Natalia Nolde

Asymptotic independence for unimodal densities

Asymptotic independence of the components of random vectors is a concept used in many applications. The standard criteria for checking asymptotic independence are given in terms of distribution functions (DFs). DFs are rarely available in an explicit form, especially in the multivariate case. Often we are given the form of the density or, via the shape of the data clouds, we can obtain a good geometric image of the asymptotic shape of the level sets of the density. In this paper we establish a simple sufficient condition for asymptotic independence for light-tailed densities in terms of this asymptotic shape. This condition extends Sibuyas classic result on asymptotic independence for Gaussian densities.

The Shape of Asymptotic Dependence

Multivariate risk analysis is concerned with extreme observations. If the underlying distribution has a unimodal density then both the decay rate of the tails and the asymptotic shape of the level sets of the density are of importance for the dependence structure of extreme observations. For heavy-tailed densities, the sample clouds converge in distribution to a Poisson point process with a homogeneous intensity. The asymptotic shape of the level sets of the density is the common shape of the level sets of the intensity. For light-tailed densities, the asymptotic shape of the level sets of the density is the limit shape of the sample clouds. This paper investigates how the shape changes as the rate of decrease of the tails is varied while the copula of the distribution is preserved. Four cases are treated: a change from light tails to light tails, from heavy to heavy, heavy to light and light to heavy tails.

Elicitability and backtesting: Perspectives for banking regulation

Conditional forecasts of risk measures play an important role in internal risk management of financial institutions as well as in regulatory capital calculations. In order to assess forecasting performance of a risk measurement procedure, risk measure forecasts are compared to the realized financial losses over a period of time and a statistical test of correctness of the procedure is conducted. This process is known as backtesting. Such traditional backtests are concerned with assessing some optimality property of a set of risk measure estimates. However, they are not suited to compare different risk estimation procedures. We investigate the proposal of comparative backtests, which are better suited for method comparisons on the basis of forecasting accuracy, but necessitate an elicitable risk measure. We argue that supplementing traditional backtests with comparative backtests will enhance the existing trading book regulatory framework for banks by providing the correct incentive for accuracy of risk measure forecasts. In addition, the comparative backtesting framework could be used by banks internally as well as by researchers to guide selection of forecasting methods. The discussion focuses on three risk measures, Value-at-Risk, expected shortfall and expectiles, and is supported by a simulation study and data analysis.

Sensitivity of the limit shape of sample clouds from meta densities

The paper focuses on a class of light-tailed multivariate probability distributions. These are obtained via a transformation of the margins from a heavy-tailed original distribution. This class was introduced in Balkema et al. (J. Multivariate Anal. 101 (2010) 1738-1754). As shown there, for the light-tailed meta distribution the sample clouds, properly scaled, converge onto a deterministic set. The shape of the limit set gives a good description of the relation between extreme observations in different direction. This paper investigates how sensitive the limit shape is to changes in the underlying heavy-tailed distribution. Copulas fit in well with multivariate extremes. By Galambos’s theorem, existence of directional derivatives in the upper endpoint of the copula is necessary and sufficient for convergence of the multivariate extremes provided the marginal maxima converge. The copula of the max-stable limit distribution does not depend on the margins. So margins seem to play a subsidiary role in multivariate extremes. The theory and examples presented in this paper cast a different light on the significance of margins. For light-tailed meta distributions, the asymptotic behaviour is very sensitive to perturbations of the underlying heavy-tailed original distribution, it may change drastically even when the asymptotic behaviour of the heavy-tailed density is not affected.

