arXiv subjects

Takaaki Koike

Publications and source records attributed to Takaaki Koike.

17 recordsLinked to original sources

arXiv2026

A Characterization of Measures of Concordance of Degree Two

We characterize bivariate measures of concordance of degree at most two in the sense that the measures evaluated at copula-mixture segments are polynomials of degree at most two. Our main device is the canonical polarization, which converts quadraticity of a functional into separate affinity, thereby allowing an integral representation for affine functionals to be applied sectionwise. Using this technique, we prove that measures of degree at most two are in bijection with copula-indexed measure fields satisfying several properties including square-group equivariance. The field at independence determines the affine linearization, while its displacement determines the purely quadratic part. Moreover, exact degree two is equivalent to field non-constancy. Leveraging our characterization result, we provide a construction of degree-two measures of concordance based on affine copula operators that are square-group equivariant. As a more concrete example, we derive a necessary and sufficient condition for a weighted averaging operator of square-group transformed copulas to generate a measure of concordance of degree two.

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arXiv2026

Measuring multivariate maximal tail dependence

The classical tail dependence coefficient (TDC) may fail to capture non-exchangeable features of bivariate tail dependence since it evaluates the underlying copula only along the diagonal. To address this limitation, several measures of strongest manifestation of tail dependence have been proposed in the bivariate case, including a measure based on the tail copula of the underlying bivariate copula. This paper introduces and investigates the multivariate maximal tail concordance measure (MTCM) which extends the bivariate measure to the multivariate case. The MTCM quantifies the largest tail mass over lower hyperrectangles of common unit volume, while the associated maximizer identifies the direction of maximal tail probability. We establish fundamental properties of the MTCM in the multivariate case, including existence of an optimal direction. We also derive analytical representations for several important model classes. Closed-form expressions are further obtained for survival Marshall-Olkin copulas, Archimax and nested Archimedean copulas with regularly varying Archimedean generators. An application to trivariate annual sea-level maxima in England shows that the MTCM can reveal off-diagonal stress directions and substantial differences in the underlying extremal dependence not detected by likelihood- or TDC-based comparisons.

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arXiv2026

Tail copula representation of path-based maximal tail dependence

The classical tail dependence coefficient (TDC) may fail to capture non-exchangeable features of tail dependence due to its restrictive focus on the diagonal of the underlying copula. To address this limitation, the framework of path-based maximal tail dependence has been proposed, where a path of maximal dependence is derived to capture the most pronounced feature of dependence over all possible paths, and the path-based maximal TDC serves as a natural analogue of the classical TDC along this path. However, the theoretical foundations of path-based tail analyses, in particular the existence and analytical tractability, have remained limited. This paper addresses this issue in several ways. First, we prove the existence of a path of maximal dependence and the path-based maximal TDC when the underlying copula admits a non-degenerate tail copula. Second, we obtain an explicit characterization of the maximal TDC in terms of the tail copula. Third, we show that the first-order asymptotics of a path of maximal dependence is characterized by a one-dimensional optimization involving the tail copula. These results improve the analytical and computational tractability of path-based tail analyses. As an application, we derive the asymptotic behavior of a path of maximal dependence for the bivariate t-copula and the survival Marshall--Olkin copula.

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arXiv2026

Robust risk evaluation of joint life insurance under dependence uncertainty

Dependence among multiple lifetimes is a key factor for pricing and evaluating the risk of joint life insurance products. The dependence structure can be exposed to model uncertainty when available data and information are limited. We address robust pricing and risk evaluation of joint life insurance products against dependence uncertainty between two lifetimes. We first show that, for some class of standard contracts, the risk evaluation based on a distortion risk measure is monotone with respect to the concordance order of the underlying copula. Based on this monotonicity, we then study the most conservative and anti-conservative risk evaluations for this class of contracts. We prove that the bounds for the mean, Value-at-Risk and Expected Shortfall are computed by combinations of linear programs when the uncertainty set is defined by a norm-ball centered around a reference copula. Our numerical analysis reveals that the sensitivity of the risk evaluation against the choice of the copula differs depending on the risk measure and the type of the contract, and our proposed bounds can improve the existing bounds based on the available information.

