Fat Tails and Traditional Capital Market Theory - Gilles Daniel

Fat Tails and
Traditional Capital Market Theory
Günter Bamberg
Gregor Dorfleitner
Heft 177/2001
(3., abermals überarbeitete Auflage August 2001)
Institut für Statistik
und Mathematische Wirtschaftstheorie
der Universität Augsburg
Prof. Dr. Günter Bamberg
Dr. Gregor Dorfleitner
Universitätsstr. 16, D-86159 Augsburg
Telefon: 0821/598-4151
Telefax: 0821/598-4227
E-mail: guenter.bamberg@wiso.uni-augsburg.de

Fat Tails and Traditional Capital Market
Theory
Günter Bamberg, Gregor Dorfleitner
Overview
Many empirical studies of stock price data conclude that log returns are fat-tailed.
Statisticians mainly interested in time series analysis or curve fitting are not aware
or do not bother about the consequences of fat-tailed log returns: expected stock
prices fail to exist, expected returns (in the sense of percentage price changes) do
not exist as well.
If expected returns do not exist, then important theoretical underpinnings of capital
market theory like mean-variance portfolio selection, CAPM, security market line,
APT are no longer valid.
These serious consequences are known to be true in the framework of stable (nonnormal)
distributions. However, they are to be faced for arbitrary fat-tailed nonstable
distributions.
Acknowledgements. The authors would like to thank S.T. Rachev and two anonymous referees
for valuable suggestions and improvements.
1

A. Introduction and clarification of important notions
This paper investigates the consequences of fat-tailed log returns for the percentage returns,
on which traditional capital market theory is based. Before we come to our main
statement in Section B and the conclusions in Section C, the different notions of returns
and fat tails shall be clarified in this section.
I. Percentage returns and log returns
Let P t be the price of a certain security (adjusted for dividends, stock splits etc.) or the
value of a stock index. The log return with respect to period t is defined as
( )
Pt
R t = ln . (1)
P t−1
Hence the end-of-period price P t is
P t = P t−1 e R t
. (2)
from which one can see that R t is the continuously compounded rate of return.
equation
The
P t = P 0 e R 1+···+R t
= P 0 e R 0t
, (3)
where the log return R 0t corresponds to the time interval [0,t], is valid from the definition
of R t . Hence we have the additivity property
R 1 + ···+ R t = R 0t . (4)
Due to this property log returns are favorable if developments along the time axis are the
topic of interest. The log return Rt
P of a portfolio is related to the log returns R (i)
t of the
single stocks by the nonlinear formula
R P t = ln
( n∑
a i e R(i) t
i=1
)
, (5)
where a 1 ,...,a n are the fractions of the invested capital. Because of this nonlinearity (5)
traditional capital market theory prefers to work with discrete returns (or percentage price
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changes, or shortly: returns)
DR t = P t − P t−1
P t−1
= P t
P t−1
− 1 = e R t
− 1 . (6)
With this notion of returns, we have a linear relation between the returns of the single
stocks and the portfolio return, i.e.
DR P t =
n
∑
i=1
a i DR (i)
t . (7)
Markowitz portfolio models (CAPM etc.) make full use of this simple linear structure.
Obviously, the support of R t is the entire real line IR, whereas DR t is subject to the restriction
−1 ≤ DR t < ∞ . (8)
Therefore, the textbook approach to portfolio analysis of assuming discrete returns as
normally distributed can only be considered as a rough approximation since it does not
only contradict the property (8), but also ignores that the normal distribution is not closed
under multiplications. The latter implies that the two-period return
(1 + DR 1 ) · (1 + DR 2 ) − 1
cannot be normally distributed simultaneously with DR 1 and DR 2 . Furthermore, the conventional
techniques for annualizing parameters become very clumsy. 1
Additionally, there are numerous papers (compare, for instance, the great amount of references
given by Rachev/Mittnik (2000)) concluding that there is too much empirical
evidence against normally distributed discrete returns. But what are the alternative distributions
Of course, it is also heroic to consider the discrete return as a stable (non-normal) random
variable though some eminent researchers (for instance Fama (1965, 1971), Mandelbrot/Taylor
(1967), Samuelson (1967)) worked with this hypothesis.
