The tail index of a density has been widely used as an indicator of the probability of getting a large deviation in a random variable. Most of the theory underlying popular estimators of it assume that the data are independently and identically distributed (i.i.d.). However, many recent applications of the estimator have been to financial data, and such data tend to exhibit long-range dependence. We show, via Monte Carlo simulations, that conventional measures of the precision of the estimator, which are based on the i.i.d. assumption, are greatly exaggerated when such dependent data are used. This conclusion also has implications for estimates of the likelihood of getting some extreme values, and we illustrate the changed conclusions one would get using equity return data.