same 1% standard deviation (a theory card two screens on shows how), so only the tails change. The last screen used ν = 6. Slide to the fattest tails, ν = 3, and to the bell.

One ten-year series at each setting (simulated)

Standard deviation 1% at every setting. Colour: 5% or more either way.

5-SD days in 1,000 series2,190
Series with one or more891
4-SD days in 1,000 series6,824

Across 1,000 ten-year series at each setting, 5-standard-deviation days numbered 8,201 at ν = 3, 2,190 at ν = 6 and 2 with bell days. The standard deviation was the same at every setting: the tails alone decided how often an extreme day came.

For the preciseIn between: 641 at ν = 10 and 92 at ν = 20. At 4 standard deviations, where the bell predicts about 159.6 in 1,000 ten-year series, ν = 6 gave 6,824.

The bell and the fat tail, side by side

Here are all 2,520,000 fat-tailed days (ν = 6) of the 1,000 series, counted into columns 0.25% wide, with the bell curve of the same standard deviation drawn over them. Build the picture, then look at the tails, where the far days are grouped by whole standard deviations.

2,520,000 fat-tailed days and the bell (simulated)

Columns: the fat-tailed days. Line: the bell with the same standard deviation.

Within 1 SD, bell–
Within 1 SD, fat tail–
3 SD or more, fat tail–

The same standard deviation, a different shape. The fat-tailed curve is taller in the middle: 73.1% of its days landed within one standard deviation, against the bell's 68.3%. It is lower in between, and fatter far out: 1.05% of its days were 3 standard deviations or more out, against the bell's 0.27%. The further out you look, the wider the gap.

Theory

A fat-tailed curve with a fixed standard deviation

To compare tails fairly, hold the standard deviation fixed. The t distribution with ν degrees of freedom has a standard deviation of √(ν ÷ (ν − 2)), for ν above 2. Divide each draw by that, multiply by the standard deviation you want, and the days have exactly that standard deviation, whatever ν is. This is a scaled t distribution.

r = σ × t ÷ √(ν ÷ (ν − 2))
t
a draw from Student's t distribution with ν degrees of freedom
ν
"nu", the degrees of freedom, above 2: small for fat tails, large for close to a bell
σ
the standard deviation the days should have (1% in this lesson)
r
the day's return, with standard deviation σ
Worked example: the curve you played

ν = 6: 6 ÷ (6 − 2) = 6 ÷ 4 = 1.5, and √1.5 ≈ 1.22. A draw of t = 6.1 becomes 1% × 6.1 ÷ 1.22 = 5.0%: a 5-standard-deviation day.

For the precise: where a t draw comes fromOne t draw with ν degrees of freedom is a bell draw divided by √((z1² + … + zν²) ÷ ν), where z1 … zν are ν more bell draws. When those happen to be small together, the day is huge. The fewer there are, the more often that happens: fat tails. Student (1908) found the curve.
Holds when
  • ν is above 2 (at 2 or below the curve has no finite standard deviation);
  • each day is a fresh draw, independent of the others.
Breaks when

Real returns can lean to one side, with a fatter tail for losses than for gains (skew), and come in calm and wild spells (a later course shows these). One scaled t curve has neither.

New numbers. ν = 10, and the days should have a standard deviation of 1.5%. A draw of t = 3.36 becomes what return, in percent?

10 ÷ (10 − 2) = 1.25, and √1.25 ≈ 1.12. Then 1.5% × 3.36 ÷ 1.12 = 1.5% × 3 = 4.5%: three standard deviations.
Theory

One number for the tails: excess kurtosis

Put each day on lesson A5's z-score scale, its distance from the average in standard deviations, and raise it to the fourth power (multiply four copies together). A day 1 standard deviation out gives 1⁴ = 1; a day 5 out gives 5⁴ = 625. So the far days count overwhelmingly. The average of these fourth powers is the kurtosis. A bell's is 3, so the usual measure takes 3 off: the excess kurtosis, 0 for a bell, above 0 for fatter tails than a bell with the same standard deviation.

z = (x − μ) ÷ σ
K = (z1⁴ + z2⁴ + … + zn⁴) ÷ n − 3
x
one return; there are n of them
μ
"mu", the mean (average) of the returns
σ
their standard deviation (given in this lesson's examples; a later lesson, Volatility, works it out)
z
the return's distance from the mean, in standard deviations
K
the excess kurtosis: 0 for a bell, above 0 for fatter tails
Worked example: eight made-up days

0%, 0%, 0%, 0%, 0%, 0%, +2% and −2%. Mean 0, standard deviation 1%. The z-scores: six 0s, +2 and −2. Fourth powers: 2⁴ = 16, twice: 32. 32 ÷ 8 = 4, and 4 − 3 = 1. All of it comes from the two big days.

