The Lognormal Distribution: Why Prices Are Modelled This Way

A price cannot be described by a shape that folds onto itself either side of its centre, for two reasons. Such a shape puts weight below zero, where a price cannot go. And it treats a fall and a rise of the same size as cancelling. A fall and a rise do not cancel: Rs 100/- down a fifth then up a fifth is Rs 96/-. Put the shape on the log change instead and both faults go.

Underneath that answer sits one idea, and it is worth holding on to before any arithmetic arrives. A quantity whose changes are multiplicativeChanging by being multiplied by something, rather than by having something added on. Doubling is multiplicative; adding ten rupees is not. is a different animal from a quantity that moves by having something added to it. A person's height in a year is this year plus a bit. A price in a year is this price times something. The shapes that work beautifully on the first kind break on the second, and they break in two specific, checkable ways rather than vaguely. Both breaks have a single cause, one change repairs both, and the change leaves the shape's behaviour far out in the tails exactly as it was.

A shape over an uncertain quantity, its centre, its spread and the running totalThe chance of landing at or below a stated value, built by adding up all the weight from the far left of the shape up to that point. it gives at any value are the standard equipment, and so is the smooth bell and the reading of a distance from its centre. Only one move is new, and it is made once: a logarithm is taken before the shape is applied, and everything else follows from that.

What goes wrong first when a folding shape is put on a price?

Start with the object. The Nakshatra unit is an invented traded unitSomething with a price that can be bought and sold. priced once a month, and it starts at Rs 100/-. Suppose somebody supplies a symmetricFolding onto itself either side of a centre, so that the two halves of the picture land on each other exactly when it is folded down the middle. bell over the monthly change. Its centre is 1.00 per cent and its spread is 40.00 per cent. The fault shows up more clearly at a big spread, so 40.00 per cent is chosen deliberately.

Now ask the shape a question it will happily answer. What chance does it give a monthly change worse than minus 100.00 per cent? A change of minus 100.00 per cent takes Rs 100/- to Rs 0/-. Anything worse takes it below Rs 0/-. Measure the distance from the centre: minus 100.00 is 101.00 below a centre of 1.00, and 101.00 divided by a spread of 40.00 is 2.525 spreads out. The bell answers 0.5785 per cent. One month in about a hundred and seventy three.

A shape that gives a price a 0.5785 per cent chance each month of ending below nothing is not slightly miscalibrated, it is describing an event that cannot occur. And a monthly fault compounds into an annual one. Run it twelve times and the chance of it happening at least once is one less 0.994215 raised to the twelfth. The answer is 6.73 per cent. Roughly one year in fifteen, this model has the Nakshatra unit finishing somewhere below zero rupees, and a reader who never asks the model that particular question will never find out.

Why does a fall and a rise of the same size not cancel out?

Here is the second break, and it needs no probability at all. Take the Nakshatra unit at Rs 100/-. In the first month it falls 20.00 per cent, so it lands at Rs 80/-. In the second month it rises 20.00 per cent. Where does it finish?

Not at Rs 100/-. The unit finishes at Rs 96/-, 4.00 per cent below where it began. The rise is 20.00 per cent of a smaller number than the fall was, so the two moves are equal as percentages and unequal in rupees. Twenty per cent of Rs 100/- is Rs 20/-. Twenty per cent of Rs 80/- is Rs 16/-. A shortfall of Rs 4/- is left behind, and nothing recovers it.

The same arithmetic turns up in ordinary life. A tea stall on a busy corner loses a fifth of its regulars when a road is dug up. The road reopens and the stall wins back a fifth of the customers it has left. The fifth it won back was a fifth of a thinner queue, so the stall is not back where it started. The arithmetic that hurts a stall is the same arithmetic that hurts a price, and it is why every fall needs a bigger percentage rise to undo it than the percentage it fell by.

What is the log change, and what does taking it do?

Both faults come from the same source: the shape was laid on the wrong quantity. Percentage changes on a price multiply. A month of minus 20.00 per cent multiplies by 0.80, a month of plus 20.00 per cent multiplies by 1.20, and 0.80 times 1.20 is 0.96, the Rs 96/- above. Adding percentages was never the right operation, and a shape built for adding was always going to misbehave.

So change the quantity. A logarithm is the tool that turns multiplying into adding, and that is the entire reason it appears here. Written out with no assumed background: if the price now is Rs 100/- and the price next month is Rs 105/-, form the ratio 105 divided by 100. The ratio is 1.05. The log change is the logarithm of that ratio, and here it is 0.04879. If the price instead fell to Rs 95/-, the ratio is 0.95 and the log change is minus 0.05129.

