The two-sigma promise · Market Vidyalaya

You do something almost nobody does, which is to check. Over the last 250 sessions you count how often a midcap closed outside its two-sigma Bollinger Bands, and the answer is not the 5% the textbook implies — it is well over twice that, and worse, the closes outside the band arrive in clusters of four and five consecutive days rather than scattered one at a time. Then you look at the three largest single-day falls of the year, express each in standard deviations of the daily move, and find that the model in your head assigns them frequencies like once a century. The band arithmetic is right. The stock is ordinary. The 95% is the problem, and it was never a fact about the stock at all.

Think of it like this

The average bus and the actual wait

A route is advertised every ten minutes. Over a month you note the waits: mostly two or three minutes because two buses are bunched, and then twice a fortnight, nothing for forty minutes. The average is honestly about ten. Almost no wait is ten. Plan around the average and you are early most days and catastrophically late twice a fortnight — and the timetable, which was not lying, gave you no way to see that coming.

In the market

Daily returns behave the same way: far more very small days than a bell curve predicts, far more enormous days, and a shortage of the ordinary middling days that would justify the average. Standard deviation is computed honestly and describes the crowd of tiny days. The days that decide your year are in the part of the distribution the number does not describe.

What the model actually promises, in sessions

A move beyond	Probability under a normal model, both directions	Expected frequency at 252 sessions a year
1σ	31.7%	About 80 days a year
2σ	4.6%	About 11 days a year — roughly one a month
3σ	0.27%	One day in 370 — about once in 18 months
4σ	0.0063%	One day in about 15,800 — about once in 63 years
5σ	0.00006%	One day in about 1.7 million — about once in 7,000 years

These are arithmetic, not observations: they are what the normal distribution says, exactly, and they are the yardstick against which the counting is done. Note how violently the last two rows fall away. That collapse is the whole issue — the model does not merely mis-estimate large moves, it declares them impossible on any human timescale, and then they happen.

The maths

Take a smallcap whose daily σ is 1.5%. A 6% fall is four standard deviations. Read the 4σ row with its heading, because the heading is doing work: the probabilities in that table count moves in both directions, so the row schedules a 4σ day of either sign about once in sixty-three years — and a 4σ fall specifically about half as often, roughly once in a hundred and twenty-five. Now ask anybody who has held Indian smallcaps for three years how many 6% down-days they have sat through. The answer is not "one or two, so the model is a bit optimistic". The model is wrong by orders of magnitude, and it is wrong in the only region where being wrong costs money. Halving the two-sided figure when the question is one-directional is worth doing out loud every time — forgetting it is the commonest arithmetic slip in this whole subject, and it is the trap set in this lesson’s quiz.

The mechanism, which is duller than the mystique

It is tempting to treat Fat tails as an exotic law of markets. The main mechanism is far more ordinary and it survives every rule change and every decade: σ is not a constant. Volatility clusters — quiet weeks follow quiet weeks and violent days arrive next to other violent days. When you pool a calm stretch and a panic stretch into one sample and compute one standard deviation, you have not measured a distribution. You have measured a mixture of two, and a mixture of two normal distributions with different widths has fat tails even though neither component does.

Worked example

One stock, one number for σ, two answers 27 times apart

A stock whose days come from two regimes: four days in five behave with a σ of 0.9%, and one day in five with a σ of 3.5%. We ask both ways how often it should fall 6% or more, or rise 6% or more

The single σ you would compute from the pooled dataThis is what a volatility function returns if you hand it the whole sample. It is not wrong — it is the standard deviation of that sample√(0.8 × 0.9² + 0.2 × 3.5²) = √3.098 = 1.76%

A 6% move, expressed in that σComfortably into the region the table above calls rare6 ÷ 1.76 = 3.4 standard deviations

Frequency, if the pooled σ describes a single normal distributionRoughly once in six years of trading. This is the answer a σ-scaled tool gives youAbout one session in 1,540

Now the same question, regime by regime — the calm daysEffectively never. Calm days contribute nothing to the count, correctly6% is 6.7σ of a 0.9% day

The violent daysOn a violent day a 6% move is unremarkable — not a tail event at all, just a normal day in that regime6% is 1.7σ of a 3.5% day, which happens about 8.7% of the time

Weighted by how often violent days occurAbout one session in 58, or four to five times a year0.2 × 8.7% = 1.7% of all sessions

Same stock, same underlying behaviour, same data: once in six years against four or five times a year, a factor of about twenty-seven, and the entire difference is the decision to describe a changing σ with a single number. Nothing here required a strange distribution or a market conspiracy — only the fact, visible on any chart, that volatility comes in regimes. This is also why closes outside a two-sigma band arrive in clusters rather than scattered: they are not independent rare events, they are the violent regime showing up, several days in a row, while a twenty-day σ is still averaging in the calm fortnight behind it.

