Kurtosis: Why Fat Tails Matter More Than Volatility

Kurtosis is one number for how much of a shape sits far out in its tails. The normal shape reads 3.00, so every kurtosis reading is really being compared against 3. The fifty month record of the Nakshatra unit reads 2.9998. The population behind those months takes five values, reads 2.92, and is nothing like the normal shape. A reading near 3.00 is not evidence of normality.

There is a particular kind of number that is worse than a wrong one. A wrong number disagrees with something, so sooner or later it gets caught. The dangerous number is the one that comes out exactly where it was hoped for, on a quantity nobody looked at closely, and quietly licenses everything that follows. Kurtosis is one of those numbers. Anybody could compute the reading in an afternoon, on fifty months of figures, and it hands back the right answer to a question nobody asked.

Two invented objects carry everything below. The Nakshatra unit is a traded unitShorthand for anything carrying a price that moves, so each month yields exactly one number. The sort of object it is stays deliberately unsaid, and nothing in the arithmetic depends on it. whose monthly changeThe distance a price travelled over one month, written against where that price stood when the month opened. A month that goes from Rs 100/- to Rs 111/- is a monthly change of 11.00 per cent. can land on exactly five values and no others: minus 9.00, minus 4.00, 1.00, 6.00 and 11.00 per cent, weighted in that order 0.08, 0.18, 0.48, 0.18 and 0.08. Fifty months pulled out of that rule give the fifty month record used throughout here, in which the five values turned up 5, 9, 25, 8 and 3 times.

Unusually, the whole populationEvery outcome the rule behind a record is capable of producing, together with how heavily each one is favoured. A record is a sample of the population, and the population is set out under its own name elsewhere. is known, because somebody typed it out before a single month was drawn. That means when the record reports a shape reading, the true reading it was aiming at is already sitting on the table beside it. Nobody's word has to be taken for whether the record got it right. The true reading can simply be read off.

Before any figure arrives. Five values, five weights and a tally of fifty months were picked to keep the arithmetic small enough for a pen, and every reading below falls out of those choices.

What part of a shape is kurtosis actually about?

Kurtosis gets described almost everywhere as peakedness, and that description costs its readers in understanding long before it costs them anything else. The arithmetic behind kurtosis takes every value, measures how far it sits from the centre, and then raises that distance to the fourth power before averaging. Raising to the fourth power is a brutal amplifier. A value one spread away contributes one unit. A value three spreads away contributes eighty one. A value four spreads away contributes two hundred and fifty six. The middle of the shape, where the distances are small, contributes almost nothing at all.

So kurtosis is a statement about the far tails, about how much of a shape sits a long way from its own centre, and the peak is very nearly irrelevant to it. The peak looks tall or flat as a side effect. If a shape has plenty of weight parked far out, and its total spread is held fixed, then the weight has to come from somewhere, and it comes from the shoulders. Thinning the shoulders makes the middle look narrow and pointy. The pointiness is a consequence. The tails are the subject.

An everyday version helps. Consider how long a commute takes. Two people can both report an average of forty minutes and both report the same typical variation of ten minutes. One of them has commutes that run between twenty and sixty minutes and never anything else. The other is at forty five minutes most days, occasionally thirty, and about twice a year sits in a jam for four hours. Their average is the same. Their standard deviationOne figure summarising the typical distance between an entry and the middle of the list it belongs to, kept in the units of the entries themselves. Its construction, and what it can and cannot settle, is handled elsewhere. is the same. Nobody would call those two commutes the same commute, and no summary of the centre or the spread can tell them apart. Kurtosis is the number built to notice the difference.

What is the number 3 doing in every discussion of this?

Open any treatment of kurtosis and the number 3 turns up within two paragraphs, usually without explanation, as though everybody already agreed what it was for. So here is what it is for, in one line. The normal shape has a kurtosis of exactly 3.00, and the number is quoted so often because that particular shape is the one everything else gets held up against.

Three is not a threshold, not a boundary and not a natural dividing line; it is simply the value one specific shape happens to take, and it earned its place in the literature by being the shape everybody compares things to. Nothing changes character as a reading crosses it. There is no rule that gets violated at 3.01 and respected at 2.99. If the reference shape had been some other shape, the number quoted would have been some other number, and every sentence written about kurtosis would have been rearranged around that one instead.