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arXiv2024

Invariant correlation under marginal transforms

A useful property of independent samples is that their correlation remains the same after applying marginal transforms. This invariance property plays a fundamental role in statistical inference, but does not hold in general for dependent samples. In this paper, we study this invariance property on the Pearson correlation coefficient and its applications. A multivariate random vector is said to have an invariant correlation if its pairwise correlation coefficients remain unchanged under any common marginal transforms. For a bivariate case, we characterize all models of such a random vector via a certain combination of comonotonicity -- the strongest form of positive dependence -- and independence. In particular, we show that the class of exchangeable copulas with invariant correlation is precisely described by what we call positive Fréchet copulas. In the general multivariate case, we characterize the set of all invariant correlation matrices via the clique partition polytope. We also propose a positive regression dependent model that admits any prescribed invariant correlation matrix. Finally, we show that all our characterization results of invariant correlation, except one special case, remain the same if the common marginal transforms are confined to the set of increasing ones.

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arXiv2024

Measuring and testing tail equivalence

We call two copulas tail equivalent if their first-order approximations in the tail coincide. As a special case, a copula is called tail symmetric if it is tail equivalent to the associated survival copula. We propose a novel measure and statistical test for tail equivalence. The proposed measure takes the value of zero if and only if the two copulas share a pair of tail order and tail order parameter in common. Moreover, taking the nature of these tail quantities into account, we design the proposed measure so that it takes a large value when tail orders are different, and a small value when tail order parameters are non-identical. We derive asymptotic properties of the proposed measure, and then propose a novel statistical test for tail equivalence. Performance of the proposed test is demonstrated in a series of simulation studies and empirical analyses of financial stock returns in the periods of the world financial crisis and the COVID-19 recession. Our empirical analysis reveals non-identical tail behaviors in different pairs of stocks, different parts of tails, and the two periods of recessions.

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arXiv2024

Forecasting and Backtesting Gradient Allocations of Expected Shortfall

Capital allocation is a procedure for quantifying the contribution of each source of risk to aggregated risk. The gradient allocation rule, also known as the Euler principle, is a prevalent rule of capital allocation under which the allocated capital captures the diversification benefit of the marginal risk as a component of overall risk. This research concentrates on Expected Shortfall (ES) as a regulatory standard and focuses on the gradient allocations of ES, also called ES contributions (ESCs). We present the comprehensive treatment of backtesting the tuple of ESCs in the framework of the traditional and comparative backtests based on the concepts of joint identifiability and multi-objective elicitability. For robust forecast evaluation against the choice of scoring function, we also extend the Murphy diagram, a graphical tool to check whether one forecast dominates another under a class of scoring functions, to the case of ESCs. Finally, leveraging the recent concept of multi-objective elicitability, we propose a novel semiparametric model for forecasting dynamic ESCs based on a compositional regression model. In an empirical analysis of stock returns we evaluate and compare a variety of models for forecasting dynamic ESCs and demonstrate the outstanding performance of the proposed model.

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arXiv2024

Tail risk forecasting with semi-parametric regression models by incorporating overnight information

This research incorporates realized volatility and overnight information into risk models, wherein the overnight return often contributes significantly to the total return volatility. Extending a semi-parametric regression model based on asymmetric Laplace distribution, we propose a family of RES-CAViaR-oc models by adding overnight return and realized measures as a nowcasting technique for simultaneously forecasting Value-at-Risk (VaR) and expected shortfall (ES). We utilize Bayesian methods to estimate unknown parameters and forecast VaR and ES jointly for the proposed model family. We also conduct extensive backtests based on joint elicitability of the pair of VaR and ES during the out-of sample period. Our empirical study on four international stock indices confirms that overnight return and realized volatility are vital in tail risk forecasting.