The mentioned distributions, which are defined on the whole real line, are surely better
suited for modeling the log (and not the discrete) returns. Moreover, due to property (4)
3

log returns can more easily be handled in statistical models. For these reasons, it has
become very common to model the log returns with
• normal distributions (i.e. the classical Black/Scholes world),
• stable non-normal distributions (see e.g. Rachev/Mittnik (2000)),
• (G)ARCH processes (see e.g. Taylor (1986)),
• student distributions (see e.g Praetz (1972) or Blattberg/Gonades (1974)),
to mention only a few.
From (6) it is clear that DR is fully determined by R. However, the properties of the discrete
return distribution are completely different from those of the log return distribution.
This has severe consequences, as will be be pointed out in Section B. But firstly the term
“fat tail” has to be defined.
II. Fat Tails
Besides the considerable linguistic variety (fat tails, heavy tails, thick tails, long tails)
there is also a variety of attempts to define fat tailedness exactly. 2
Some definitions focus on a single moment, for instance
R is fat-tailed ⇔ Var(R)=∞
or
R is fat-tailed ⇔ R is leptokurtic (i.e. kurtosis> 3) .
Bryson (1982) states that these definitions are too crude and have to be replaced by approaches
which take the tail behavior more explicitly into account, for instance the criterion
of increasing conditional mean exceedance or the limiting distribution of the maximum
of a random sample. The following five definitions seem to be rather important in
the fields of insurance and finance. The resulting classes of fat-tailed distributions can be
ordered with respect to inclusion as illustrated by Figure 1. Note that the family of stable
(non-normal) distributions (set A in Figure 1) is a small subset of all those sets.
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Figure 1:
The different classes of fat-tailed distributions
F
E
D
C
B
A
A: stable (non-normal) distributions
B: Pareto tails with α > 0
C: regular variation with tail index α > 0
D: subexponential distributions
E: nonexistence of all exponential moments
F: nonexistence of the exponential
moment of order 1
Corresponding to class B of Figure 1, a random variable R is said to have an (upper 3 )
Pareto tail iff
P(R > x) ∼ c
x α (x → ∞) for some α > 0 . (9)
Corresponding to class C of Figure 1, a random variable R is of regular variation iff
P(R > x) ∼ L(x)
x α (x → ∞) for some α > 0 , (10)
where L(x) is a slowly varying function. 4 This definition of fat tailedness is rather
widespread; compare, for instance, Shiryaev (1999), p. 334.
The next section shows that both fat tail definitions prevent the existence of expected
stock prices and expected (discrete) returns. Actually, the nonexistence of E ( e R) holds
for a huge class of log return distributions.
A third notion of fat tailedness is provided by the subexponential distributions. 5 A recent
survey on this class is Goldie/Klüppelberg (1998). The survey makes it perfectly clear
that, firstly, regular variation distributions with any tail index are subexponential, and
secondly, that subexponential distributions have no finite exponential moments of any
order s, i.e.
E ( e sR) = ∞ for all s > 0. (11)
Property (11) can also be used as the fourth fat tail definition, which is wider
5

than the requirement of a subexponential distribution. The monographs Rolski/Schmidili/Schmidt/Teugels
(2000), p. 49, and Embrechts/Klüppelberg/Mikosch
(1997), p. 405, define fat tailedness this way.
In section B it is shown that some exponential distributions (though neither subexponential
nor fat-tailed according to (11)) lead to a nonexisting exponential moment.
Therefore, we need another relaxation of the term fat tailedness. Obviously, this definition
is
R is fat-tailed :⇔ E ( e R) = ∞ . (12)
The definition corresponds to the biggest set F in Figure 1. Whereas definition (11) demands
the nonexistence of the moment generating function at all points s > 0definition
(12) only calls for the nonexistence of this function at the point s = 1.
B. Fat tails and traditional capital market theory
I. Nonexistence of Expected Prices and Returns
The scope of this section is to show how a fat-tailed log return distribution of any kind
leads to nonexisting expected prices and discrete returns. According to (2) and (6) both
expected returns and expected stock prices are affine functions of the exponential moment
E ( e R) of order 1 and thus their existence directly depends on a finite expectation of e R .
Now we will carry out the proof for the case of a log return distribution with a Pareto
tail (class B in Figure 1). We have to borrow only a little bit from probability theory (for
instance Rényi (1966), p. 179), which tells us that if a random variable X ≥ 0 has finite
expectation E(X), then the upper tail behavior must be of the following type:
In the sequel, we use the negation of (13), i.e.
lim x · P(X > x)=0 . (13)
x→∞
x · P(X > x) ↛ 0 for x → ∞ ⇒ E(X)=∞ . (14)
With
X = e R
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and R fat-tailed in the sense of formula (9), the left-hand side of (13) yields
lim xP( e R > x ) = lim xP(R > lnx)=lim e z P(R > z)=lim e z c
x→∞ x→∞ z→∞ z→∞ z α = ∞ . (15)
The last equation results from the fact that the exponential function grows faster than any
power function.