The t curve with 6 degrees of freedom has a true excess kurtosis of 3 (a textbook result: 6 ÷ (ν − 4), for ν above 4).

Holds when
  • there are enough days for the few far ones to have come;
  • one number for both tails together is what you want.
Breaks when

A few extreme days decide it, so short samples give wildly different values (the next screen counts it). It can't tell a fat loss tail from a fat gain tail (skew).

New numbers. Ten made-up days: eight of 0%, one of +3% and one of −3%. Their mean is 0 and their standard deviation √1.8 ≈ 1.342%. What is their excess kurtosis?

The z-scores of the big days: ±3 ÷ 1.342 ≈ ±2.236, and 2.236² = 5, so z⁴ = 25. Twice: 50. 50 ÷ 10 = 5, and 5 − 3 = 2.

1,000 measurements

The ten years you played measured 9.36, about three times the true 3. Its biggest day, +11.94%, supplied 47.7% of the sum of fourth powers on its own. Guess first: of 1,000 ten-year series from this curve, how many measure between 2 and 4?

Measured excess kurtosis, 1,000 series (simulated)

Each column counts the series in a range 0.5 wide; the last holds 12 or more. Dashed: the true value. Shaded: 2 to 4.

Your guess:
Series:
Between 2 and 4–
Middle value–
Lowest · highest–

Only 515 of the 1,000 measured between 2 and 4, and they ran from 0.93 to 246.21. Over one year, the first 252 days of each series, the middle value when sorted (the median) was 1.47, half the true 3. Bell days over ten years all measured between −0.33 and 0.40.

Excess kurtosis is driven by the few most extreme days, and in a short sample they may not have come yet. In this simulation a low measured value did not show thin tails.

For the preciseTen years: 760 of 1,000 below 3, a median of 2.28 and an average of 3.03, near 3 only because a few huge values (12 at 10 or more) make up for the many low ones. One year: 247 between 2 and 4, from −0.37 to 40.28.

What the bell gets wrong

Two of the bell's answers, checked against this lesson's fat-tailed days (ν = 6, standard deviation 1%).

The bell's figure against fat-tailed days (simulated)

Grey: the bell's own figure. Colour: the fat-tailed days.

The 68–95–99.7 rule. For these days the middle number happened to hold: 95.0% within two standard deviations. The others did not: 73.1% within one, and 1.05% three or more out, about four times the bell's 0.27%. The rule is most wrong far from the average, where the bell expects the fewest days.

The bell's loss lines. How big a loss comes on only 1 day in 100? The bell's answer is 2.33 standard deviations below the average, and 3.09 for 1 day in 1,000 (a later course calls such a line value-at-risk). Fat-tailed days fell to −2.33% or below on 1.46 days in 100, and to −3.09% or below on 4.6 in 1,000. Their own lines sat at −2.56% and −4.24%. The bell's lines understated the loss, and more so the rarer the day.

Crises and model risk

Model risk is the risk of acting on a model, a simplified description of how something behaves, that is wrong. Here the model is a curve that says how often each size of day comes. Studies of real daily returns since the 1960s have found more extreme days than a bell allows (sources at the end), so a risk model built on the bell can call a real day next to impossible.

Real markets have had such days. On 19 October 1987 a leading US stock index fell 22.6% in a single day (Federal Reserve History). In 2012 the Basel Committee on Banking Supervision, which sets international standards for banks, proposed moving banks' measure of trading risk from value-at-risk (a loss line that ignores how bad losses beyond it are) to a measure that "better captures tail risk", the risk of rare, large losses; it adopted the change in 2016.

In this lesson's simulation, where was the bell-based model most wrong?

It missed there too, 73.1% against 68.3%, but only a little. At the 1-in-1,000 loss line the fat-tailed days came 4.6 times as often, and at 5 standard deviations about 1,500 times: the rarest, largest days.
At the 1-in-1,000 loss line the fat-tailed days came 4.6 times as often as the bell said, and at 5 standard deviations about 1,500 times. The model was most wrong about the days that cost the most.
The miss grew with the size of the day: 73.1% against 68.3% within one standard deviation, 4.6 times as often at the 1-in-1,000 loss line, about 1,500 times at 5 standard deviations. The answer is the rarest, largest days.

