On the distribution tail of an integrated risk model: A numerical approach

We consider an insurance risk process with the possibility to invest the capital reserve into a portfolio consisting of a risky asset and a riskless asset. The stock price is modelled by an exponential Lévy process and the riskless interest rate is assumed to be constant. We aim at the risk assessment of the integrated risk process in terms of a high quantile or the far out distribution tail. We indicate an application to an optimal investment strategy of an insurer.     

On a family of risk measures based on proportional hazards models and tail probabilities

In this paper, we explore a class of tail variability measures based on distances among proportional hazards models. Tail versions of some well-known variability measures, such as the Gini mean difference, the Wang right tail deviation and the cumulative residual entropy are, up to a scale factor, in this class. These tail variability measures are combined with tail conditional expectation to generate premium principles that are especially useful to price heavy-tailed risks. We study their properties, including stochastic consistency and bounds, as well as the coherence of the associated premium principles.     

Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures

In this paper, we extend the concept of tail subadditivity (Belles-Sampera et al., 2014a; Belles-Sampera et al., 2014b) for distortion risk measures and give sufficient and necessary conditions for a distortion risk measure to be tail subadditive. We also introduce the generalized GlueVaR risk measures, which can be used to approach any coherent distortion risk measure. To further illustrate the applications of the tail subadditivity, we propose multivariate tail distortion (MTD) risk measures and generalize the multivariate tail conditional expectation (MTCE) risk measure introduced by Landsman et al. (2016). The properties of multivariate tail distortion risk measures, such as positive homogeneity, translation invariance, monotonicity, and subadditivity, are discussed as well. Moreover, we discuss the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks and explore the impacts of the dependence between risks in a portfolio and extreme tail events of a risk portfolio in capital allocations.     

Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error

Efficient estimation of tail probabilities involving heavy tailed random variables is amongst the most challenging problems in Monte-Carlo simulation. In the last few years, applied probabilists have achieved considerable success in developing efficient algorithms for some such simple but fundamental tail probabilities. Usually, unbiased importance sampling estimators of such tail probabilities are developed and it is proved that these estimators are asymptotically efficient or even possess the desirable bounded relative error property. In this paper, as an illustration, we consider a simple tail probability involving geometric sums of heavy tailed random variables. This is useful in estimating the probability of large delays in M/G/1 queues. In this setting we develop an unbiased estimator whose relative error decreases to zero asymptotically. The key idea is to decompose the probability of interest into a known dominant component and an unknown small component. Simulation then focuses on estimating the latter ‘residual’ probability. Here we show that the existing conditioning methods or importance sampling methods are not effective in estimating the residual probability while an appropriate combination of the two estimates it with bounded relative error. As a further illustration of the proposed ideas, we apply them to develop an estimator for the probability of large delays in stochastic activity networks that has an asymptotically zero relative error.      

Mortality risk modeling: Applications to insurance securitization

This paper proposes a stochastic mortality model featuring both permanent longevity jump and temporary mortality jump processes. A trend reduction component describes unexpected mortality improvement over an extended period of time. The model also captures the uneven effect of mortality events on different ages and the correlations among them. The model will be useful in analyzing future mortality dependent cash flows of life insurance portfolios, annuity portfolios, and portfolios of mortality derivatives. We show how to apply the model to analyze and price a longevity security.     

On tail dependence: A characterization for first-order max-autoregressive processes

In this paper, we consider first-order MARMA or ARMAX processes and a modified version of these involving a power transformation, denoted pARMAX. We assume Pareto-type tails, the most interesting case for inference within these processes. Some well-known dependence measures of multivariate extreme value theory are considered in a time series framework. In calculating these measures, we find that ARMAX and pARMAX have opposite behavior in concomitant extremes, covering all types of tail dependence. This characterization will serve modeling purposes.     

