Multivariate Risk Measures Research Articles

Estimation of extreme multivariate expectiles with functional covariates

The present article is devoted to the semi-parametric estimation of multivariate expectiles for extreme levels. The considered multivariate risk measures also include the possible conditioning with respect to a functional covariate, belonging to an infinite-dimensional space. By using the first order optimality condition, we interpret these expectiles as solutions of a multidimensional nonlinear optimum problem. Then the inference is based on a minimization algorithm of gradient descent type, coupled with consistent kernel estimations of our key statistical quantities such as conditional quantiles, conditional tail index and conditional tail dependence functions. The method is valid for equivalently heavy-tailed marginals and under a multivariate regular variation condition on the underlying unknown random vector with arbitrary dependence structure. Our main result establishes the consistency in probability of the optimum approximated solution vectors with a speed rate. This allows us to estimate the global computational cost of the whole procedure according to the data sample size. The finite-sample performance of our methodology is provided via a numerical illustration of simulated datasets.

Measuring Systemic Governmental Reinsurance Risks of Extreme Risk Events

This study presents an easy-to-handle approach to measuring the severity of reinsurance that faces a system of dependent claims, where the reinsurance contracts are of excess loss or proportional loss. The proposed approach is a natural generalization of common reinsurance methodologies providing a conservative framework that deals with the fundamental question of how much money should a government hold to prepare for natural or human-made extreme risk events that the government will cover? Although the ruin theory is commonly used for extreme risk events, we suggest a new risk measure to deal with such events in a new framework based on multivariate risk measures. We analyze the results for the log-elliptical model of dependent claims, which are commonly used in risk analysis, and illustrate our novel risk measure using a Monte Carlo simulation.

Modelling and forecasting risk dependence and portfolio VaR for cryptocurrencies

In this paper, we investigate the co-dependence and portfolio value-at-risk of cryptocurrencies, with the Bitcoin, Ethereum, Litecoin and Ripple price series from January 2016 to December 2021, covering the crypto crash and pandemic period, using the generalized autoregressive score (GAS) model. We find evidence of strong dependence among the virtual currencies with a dynamic structure. The empirical analysis shows that the GAS model smoothly handles volatility and correlation changes, especially during more volatile periods in the markets. We perform a comprehensive comparison of out-of-sample probabilistic forecasts for a range of financial assets and backtests and the GAS model outperforms the classic DCC (dynamic conditional correlation) GARCH model and provides new insights into multivariate risk measures.

Capital allocation with multivariate convex risk measures

Capital allocation is of fundamental importance in risk management. Capital allocation problems with univariate risk measures have been extensively studied in the finance literature. In contrast, the studies about capital allocations with multivariate risk measures are quite insufficient in the literature. In this paper, we study capital allocations with multivariate coherent and convex risk measures. First, we propose the definitions of capital allocation problem and principle for general multivariate risk measures. Then based on the dual representations proved by [41], we provide the capital allocation principles with multivariate coherent and convex risk measures. Their properties are also investigated. For the case of multivariate convex risk measures, we also suggest a reasonable and effective capital allocation scheme. Finally, examples are given to illustrate the effectiveness of the proposed capital allocation scheme, where the involved convex risk measures are multivariate and univariate entropic risk measures, respectively.

Multi-Variate Risk Measures under Wasserstein Barycenter

When the uni-variate risk measure analysis is generalized into the multi-variate setting, many complex theoretical and applied problems arise, and therefore the mathematical models used for risk quantification usually present model risk. As a result, regulators have started to require that the internal models used by financial institutions are more precise. For this task, we propose a novel multi-variate risk measure, based on the notion of the Wasserstein barycenter. The proposed approach robustly characterizes the company’s exposure, filtering the partial information available from individual sources into an aggregate risk measure, providing an easily computable estimation of the total risk incurred. The new approach allows effective computation of Wasserstein barycenter risk measures in any location–scatter family, including the Gaussian case. In such cases, the Wasserstein barycenter Value-at-Risk belongs to the same family, thus it is characterized just by its mean and deviation. It is important to highlight that the proposed risk measure is expressed in closed analytic forms which facilitate its use in day-to-day risk management. The performance of the new multi-variate risk measures is illustrated in United States market indices of high volatility during the global financial crisis (2008) and during the COVID-19 pandemic situation, showing that the proposed approach provides the best forecasts of risk measures not only for “normal periods”, but also for periods of high volatility.

