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Expected Shortfall (expected + shortfall)
Selected AbstractsAsymmetric power distribution: Theory and applications to risk measurementJOURNAL OF APPLIED ECONOMETRICS, Issue 5 2007Ivana Komunjer 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. [source] Analysis of Participating Life Insurance Contracts: A Unification ApproachJOURNAL OF RISK AND INSURANCE, Issue 3 2007Nadine Gatzert Fair pricing of embedded options in life insurance contracts is usually conducted by using risk-neutral valuation. This pricing framework assumes a perfect hedging strategy, which insurance companies can hardly pursue in practice. In this article, we extend the risk-neutral valuation concept with a risk measurement approach. We accomplish this by first calibrating contract parameters that lead to the same market value using risk-neutral valuation. We then measure the resulting risk assuming that insurers do not follow perfect hedging strategies. As the relevant risk measure, we use lower partial moments, comparing shortfall probability, expected shortfall, and downside variance. We show that even when contracts have the same market value, the insurance company's risk can vary widely, a finding that allows us to identify key risk drivers for participating life insurance contracts. [source] On some measures and distances for positive random variablesAPPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 2 2003M. S. FinkelsteinArticle first published online: 2 MAY 200 Abstract A number of conventional measures of risk as real-valued functions on the space of positive random variables are considered: the expected shortfall, the mean excess over the threshold, the stop-loss and some others. Ordering of risks, based on these measures and the distances between corresponding distribution functions, are described. The perturbed measures, describing the effect of changing environment, are discussed. These measures are defined by the accelerated life and proportional hazards models widely used in reliability and survival analysis. The case of a random environment is of a prime interest in the paper. The main result states that if, for instance, the stochastic environment is ,neutral in expectation' with respect to the baseline one, the distance between the corresponding distribution functions can be still sufficiently large. Copyright © 2003 John Wiley & Sons, Ltd. [source] | ||||||||||