The comparison buys a direction. Above 3.00, a shape carries more of itself out in the far tails than the normal shapeOne particular curve, with a single rise and a single fall, which statistics adopted long ago as its default point of comparison. The normal shape is built under its own name; here it works purely as the yardstick. does. A reading below 3.00 says it carries less. The direction is genuinely useful, and it is the entire content of the comparison. Notice what it does not say: it says nothing about whether the shape resembles the normal shape in any other respect, and nothing about whether the shape is one hump or three spikes or a wall with a gap in the middle.

What does the fifty month record report?

Take the fifty months. Their mean is 0.50 per cent. Measure how far each month sits from that mean, raise each of those distances to the fourth power, average the fifty results, and divide by the record's own spread raised to the fourth power. Every step is arithmetic on the counts 5, 9, 25, 8 and 3.

The answer is 2.9998.

A reading of 2.9998 is the normal shape's value of 3.00, agreeing to four decimal places. The conclusion that suggests itself, to anybody who has met kurtosis before, is that the fifty months look normally distributed. The conclusion is natural, it is what the number appears to be saying, and almost everybody who computes this reading gets exactly that far.

Why is that reading a trap?

Here is what those fifty months actually came out of. Not a bell curve. Not a curve of any kind. A rule with five entries on it, handing back one of five numbers and nothing else, ever. The Nakshatra unit cannot produce a month of minus 9.01 per cent. The unit cannot produce a month of 11.5 per cent. A month of 3.00 per cent sits comfortably inside its range and is not on the list, so the unit cannot produce that one either. Computed from the five values and the five weightsThe numbers saying how heavily each outcome counts, which have to total one across the whole list. The weights turn a list of possible values into a shape. rather than from any record, its own kurtosis is 2.92.

A record drawn from a five value rule that is nothing like the normal shape reported the normal shape's kurtosis to four decimal places, and a reading near 3.00 is therefore not evidence of normality. The record did not malfunction. The arithmetic was not done wrongly. Fifty months from a shape whose true reading is 2.92 came out at 2.9998, drifting 0.0798 above the truth on ordinary sampling errorThe gap between what one record reports and what the rule behind it would report, arising for no reason beyond the record being a handful of cases rather than all of them. Where it comes from and how fast it shrinks is worked through separately in these notes., and landed by accident on a number that looks like a certificate.

Seeing the five value rule settles the argument faster than any amount of prose, so it is worth looking at directly. Its whole existence is bounded. Its lowest value sits exactly two spreads below the centre and its highest sits exactly two spreads above, and there is nothing outside those two points at any weight at all. Sixteen per cent of its weight sits precisely on the two spread line and none of it sits beyond. A shape like that has no far tail to speak of. The rule has an edge.

So what does a summary number actually establish?

The lesson runs larger than kurtosis. A statistic can take a value that looks like evidence of something it is not evidence of. That is the sentence to carry away, and it does not need decorating.

The reason is not mysterious. Kurtosis measures one quantity: how much of a shape sits far out. The normal shape is one of countless shapes for which that quantity comes to about 3. Learning that a record reads near 3.00 rules out the shapes that read 1.8 and the shapes that read 10. Ruling those out is real information, and it leaves standing every single shape in the enormous set that reads near 3. The normal shape is a member of that set. So is a rule with three spikes on it. So is a badly lopsided shape. So is the five value rule set out above.

A summary number is evidence about the quantity it measures and about nothing else, and a shape is not settled by one number about it. The five shapes below all read between 2.92 and 3.00. Two of them read exactly 3.0000, the same as the normal shape does, and neither of them is remotely the normal shape. Handed only the readings, nobody could sort the pictures.

What does a genuinely fat tailed shape read?

So far every shape above has read below or barely at 3.00, and that risks leaving the impression that the reading never moves. The reading moves a great deal. Here is a shape built to sit at the other end.

Something that spends most of its months quiet, with monthly changes wandering in a narrow band, then occasionally has a month where everything is much more violent than usual. Not a different thing, just the same thing in a different state. Nineteen months in twenty come out of the quiet state with a spread of 4.00 per cent, and one month in twenty comes out of the stressed state with a spread of 14.00 per cent. Combined, the resulting shape has a centre of 1.00 per cent and a combined spread of exactly 5.00 per cent. The arithmetic checks out: nineteen twentieths of sixteen plus one twentieth of one hundred and ninety six comes to twenty five, and the square root of twenty five is five.

