Comparing tail variabilities of risks by means of the excess wealth order

There is a growing interest in the actuarial community to employ certain tail conditional characteristics as measures of risk, which are informative about the variability of the losses beyond the value-at-risk (one example is the tail conditional variance, introduced by Furman and Landsman, 2006). However, comparisons of tail risks based on different measures may not always be consistent. In addition, conclusions based on these conditional characteristics depend on the choice of the tail probability p, so different p's also may produce contradictory conclusions. In this note, we suggest to compare tail variability of risks by means of the excess wealth order, which makes judgements only if large classes of tail conditional characteristics imply the same conclusion, independently of the choice of p.Ministerio de Ciencia e Innovación (grant MTM2009-08326

On a class of poverty measures based on weighted poverty gaps

We consider a class of poverty measures based on ranks that generalizes, among other measures, the class of linear indices discussed by Hagenaars (1987) and the class of equally distributed equivalent (EDE) poverty gaps considered by Duclos and Gregoire (2002). This class, introduced by Sordo, Ramos and Ramos (2007), is closely related to the TIP (Three I's of poverty) curve, a graphical device used to describe distributional poverty (Jenkins and Lambert, 1997). In this paper, we study the consistency of this class of measures with the most commonly accepted axioms for poverty measures and illustrate, with Spanish real data, the connection between stochastic dominance of TIP areas and orderings of income distributions according to these measures.Miguel A. Sordo acknowledges the support of Ministerio de Ciencia e Innovación (grant MTM2009-08326) and Consejería de Economía Innovación y Ciencia (grant P09-SEJ-4739). Carmen D. Ramos acknowledges the support of Consejería de Economía Innovación y Ciencia (grant P09-SEJ-4739)

Weak Dependence Notions and Their Mutual Relationships

New weak notions of positive dependence between the components X and Y of a random pair (X,Y) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The purpose of this paper is to provide a structured framework for the definition and description of these notions, and other new ones, and to describe their mutual relationships. An exhaustive review of some well-know notions of dependence, with a complete description of the equivalent definitions and reciprocal relationships, some of them expressed in terms of the properties of the copula or survival copula of (X,Y), is also provided

Stochastic orders and multivariate measures of risk contagion

Co-risk measures and risk contributions measures are used in portfolio risk analysis to assess and quantify the risk of contagion, given that one or more assets in the portfolio are in distress. In this paper, given two random vectors X and Y that represent two portfolios of n assets (n ≥ 2) and exhibit some kind of positive dependence, we give sufficient conditions based on stochastic orders to compare the risk of contagion of the portfolios. The measures of risk contagion that we consider are the conditional value at risk (CoVaR), the conditional expected shortfall (CoES) and the recently introduced marginal mean excess (MME)

On the Increasing Convex Order of Relative Spacings of Order Statistics

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex order of relative spacings of two distributions from the case of consecutive spacings to general spacings. The sufficient conditions are given in terms of the expected proportional shortfall order. As an application, we compare relative deprivation within some parametric families of income distributions

Stochastic Comparisons of Some Distances between Random Variables

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given

Poverty comparisons when TIP curves intersect

Non-intersection of TIP curves is recognized as a criterion to compare two income distributions in terms of poverty. The purpose of this paper it to obtain comparable poverty results for income distributions whose TIP curves intersect (possibly more than once). To deal with such situations, a sequence of higher-degree dominance criteria between TIP curves is introduced. The normative significance of these criteria is provided in terms of a sequence Cn of nested classes of linear poverty measures with the property that, as the order n of the class increases, the measures become more and more sensitive to the distribution of income among the poorest

Stochastic orders and co-risk measures under positive dependence

Conditional risk measures (or co-risk measures) and risk contribution measures are increasingly used in actuarial portfolio analysis to evaluate the systemic risk, which is related to the risk that the failure or loss of a component spreads to another component or even to the whole portfolio: while co-risk measures are risk-adjusted versions of measures usually employed to assess isolate risks, risk contribution measures quantify how a stress situation for a component affects another one. In this paper, we provide sufficient conditions under which two random vectors could be compared in terms of CoVaR (conditional value-at- risk), CoES (conditional expected shortfall) and different risk contribution measures. Conditions are given in terms of the increasing convex order, the dispersive order and the excess wealth order of the marginals under some assumptions of positive dependence

Stochastic Bounds for Conditional Distributions Under Positive Dependence

We provide stochastic bounds for conditional distributions of individual risks in a portfolio, given that the aggregate risk exceeds its value at risk. Expectations of these conditional distributions can be interpreted as marginal risk contributions to the aggregate risk as measured by the tail conditional expectation. We first provide general lower and upper stochastic bounds and then we obtain further improvements of the bounds in the case of a portfolio consisting of dependent risks. We also derive new characterizations of comonotonic random vectors.Miguel A. Sordo and Alfonso Suarez-Llorens acknowledge the support of Ministerio de Ciencia e Innovación (grant MTM2009-08326) and Consejería de Economía Innovación y Ciencia (grant P09-SEJ-4739)