  Power-law tail behavior and the summation scheme of Levy-stable distributions
is the basis for their frequent use as models when fat tails above a Gaussian
distribution are observed. However, recent studies suggest that financial asset
returns exhibit tail exponents well above the Levy-stable regime ($0<\alpha\le
2$). In this paper we illustrate that widely used tail index estimates (log-log
linear regression and Hill) can give exponents well above the asymptotic limit
for $\alpha$ close to 2, resulting in overestimation of the tail exponent in
finite samples. The reported value of the tail exponent $\alpha$ around 3 may
very well indicate a Levy-stable distribution with $\alpha\approx 1.8$.