Optimal scenario-dependent multivariate shortfall risk measure and its application in risk capital allocation

In this paper, we propose a novel multivariate shortfall risk measure to evaluate the systemic risk of a financial system, where the allocation weight is scenario-dependent and optimally chosen from a predetermined feasible set, and examine its properties such as (quasi-)convexity and translation invariance. To compute the proposed risk measure, we reformulate it as a two-stage stochastic program. When the underlying risk is discretely distributed, the second-stage program is a finite convex program while for the continuous case, is a semi-infinite program. To tackle the latter, we use the polynomial decision rule to approximate it and reformulate it as a tractable optimization program via the standard sums-of-squares techniques. Some convergence results are established for the approximation scheme. Moreover, we apply the proposed risk measure to the risk capital allocation problem and introduce the scenario-dependent allocation strategy. In contrast to the existing allocation methods, the new approach considers losses of all scenarios and minimizes the systemic risk. We then carry out some numerical tests on the proposed model and computational schemes for a continuous system, a discrete system, and a risk capital allocation problem in life insurance. The results show that our allocation strategy performs better than the Euler allocation rule based on the expected shortfall and the method by Armenti et al., 2018, and is robust to the (un-)systemic changes of the considered dataset. Finally, we extend our model by incorporating the cost of risk capital and investigate its impact on the optimal total amount of risk capital.

Multivariate matrix-exponential affine mixtures and their applications in risk theory

In this paper, a class of multivariate matrix-exponential affine mixtures with matrix-exponential marginals is proposed. The class is shown to possess various attractive properties such as closure under size-biased Esscher transform, order statistics, residual lifetime and higher order equilibrium distributions. This allows for explicit calculations of various actuarial quantities of interest. The results are applied in a wide range of actuarial problems including multivariate risk measures, aggregate loss, large claims reinsurance, weighted premium calculations and risk capital allocation. Furthermore, a multiplicative background risk model with dependent risks is considered and its capital allocation rules are provided as well. We finalize by discussing a calibration scheme based on complete data and potential avenues of research.

The location of a minimum variance squared distance functional

In this paper, we introduce a novel multivariate functional that represents a position where the intrinsic uncertainty of a system of mutually dependent risks is maximally reduced. The proposed multivariate functional defines the location of the minimum variance of squared distance (LVS) for some n-variate vector of risks X. We compute the analytical representation of LVS(X), which consists of the location of the minimum expected squared distance, LES(X), covariance matrix A, and a matrix B of the multivariate central moments of the third order of X. From this representation it follows that LVS(X) coincides with LES(X) when X has a multivariate symmetric distribution, but differs from it in the non-symmetric case. As LES(X) is often considered a neutral multivariate risk measure, we show that LVS(X) also possesses the important properties of multivariate risk measures: translation invariance, positive homogeneity, and partial monotonicity. We also study the mean-variance approach based on the balanced sum of an expectation and a variance of the square of the aforementioned Euclidean distance and control for the closeness of LES(X) and LVS(X). The proposed theory and the results are distribution free, meaning that we do not assume any particular distribution for the random vector X. The results are demonstrated with real data of Danish fire losses.

Multivariate tail covariance risk measure for generalized skew-elliptical distributions

In this paper, the multivariate tail covariance (MTCov) for generalized skew-elliptical distributions is considered. Some special cases for this distribution, such as generalized skew-normal, generalized skew student-t, generalized skew-logistic and generalized skew-Laplace distributions, are also considered. In order to test the theoretical feasibility of our results, the MTCov for skewed and non skewed normal distributions is computed and compared. Finally, we give a special formula of the MTCov for generalized skew-elliptical distributions.

Capital allocation with multivariate risk statistics with positive homogeneity and subadditivity

In this paper, we aim to discuss an axiom system for capital allocation with multivariate risk statistics. First, we introduce a class of multivariate risk statistics satisfying the properties of positive homogeneity and subadditivity, and provide the representations. Second, the axiom system of capital allocation with multivariate risk statistics are discussed. Specifically, the existence and the uniqueness of the capital allocation principles with positively homogeneous and subadditive risk statistics are studied. Finally, examples are also given to illustrate the proposed axioms for capital allocation with multivariate risk statistics, where the explicit capital allocation principles are derived for two kinds of multivariate risk statistics based on mean and standard deviation. The proposed capital allocations with multivariate risk statistics directly base on the samples of the random vectors, and thus they are more tractable than those with multivariate risk measures.

