Behavioral finance FAQ / Glossary (Fat tails)

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Dates of related message(s) in the
Behavioral-Finance group (*):

Year/month, d: developed / discussed,
i: incidental

Fat tails / wings (in distribution curves)

00/6i,8i,10i -01/9i,10i,12i -
02/4i,7i -03/7i - 04/2d,9i,11i
06/7d - 07/4i - 08/9i + see
distribution curve, kurtosis,
rare event, (law of) extremes
+ bfdef3

When bells grow tails, beware!

Rare things and surprises will happen!


Wandering far from the normal lanes,
not matching fully standard probabilities

To put it simply, fat tails are seen when too many events or values

stray widely from the average.

Rare events (see that phrase), not only occur,
but they are ...not so rare,
although people
tend to neglect their possibility.

For example, in asset markets, there are often more days of
spectacular price rises or falls than is expected under the "normal"
statistical distribution laws

=> As a consequence investors can be caught in unsuspected risks.

To give a general definition, fat tails
(aka fat distribution tails, heavy tails, fat wings ...)
are statistical phenomena in which:

Extreme values (low and high values)
are more frequent than the "normal law" (*) predicts.

(*) The "normal law" (aka Gauss-Laplace, see distribution) of random
      distribution is routinely
applied in various probability calculations,

 for example in financial mathematics.

The phrase "fat tails" means that visually on a chart (see below) the two
"tails" are bigger than those of a "normal" distribution of data.

The chart looks like a Cadillac Eldorado from the 60's, with a
tail and a front that span the whole city block

The issue is that in some areas (finance among others)
real data seem distributed according to a fat tail pattern
of a pure random law


Your Highness, you are largely above average.

Let us take the example of an adult population with a is 1.75 meter
average height

Fat tails would be seen if the proportion of people who are over
2.1 m high or under 1,5 m high
is larger than the normal law
would predict.

The obvious consequence would be less people than predicted in
the "central" 1.5 - 2.1 range and even in the "hyper-central"
1.65 - 1.85 range, in which the bulk of the population would
theoretically be found.

Another example below shows how this is expressed visually on a chart.

Bell and tails

The fat tails allegory refers visually to statistical distribution
in which:

The horizontal scale gives the values (example: people heights)

The vertical scale gives the number of times each value occurs
    (frequency, example: number of people)

Let us remind that the normal law (Gaussian statistical distribution) -
shown as the "bell curve" on that kind of charts - represents perfect
randomness around a mean.

If the data match that law, the farther from the mean you look, the less
occurrences you see.

=> But this is not what happens in the case of fat tails.

      Here is the difference:

In a "normal" distribution the two "tails" (or wings), on the far

        right and the far left of the curve,

Get slimmer and slimmer, until they reach zero occurrences at
    each end.

Also, are more or less symmetrical.

But in real life, various statistical series show rather fat "tails",

<=> This means that there are more occurrences of low and high

      values than theoretically expected. We  have a "leptokurtic
as explained in a section below

To use the above example (people size distribution), some biological
accident might cause fat tail(s) at one end, or at both ends, of the
curve with many more people above 2.1 m and/or below 1.5 m than
the Gaussian law would predict.

Gaussian distribution (= blue X)

<=> Leptokurtic distributions / fat tails (= red O)

            X   X            
            O   O            
          X   X        
          X       X          
          O   O          
        X           X        
O     X O           O X     O
O O O               O O O
  X X                   X X  
X                       X

Horizontal scale = values (i.e. shoe sizes)

Vertical scale = frequencies (number of times each value occurs)

Other strange shapes

Other distortions might also appear sometimes - together with fat tails or
without them - such as:

Asymmetries / skews (see those words):

One tail is fatter than the other.

=> It means that one side is more prone to "rare events".

For example, in financial market return distributions, there are
more sizable price falls than sizable price rises: the negative tail is often
than the positive one.

=> There is more volatility, thus more prices wandering far from the average
       trend, in a crash than in a bubble because of higher perceived

Clusters (see that word):

They are bumps seen at some places of a distribution curve, warts on the
bell's skin, traffic jams in market streets

Those "data clusters" can be a result of pure randomness. But they could
also signal specific circumstances (or biased behaviors when the distribution
relates to social data).

Long tails,

They are less related to a higher than anticipated frequency  
of rare events like in fat tails than to the existence and size of
extremely rare (and often disastrous) events

The frequency of those exceptional / singular events is extremely
small (very far from the mean) but their magnitudes are exceptionally
low or high,
, centennial catastrophes for example).

Those "rare events" are a crucial phenomenon that might have
huge consequences: see the "rare event" article in the glossary.

