Asymptotic Expansions of Generalized Quantiles and Expectiles for Extreme Risks (original) (raw)

Generalized quantiles as risk measures

Insurance: Mathematics and Economics, 2014

In the statistical and actuarial literature several generalizations of quantiles have been considered, by means of the minimization of a suitable asymmetric loss function. All these generalized quantiles share the important property of elicitability, which received a lot of attention recently since it corresponds to the existence of a natural backtesting methodology. In this paper we investigate the case of M-quantiles as the minimizers of an asymmetric convex loss function, in contrast to Orlicz quantiles that have been considered in Bellini and Rosazza Gianin (2012). We discuss their properties as risk measures and point out the connection with the zero utility premium principle and with shortfall risk measures introduced by Föllmer and Schied (2002). In particular, we show that the only M-quantiles that are coherent risk measures are the expectiles, introduced by Newey and Powell (1987) as the minimizers of an asymmetric quadratic loss function. We provide their dual and Kusuoka representations and discuss their relationship with CVaR. We analyze their asymptotic properties for α → 1 and show that for very heavy tailed distributions expectiles are more conservative than the usual quantiles. Finally, we show their robustness in the sense of lipschitzianity with respect to the Wasserstein metric.

Model Efficiency and Uncertainty in Quantile Estimation of Loss Severity Distributions

Risks, 2019

Quantiles of probability distributions play a central role in the definition of risk measures (e.g., value-at-risk, conditional tail expectation) which in turn are used to capture the riskiness of the distribution tail. Estimates of risk measures are needed in many practical situations such as in pricing of extreme events, developing reserve estimates, designing risk transfer strategies, and allocating capital. In this paper, we present the empirical nonparametric and two types of parametric estimators of quantiles at various levels. For parametric estimation, we employ the maximum likelihood and percentile-matching approaches. Asymptotic distributions of all the estimators under consideration are derived when data are left-truncated and right-censored, which is a typical loss variable modification in insurance. Then, we construct relative efficiency curves (REC) for all the parametric estimators. Specific examples of such curves are provided for exponential and single-parameter Par...

After VaR: The Theory, Estimation, and Insurance Applications of Quantile-Based Risk Measures

Journal of Risk and Insurance, 2006

We discuss a number of quantile-based risk measures (QBRMs) that have recently been developed in the financial risk and actuarial/insurance literatures. The measures considered include the Value-at-Risk (VaR), coherent risk measures, spectral risk measures, and distortion risk measures. We discuss and compare the properties of these different measures, and point out that the VaR is seriously flawed. We then discuss how QBRMs can be estimated, and discuss some of the many ways they might be applied to insurance risk problems. These applications are typically very complex, and this complexity means that the most appropriate estimation method will often be some form of stochastic simulation.

Parameter and quantile estimation for the generalized extreme-value distribution

Environmetrics, 1994

The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.

Bias reduction in risk modelling: Semiparametric quantile estimation

Test, 2006

InStatistics of Extremes we are mainly interested in the estimation of quantities related to extreme events. In many areas of application, like for instanceInsurance Mathematics, Finance andStatistical Quality Control, a typical requirement is to find a value, high enough, so that the chance of an exceedance of that value is small. We are then interested in the estimation of ahigh quantile X p , a value which is overpassed with a small probabilityp. In this paper we deal with the semi-parametric estimation ofX p for heavy tails. Since the classical semi-parametric estimators exhibit a reasonably high bias for low thresholds, we shall deal with bias reduction techniques, trying to improve their performance.

The use of flexible quantile-based measures in risk assessment

Communications in Statistics, 2015

A new family of distortion risk measures-GlueVaR-is proposed in Belles-Sampera et al. (2014) to procure a risk assessment lying between those provided by common quantile-based risk measures. GlueVaR risk measures may be expressed as a combination of these standard risk measures. We show here that this relationship may be used to obtain approximations of GlueVaR measures for general skewed distribution functions using the Cornish-Fisher expansion. A subfamily of GlueVaR measures satisfies the tail-subadditivity property. An example of risk measurement based on real insurance claim data is presented, where implications of tail-subadditivity in the aggregation of risks are illustrated.

A Log Probability Weighted Moment Estimator of Extreme Quantiles

In this paper we consider the semi-parametric estimation of extreme quantiles of a right heavy-tail model. We propose a new Log Probability Weighted Moment estimator for extreme quantiles, which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-order regular variation condition on the tail, of the underlying distribution function, we deduce the non degenerate asymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimal levels. In addition, the performance of the estimators is illustrated through an application to real data.

Bounds for Quantile-Based Risk Measures of Functions of Dependent Random Variables

Economic Quality Control, 2008

This paper introduces two techniques for computing bounds for several quantilebased risk measures based on distortion functions. Knowledge about the marginal distribution of the involved random variables is assumed with the optional assumption of some partial information about the structure of dependence. The aim is to derive bounds for risk measures of functions of dependent random variables. Several examples taken from an insurance context are given. We use Embrechts et al. (2003) methodology and the stochastic ordering approach to derive bounds for various risk measures in the bi-dimensional and the multidimensional cases.

The Maximum Lq-Likelihood Method: An Application to Extreme Quantile Estimation in Finance

Methodology and Computing in Applied Probability, 2008

Estimating financial risk is a critical issue for banks and insurance companies. Recently, quantile estimation based on Extreme Value Theory (EVT) has found a successful domain of application in such a context, outperforming other approaches. Given a parametric model provided by EVT, a natural approach is Maximum Likelihood estimation. Although the resulting estimator is asymptotically efficient, often the number of observations available to estimate the parameters of the EVT models is too small in order to make the large sample property trustworthy. In this paper, we study a new estimator of the parameters, the Maximum Lq-Likelihood estimator (MLqE), introduced by Ferrari and Yang (2007). We show that the MLqE can outperform the standard MLE, when estimating tail probabilities and quantiles of the Generalized Extreme Value (GEV) and the Generalized Pareto (GP) distributions. First, we assess the relative efficiency between the the MLqE and the MLE for various sample sizes, using Monte Carlo simulations. Second, we analyze the performance of the MLqE for extreme quantile estimation using real-world financial data. The MLqE is characterized by a distortion parameter q and extends the traditional log-likelihood maximization procedure. When q → 1, the new estimator approaches the traditional Maximum Likelihood Estimator (MLE), recovering its desirable asymptotic properties; when q = 1 and the sample size is moderate or small, the MLqE successfully trades bias for variance, resulting in an overall gain in terms of accuracy (Mean Squared Error).