Clearly, (13) is violated and hence with (14) we have
E(X)=E ( e R) = ∞ . (16)
So far we demonstrated the nonexistence of E ( e R) only for fat-tailed log returns in the
sense (9) of a Pareto tail. However, the conclusion is valid for all types of fat tailedness
illustrated by Figure 1. First, notice that neither the stability property nor the restriction
0 < α < 2 has been used. Therefore, the nonexistence is also true for α ≥ 2 (despite the
fact that the corresponding distributions belong to the domain of attraction of the normal
law). Furthermore, the proof remains valid if the constant c is replaced by a slowly varying
function L(z). One has to take into account the property (compare Ibragimov/Linnik
(1971), p. 397):
lim
z→∞ zε · L(z)=∞ for all ε > 0 .
Hence, (16) is also true for distributions belonging to set C of Figure 1. Moreover, (16) is
valid for arbitrary subexponential distributions (set D in Figure 1). The first who discovered
this fact was Chistyakov (1964). But his work has no relation whatsoever to finance.
He was interested in technical applications (queuing theory, reliability etc.). Finally, for
fat-tailed distributions according to (11) and (12) the validity of (16) is given by definition.
The first who pointed to the consequence (of fat-tailed log returns) that expected returns
do not exist was Agnew (1971). Agnew formulated his findings only in the framework
of stable non-normal distributions (set A in Figure 1). Maybe this was the reason why
his short note remained unnoticed by most researchers and textbook writers. To the
best knowledge of the authors there exists only one additional reference pointing to (16),
namely Smith (1976). His remark is hidden in a footnote. Like Agnew he confined himself
to stable non-normal distributions. He was worried about the strange consequences
for option pricing. But he did not mention (or was not aware of) the rather general validity
of (16) and the far-reaching consequences for traditional capital market theory.
7

Figure 2 shows some examples of log return distributions which are incompatible with finite
expected prices. Some of the distributions (e.g. χ 2 , gamma, Weibull) are only defined
on the positive axis. Of course, these distributions are not supposed to model the log return
distribution directly. Rather, it is meant that the (maybe piecewise defined) log return
distribution has an upper tail which is equivalent 6 to the tail of the respective distribution.
Figure 2: Examples for fat-tailed (in the sense of nonexisting prices, grey area) and
light-tailed (white area) log return distributions
• χ 2 and Cauchy distr.
• lognormal and loggamma distr.
• student and F distr.
• empirical fat-tailed distr. with α ≥ 2
• unconditional distributions of (G)ARCH models
• exponential distr. with intensity parameter ≤ 1
• gamma and Weibull distr. with appropriate
values of the shape parameter
• normal distr.
• triangular distr.
• truncated distr. of
any kind
• exp., gamma
Weibull distr.
with certain
parameter
values
For the empirically found fat-tailed distributions mentioned in Figure 2 see e.g. Guillaume
et al. (1997) where foreign exchange data are studied and tail indices of about 2.7 to3.5
are estimated. A reference for the unconditional distributions of certain (G)ARCH models
is Embrechts/Klüppelberg/Mikosch (1997), p. 461 ff, who proved that the unconditional
distribution of the ARCH(1)-process
R t = Z t
√β + λRt−1 2 t ∈{1,2,...} ,
where λ varies between 0 and about 3.56, has a Pareto tail. For instance a value for λ of
0.1 (resp. 3.0) yields a tail index of α = 26.48 (resp. α = 0.15).
II. Distribution families on the borderline between fat and light tails
In Figure 2 the families of gamma and Weibull distributions and their cutting set, the
familiy of exponential distributions, were mentioned. Indeed these families are on the
borderline between fat-tailed and light-tailed distributions. We now want to exemplify
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this by the exponential distributions, which were already suggested by Agnew (1971) as a
model for log returns. Moreover, the following proves that there are fat-tailed distributions
which are not subexponential.
From (15) we see, that if the tail P(R > z) multiplied by e z does not converge to zero, the
exponential moment fails to exist. We now apply this criterion directly on exponential
distributions.
