Skewness: When Returns Are Not Symmetric and What It Costs

Skewness reports which side of a shape has been stretched further out. A negative reading says the long side runs left, a positive one says it runs right. The Nakshatra population used here is exactly symmetric, so its true skewness is 0.0000 by construction, and the fifty month record drawn out of it still reports minus 0.0502. Separating a reading like that from a genuine lean is the whole job.

Two objects carry the arithmetic below. The first is the Nakshatra unit, a fictional traded unitAnything bought and sold at a price somebody quotes. The Nakshatra unit is the made up object that fifty months of figures belong to.. The second is the fifty month record, a tally of what the Nakshatra unit's monthly changeHow much a price moved over one month, stated as a percentage of what it was at the start of that month. A return means that and nothing more. came to in each of fifty months. Not one reading below calls for any knowledge of markets. The word return means a monthly change in price and nothing beyond it.

Almost no record comes with an exactly known population behind it. Somebody wrote this one down before drawing a single month out of it. Its five possible values are minus 9.00, minus 4.00, 1.00, 6.00 and 11.00 per cent, and the weights sitting on those five run 0.08, 0.18, 0.48, 0.18 and 0.08. Work that definition through and out comes a mean of 1.00 per cent and a standard deviationOne figure for how far a typical case sits from the middle of a set, carried in the same units as the cases themselves. The standard deviation is built in full separately, and a lean gets divided by it. of 5.00 per cent, neither of them measured off anything, both of them consequences of what was typed. So whenever the record disagrees with the truth, the disagreement can be put on screen and its exact size printed. Hardly any real record allows that.

What is skewness actually measuring?

Take any set of outcomes, find its middle, and look at how far the cases on each side run out from it. If the cases on the right reach further than the cases on the left, the shape is stretched to the right. If the left side reaches further, it is stretched to the left. Skewness is a single number that reports which of those two is happening and by how much. Skewness is a statement about the geometry of a shape, and about nothing else.

Here is the everyday version. A household running on one salary spends a fairly steady amount most months, and then in one month the fridge dies. Its spending has a long right side: many ordinary months clustered together, and a few sitting far out above them. Nothing in that sentence says spending is rising or falling. The long right side says the departures from the ordinary are lopsided, running in one direction more than the other.

A shape can lean one way while its centre sits the other way, and the fifty month record does exactly that. The fifty month record leans a hair to the left and its average change is positive at 0.50 per cent a month. Lean is not direction of travel. Read a negative skewness as a statement that the far outcomes on the left run further from the middle than the far outcomes on the right, and stop there.

The arithmetic underneath is short and worth seeing once. Every case has its distance from the mean measured, each of those distances is cubed, the cubes are added up, and the total is divided by the cube of the standard deviation. Cubing is the whole trick. Squaring a distance destroys its sign. A spread figure therefore never says which side anything is on. Cubing keeps the sign, so a case sitting far below the middle contributes a large negative amount and a case sitting far above it contributes a large positive one. If the two sides balance, the contributions cancel and the answer is nil.

What does the sign of the reading do to the picture?

A negative reading means the left side of the shape is the long one. In plain terms, most outcomes sit in an ordinary huddle and the departures that do occur are worse on the downside than they are good on the upside. A positive reading is the mirror of that: an ordinary huddle, and the departures that occur run further above the huddle than below it. The sign says which side has the room, never which side is more likely.

A second consequence matters more in practice, and it involves two figures already to hand. Every case enters the mean at its full size, so the mean is pulled by distance. The modeThe value that turned up more often than any other. Three ordinary answers describe a typical case, and all three are set out in full separately. and the middle case are not, because a case sitting far out counts once wherever it sits. So a long left side drags the mean below the middle case, and a long right side drags it above. On every shape drawn in this guide the two move together in exactly that way. The pairing is a strong companion signal rather than a law that can never fail.

Watch it happen on two of the three shapes above. Take the left leaning arrangement, with counts of 8, 8, 25, 9 and none. Its mean is minus 0.50 per cent and its middle case is 1.00 per cent, so the mean sits below. Now take the right leaning arrangement, with counts of none, 9, 25, 8 and 8. Its mean is 2.50 per cent and its middle case is still 1.00 per cent, so the mean has crossed to the other side. The middle case did not move at all between the two. The mean did all the travelling, and it travelled in the direction the stretch pointed.

Why is this population's skewness exactly zero?

Look at the weights again, in order: 0.08, 0.18, 0.48, 0.18 and 0.08. Fold that list in half at the middle entry and the two halves land on top of each other. The outer pair match at 0.08 apiece and the inner pair match at 0.18 apiece. Now look at the values those weights sit on, minus 9.00, minus 4.00, 1.00, 6.00 and 11.00 per cent. The five values are spaced five percentage points apart all the way along, so the middle value at 1.00 per cent sits exactly halfway between its neighbours on both sides and exactly halfway between the two ends.