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arXiv2024

Joint mixability and notions of negative dependence

A joint mix is a random vector with a constant component-wise sum. The dependence structure of a joint mix minimizes some common objectives such as the variance of the component-wise sum, and it is regarded as a concept of extremal negative dependence. In this paper, we explore the connection between the joint mix structure and popular notions of negative dependence in statistics, such as negative correlation dependence, negative orthant dependence and negative association. A joint mix is not always negatively dependent in any of the above senses, but some natural classes of joint mixes are. We derive various necessary and sufficient conditions for a joint mix to be negatively dependent, and study the compatibility of these notions. For identical marginal distributions, we show that a negatively dependent joint mix solves a multi-marginal optimal transport problem for quadratic cost under a novel setting of uncertainty. Analysis of this optimal transport problem with heterogeneous marginals reveals a trade-off between negative dependence and the joint mix structure.

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arXiv2023

Comparison of correlation-based measures of concordance in terms of asymptotic variance

We compare measures of concordance that arise as Pearson's linear correlation coefficient between two random variables transformed so that they follow the so-called concordance-inducing distributions. The class of such transformed rank correlations includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient. When only the standard axioms of measures of concordance are required, it is not always clear which transformed rank correlation is most suitable to use. To address this question, we compare measures of concordance in terms of their best and worst asymptotic variances of some canonical estimators over a certain set of dependence structures. A simple criterion derived from this approach is that concordance-inducing distributions with smaller fourth moment are more preferable. In particular, we show that Blomqvist's beta is the optimal transformed rank correlation in this sense, and Spearman's rho outperforms van der Waerden's coefficient. Moreover, we find that Kendall's tau, although it is not a transformed rank correlation of that nature, shares a certain optimal structure with Blomqvist's beta.

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arXiv2023

Measuring non-exchangeable tail dependence using tail copulas

Quantifying tail dependence is an important issue in insurance and risk management. The prevalent tail dependence coefficient (TDC), however, is known to underestimate the degree of tail dependence and it does not capture non-exchangeable tail dependence since it evaluates the limiting tail probability only along the main diagonal. To overcome these issues, two novel tail dependence measures called the maximal tail concordance measure (MTCM) and the average tail concordance measure (ATCM) are proposed. Both measures are constructed based on tail copulas and possess clear probabilistic interpretations in that the MTCM evaluates the largest limiting probability among all comparable rectangles in the tail, and the ATCM is a normalized average of these limiting probabilities. In contrast to the TDC, the proposed measures can capture non-exchangeable tail dependence. Analytical forms of the proposed measures are also derived for various copulas. A real data analysis reveals striking tail dependence and tail non-exchangeability of the return series of stock indices, particularly in periods of financial distress.

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arXiv2023

Matrix compatibility and correlation mixture representation of generalized Gini's gamma

Representations of measures of concordance in terms of Pearson' s correlation coefficient are studied. All transforms of random variables are characterized such that the correlation coefficient of the transformed random variables is a measure of concordance. Next, Gini' s gamma is generalized and it is shown that the resulting generalized Gini' s gamma can be represented as a mixture of measures of concordance that are Pearson' s correlation coefficients of transformed random variables. As an application of this correlation mixture representation of generalized Gini' s gamma, lower and upper bounds of the compatible set of generalized Gini' s gamma, which is the collection of all possible square matrices whose entries are pairwise bivariate generalized Gini' s gammas, are derived.

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arXiv2022

Avoiding zero probability events when computing Value at Risk contributions

This paper is concerned with the process of risk allocation for a generic multivariate model when the risk measure is chosen as the Value-at-Risk (VaR). We recast the traditional Euler contributions from an expectation conditional on an event of zero probability to a ratio involving conditional expectations whose conditioning events have strictly positive probability. We derive an analytical form of the proposed representation of VaR contributions for various parametric models. Our numerical experiments show that the estimator using this novel representation outperforms the standard Monte Carlo estimator in terms of bias and variance. Moreover, unlike the existing estimators, the proposed estimator is free from hyperparameters under a parametric setting.