The quiet plus stressed shape agrees with the normal shape on the centre and agrees with it on the spread, disagrees violently about the far tail, and reads a kurtosis of 10.3872 against the normal shape's 3.00. Both of the two headline summary numbers are blind to the difference. Kurtosis is one of the very few common summary figures that registers it at all.

The size of the disagreement is worth stating in months rather than in a reading. Beyond two spreads from the centre the two shapes are almost the same, one month in twenty two against one month in twenty eight. Beyond three spreads they part company: one month in three hundred and seventy under the normal shape against one month in seventy. Beyond four spreads it is not close at all, one month in fifteen thousand seven hundred and eighty seven against one month in one hundred and thirty one.

How do the four readings look set out in one row?

Four objects have now been read. The arrangement makes the argument on its own, so the four go side by side.

The first three readings are indistinguishable, and they describe three completely different objects. The fourth sits more than seven away from all of them. The population and the record and the normal shape span a range of 0.0800 between them. Between the record and the normal shape the gap is 0.0002126. There is no honest reading of those three numbers that separates a five value rule with hard edges from a smooth curve with endless tails.

Why does a fat tail matter more than volatility?

The heading makes a claim, so it had better be argued rather than repeated. First a definition: volatilityVolatility is the spread of the monthly change, so a unit whose months typically land within about five per cent of their own average has a volatility of about five per cent. The word carries no meaning beyond that spread. is the spread of the monthly change and nothing more. When somebody says a thing is more volatile than another they mean its months scatter more widely around their own average.

Now the argument. The spread describes an ordinary month, and the enormous majority of months are ordinary. The spread earns its usefulness from ordinary months, and it is limited by exactly the same thing. The tail describes the months that are not ordinary. Extraordinary months are rare, so they contribute almost nothing to the spread, and they are the months in which anything irreversible happens.

Being wrong about the spread means being wrong about most months by a little, and being wrong about the tail means being wrong about a few months by a lot, and the second error is not recovered by being right the rest of the time. That asymmetry is the whole claim. A description that understates the ordinary month by half a per cent is corrected the following month, and the month after, by simply looking again. A description that had nothing to say about a single month three times worse than anything it allowed for does not get corrected by the forty nine months either side of it. Those months were fine and the damage was not.

The arithmetic here is quietly severe. Watch what eight months do to the reading. Of the fifty months in the record, eight are extreme: the five that landed on minus 9.00 per cent and the three that landed on 11.00. The eight extreme months, sixteen per cent of the record, carry 87.51 per cent of the fourth power total that the kurtosis reading is built from. The twenty five middle months, half the entire record, contribute 0.0018 per cent of it. The reading is almost entirely a statement about eight months.

What should be asked before a kurtosis reading is treated as evidence?

An analyst, a lender or a household treasurer handed a shape reading is in the same position. Somebody has compressed a record into one number and handed over the number. Four questions get most of the value back, and they work the same way whether the record is fifty months of an invented unit or a small business owner counting how often a customer pays late.

First, how many cases is the reading built on? The first question bites hardest, and it bites specifically here. The far tail is the part of a record there is least of, by definition, and the far tail is the entire subject of the reading. On the fifty month record, eight months carry 87.51 per cent of the reading. Moving a single one of those fifty months to a different value shifts the reading by between 0.1026 and 0.1814, depending on which month moves. The whole distance between the record's 2.9998 and the normal shape's 3.00 is 0.0002126. One month out of fifty moves the reading roughly five hundred to eight hundred and fifty times further than the entire agreement everybody found so persuasive.

Second, what shapes are consistent with this reading, rather than which single one? Third, what does the record say about its own worst cases, counted rather than summarised? Ask for the five worst months as five numbers. Fourth, is there a structural reason to expect a fat tail, meaning something about how the thing works rather than something in the record: a mechanism that stays quiet and then does not, a position that must be closed at a moment somebody else chooses, a supplier with no substitute.

Here is the everyday version of that last question. A household running on one salary and a household running on two salaries can have identical average income and identical month to month variation. The first one has a tail the second does not. The tail has not happened yet, so no amount of looking at last year's bank statement reveals it. The reason to expect it comes from knowing how the income is structured, not from the numbers. The far tail is the region a record knows least about, and kurtosis is built almost entirely on the far tail. Kurtosis is therefore the least reliable of the common summary numbers, and structural knowledge is often better evidence about a tail than the record is.