Fat tails in financial markets

Beware of pianos falling from the sky!
A perverse behavior that is more frequent than expected,
at least on markets.
Beware of crushed investor bones!

In asset markets, as in other economic and social phenomena,
classical theories generally take for granted that randomness and Gaussian
distribution are all over the place.

Market models that apply the efficient market theory usually admit those

But actually, in asset markets (like in most
human fields
),reality differs from perfect

random laws.

More precisely, the distribution of (monthly, daily) market returns, risks
and volatilities,
does not always follow the "normal law":

Either in static distributions (distributions observed at a precise time).

It is the case when a chart compares all stock P/Es (or other asset
price criteria) at the end of a day.

Or in time-distributions (historical series).

An example is daily returns, observed day after day.

If the normal law applied fully, they would spread nicely and equally
between, let us say, minus .4% and plus .5%, around a slightly positive

In reality the distribution is less perfect.

The curve shows more "rare / extreme events",
a larger number of extremely high and low daily
returns (*), than is theoretically
Tails are fatter than predicted.

(*) There are days when volatility can go to extremes, as

illiquidity or near-illiquidity (see liquidity squeeze) strikes

Also a violent shock can be followed by violent aftershocks.

Danger ahead, the big market earthquake can strike unexpectedly!

Why those market distortions?

Tail-pulling investors

Why and how is this "law of extremes" (see that word) generated in
asset markets?

The main explanation might lie, sometimes in exogenous shocks, but most
often in collective behavioral biases, such as under / over reactions
and herding
, which push prices to extreme rises or falls.

People are not fully independent in their thoughts and behaviors.

Thus their decisions might imitate those of other players.

Massive and prolonged collective behaviors occur and lead at
     times to extreme moves, that are incompatible with a random

law and to a "reversion to the mean".

A "leptokurtic" evolution for market returns?

Strange camel bumps

It seems that, nowadays, financial markets evolve towards leptokurtic

We can give that pet name to a camel with a vertical towering bulge in the
center, deep ditches on each side of it, and then fat tails (please, no bawdy
allegory ;-) at the end of each side.

Applied to market returns (when defined as price variations),
that kind of statistical distribution shows:

A peaking tower

where small

returns (= price
variations) concentrate
close to the mean.


Let us say that there are many days in which

prices vary within a narrow - 0.3% / + 0.4%

Volatility is small in normal time, a daily
routine of small moves, giving an impression
of efficiency around a purported "fair value".

The market is tightly controlled by
professionals who speedily correct "anomalies".

The distribution graphs shows a tower and a
data desert around.

Law and order, only dromedaries allowed at
Camel city!

Two narrow plains

aside a tower that
shows scarce medium
size variations.

Let us say that variations in the
     +/- 0,5%   to    +/- 1,5%  

are less frequent than would be seen in bell /
normal law curves.

Two small hills

on the far sides
that show sudden
of strong price
rises or falls.

Here we have clusters of days with exceptional 
     high and low
returns, like minus or plus 5% in  
a day or even much more.

They can be caused by brutal accidents, or by
general and sudden
opinion shifts.
Stampede in the Camel corral!

Unexpected fat tails: the "tail risk"

Careful! If you ignore the tail, you might step on it!

As fat tails relate to events that are the most infrequent (thus that are far
apart in time) fat tails might be undetectable in too short historical
series (*)

The problem is that those events are also the most extreme in amplitude
and can have a strong impact on people, among them investors.

The "tail risk" is the danger of undetected fat
tails / rare events
(see that phrase) that might bring
extreme losses (the 100 year storm).

Their probability, although it exists, is too small to be
on distribution curves made with recent data series.

In most current statistics, done on too short periods, the real
shape of the curve does not show.

What is worse is that those rare events are often

more frequent than the randomness theories
would expect

However invisible fat tails (or more probably, long tails) are in
thosetoo short statistics, the risk of extreme prices, returns or
volatilities (therefore of extreme losses) is often higher than a
normal distribution law would ...entail.

Therefore the models that rely on such standard random law

might underestimate the real market risk .

=> What investors should keep in mind.

As a result, investors tend to get overconfident and take unsuspected

This is notably the case for those who:

are new in the game,

do not have full information on past events,

neglect such statistical info (see "base rate fallacy", another example),

have a "short memory" (a widespread bias, see "memory"),

(*) Some say that with an infinite number of time spans, or with an infinite
     length of time, the return distributions would be purely random. This
     would prove the
RWH and EMH to be right in absolute.

Although this is impossible to check, it doesn't prevent that fat or long
tails, which can be spotted (big data might help) in long - but not infinite -
series , are undetectable in short series.

(*) To find those messages: reach that BF group and, once there,
      1) click "messages", 2) enter your query in "search archives".

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This page last update: 24/07/15  

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