Asymmetric power distribution: Theory and applications to risk measurement

Theoretical literature in finance has shown that the risk of financial time series can be well quantified by their expected shortfall, also known as the tail value‐at‐risk. In this paper, I construct a parametric estimator for the expected shortfall based on a flexible family of densities, called the asymmetric power distribution (APD). The APD family extends the generalized power distribution to cases where the data exhibits asymmetry. The first contribution of the paper is to provide a detailed description of the properties of an APD random variable, such as its quantiles and expected shortfall. The second contribution of the paper is to derive the asymptotic distribution of the APD maximum likelihood estimator (MLE) and construct a consistent estimator for its asymptotic covariance matrix. The latter is based on the APD score whose analytic expression is also provided. A small Monte Carlo experiment examines the small sample properties of the MLE and the empirical coverage of its confidence intervals. An empirical application to four daily financial market series reveals that returns tend to be asymmetric, with innovations which cannot be modeled by either Laplace (double‐exponential) or Gaussian distribution, even if we allow the latter to be asymmetric. In an out‐of‐sample exercise, I compare the performances of the expected shortfall forecasts based on the APD‐GARCH, Skew‐ t ‐GARCH and GPD‐EGARCH models. While the GPD‐EGARCH 1% expected shortfall forecasts seem to outperform the competitors, all three models perform equally well at forecasting the 5% and 10% expected shortfall. Copyright © 2007 John Wiley & Sons, Ltd.