Matching weights on evenly spaced values is a mirror, and a mirror has no long side, so the skewness of this population is 0.0000 by construction rather than by measurement. Nothing was counted to arrive at that reading. The nil follows from the definition the same way the number of corners on a square follows from the definition of a square.

The cancellation can be watched directly. Each value's distance from the mean of 1.00 per cent gets cubed and multiplied by its weight. The value at minus 9.00 sits ten percentage pointsThe plain difference between two percentages. Moving from 1 per cent to 11 per cent is ten percentage points, whatever that move might be as a proportion of where it started. below, and the value at 11.00 sits ten above. Minus ten cubed is minus a thousand; ten cubed is a thousand. Both carry the same weight of 0.08, so they contribute minus 80 and plus 80. The pair at minus 4.00 and 6.00 do the same thing on a smaller scale, contributing minus 22.5 and plus 22.5. The middle value is the mean, so its distance is nil and it contributes nothing at all.

The total of nil, divided by the cube of the true standard deviation at 5.00 cubed, or 125.00, is nil again. There is no rounding anywhere in that column and no approximation. An exactly known population is the reference point everything below rests on, and no real record can supply one.

So why does the record report minus 0.0502?

The fifty month record came out of that mirror. Its counts are 5, 9, 25, 8 and 3, adding to fifty, and its mean is 0.50 per cent. Put those counts through the identical arithmetic and the reading that emerges is minus 0.0502. The population is exactly symmetric, the record is not, and nothing whatsoever went wrong.

Fifty draws from a mirror do not arrive in mirror image proportions, any more than fifty coin tosses arrive at twenty five heads. The record happened to collect five months at the worst value and only three at the best. An imbalance of two months, out of fifty, is the entire source of the reading. Here is the same column of cubes as before, run on counts instead of weights.

Minus 300.000 divided by the fifty months is minus 6.0000. Dividing minus 6.0000 by the cube of the record's own spread, 4.9244 cubed or 119.4174, gives minus 0.0502. Every step is an ordinary multiplication and there is no judgement anywhere in it. The reading is correct arithmetic on the months that turned up, and it is a fact about which fifty months turned up rather than a fact about the unit they came from.

Two other figures on the record agree with the reading, and that agreement is what makes the trap so convincing. The record's mean of 0.50 per cent does sit below its middle case of 1.00 per cent, exactly as a leftward stretch would predict. So a reader checking the reading against the centres finds confirmation, checks the arithmetic and finds it clean, and concludes the unit leans. The reading, the check and the arithmetic are all correct. All three describe fifty months, and the claim was about the object behind them, so the conclusion is still false.

How much of a skewness reading is only noise?

Answering that turns the arithmetic from interesting into useful, and the way to answer it is to stop reasoning and start counting. Elsewhere in these notes four more tallies exist, each of them fifty months long and each pulled out of the same mirror. Putting all five through the identical arithmetic returns five readings.

Five records out of one perfectly symmetric population produced three positive readings and two negative ones, spread across a stretch of 0.1864, and not one of the five landed on the truth. They do not even agree on which way the lean runs. An analyst handed the fourth record alone would have written that the unit leans left; handed the fifth alone, that it leans right. Both notes would have been arithmetic without a fault in it, and both would have been wrong about the same object.

There is an even blunter way to feel the fragility. Take the fifty month record and move a single month from the worst value to the best. The reading goes from minus 0.0502 to plus 0.0400 and the sign flips. Move a single month the other way instead and it goes to minus 0.1544, three times its original size. One month out of fifty is enough to change what the reading appears to say. A reading of this size on a record of this length is worth very little on its own.

So the working question is not the size of the reading. The working question is whether a reading of that size is large enough to be anything, given how many cases went into it. Answering that question formally, with a threshold and a stated chance of being fooled, needs machinery that is set out separately. A known population makes a comparison possible instead: five honest records from a shape with no lean at all reported readings from minus 0.1050 to plus 0.0814, so anything inside that stretch has already been produced here by pure sampling errorThe gap between what a limited record shows and what the thing it came from is really like, caused by nothing except which cases happened to turn up. Sampling error is built in full separately..

What does a shape that really is skewed look like?

An account that stopped here would leave something false standing, namely that a lean is always the record talking. A genuine case belongs beside the false one. Elsewhere in these notes a shape is defined for where a price sits after twelve months, built from a monthly change with a spread that accumulates over the year to a spread of 0.1732 in log terms. The twelve month shape has a positive skewness of 0.5289. The reading is a property of the object rather than of any record, so no amount of extra data would make it go away.

The reason is structural and can be stated without any arithmetic at all. A price has a floor at zero and no ceiling above it. It can fall by at most everything, and there the fall has to stop, because a price cannot turn negative. Upwards there is no such wall anywhere. So the right side of the shape has room the left side does not have, and the shape spreads into that room. The lean is not a fact about a particular twelve months; it is a fact about what a price is allowed to do.

