Reducing model risk via positive and negative dependence assumptions

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence uncertainty spreads compared to the case where only marginals information is known. In more detail, we show that in our model the assumption of a positive dependence structure improves the best-possible lower estimate of a risk measure, while leaving unchanged its worst-possible upper risk bounds. In a similar way, we derive for convex risk measures that the assumption of a negative dependence structure leads to improved upper bounds for the risk while it does not help to increase the lower risk bounds in an essential way. As a result we find that additional assumptions on the dependence structure may result in essentially improved risk bounds.     

Upper comonotonicity and convex upper bounds for sums of random variables

It is well-known that if a random vector with given marginal distributions is comonotonic, it has the largest sum with respect to convex order. However, replacing the (unknown) copula by the comonotonic copula will in most cases not reflect reality well. For instance, in an insurance context we may have partial information about the dependence structure of different risks in the lower tail. In this paper, we extend the aforementioned result, using the concept of upper comonotonicity, to the case where the dependence structure of a random vector in the lower tail is already known. Since upper comonotonic random vectors have comonotonic behavior in the upper tail, we are able to extend several well-known results of comonotonicity to upper comonotonicity. As an application, we construct different increasing convex upper bounds for sums of random variables and compare these bounds in terms of increasing convex order.     

Diversification limit of quantiles under dependence uncertainty

In this paper, we investigate the asymptotic behavior of the portfolio diversification ratio based on Value-at-Risk (quantile) under dependence uncertainty, which we refer to as “worst-case diversification limit”. We show that the worst-case diversification limit is equal to the upper limit of the worst-case diversification ratio under mild conditions on the portfolio marginal distributions. In the case of regularly varying margins, we provide explicit values for the worst-case diversification limit. Under the framework of dependence uncertainty the worst-case diversification limit is significantly higher compared to classic results obtained in the literature of multivariate regularly varying distributions. The results carried out in this paper bring together extreme value theory and dependence uncertainty, two popular topics in the recent study of risk aggregation.     

On multivariate extensions of Conditional-Tail-Expectation

In this paper, we introduce two alternative extensions of the classical univariate Conditional-Tail-Expectation (CTE) in a multivariate setting. The two proposed multivariate CTEs are vector-valued measures with the same dimension as the underlying risk portfolio. As for the multivariate Value-at-Risk measures introduced by Cousin and Di Bernardino (2013), the lower-orthant CTE (resp. the upper-orthant CTE) is constructed from level sets of multivariate distribution functions (resp. of multivariate survival distribution functions). Contrary to allocation measures or systemic risk measures, these measures are also suitable for multivariate risk problems where risks are heterogeneous in nature and cannot be aggregated together. Several properties have been derived. In particular, we show that the proposed multivariate CTE-s satisfy natural extensions of the positive homogeneity property, the translation invariance property and the comonotonic additivity property. Comparison between univariate risk measures and components of multivariate CTE is provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Sub-additivity of the proposed multivariate CTE-s is provided under the assumption that all components of the random vectors are independent. Illustrations are given in the class of Archimedean copulas.     

Multivariate reinsurance designs for minimizing an insurer’s capital requirement

This paper investigates optimal reinsurance strategies for an insurer with multiple lines of business under the criterion of minimizing its total capital requirement calculated based on the multivariate lower-orthant Value-at-Risk. The reinsurance is purchased by the insurer for each line of business separately. The premium principles used to compute the reinsurance premiums are allowed to differ from one line of business to another, but they all satisfy three mild conditions: distribution invariance, risk loading and preserving the convex order, which are satisfied by many popular premium principles. Our results show that an optimal strategy for the insurer is to buy a two-layer reinsurance policy for each line of business, and it reduces to be a one-layer reinsurance contract for premium principles satisfying some additional mild conditions, which are met by the expected value principle, standard deviation principle and Wang’s principle among many others. In the end of this paper, some numerical examples are presented to illustrate the effects of marginal distributions, risk dependence structure and reinsurance premium principles on the optimal layer reinsurance.     

Large deviations bounds for estimating conditional value-at-risk

In this paper, we prove an exponential rate of convergence result for a common estimator of conditional value-at-risk for bounded random variables. The bound on optimistic deviations is tighter while the bound on pessimistic deviations is more general and applies to a broader class of convex risk measures.     

On convex risk measures on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

Much of the recent literature on risk measures is concerned with essentially bounded risks in L ^∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on L ^p spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on L ^p we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for L ^p risks.     

Improved convex upper bound via conditional comonotonicity

Comonotonicity provides a convenient convex upper bound for a sum of random variables with arbitrary dependence structure. Improved convex upper bound was introduced via conditioning by Kaas et al. [Kaas, R., Dhaene, J., Goovaerts, M., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econ. 27, 151-168]. In this paper, we unify these results in a more general context using the concept of conditional comonotonicity. We also construct an approximating sequence of convex upper bounds with nice convergence properties.     

Distorted Mix Method for constructing copulas with tail dependence

This paper introduces a method for constructing copula functions by combining the ideas of distortion and convex sum, named Distorted Mix Method. The method mixes different copulas with distorted margins to construct new copula functions, and it enables us to model the dependence structure of risks by handling the central and tail parts separately. By applying the method we can modify the tail dependence of a given copula to any desired level measured by tail dependence function and tail dependence coefficients of marginal distributions. As an application, a tight bound for asymptotic Value-at-Risk of order statistics is obtained by using the method. An empirical study shows that copulas constructed by this method fit the empirical data of SPX 500 Index and FTSE 100 Index very well in both central and tail parts.     

