There is much recent interest in distributions alternative to the classical beta distribution. In this paper, financial risk and inequality measures have been derived for the log-Lindley distribution with support on the unit interval and a robust quantile regression is also proposed. The log-Lindley distribution is parameterized in terms of its quantile function to permit the modelling of the covariate effects across the whole distribution of response, instead of restriction to the mean. The performance of the log-Lindley quantile regression is examined by Monte Carlo simulations with application to a risk management data set.