Everyday version: think of how long a train journey takes. A journey cannot take less than the time the track physically allows, so a hard floor sits a few minutes below the scheduled time. But a signal failure can add an hour. Most journeys huddle just above the floor and a handful run far out to the right, and that is a right leaning shape produced by a floor rather than by anything anybody measured.

The figures follow from the definition. Start at Rs 100.00/-. The most likely landing place after twelve months is Rs 109.4174/-, the middle case is Rs 112.7497/- and the mean is Rs 114.4537/-. The ordering, most likely below middle below mean, is what a right leaning shape does to its three centres, and the definition fixes it rather than any counting. Measure Rs 40.00/- either side of the middle case and the asymmetry is plain: the chance of ending below Rs 72.7497/- is 0.5709 per cent while the chance of ending above Rs 152.7497/- is 3.9800 per cent, about seven times as much room on the right.

What do the three readings say when they are lined up?

The three readings sit together below, and the comparison is the whole point. A true nil, a noisy near nil and a real lean look like three positions on one scale, and they are three completely different kinds of claim. The first is a consequence of a definition. The second is a consequence of which fifty months turned up. The third is a consequence of what a price is allowed to do. Only the first and third would survive being collected again.

What does the lean do to the centre being quoted?

On a shape with no lean the mean and the middle case land in the same place, so a summary can quote either one and nobody is misled. On a leaning shape they separate, and a summary that quotes only one of them has quietly made a choice on the reader's behalf. On a right leaning shape most outcomes fall below the mean, so a mean quoted on its own describes an outcome that is less common than it sounds.

The price shape shows this cleanly. Its mean is Rs 114.4537/- and its middle case is Rs 112.7497/-. By definition half of all outcomes land below Rs 112.7497/-, and Rs 112.7497/- sits below the mean. Quote the mean alone and a reader forms a picture of a typical year that more than half of all years will fall short of. Nothing dishonest has happened. One figure was quoted where the shape needed two.

The everyday version is the one everybody has already met. A street food stall's takings across thirty days are a huddle of ordinary days plus three festival days at four times the usual. Three days did all the lifting, so the mean daily takings sit well above what the stall collects on a typical day. Sizing the stall's rent against the mean would size it against a day the stall almost never has. Which centre a leaning record deserves is settled separately and in full; the lean is what creates the question.

How does a reader test a skewness reading before using it?

Analysts, lenders and anybody writing a note about a set of outcomes meet this figure the same way: it arrives already computed, sitting in a summary table beside a mean and a spread, with no indication of how much weight it can carry. Four questions turn that figure back into something that can be used or discarded, and they take about a minute.

How many cases went into it? A skewness reading is built from cubed distances, and cubing means the few cases furthest out dominate the total. Look back at the record's own column of cubes. The two outer values, holding eight months between them, account for nearly all of the minus 300.000. The twenty five months piled on the middle value manage 3.125 between them. A reading built on fifty cases is really a reading built on a handful of them.

Does anything structural force a lean? Ask what an outcome is forbidden to do. Does a floor exist, like a price at nil or a delay that cannot be negative? Does a ceiling exist, like a share that cannot exceed the whole? A floor is still there next year and a record is not, so a reason of that kind separates a lean that can be relied on from a lean that merely turned up.

How far is the mean from the middle case on this same record? It is a free cross check on figures that are already to hand. If the reading says the shape leans left and the mean sits above the middle case, something is inconsistent and worth opening up. If the two agree, that is two weak signals rather than one. Two weak signals are a little better than one, and not by much.

Would a shape with no lean at all have produced this reading? A shape with no lean produced five readings from minus 0.1050 to plus 0.0814, so the question can be answered flat out. Away from a made up population it cannot be answered by inspection, and the machinery that answers it properly is covered separately.

A structural reason survives resamplingCollecting the record again from scratch and seeing what the second attempt reports. These fifty months never happened, so resampling is a thought experiment rather than an instruction. and a bare reading may not, so the second question separates a finding from a reading. A lean with a reason behind it will be there in the next record, and the one after that. A lean with no reason behind it is usually the record talking, and the honest way to write it up is to say what the reading was, how many cases produced it, and that nothing about the object has been established.

Who vouches for the three readings?

Nobody, and asking the question is more useful than the answer. A skewness reading is a sum of cubed distances divided by a cubed spread. The arithmetic leaves nothing for an institution to have an opinion about. Put five values, five weights and five counts in front of any two people and they will land on the identical figure. So the column where a document would normally sit is all but empty, and the column beside it carries the check that can be run in its place.

The Nakshatra unit and its five tallies of fifty months are invented.
Educational material. Not advice on any investment, tax, budget or market position.