What this does to the band specifically

The window is short and the target moves. A 20-day σ is estimated from twenty observations, which is a small sample even if the underlying σ were constant — and it is not. The band is a fast estimate of a moving quantity, not a stable envelope.
The window lags exactly when it matters. σ rises after the violent days have already been included, so the band is at its narrowest going into a volatility expansion and widens after the damage. The squeeze taught earlier in this track is the same observation used deliberately; the surprise here is the same observation suffered accidentally.
"Walking the band" is this fact from the other side. Price riding the upper band for weeks is not a statistical impossibility that keeps recurring. It is a demonstration that the 95% was never a property of the stock.
Nothing above is an argument against the bands. They are a good adaptive measure of what is ordinary for this stock right now, which is what the earlier lesson claims for them. The claim that does not survive is the frequency promise stapled to the side of it.

The replacement, which needs no distribution at all

1
Count, do not model
Over the last 500 sessions, sort the daily moves and read off the worst five. That is your worst one per cent, actually observed, with no assumption of any kind inside it. It is a smaller number of observations than you would like and it is still enormously better than a formula that says the observation cannot happen.
2
Keep two volatility numbers, not one
A calm-regime σ and a stressed-regime σ, computed over the quietest and the most violent stretches you can identify in the history you have. A single blended number describes neither state, which is exactly what the worked example above demonstrates.
3
Size against the observed tail, not against 2σ
Position sizing that survives is sized so that the worst day you have actually seen in this name is survivable — not so that a two-sigma day is survivable. The gap between those two instructions is the whole of Tail risk.
4
Expect the exceedances in clusters, and plan the second one
Because volatility clusters, the day after a band break is a likely candidate for another. A plan that assumes one bad day at a time is assuming independence — which is the subject of the next lesson, and which you can test rather than assume.

Loading interactive demo…

Move the standard-deviation multiple and the lookback. What changes is the width of the band, computed from this stock. What never appears anywhere in the calculation is the 95%.

Check yourself

A stock’s daily σ is 1.2%. Under a normal model, how often should it fall 3.6% or more in a session — and what should you conclude if it has done so four times this year?

Simple bhasha mein

Das minute waali bus

Route pe likha hai "har das minute". Mahine bhar note karo: zyadatar do-teen minute ka wait (do bus saath aa gayi), aur pandrah din mein do baar — chalis minute kuch nahi. Average imaandari se das ke aas-paas hai, aur das minute ka wait lagbhag kabhi hota hi nahi. Rozana return bhi aise hi hain: bell curve se zyada chhote din, zyada bade din, aur beech ke aam din kam. "Do sigma mein 95%" bell curve ki baat hai, aapke share ki nahi — band ki chaudai data se aati hai, 95% kitaab se. Normal model ke hisaab se kisi bhi taraf ka 4σ din 63 saal mein ek baar aata hai — aur sirf girawat waala 4σ din uska aadha, yaani lagbhag 125 saal mein ek (do taraf ka number ek taraf ke sawaal pe lagana sabse aam galti hai). 1.5% σ waale smallcap pe 4σ matlab 6% ki girawat. Teen saal smallcap rakhne waale se poochh lo, usne kitne 6% waale din dekhe hain. Wajah jaadui nahi hai: σ badalta rehta hai. Maano paanch mein chaar din σ 0.9% ke hain aur ek din 3.5% ka. Poora data mila ke ek σ nikaalo toh 1.76% aata hai, aur 6% ka move 3.4σ — yaani 1,540 session mein ek, lagbhag chhe saal mein ek. Ab regime se poochho: shaant din pe 6% matlab 6.7σ, kabhi nahi; toofani din pe 6% sirf 1.7σ, jo 8.7% baar hota hai — aur toofani din paanch mein ek hain, toh 0.2 × 8.7% = 1.7% session, matlab 58 session mein ek, saal mein chaar-paanch baar. Ek hi share, ek hi data, jawab 27 guna alag. Isiliye band ke bahar waale close jhund mein aate hain, ek-ek karke nahi. Model ki jagah ginti karo: pichhle 500 session ke sabse kharab paanch din — bina kisi assumption ke.

What to remember

A two-sigma band takes its width from the stock and its 95% from the normal distribution.
Under that model a 4σ day is a once-in-63-years event and a 5σ day a once-in-7,000-years event.
The main mechanism behind fat tails is not exotic: σ changes, and pooling regimes into one number produces them.
A single blended σ can understate the frequency of a large move by more than an order of magnitude.
Exceedances cluster, because volatility clusters — they are not independent rare events.
Replace the model with a count: the worst five days in 500 sessions, and size so those are survivable.

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