What this means

  1. Count extreme days; don't take the bell's word.

    Under a bell, a 5-standard-deviation day comes about once in 6,922 years. In this simulation, days with fat tails and the same standard deviation had 2.19 per ten years. Where there is data, count how often its extreme days actually came.

Practice 1 of 2

Ten months, one bad one

The situationA made-up investment's ten monthly returns are nine months of +1% and one month of −9%. Their mean is 0 and their standard deviation 3%.

What is their excess kurtosis?

Hint: find each month's z-score, then the average of the fourth powers, minus 3.

The z-scores: +1 ÷ 3 ≈ 0.333 nine times, and −9 ÷ 3 = −3. Fourth powers: 0.333⁴ ≈ 0.0123, nine times ≈ 0.111, and (−3)⁴ = 81. Total 81.111, ÷ 10 = 8.111, − 3 = 5.11. The one bad month supplies 81 of the 81.111, and it is all on the loss side, which the number can't show.
Practice 2 of 2

"Once in thousands of years"

The situationA made-up risk note says: "This market's daily standard deviation is 1%, so a 5% day either way is a once-in-thousands-of-years event." The market's last ten years of daily returns hold three such days.

What do the three days tell you?

A bad price in the data is worth ruling out, but three days where the bell expects 0.0014 mostly say the bell doesn't fit. Fat-tailed days in this lesson's simulation had 2.19 per ten years.
Three where the bell expects 0.0014 is far more than a bell allows. One ten-year sample can't pin down how fat the tails are, though, and a bad price in the data is still worth ruling out.
Three big days push the measured value up, but how far depends on every other day, and in this lesson measured values jumped from below 1 to over 200 for one curve. Three days can't fix the true value.
✓

Lesson done

Under a bell curve, a day 5 standard deviations out comes about once in 6,922 years of 252 trading days. In this lesson's simulation, independent days with fat tails and the same 1% standard deviation (a scaled t distribution with 6 degrees of freedom) had 2.19 such days per ten years. Excess kurtosis, the average fourth power of the z-scores minus 3, measures the tails, but it is driven by the few most extreme days and jumped widely on short samples. In this simulation, models built on the bell were most wrong about the rarest, largest days.

Key ideas

Fat tails
more days far from the average than a bell curve with the same standard deviation has
5-standard-deviation day
a day 5 or more standard deviations from the average, either way; a bell puts about 0.57 days in a million there, about once in 6,922 years of 252 trading days
Student's t distribution
a family of hill-shaped curves set by its degrees of freedom ν: a small ν gives fat tails, a large one is close to the bell
Excess kurtosis
the average fourth power of the z-scores, minus 3: 0 for a bell, above 0 for fatter tails; driven by the few most extreme days, so it varies widely on short samples
Model risk
the risk of acting on a model that is wrong; in this lesson's simulation a bell-based model was most wrong about the rarest, largest days

Formulas

r = σ × t ÷ √(ν ÷ (ν − 2))

z = (x − μ) ÷ σ · K = (z1⁴ + … + zn⁴) ÷ n − 3

The first gives days with standard deviation σ for ν above 2. The second is the excess kurtosis; K is 0 for a bell.

Lesson notes

A bell curve with a standard deviation of 1% puts a day of 5% or more, either way, about once in 6,922 years of trading days. In this lesson's simulation, days, each a fresh draw, with the same 1% standard deviation but fat tails had 2.19 such days per ten years. The tails, not the standard deviation, decide how often an extreme day comes. Excess kurtosis measures them, but it is driven by the few most extreme days and jumped widely on short samples. In the simulation, models built on the bell were most wrong about the rarest, largest days. Every count below is from the lesson's simulation; it makes no claim about any real market's numbers.

The bell's answer

A day's return is what a market gained or lost that day, in percent, and the standard deviation is how far a typical day lands from the average day: the width of the bell. The bell curve, the normal distribution, puts about 68 days in 100 within one standard deviation of the average, 95 within two and 99.7 within three. Further out it is far stingier: about 0.57 days in a million at 5 standard deviations or more, either way. That is 1 day in about 1,744,278, and with 252 trading days a year, 1,744,278 ÷ 252 ≈ 6,922 years. In ten years, 2,520 days, the bell expects 2,520 × 0.57 ÷ 1,000,000 ≈ 0.0014 such days. At 4 standard deviations it is about once in 63 years.