On some relative convexities

We study two types of relative convexities of convex functions f and g. We say that f is convex relative to g   in the sense of Palmer (2002, 2003), if f=h(g)$f=h(g)$, where h   is strictly increasing and convex, and denote it by f_?(1)g$f{?}_{(1)}g$. Similarly, if f is convex relative to g   in the sense studied in Rajba (2011), that is if the function f−g$f-g$ is convex then we denote it by f_?(2)g$f{?}_{(2)}g$. The relative convexity relation _?(2)${?}_{(2)}$ of a function f   with respect to the function g(x)=cx²$g(x)=c{x}^2$ means the strong convexity of f. We analyze the relationships between these two types of relative convexities. We characterize them in terms of right derivatives of functions f and g, as well as in terms of distributional derivatives, without any additional assumptions of twice differentiability. We also obtain some probabilistic characterizations. We give a generalization of strong convexity of functions and obtain some Jensen-type inequalities.     

On relative normal modes

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. In particular, we prove that under appropriate hypotheses there exist relative periodic orbits near relative equilibria even when these relative equilibria are singular points of the corresponding moment map, i.e. when the reduced spaces are singular.     

On the relative connections

Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and Kähler.     

Integrated bank risk modeling: A bottom-up statistical framework

In the last few years, according to the evolution of financial markets and the enforcement of international supervisory requirements, an increasing interest has been devoted to risk integration. The original focus on individual risk estimation has been replaced by the growing prominence of top-down and bottom-up risk integration perspectives. Following this latter way, we bring together different approaches developed in the recent literature elaborating a general model to assess banking solvency in both the long-run (economic capital) as well as in the short period (liquidity mismatching). We consider banking capability to face credit, interest rate and liquidity risks associated to macro-economic shocks affecting both assets and liabilities. Following the perspective of commercial banks, we concentrate on information available in the risk management practice to propose an easy to implement statistical framework. We put in place this framework estimating its scenario generation parameters on Italian macro-economic time series from 1990 to 2009. Once applied to a stylized commercial bank, we compare the results of our approach to regulatory capital requirements. We emphasize the need for policy makers as well as risk managers, to take into account the entire balance sheet structure to assess banking solvency.     

On a disparity between relative cliquewidth and relative NLC-width

Cliquewidth and NLC-width are two closely related parameters that measure the complexity of graphs. Both clique- and NLC-width are defined to be the minimum number of labels required to create a labelled graph by certain terms of operations. Many hard problems on graphs become solvable in polynomial-time if the inputs are restricted to graphs of bounded clique- or NLC-width. Cliquewidth and NLC-width differ at most by a factor of two.The relative counterparts of these parameters are defined to be the minimum number of labels necessary to create a graph while the tree-structure of the term is fixed. We show that Relative Cliquewidth and Relative NLC-width differ significantly in computational complexity. While the former problem is NP-complete the latter is solvable in polynomial time. The relative NLC-width can be computed in O(n³) time, which also yields an exact algorithm for computing the NLC-width in time O(3ⁿn). Additionally, our technique enables a combinatorial characterisation of NLC-width that avoids the usual operations on labelled graphs.     

Problem-driven scenario generation: an analytical approach for stochastic programs with tail risk measure

Mathematical Programming - Scenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program. Most approaches to scenario...     

On modeling evaporation

We develop a theoretical model for evaporation of a pure liquid drop on a thermally conductive solid substrate. We discuss a variety of effects regarding evaporation regime, the state of the liquid/gas interface and the content of gas phase. Then, we further consider two models: the one resulting from the one-sided non-equilibrium assumption and the other that assumes diffusion-limited regime and equilibrium at the liquid/gas interface. A single governing equation for the evolution of drop thickness is derived for both models. We show that although the model predicts qualitatively different temperature along liquid/gas and liquid/solid interface, the dynamics of the drops is almost the same.      

Conditional tail risk measures for the skewed generalised hyperbolic family

This paper deals with the estimation of loss severity distributions arising from historical data on univariate and multivariate losses. We present an innovative theoretical framework where a closed-form expression for the tail conditional expectation (TCE) is derived for the skewed generalised hyperbolic (GH) family of distributions. The skewed GH family is especially suitable for equity losses because it allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to the widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its special cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. The GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on equity losses. The TCE risk measure computed for the skewed family of GH distributions provides a conservative estimator of risk, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolios of losses, allowing the multivariate GH distribution to capture the combination of correlated risks and demonstrate how the TCE of the portfolio can be decomposed into individual components, representing individual risks in the aggregate (portfolio) loss.