Bounds on stochastic convex allocation problems

We consider a stochastic convex program arising in a certain resource allocation problem. The uncertainty is in the demand for a resource which is to be allocated among several competing activities under convex inventory holding and shortage costs. The problem is cast as a two–period stochastic convex program and we derive tight upper and lower bounds to the problem using marginal distributions of the demands, which may be stochastically dependent. It turns out that these bounds are tighter than the usual bounds in the literature which are based on limited moment information of the underlying random variables. Numerical examples illustrate the bounds.     

Reflected Backward Stochastic Differential Equations,Convex Risk Measures and American Options

We use convex risk measures to assess unhedged risks for American-style contingent claims in a continuous-time non-Markovian economy using reflected backward stochastic differential equations (RBSDEs). A two-stage approach is adopted to evaluate the risk. We formulate the evaluation problem as an optimal stopping-control problem and discuss the problem using reflected BSDEs. The convex risk measures are represented as solutions of RBSDEs. In the Markov case, we relate the RBSDE solutions to the unique viscosity solutions of related obstacle problems for parabolic partial differential equations.     

Estimates for the finite-time ruin probability with insurance and financial risks

The paper gives estimates for the finite-time ruin probability with insurance and financial risks. When the distribution of the insurance risk belongs to the class L(??) for some ?? > 0 or the subexponential distribution class, we abtain some asymptotic equivalent relationships for the finite-time ruin probability, respectively. When the distribution of the insurance risk belongs to the dominated varying-tailed distribution class, we obtain asymptotic upper bound and lower bound for the finite-time ruin probability, where for the asymptotic upper bound, we completely get rid of the restriction of mutual independence on insurance risks, and for the lower bound, we only need the insurance risks to have a weak positive association structure. The obtained results extend and improve some existing results.     

Market price-based convex risk measures: A distribution-free optimization approach

In Cont (2006) [1], a convex risk measure was proposed to measure the impact of uncertainty resulting from the misspecification of derivative models. Evaluation of the risk measures was illustrated on finite families of probability measures. In this paper, we consider the case of infinite families of measures that share common moments, e.g. mean and variance for European-style options. We show that the risk measure can still be efficiently evaluated based on semi-infinite programming. Examples are given that illustrate the benefits of evaluating the risk measure with infinite families of measures and shed light on the limitations of considering only finite families of measures.     

Comonotonic convex upper bound and majorization

When the dependence structure among several risks is unknown, it is common in the actuarial literature to study the worst dependence structure that gives rise to the riskiest aggregate loss. A central result is that the aggregate loss is the riskiest with respect to convex order when the underlying risks are comonotonic. Many proofs were given before. The objective of this article is to present a new proof using the notions of decreasing rearrangement and the majorization theorem, and give clear explanation of the relation between convex order, the theory of majorization and comonotonicity.     

A class of problems where dual bounds beat underestimation bounds

We investigate the problem of minimizing a nonconvex function with respect to convex constraints, and we study different techniques to compute a lower bound on the optimal value: The method of using convex envelope functions on one hand, and the method of exploiting nonconvex duality on the other hand. We investigate which technique gives the better bound and develop conditions under which the dual bound is strictly better than the convex envelope bound. As a byproduct, we derive some interesting results on nonconvex duality.     

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.     

Spectral Gap for Stable Process on Convex Planar Double Symmetric Domains

We study the semigroup of the symmetric α-stable process in bounded domains in R ². We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the spectral gap for convex planar domains which are symmetric with respect to both coordinate axes. For rectangles, using “midconcavity” of the first eigenfunction (Bañuelos et al., Potential Anal. 24(3): 205–221, 2006), we obtain sharp upper and lower bound estimates of the spectral gap.     

Portfolio optimization with linear and fixed transaction costs

We consider the problem of portfolio selection, with transaction costs and constraints on exposure to risk. Linear transaction costs, bounds on the variance of the return, and bounds on different shortfall probabilities are efficiently handled by convex optimization methods. For such problems, the globally optimal portfolio can be computed very rapidly. Portfolio optimization problems with transaction costs that include a fixed fee, or discount breakpoints, cannot be directly solved by convex optimization. We describe a relaxation method which yields an easily computable upper bound via convex optimization. We also describe a heuristic method for finding a suboptimal portfolio, which is based on solving a small number of convex optimization problems (and hence can be done efficiently). Thus, we produce a suboptimal solution, and also an upper bound on the optimal solution. Numerical experiments suggest that for practical problems the gap between the two is small, even for large problems involving hundreds of assets. The same approach can be used for related problems, such as that of tracking an index with a portfolio consisting of a small number of assets.     

Measuring the coupled risks: A copula-based CVaR model

Integrated risk management for financial institutions requires an approach for aggregating risk types (such as market and credit) whose distributional shapes vary considerably. The financial institutions often ignore risks’ coupling influence so as to underestimate the financial risks. We constructed a copula-based Conditional Value-at-Risk (CVaR) model for market and credit risks. This technique allows us to incorporate realistic marginal distributions that capture essential empirical features of these risks, such as skewness and fat-tails while allowing for a rich dependence structure. Finally, the numerical simulation method is used to implement the model. Our results indicate that the coupled risks for the listed company’s stock maybe are undervalued if credit risk is ignored, especially for the listed company with bad credit quality.