Days with fat tails

A curve has fat tails when it has more days far from the average than a bell with the same standard deviation. The lesson drew 1,000 ten-year series of independent days from such a curve, each with a standard deviation of 1%. Together they held 2,190 days of 5% or more either way: 2.19 per ten years on average, and 891 of the 1,000 series had at least one. The first series held 9, and its biggest day was +11.94%, so one series is a poor guide. 1,000 ten-year series of bell-curve days held 2, against the 1.44 the bell predicts. In this simulation the fat-tailed series' 2,190 were about 1,500 times the bell's 1.44.

Tail thickness

The fat-tailed curve is Student's t distribution, a family of hill shapes set by its degrees of freedom ν: a small ν gives fat tails, and as ν grows the curve closes in on the bell. The series above used ν = 6. Scaled to the same 1% standard deviation, 1,000 ten-year series at each setting held 8,201 days of 5 standard deviations or more at ν = 3, 641 at ν = 10 and 92 at ν = 20. At 4 standard deviations, where a bell predicts about 159.6, ν = 6 gave 6,824. Pooled, the ν = 6 days were taller in the middle than the bell, 73.1% within one standard deviation against 68.3%, lower in between, and fatter far out: 1.05% three or more out against 0.27%.

Measuring the tails, and why the measure jumps

Excess kurtosis (formula below) averages the fourth power of each day's z-score, its distance from the mean in standard deviations, and takes 3 off, so the far days count overwhelmingly and a bell gives 0. The t curve with ν = 6 has a true excess kurtosis of 3 (the textbook formula is 6 ÷ (ν − 4), which gives 6 ÷ 2 = 3 for ν = 6). The first ten-year series measured 9.36, about three times the true 3; its biggest day supplied 47.7% of the sum of fourth powers. Across all 1,000 ten-year series, 515 measured between 2 and 4 and 760 below 3; the median, the middle value when sorted, was 2.28, and the values ran from 0.93 to 246.21. Over one year (the first 252 days of each series) only 247 landed between 2 and 4, with a median of 1.47. Bell days over ten years all measured between −0.33 and 0.40. In a short sample the extreme days may not have come yet, so in this simulation a low measured value did not show thin tails.

What the bell gets wrong

With the ν = 6 days, two of the bell's answers fared differently. The 68–95–99.7 rule: the middle number held, 95.0% within two standard deviations, but not the others (73.1% within one; 1.05% three or more out, about four times the bell's 0.27%). The bell's loss lines: the bell puts 1 day in 100 at a loss of 2.33 standard deviations or more below the average, and 1 in 1,000 at 3.09 (a later course calls such a line value-at-risk). The fat-tailed days fell to −2.33% or below on 1.46 days in 100 and to −3.09% or below on 4.6 in 1,000; their own lines were −2.56% and −4.24%. The rarer the day, the more the bell understated it.

Crises and model risk

Model risk is the risk of acting on a model, a simplified description of how something behaves (here, a curve for how often each size of day comes), that is wrong. In the simulation the bell-based model missed ordinary days only a little (73.1% within one standard deviation against 68.3%) but the rarest, largest days by far: 4.6 times as many at the 1-in-1,000 loss line and about 1,500 times as many at 5 standard deviations. Studies of real daily returns since the 1960s have found more extreme days than a bell allows. On 19 October 1987 a leading US stock index fell 22.6% in a single day. In 2012 the Basel Committee on Banking Supervision, which sets international standards for banks, proposed moving banks' measure of trading risk from value-at-risk (a loss line that ignores how bad losses beyond it are) to a measure that "better captures tail risk", the risk of rare, large losses; it adopted the change in 2016. A risk model built on the bell can call a real day next to impossible; one ten-year sample can't pin down how fat the tails are. The simulation's counts say nothing about how often real markets have extreme days.

Formula: the scaled t distribution

Dividing a t draw by the t distribution's own standard deviation, √(ν ÷ (ν − 2)), and multiplying by σ gives days with standard deviation σ, whatever ν is above 2.

r = σ × t ÷ √(ν ÷ (ν − 2))
t
a draw from Student's t distribution with ν degrees of freedom
ν
"nu", the degrees of freedom, above 2: small for fat tails, large for close to a bell
σ
the standard deviation the days should have (1% in the lesson)
r
the day's return, with standard deviation σ
Worked example

ν = 6: 6 ÷ 4 = 1.5, and √1.5 ≈ 1.22. A draw of t = 6.1 becomes 1% × 6.1 ÷ 1.22 = 5.0%. With ν = 10 and σ = 1.5%: 10 ÷ 8 = 1.25, √1.25 ≈ 1.12, and a draw of 3.36 becomes 1.5% × 3.36 ÷ 1.12 = 4.5%.

Holds whenBreaks when

Real returns can have a fatter tail for losses than for gains (skew) and come in calm and wild spells (a later course shows these). One scaled t curve has neither.