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arXiv2020

Modality for Scenario Analysis and Maximum Likelihood Allocation

We study the variability of a risk from the statistical viewpoint of multimodality of the conditional loss distribution given that the aggregate loss equals an exogenously provided capital. This conditional distribution serves as a building block for calculating risk allocations such as the Euler capital allocation of Value-at-Risk. A superlevel set of this conditional distribution can be interpreted as a set of severe and plausible stress scenarios the given capital is supposed to cover. We show that various distributional properties of this conditional distribution, such as modality, dependence and tail behavior, are inherited from those of the underlying joint loss distribution. Among these properties, we find that modality of the conditional distribution is an important feature in risk assessment related to the variety of risky scenarios likely to occur in a stressed situation. Under unimodality, we introduce a novel risk allocation method called maximum likelihood allocation (MLA), defined as the mode of the conditional distribution given the total capital. Under multimodality, a single vector of allocations can be less sound. To overcome this issue, we investigate the so-called multimodalty adjustment to increase the soundness of risk allocations. Properties of the conditional distribution, MLA and multimodality adjustment are demonstrated in numerical experiments. In particular, we observe that negative dependence among losses typically leads to multimodality, and thus a higher multimodality adjustment can be required.

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arXiv2020

Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations

We propose a novel framework of estimating systemic risk measures and risk allocations based on Markov chain Monte Carlo (MCMC) methods. We consider a class of allocations whose jth component can be written as some risk measure of the jth conditional marginal loss distribution given the so-called crisis event. By considering a crisis event as an intersection of linear constraints, this class of allocations covers, for example, conditional Value-at-Risk (CoVaR), conditional expected shortfall (CoES), VaR contributions, and range VaR (RVaR) contributions as special cases. For this class of allocations, analytical calculations are rarely available, and numerical computations based on Monte Carlo (MC) methods often provide inefficient estimates due to the rare-event character of the crisis events. We propose an MCMC estimator constructed from a sample path of a Markov chain whose stationary distribution is the conditional distribution given the crisis event. Efficient constructions of Markov chains, such as Hamiltonian Monte Carlo and Gibbs sampler, are suggested and studied depending on the crisis event and the underlying loss distribution. The efficiency of the MCMC estimators is demonstrated in a series of numerical experiments.

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arXiv2019

Compatibility and attainability of matrices of correlation-based measures of concordance

Measures of concordance have been widely used in insurance and risk management to summarize non-linear dependence among risks modeled by random variables, which Pearson's correlation coefficient cannot capture. However, popular measures of concordance, such as Spearman's rho and Blomqvist's beta, appear as classical correlations of transformed random variables. We characterize a whole class of such concordance measures arising from correlations of transformed random variables, which includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases. Compatibility and attainability of square matrices with entries given by such measures are studied, that is, whether a given square matrix of such measures of concordance can be realized for some random vector and how such a random vector can be constructed. Compatibility and attainability of block matrices and hierarchical matrices are also studied due to their practical importance in insurance and risk management. In particular, a subclass of attainable block Spearman's rho matrices is proposed to compensate for the drawback that Spearman's rho matrices are in general not attainable for dimensions larger than four. Another result concerns a novel analytical form of the Cholesky factor of block matrices which allows one, for example, to construct random vectors with given block matrices of van der Waerden's coefficient.

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arXiv2019

Estimation of Risk Contributions with MCMC

Determining risk contributions of unit exposures to portfolio-wide economic capital is an important task in financial risk management. Computing risk contributions involves difficulties caused by rare-event simulations. In this study, we address the problem of estimating risk contributions when the total risk is measured by value-at-risk (VaR). Our proposed estimator of VaR contributions is based on the Metropolis-Hasting (MH) algorithm, which is one of the most prevalent Markov chain Monte Carlo (MCMC) methods. Unlike existing estimators, our MH-based estimator consists of samples from conditional loss distribution given a rare event of interest. This feature enhances sample efficiency compared with the crude Monte Carlo method. Moreover, our method has the consistency and asymptotic normality, and is widely applicable to various risk models having joint loss density. Our numerical experiments based on simulation and real-world data demonstrate that in various risk models, even those having high-dimensional (approximately 500) inhomogeneous margins, our MH estimator has smaller bias and mean squared error compared with existing estimators.

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