Article 3

losses such as value-at-risk may not contain enough, or the right, information for risk managers. This paper presents a comprehensive empirical analysis of a set of left-tail measures (LTMs): the mean and standard deviation of a loss larger than the VAR (MLL and SDLL) and the VAR. We investigate the empirical dynamics of the LTMs. We present a robust and unified framework, the Arch quantile regression approach, in estimating the LTMs. Our Monte Carlo simulation shows that the VAR is appropriate for risk management when returns follow Gaussian processes, but the MLL strategy and strategies accounting for the SDLL are useful in reducing the risk of large losses under non-normal distributions and when there are jumps in asset prices.