Formula: excess kurtosis

The average of the fourth powers of the z-scores, minus a bell's 3.

z = (x − μ) ÷ σ
K = (z1⁴ + z2⁴ + … + zn⁴) ÷ n − 3
x
one return; there are n of them
μ
"mu", the mean of the returns
σ
their standard deviation, dividing by n (given in the lesson's examples; a later lesson, Volatility, works it out)
z
the return's distance from the mean, in standard deviations
K
the excess kurtosis: 0 for a bell, above 0 for fatter tails
Worked example: eight made-up days

0%, 0%, 0%, 0%, 0%, 0%, +2% and −2%: mean 0, standard deviation 1%. The z-scores are six 0s, +2 and −2; fourth powers 16 + 16 = 32; 32 ÷ 8 = 4, and 4 − 3 = 1. Ten days, eight of 0% and one each of +3% and −3%: mean 0, standard deviation √1.8 ≈ 1.342%, so z ≈ ±2.236, z⁴ = 25, and 50 ÷ 10 − 3 = 2.

Holds whenBreaks when

A few extreme days decide it, so short samples give wildly different values; and it can't tell a fat loss tail from a fat gain tail (skew).

One more worked example

Made-up numbers: no tails at all

Six days: +1%, −1%, +1%, −1%, +1% and −1%. Mean 0, and every day sits exactly 1% from it, so the standard deviation is 1%. Every z-score is +1 or −1, and every fourth power is 1. Average: 6 ÷ 6 = 1, and 1 − 3 = −2: every day sits exactly one standard deviation out, so nothing is far, and the number is below a bell's 0.

Common mistakes

Check yourself

  1. Twenty made-up days: eighteen of 0%, one of +3% and one of −3%, with mean 0 and standard deviation √0.9 ≈ 0.949%. What is their excess kurtosis, and why is it higher than for eight days of 0% with the same two big days?
    Answer

    Each big day's z² = 9 ÷ 0.9 = 10, so z⁴ = 100; twice, 200. 200 ÷ 20 = 10, and 10 − 3 = 7. With ten days the answer was 2: the more quiet days around the same two big ones, the smaller σ, the further out the big days sit, and the higher the number.

  2. With ν = 6, what t draw makes a 5% day when the days have a standard deviation of 1%?
    Answer

    The t distribution's standard deviation is √(6 ÷ 4) = √1.5 ≈ 1.22. So 1% × t ÷ 1.22 = 5% needs t = 5 × 1.22 = 6.1.

  3. In the lesson's simulation, which of the 68–95–99.7 rule's numbers held for the fat-tailed days, and which did not?
    Answer

    Two standard deviations held: 95.0% of the days within, against the bell's 95.4%. One did not: 73.1% within, against 68.3%. Three did not: 1.05% of days three or more out, against 0.27%.

  4. (from lesson A5) A bell puts about 0.27% of days three or more standard deviations out. In 2,520 days, about how many is that?
    Answer

    2,520 × 0.27 ÷ 100 ≈ 6.8 days, for days that follow a bell. The lesson's fat-tailed days put 1.05% there: about 26 in 2,520.

  5. For days with a 1% standard deviation, the bell puts 1 day in 1,000 at a loss of 3.09% or worse. In the lesson's simulation, how many fat-tailed days in 1,000 fell that far, and where was the bell-based model most wrong?
    Answer

    4.6 in 1,000, about 4.6 times the bell's 1. The model missed ordinary days only a little (73.1% within one standard deviation against 68.3%) and the rarest, largest days most: about 1,500 times as many 5-standard-deviation days as the bell expects.

Try it yourself

Run the script. Every number in this lesson comes from one Python script (below). It draws 1,000 ten-year series at each tail setting, counts the extreme days, measures every excess kurtosis, and counts the days past the bell's loss lines. Change the degrees of freedom, the length of a series or the size of an extreme day, and see what moves.

By hand, in a spreadsheet. Put daily returns in column A. =KURT(A:A) gives an excess kurtosis (with a small-sample adjustment, so it differs a little from this lesson's formula on few days). For this lesson's version: the mean in C1 with =AVERAGE(A:A), σ in C2 with =STDEV.P(A:A), then =((A1-$C$1)/$C$2)^4 down column B, and =AVERAGE(B:B)-3. A scaled t draw with ν in C3: =T.INV(RAND(),$C$3)/SQRT($C$3/($C$3-2)).

The code behind every number (Python, numpy; seed 2026111001)
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Honest notes

Where this comes from