Challenging the standard dike freeboard: Methods to quantify statistical uncertainties in river flood protection

In most developed and developing nations, a freeboard is being applied to flood defense structures as a margin of uncertainty in the estimated flood stage. In some jurisdictions, practice is shifting towards the use of confidence intervals based on the fitted flood probability distribution, albeit often relying on only one statistical distribution and on only annual maximum flows. In this paper, we argue that, independent of geotechnical, geomorphological or hydrological uncertainties pertaining to the estimation of flood stage, the application of standard freeboards ignores stochastic uncertainty, the inclusion of which would provide a more scientifically defensible measure for allowable freeboard. The river stage estimate is subject to multiple sources of uncertainty, including but not limited to model and parameter uncertainty. Consequently, freeboards should be determined via a frequency analysis that explicitly takes into consideration, as well as minimizes, the uncertainty of the estimate due to known factors. Confidence intervals are a common way to represent uncertainty of a statistical estimate such as for the river stage. The choice of the confidence level will be critical, and in many cases will be associated with significant cost implications for the construction or upgrade of flood defense structures. Quantitative flood risk assessments, which are emerging as a standard in developed nations, are well suited to address this issue by allowing loss and mitigation cost comparisons for different flood scenarios. Our paper provides guidance for confidence interval calculations of river stage using an extension of the classical peaks-over-threshold method for daily river levels.

Rejoinder: “Elicitability and backtesting: Perspectives for banking regulation”

Robust traditional and comparative backtests. Chen Zhou clarifies the relation between the notion of identifiability of a risk measure and the ability to perform traditional backtests in the form of conditional calibration tests. We fully agree with him that in the absence of an identification function it is still possible to perform traditional backtests by assuming common properties of the conditional distributions across time. In our work, we have entirely focused on robust backtests as Zhou has phrased it, where robustness refers to robustness with respect to model uncertainty. We would like to add that the same clarifications are in order for comparative backtests. Both elicitability and identifiability are only meaningful concepts when stated with respect to which class of distributions P they hold; cf. Definitions 1 and 2. Broadly speaking, the smaller the class P , the weaker the condition for existence of an identification function or a strictly consistent scoring function for a given functional T . Let us give the following simple example: Suppose that Ps is a class of symmetric distributions. Then, for each P ∈ Ps , the mean and the median coincide. Therefore, all consistent scoring functions for the median are also consistent scoring functions for the mean relative to Ps , and the same holds for the respective identification functions. Relative to a class Pc of distribution functions such that all distributions have the same α-quantile, say VaRα(P ) = c for all P ∈ Pc, ES is identifiable and elicitable. Strictly consistent scoring functions can be obtained by setting r1 = c in equation (2.4). Similarly, the second component of the identification function at (2.7) with r1 = c identifies ESα relative to Pc. This is reflected in the ES backtest given by Zhou: The assumptions on the data-generating process allow to estimate c well enough that asymptotically we can work as if c was known. Hajo Holzmann and Berhard Klar suggest comparative backtests for the entire tail of the P&L distribution instead of a specific risk measure; let us term them

Conditional Extremes in Asymmetric Financial Markets

ABSTRACT The global financial crisis of 2007–2009 revealed the great extent to which systemic risk can jeopardize the stability of the entire financial system. An effective methodology to quantify systemic risk is at the heart of the process of identifying the so-called systemically important financial institutions for regulatory purposes as well as to investigate key drivers of systemic contagion. The article proposes a method for dynamic forecasting of CoVaR, a popular measure of systemic risk. As a first step, we develop a semi-parametric framework using asymptotic results in the spirit of extreme value theory (EVT) to model the conditional probability distribution of a bivariate random vector given that one of the components takes on a large value, taking into account important features of financial data such as asymmetry and heavy tails. In the second step, we embed the proposed EVT method into a dynamic framework via a bivariate GARCH process. An empirical analysis is conducted to demonstrate and compare the performance of the proposed methodology relative to a very flexible fully parametric alternative.

Asymptotic dependence for light-tailed homothetic densities

Dependence between coordinate extremes is a key factor in any multivariate risk assessment. Hence, it is of interest to know whether the components of a given multivariate random vector exhibit asymptotic independence or asymptotic dependence. In the latter case the structure of the asymptotic dependence has to be clarified. In the multivariate setting it is common to have an explicit form of the density rather than the distribution function. In this paper we therefore give criteria for asymptotic dependence in terms of the density. We consider distributions with light tails and restrict attention to continuous unimodal densities defined on the whole space or on an open convex cone. For simplicity, the density is assumed to be homothetic: all level sets have the same shape. Balkema and Nolde (2010) contains conditions on the shape which guarantee asymptotic independence. The situation for asymptotic dependence, treated in the present paper, is more delicate.