The two-sided exponential distribution has the density
λ
2 e−λ|z|
and the upper tail
P(R > z)= 1 2 e−λz .
We get
e z · P(R > z)= 1 2 ez · e −λz = 1 2 ez(1−λ) . (17)
Obviously, (17) tends to ∞ if λ < 1 and to 1 2
if λ = 1 such that
E ( e R) = ∞ if λ ≤ 1 .
For λ > 1, (17) tends to zero. The theorem from Section B.I gives no answer.
But ( straightforward calculations ) show that in this case the exponential moment exists
and is equal to . Thus the class of exponential distributions lies on the borderline
λ2
λ 2 −1
between fat-tailed and thin-tailed distributions. Nevertheless, with respect to the existence
of the exponential moment the class has to be subdivided into the two subclasses
corresponding to λ ≤ 1 (in our sense fat-tailed) and λ > 1 (existence of the expected stock
prices and returns).
More generally, the class of Weibull distributions, of which the exponential distributions
are a subclass, is also to be divided into fat-tailed and non-fat-tailed distributions. The
same is true for the class of gamma distributions, the class of hyperbolic and the class
of logistic distributions. However, the two latter classes do not contain the exponential
distributions family.
Figure (3) illustrates the situation.
9

Figure 3:
area)
Distribution families on the borderline between fat (grey area) and light (white
hyperbolic distr.
exp. distr.
Weibull distr.
gamma distr.
logistic distr.
III. Markowitz compatibility of log return distributions
Many empirical studies try to explain the expected (discrete) return through specified
factors like the expected excess return of the market, the firm size, the industrial sector or
certain ratios of the balance sheet.
These studies are meaningful if the log return distribution does not belong to any of the
fat-tailed distributions of Figure 1, which lead to nonexisting expected discrete returns.
For other issues of traditional capital market theory (derivation of the efficient frontier,
the CAPM, the security market line etc.) one needs the existence of the (discrete) return
variance and the relevant covariances. These notions are meaningful if and only if E ( DR 2)
is finite, or equivalently
E ( e 2R) < ∞ . (18)
For lack of an established technical term we call log returns satisfying (18) Markowitz
compatible, and otherwise Markowitz incompatible. Since
E ( e R) = ∞
implies
E ( e 2R) = ∞ ,
the set of all Markowitz incompatible distributions includes all the sets in Figure 1 as
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subsets. Thus R is Markowitz incompatible if its distribution belongs to any of the sets in
Figure 1.
In terms of the density function f of R Markowitz compatibility is equivalent to
∞
lim
z→∞ z
e 2x f (x)dx = 0 (19)
and to
∞
lim f
z→∞ z
( ) lnx
dx = 0 . (20)
2
If we apply (20) to the two-sided exponential distribution (already discussed in Section
B.II) we get
( ) lnx
f
2
= λ lnx
e−λ 2 = λ 2 2 x− λ 2
and
λ
lim
z→∞ 2
∞
z
x − λ 2 dx = 0 ⇔ λ > 2 .
Summarizing the findings with respect to exponential distributions, we have
• 0 < λ ≤ 1: fat-tailed in the sense E ( e R) = ∞
• 0 < λ ≤ 2: fat-tailed in the sense E ( e 2R) = ∞
• λ > 2: light-tailed in the sense of Markowitz compatibility.
Figure (4) illustrates the situation. The light grey area corresponds to the set F of Figure 1.
The dark grey area corresponds to those Markowitz incompatible log return distributions
for which E ( e R) is finite but E ( e 2R) is infinite. Of course, the white area (Markowitz
compatibility) includes normal distributions and all distributions with finite support.
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Figure 4:
The area of Markowitz incompatibility (of log returns)
hyperbolic distr.
Weibull distr.
border between Markowitz incompatible
and Markowitz compatible distributions
gamma distr.
logistic distr.
C. Conclusions
Curve fitting does not require restrictive assumptions with respect to the underlying distribution.
Things are different if one intends to model the log return process R t (in discrete
or continuous time). Then several assumptions are plausible or desirable, for instance:
Assumption 1: The log returns R 1 ,R 2 ,... are independent (this corresponds to Fama’s
weak efficiency hypothesis).
Assumption 2: The log returns R 1 ,R 2 ,... (related to time intervals of equal length) have
the same probability distribution (thus ruling out calendar anomalies).
