statistique, théorie et gestion de portefeuille - Docs at ISFA

tail (positive or negative) of the Dow Jones data. Thus, we confirm previous results (Lux 1996, Jondeau and
Rockinger 2001, for instance) according to which the extreme tails can be considered as symmetric, at least
for the Dow Jones data.
These are the evidence in favor of the existence of an asymptotic power law tail. Balancing this, many
of our tests have shown that the power law model is not as powerful compared with the SE model, even
arbitrarily far in the tail (as far as the available data allows us to probe). In addition, our attempts for a
direct estimation of the exponent b of a possible power law tail has failed to confirm the existence of a
well-converged asymptotic value (except maybe for the positive tail of the Nasdaq). In constrast, we have
found that the exponent b of the power law model systematically increases when going deeper and deeper
in the tails, with no visible sign of exhausting this growth. We have proposed tentative parameterization of
this growth of the apparent power law exponent. We note again that this behavior is expected from models
such as the GARCH or the Multifractal Random Walk models which predict asymptotic power law tails but
with exponents of the order of 20 or larger, that would be sampled at unattainable quantiles.
Attempting to wrap up the different results obtained by the battery of tests presented here, we can offer
the following conservative conclusion: it seems that the four tails examined here are decaying faster than
any (reasonable) power law but slower than any stretched exponentials. Maybe log-normal or log-Weibull
distributions could offer a better effective description of the distribution of returns 9 . Such a model has
already been suggested by (Serva et al. 2002).
The correct description of the distribution of returns has important implications for the assessment of large
risks not yet sampled by historical time series. Indeed, the whole purpose of a characterization of the functional
form of the distribution of returns is to extrapolate currently available historical time series beyond
the range provided by the empirical reconstruction of the distributions. For risk management, the determination
of the tail of the distribution is crucial. Indeed, many risk measures, such as the Value-at-Risk
or the Expected-Shartfall, are based on the properties of the tail of the distributions of returns. In order to
assess risk at probability levels of 95% or more, non-parametric methods have merits. However, in order to
estimate risks at high probability level such as 99% or larger, non-parametric estimations fail by lack of data
and parametric models become unavoidable. This shift in strategy has a cost and replaces sampling errors
by model errors. The considered distribution can be too thin-tailed as when using normal laws, and risk will
be underestimated, or it is too fat-tailed and risk will be over estimated as with Lévy law and possibly with
Pareto tails according to the present study. In each case, large amounts of money are at stake and can be lost
due to a too conservative or too optimistic risk measurement.
Our present study suggests that the Paretian paradigm leads to an overestimation of the probability of large
events and therefore leads to the adoption of too conservative positions. Generalizing to larger time scales,
the overly pessimistic view of large risks deriving from the Paretian paradigm should be all the more revised,
due to the action of the central limit theorem. Finally, an additional note of caution is in order. This study has
focused on the marginal distributions of returns calculated at fixed time scales and thus neglects the possible
occurrence of runs of dependencies, such as in cumulative drawdowns. In the presence of dependencies
between returns, and especially if the dependence is non stationary and increases in time of stress, the
characterization of the marginal distributions of returns is not sufficient. As an example, Johansen and
Sornette (2002) have recently shown that the recurrence time of very large drawdowns cannot be predicted
from the sole knowledge of the distribution of returns and that transient dependence effets occurring in time
of stress make very large drawdowns more frequent, qualifying them as abnormal “outliers.”
9 Let us stress that we are speaking of a log-normal distribution of returns, not of price! Indeed, the standard Black and Scholes
model of a log-normal distribution of prices is equivalent to a Gaussian distribution of returns. Thus, a log-normal distribution of
returns is much more fat tailed, and in fact bracketed by power law tails and stretched exponential tails.
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