Depth level set estimation and associated risk measures

Depth functions have become increasingly powerful tools in non-parametric inference for multivariate data as they measure a degree of centrality of a point with respect to a distribution. A multivariate risk scenario is then represented by a depth-based lower level set of the risk factors, meaning that we consider a non-compact setting. The aim of this paper is to study the asymptotic behavior of level sets of a general multivariate depth function and a particular multivariate risk measure, the Covariate-Conditional-Tail-Expectation (CCTE) based on a depth function. More precisely, given a probability measure P on Rd and a depth function D(⋅,P), we are interested in the α-lower level set LD(α):=z∈Rd:D(z,P)≤α. First, we present a plug-in approach in order to estimate LD(α), then we derive consistency of its estimator under some regularity conditions. In a second part, we provide a consistent estimator of the CCTE for a general depth function with a rate of convergence and we consider the particular case of Mahalanobis depth. Finally, a simulation study complements the performances of our estimator and an application on real data is presented.

Scalar Multivariate Risk Measures with a Single Eligible Asset

In this paper, we present results on scalar risk measures in markets with transaction costs. Such risk measures are defined as the minimal capital requirements in the cash asset. First, some results are provided on the dual representation of such risk measures, with particular emphasis given on the space of dual variables as (equivalent) martingale measures and prices consistent with the market model. Then, these dual representations are used to obtain the main results of this paper on time consistency for scalar risk measures in markets with frictions. It is well known from the superhedging risk measure in markets with transaction costs that the usual scalar concept of time consistency is too strong and not satisfied. We will show that a weaker notion of time consistency can be defined, which corresponds to the usual scalar time consistency but under any fixed consistent pricing process. We will prove the equivalence of this weaker notion of time consistency and a certain type of backward recursion with respect to the underlying risk measure with a fixed consistent pricing process. Several examples are given, with special emphasis on the superhedging risk measure.

A new coherent multivariate average-value-at-risk

A new operator for handling the joint risk of different sources has been presented and its various properties are investigated. The problem of risk evaluation of multivariate risk sources has been studied, and a multivariate risk measure, so-called multivariate average-value-at-risk, , is proposed to quantify the total risk. It is shown that the proposed operator satisfies the four axioms of a coherent risk measure while reducing to one variable average-value-at-risk, , in case N = 1. In that respect, it is shown that is the natural extension of to N-dimensional case maintaining its axiomatic properties. We further show is flexible by giving the investor the option to choose the risk level of each random loss i differently. This flexibility is novel and can not be achieved applying univariate with corresponding risk level α to the sum of the risk marginals. The framework is applicable for Gaussian mixture models with dependent risk factors that are naturally used in financial and actuarial modelling. A multivariate tail variance and its connection with is also presented via Chebyshev inequality for tail events. Examples with numerical simulations are also illustrated throughout.

A multivariate CVaR risk measure from the perspective of portfolio risk management

In this paper, we define a new multivariate conditional Value-at-Risk (MCVaR) risk measure. This MCVaR considers both individual risks and the aggregate risk of a portfolio, but prioritizes the aggregate risk. The new MCVaR risk measure is based on the minimization of the expectation of a multivariate loss function, which balances the shortfall and surplus risks of the aggregate risk and the individual risks in an overall risk of a portfolio. It is shown that the MCVaR risk measure holds the properties of positive homogeneity, translation invariance, subadditivity, and monotonicity under certain conditions. Numerical examples of the MCVaR risk measure are presented to illustrate the effect of dependence among individual risks on the MCVaR.

Multivariate systemic risk measures and computation by deep learning algorithms

In this work, we propose deep learning-based algorithms for the computation of systemic shortfall risk measures defined via multivariate utility functions. We discuss the key related theoretical aspects, with a particular focus on the fairness properties of primal optima and associated risk allocations. The algorithms we provide allow for learning primal optimizers, optima for the dual representation and corresponding fair risk allocations. We test our algorithms by comparison to a benchmark model, based on a paired exponential utility function, for which we can provide explicit formulas. We also show evidence of convergence in a case in which explicit formulas are not available.