Assumption 3: The log returns R 1 ,R 2 ,... and the log return R 0t for the time span [0,t]
differ only by location and scale parameters, i.e. the log return R 0t for the time span
[0,t] has the same probability distribution as b t R 1 + a t , where b t > 0 and a t are suitable
constants (this is required to generalize the empirical findings to a shorter or longer time
horizon).
Assumption 4: Expected stock prices are finite, i.e.
E(P t )=P t−1 · E ( e R t ) < ∞ . (21)
It is well known that the only class of probability distributions satisfying assumptions 1
to 3 is the class of stable distributions.
Under assumptions 1 to 3, stockprices and discrete returns are logstable or lognormal
(shifted by −1).
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If we complement assumptions 1 to 3 with assumption 4, we end up with a unique and
well-known alternative, namely with normally distributed log returns and lognormally
distributed stock prices, i.e. with the classical Black/Scholes world. Despite this clear-cut
characterization the uniquely determined alternative is open to criticism. Indeed, problems
related to lognormal returns are addressed in the papers by Feldstein (1969), Tobin
(1969), Levy (1974) and others. The class of lognormal distributions is not closed under
the formation of linear blends. Even if all basic securities have lognormal discrete returns,
then portfolio discrete returns will not be lognormal. In order to be consistent with
expected utility, one has to take resort to the implausible quadratic utility functions.
On the other hand, if assumptions 1 to 3 are called in question 7 one falls out of the frying
pan into the fire. Dropping the i.i.d. assumptions 1 and 2 hampers statistical inference
considerably. As already mentioned, dropping assumption 3 the shape of log return distributions
corresponding to longer periods could be completely different from those corresponding
to shorter periods. The conventional techniques for annualizing parameters
(like volatility) become obsolete.
It seems to the authors that no ideal model exists which at the same time
• makes sense from the perspective of probability theory,
• is founded on decision theoretic roots,
• is handy even for practitioners and students,
• is not empirically rejectable.
Especially the last point is of some importance in our context. If fat tails of the log return
distribution were actually beyond doubt from an empirical point of view, then portfolio
analysis and capital market theory in the traditional sense would have to be fully rejected,
since even expected prices (and also µ and σ values) would not exist. In such a case it
would not make much sense to try to estimate these parameters by statistical procedures.
The fact that statistical estimation of discrete µ and σ values often works pretty fine, may
be a hint that the log return distribution could be light-tailed.
Perhaps truncated fat tail distributions, which still have a fat tail behavior up to a certain
value but vanish beyond that point, are the key to a better understanding of the observed
phenomenon that the central peak is higher and tails are “fatter” than those of the normal
law. Rachev/Mittnik (2000), Section 2.5, give some hints in this direction. Truncated fat
13

tail distributions belong to the domain of attraction of the normal law, which implies an
approximately normal long-term distribution (of log returns). The latter fits the remarks
of Poddig/Dichtl/Petersmeier (2000) that monthly log returns are closer to a normal distribution
than those of weekly or daily data, which usually have “fat tails”. If this is true in
the sense of Markowitz incompatibility then the consequence is that for these time intervals
no traditional capital market theory can be applied, whereas for longer time intervals
this still is possible.
Endnotes
1 See Dorfleitner (2001) in this regard.
2 Compare, for instance, Adler/Feldman/Taqqu (1998).
3 Since we are primarily interested in the finiteness of expected stock prices or expected (discrete)
returns, only the upper tail matters as will be completely clear in the next section. For
other purposes (e.g. calculation of VaR) the lower tail is more important. For many distributions
(normal, student, Cauchy etc.) the upper and the lower tail have the same properties.
4 For the definition of a slowly varying function see Ibragimov/Linnik (1971), p. 394ff. An
equivalent definition of (10) is given through
1 − F(tx)
lim
t→∞ 1 − F(t) = x−α (x > 0) ,
where F denotes the distribution function. The parameter α > 0 is called the tail index, which
can be interpreted as an inverse measure of fat tailedness. Note that the power moment E ( |R| β)
exists only for β < α.
5 Subexponential distributions have tails which decrease more slowly than any exponential tail.
For some equivalent definitions of this class we refer the reader to Goldie/Klüppelberg (1998).
6 Two distribution functions F and G are called tail-equivalent if
1 − F(x)
lim
x→∞ 1 − G(x) = c
for some constant c > 0.
7 See for instance Mittnik/Paolella (2000), p. 313.
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