The property that matters is this. Multiply two ratios together and the logarithms add. The fall of a fifth has ratio 0.80 and log change minus 0.22314. The rise of a fifth has ratio 1.20 and log change plus 0.18232. Add them and the total is minus 0.04082, exactly the logarithm of 0.96. Two months of a price stopped being a multiplication problem and became an addition problem, so the arithmetic that misbehaved before now behaves itself.

The folding shape is put on the log change, not on the price, and that single relocation is the whole of the method. Everything after this is a consequence of it. The name lognormal is simply the label for what appears on the price side once a normal bell has been laid on the log change: the price whose logarithm is normally shaped.

Why can a price built this way never reach zero?

Recovering the price from a log change means raising a fixed number to that power. Take the log change, raise the base of natural logarithms to it, and multiply by the starting price. The step is called taking the exponential, and it has one property that settles the first fault permanently: raising a positive base to any real power at all produces a positive result.

Work it downwards and watch it refuse to arrive. A log change of minus 0.70 gives Rs 49.66/-. Minus 1.50 gives Rs 22.31/-. Minus 3.00 gives Rs 4.98/-. Minus 6.00 gives Rs 0.25/-. Minus 10.00 gives Rs 0.0045/-. Halving something repeatedly never reaches nothing. The price is crushed towards zero and never lands there, however far down the log change is pushed.

The floor at zero is not a rule bolted on afterwards and it is not a clamp that catches bad values. The floor falls out of the arithmetic of the exponential, and arithmetic is what makes the floor trustworthy. A rule added by hand can be forgotten, mis-specified or switched off. A property of the operation cannot be. Nobody has to remember to enforce this floor, and no setting of any input can breach it.

Why does the log change handle a fall and a rise correctly?

Take two log changes of equal size in opposite directions and turn each back into a price. A log change of minus 0.20 from Rs 100/- gives Rs 81.87/-. A log change of plus 0.20 gives Rs 122.14/-. Equal in logs, and utterly unequal in rupees: a fall of Rs 18.13/- against a rise of Rs 22.14/-, a difference of Rs 4.01/- between two moves the log axis calls identical.

The pair really does cancel, and this is exactly where the earlier attempt failed. The two ratios multiply out: 0.81873 times 1.22140 is 1.00000. The two log changes add: minus 0.20 plus 0.20 is 0. A fall of Rs 18.13/- followed by a rise of Rs 22.14/- lands exactly back at Rs 100/-. A fall and a rise of equal size ought to do exactly that, and plus and minus 20.00 per cent never did.

The lopsidedness in rupees is not a distortion the model introduced, it is the lopsidedness prices genuinely have, now written down instead of assumed away. A price can only fall by 100.00 per cent and it can rise without any limit at all. Any description that treats those two directions as mirror images has already got the object wrong, and the log change is simply the coordinate in which the mirror is honest.

What happens to the shape over twelve months?

One month at a time is settled. Now stretch the horizonHow far ahead the question looks. Here it is counted in months, from one month up to twenty four.. Give the Nakshatra unit a monthly log change centred on 1.00 per cent, spread 5.00 per cent, and run it forward twelve times.

With that assumption on the table, the arithmetic is short. Log changes add, so twelve of them give a twelve-month log change centred on twelve times 1.00 per cent. The centre is 12.00 per cent. The spread does not add the same way. When the pieces are independent, spreads combine through their squares. Twelve squared spreads of 5.00 per cent give a squared spread of 0.0025 times twelve, or 0.03, and the square root of 0.03 is 0.173205. The twelve-month spread is 17.32 per cent.

The twelve-month spread is the monthly spread multiplied by the square root of twelve, 3.4641, and not by twelve. The spread lands at 17.32 per cent rather than 60.00 per cent. The reason is worth one sentence: independent moves partly cancel each other, a bad month often meeting a good one, so uncertainty grows more slowly than time does. Ten independent bad months in a row is a far rarer thing than ten bad months would be if they marched together.

Why are the average price and the middle price different numbers?

Where a shape folds onto itself either side of its centre, the three ways of naming a typical value collapse into one number: whichever one is asked for, the answer is the same. The collapse held for the monthly Nakshatra generator, where all three answered 1.00 per cent, and the coincidence was a property of that particular shape rather than a fact about shapes in general. Here the folding is gone from the price side, and the three come apart.