Time consistency for scalar multivariate risk measures

Abstract In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)).

Multivariate Tail Moments for Log-Elliptical Dependence Structures as Measures of Risks

The class of log-elliptical distributions is well used and studied in risk measurement and actuarial science. The reason is that risks are often skewed and positive when they describe pure risks, i.e., risks in which there is no possibility of profit. In practice, risk managers confront a system of mutually dependent risks, not only one risk. Thus, it is important to measure risks while capturing their dependence structure. In this short paper, we compute the multivariate risk measures, multivariate tail conditional expectation, and multivariate tail covariance measure for the family of log-elliptical distributions, which captures the dependence structure of the risks while focusing on the tail of their distributions, i.e., on extreme loss events. We then study our result and examine special cases, as well as the optimal portfolio selection using such measures. Finally, we show how the given multivariate tail moments can also be computed for log-skew elliptical models based on similar approaches given for the log-elliptical case.

Optimal Scenario-Dependent Multivariate Shortfall Risk Measure and its Application in Capital Allocation

In this paper, we consider a multivariate shortfall risk measure with scenario-dependent allocation weights and examine its properties such as convexity and quasi-convexity. For fixed allocation weights, we show that the resulting risk measure is a convex systemic risk measure in which case the property of translation invariance is dependent on the allocation weights. However, if the allocation weights are chosen optimally on a feasible set, then the resulting risk measure is a quasi-convex systemic risk measure. To evaluate the systemic risk of a financial system with the proposed risk measure computationally, we reformulate it as a two-stage stochastic program which is a finite convex program when the underlying uncertainty is discretely distributed. In the case when the uncertainty is continuously distributed, we propose a polynomial decision rule to tackle the semi-infinite two-stage stochastic program which restricts the scenario-dependent allocation weights to be a class of polynomials of the underlying uncertainty parameters and subsequently reformulate the evaluation problem as a tractable optimization problem. Some convergence results are established for the approximation scheme. Moreover, we apply the proposed risk measure to capital allocation problem and introduce scenario-dependent allocation strategy and deterministic allocation strategy. Finally, we carry out some preliminary tests on the proposed computational schemes for a continuous system, a discrete system and a capital allocation example for a life insurance company.

Preference robust models in multivariate utility-based shortfall risk minimization

Utility-based shortfall risk measure (SR) has received increasing attentions over the past few years. Recently Delage et al. [Shortfall Risk Models When Information of Loss Function Is Incomplete, GERAD HEC, Montréal, 2018] consider a situation where a decision maker's true loss function in the definition of SR is unknown but it is possible to elicit a set of plausible utility functions with partial information and consequently propose a robust formulation of SR based on the worst-case utility function. In this paper, we extend this new stream of research to a multi-attribute prospect space since multi-attribute decision-making problems are ubiquitous in practical applications. Specifically, we introduce a preference robust multivariate utility-based shortfall risk measure (PRMSR) and demonstrate that it is law invariant and convex. We then apply the PRMSR to an optimal decision-making problem where the objective is to minimize the PRMSR of a vector-valued cost function and propose some numerical scheme for solving the resulting optimization problem in the case when the underlying exogenous uncertainty is finitely distributed. Finally, we discuss statistical robustness of the PRMSR based optimization model by examining qualitative stability of the estimator of the optimal value obtained with potentially contaminated data. A case study is carried out to examine the performance of the proposed robust model and numerical scheme.

Multivariate risk measures based on conditional expectation and systemic risk for Exponential Dispersion Models

Exponential dispersion models are well used and studied in quantitative risk management and actuarial science. One of the main interests is the risk measurement analysis of such models when facing extreme loss events. In this paper, we propose two multivariate risk measures based on conditional expectation and derive the explicit formulae for exponential dispersion models. In particular, our multivariate risk measures could facilitate a systemic risk measure with explicit expressions for exponential dispersion models subject to any pre-specified “systemic event.” We provide two numerical examples based on practical data to show the advantages of our approach in the context of exponential dispersion models.