Morningstar - Q2 2020 - 43

EXHIBIT 1

Different Views of Returns The lognormal model, which is the standard way of modeling returns distributions, fails to account
for the extreme returns found in the real return data and that the log-stable model picks up. In other words, returns have fatter
tails than the thin-tailed model shows.
Alternative Distributions for U.S. Monthly Real Stock Market Returns, January 1886-March 2020

Real Return Data

Lognormal

Log-Stable

Probability Distribution Function
10

The log-stable model, a version of which Mandelbrot
used to accurately model cotton prices in the
1960s, accurately ﬁts the real return data of 150 years
of U.S. stock market returns.

8

When we zoom in on the extreme left side
of the distributions, we can see that the tail of
the log-stable distribution is fatter than
that of the lognormal distribution and better
ﬁts the real return data.

6
4

-10

2
0

-30%
Monthly Real Return

-15

0

15

30

45

Source: Morningstar Direct.

Return
Data
Lognormal
Log-Stable
means
that
Markowitz's
mean-variance
model
,
of portfolio construction (Markowitz
) is invalid. It also means that asset-pricing
theories such as the capital asset pricing model
, Lintner
) and the arbitrage
(Sharpe
well.
pricing
theory
(Rossleft side) ofare
In Canada,
the extreme
theinvalid,
log-stableasmodel
ﬁts the return data, while the lognormal distribution
misses it entirely.

sweep under the carpet until now ... but surely
before consigning centuries of work to the
ash pile, we should like to have some assurance
that all of our work is truly useless."
As Greg Satell (

Probability
Paretian model to asset
returnsDistribution
and the Function
eﬁc e c es o the lo ormal mo el, e ca look12
at themodel
stable
Paretian's
two
shape parameters:
The log-stable
for Canada's
market
returns
also accurately
ﬁts the market's
return data
peakedness
and skewness.
better than the lognormal model.

) said:

The peakedness parameter is between and
. It indicates the degree to which the peak of
the distribution curve is higher than that
of a normal or lognormal distribution (as is the
case in E X H I B I T S 1 and 2 ) and how far the fat
tails extend out on both sides.

In other words, to accept the log-stable model
of returns, we must reject nearly all of standard
ﬁ a c al eco om cs

"In other words, the locomotive was heading down
the tracks at full steam and Mandelbrot would
be left at the station. The attractions of financial
engineering were too great, the potential profits
too gargantuan. While it might be interesting
The uncomfortable implications of returns
-10
for traders to discuss Mandelbrot's findings over a
following log-stable distributions were raised in
,
a ery ue t al ook at the t me
beer after work, his brand of uncertainty did
-15
-5 nor did they create
0 multibillion-dollar
5
-25%
the ﬁ a c al eco-20om st aul
oot er u l she -10 not win clients
Monthly
Return
a collection
of papers called The Random Character
bonus pools. Sure, there were some problems,
but they tweak the models some more and
). This
of Stock Market Prices (Cootner
book included a paper by Mandelbrot and a paper
everything would work out in the end. At least they
by Fama on modeling price changes with
hoped it would. It didn't. The recent financial
crises has (sic) laid bare the lie that Mandelbrot
stable Paretian distributions.
Japan
exposed more than 40 years ago. As with
many of his seemingly outlandish ideas, he had
InPoland
his introduction to the section containing those
South Africa
been right all along."
papers,
Italy Cootner writes that Mandelbrot forced
ﬁ Mexico
a
c
al
eco
om
sts
to
ace
u
a
su
sta
t
e
Peru
United
Kingdom
Norway
France
Australia Denmark
Chile the World Turkey
Around
way
to those uncomfortable
empirical observations
Brazil
Netherlands
Hungary Canada
To
further
show the applicability of the stable
that
there
is
little
doubt
most
of
us
have
had
to
India
Belgium
Hong Kong
United States
New Zealand

These eight countries
have the most negative
possible skewness,
indicating extremely
skewed left tails.

Switzerland

Ireland

Taiwan

9

6

3

If the peakedness parameter is equal to , the
str ut o s ormal, th a ﬁ te ar a ce
0
10 than , the str
15 ut o has ﬁ20te
If it is less
variance. The more below that the peakedness
parameter is, the higher the middle and the
fatter the tails.
Peakedness
The skewness parameter indicates the
2.0
asymmetry of the distribution. The skewness
parameter
between
negative
South Korea'sis
market
index has
positive and positive
skewness, meaning its distribution tilts
1.9
.toAthevalue
of
means
that
the distribution
right. But its peakedness is below
, indicating it has
fat tails. value means that the
is2.0symmetric.
A negative
distribution is skewed to the left, and a positive 1.8
Korea
value means that it is skewed toSouth
the right.

Philippines

Thailand
Sweden

1.7
morningstar.com/lp/magazine
Indonesia

Malaysia
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Morningstar - Q2 2020

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