Run the arithmetic from Rs 100/- with a twelve-month log centre of 12.00 per cent and a log spread of 17.32 per cent. The most likely price, the one carrying the greatest weight, is Rs 100/- times the exponential of 0.12 less 0.03, or Rs 109.42/-. The middle price, with half the outcomes either side, is Rs 100/- times the exponential of 0.12, or Rs 112.75/-. The average price, the balance point of all outcomes, is Rs 100/- times the exponential of 0.12 plus half of 0.03, or Rs 114.45/-.

One number on the folding shape has become three here: Rs 109.42/-, Rs 112.75/- and Rs 114.45/-, in that order, and asking which one somebody means stops being pedantry and starts being the question. The order is not accidental. Nothing at all can stretch to the left of zero to pull the balance point down, and a handful of very large multiples stretch far out to the right to drag it up. The one-sided freedom to run is what puts the average highest.

Can the average price be Rs 114.45/- while a quarter of outcomes lose money?

Both statements are true of the same twelve-month shape, and holding them together is the single most useful habit a lopsided shape demands. The average price is Rs 114.45/-. And the chance of finishing below the Rs 100/- the Nakshatra unit started at is 24.42 per cent, close enough to one year in four to be worth saying that way.

The second number comes straight from a running total of the familiar kind. Finishing below Rs 100/- means the twelve-month log change finishing below zero. Zero sits 12.00 per cent below a centre of 12.00 per cent, and 12.00 divided by a spread of 17.32 is 0.6928 spreads below the centre. The running total at 0.6928 spreads below reads 24.42 per cent.

The average price and the chance of a loss are two answers to two different questions about one shape. Neither is the headline, and quoting either one alone misdescribes the shape. An average is where the balance point of every outcome sits, which in the long runA very large number of repetitions. The long run is the setting in which an average describes what is landed on per go, and an average is not a description of any single go. is a real and meaningful thing. The average is not what happens most often, and on this shape it is not even the halfway mark.

How does anybody actually use these three numbers?

Somebody has to make a decision with this, so here is what each of the three is good for and what it is useless for. The most likely price of Rs 109.42/- answers what single outcome carries the most weight, and it is the only one of the three that answers that. No single outcome on a smooth shape is at all probable on its own, so the most likely price is close to useless for planning.

The middle price of Rs 112.75/- answers what value has half the outcomes either side. The middle price splits the possibilities into two equal halves and gives an even bet. A household planning around a horizon wants exactly that. Somebody asking how long a queue at a government office will be faces the same distinction. The average wait is inflated by the rare morning when a system goes down; the middle wait is what a day can actually be planned around.

The average price of Rs 114.45/- answers what the balance point of every outcome is. When many independent draws are added together the pieces do average out and the total lands near the sum of the averages, so the average is the right number there. An analyst adding up a hundred separate positions wants the average, a household planning around one horizon wants the middle, and handing either person the other number gives them a defensible figure aimed at the wrong question. That is why the honest summary of a lopsided shape is never one number.

What did changing the variable not fix?

Moving the shape onto the log change repaired two things exactly, the two faults a folding shape has when it is laid on a price. The floor at zero now holds by arithmetic rather than by decree. The asymmetry between a fall and a rise is now written down correctly rather than assumed away. The repair reaches that far and no further.

The shape laid on the log change is still the folding bell, so everything that shape got wrong about extreme outcomes it still gets wrong here, unchanged and uninherited by the repair. If real months come in two temperaments, a quiet sort and a stressed sort, then the weight far out in the tails is heavier than a folding bell allows, and taking a logarithm first does nothing whatever about that. A change of coordinate cannot repair a mis-specified shape. The change of variable moved the shape onto a quantity where the floor and the asymmetry make sense, and the fat tailAn end of a shape carrying heavier weight far from the centre than a folding bell would give it, so the rare severe outcome arrives more often in reality than in the model. problem walked across the change of variable completely intact.

The second thing left standing is the assumption. Twelve months were treated as independent, and that assumption produced the square root of twelve, the spread of 17.32 per cent, and every rupee amount that followed from it. If months carry information about one another, if a bad month makes the next month more likely to be bad, then the twelve-month spread is not 17.32 per cent and none of the three prices holds. Whether that is so, and what it does to the arithmetic, belongs to the study of quantities observed in order through time and is covered separately.