Hierarchical Archimedean Copulas Research Articles

SPATIAL DEPENDENCE AND AGGREGATION IN WEATHER RISK HEDGING: A LÉVY SUBORDINATED HIERARCHICAL ARCHIMEDEAN COPULAS (LSHAC) APPROACH

AbstractAdverse weather-related risk is a main source of crop production loss and a big concern for agricultural insurers and reinsurers. In response, weather risk hedging may be valuable, however, due to basis risk it has been largely unsuccessful to date. This research proposes the Lévy subordinated hierarchical Archimedean copula model in modelling the spatial dependence of weather risk to reduce basis risk. The analysis shows that the Lévy subordinated hierarchical Archimedean copula model can improve the hedging performance through more accurate modelling of the dependence structure of weather risks and is more efficient in hedging extreme downside weather risk, compared to the benchmark copula models. Further, the results reveal that more effective hedging may be achieved as the spatial aggregation level increases. This research demonstrates that hedging weather risk is an important risk management method, and the approach outlined in this paper may be useful to insurers and reinsurers in the case of agriculture, as well as for other related risks in the property and casualty sector.

A model to account for data dependency when estimating floral cover in different land use types over a season

We propose a model to consider data dependencies and assess spatial and temporal variability in land use specific floral coverage across landscapes. Data dependence arising from repeated measurements across the flowering season is taken into account using hierarchical Archimedean copulas, where the correlation is assumed to be stronger within seasonal periods than between periods. For each seasonal period, a bounded probability distribution is assigned to capture spatial variability in floral cover. The model uses a Bayesian approach and can assess land-use-specific floral covers by integrating experts judgments and field data. The model is applied to assess floral covers in four land use types in southern Sweden, where seasonal variability is captured by dividing the season into two periods according to winter oilseed rape flowering. Floral cover is updated using Markov Chain Monte Carlo sampling based on data from 16 landscapes and 2 years, with repeated measures available from each of the two seasonal periods. Our results indicate that considering data dependence improved the estimation of floral cover based on data observed during a season. Different copula families specifying multivariate probability distributions were tested, and no family had a consistently higher performance in the four tested land use types. Uncertainty in both mode and variability of floral cover was higher when data dependence were accounted for. Posterior modes of floral covers in semi-natural grassland were higher than in field edges, but both expert’s best guesses were higher than these estimates. This confirms previous findings in expert elicitation processes that experts may fail to discriminate extreme values on a bounded range. Floral cover in flower strips were estimated to be smaller/higher than semi-natural grasslands early/late in the season. The mode of floral cover in oil seed rape was estimated to be close to 100%, and higher than estimates provided by expert judgment. Floral covers for different land use classes are key parameters when quantifying floral resources at a landscape level whose assessments rely on both expert judgment and field measurements.

Hierarchical Archimedean copula models for the analysis of binary familial data.

Archimedean copulas are commonly used in a wide range of statistical models due to their simplicity, manageable analytical expressions, rich choices of generator functions, and other workable properties. However, the exchangeable dependence structure inherent to Archimedean copulas limits its application to familial data, where the dependence among family members is often different. When response variables are binary, modeling the familial associations becomes more challenging due to the stringent constraints imposed on the dependence parameters. This paper proposes hierarchical Archimedean copulas to account for the natural hierarchical dependence structure in familial data and addresses the details in the modeling of binary familial data and the inference based on maximum likelihood estimate. An example showing the flexibility of this powerful tool is also presented with possible extension to other similar studies.

Rain Attenuation Prediction Model Based on Hyperbolic Cosecant Copula for Multiple Site Diversity Systems in Satellite Communications

This paper presents a novel model for prediction of site diversity system performance in millimeter-wave satellite communications. The model predicts the joint exceedance probability of rain attenuation based on a new family of Archimedean copulas, called the hyperbolic cosecant copula. Furthermore, employing the hierarchical Archimedean copula structures, the proposed methodology can be used for the prediction of joint first-order statistics of rain attenuation for dual- and multiple-site diversity satellite communication systems. The proposed method has been tested on an extensive data set of site diversity experiments with various system configurations. Moreover, in addition to the common approaches, where the dependence of rain attenuation induced on multiple links is based solely on the distance between ground stations, elevation and baseline angle are also taken into account for the modeling of the dependence parameter. Both versions of the model outperform the existing model for two-site diversity systems given in the ITU-R Recommendation P.618-12. The version, which accounts for the system’s angles, shows a better performance also in comparison with the state-of-the-art models such as the models based on the Archimedean Clayton copula and the Gaussian copula.

On structure, family and parameter estimation of hierarchical Archimedean copulas

ABSTRACTResearch on structure determination and parameter estimation of hierarchical Archimedean copulas (HACs) has so far mostly focused on the case in which all appearing Archimedean copulas belong to the same Archimedean family. The present work addresses this issue and proposes a new approach for estimating HACs that involve different Archimedean families. It is based on employing goodness-of-fit test statistics directly into HAC estimation. The approach is summarized in a simple algorithm, its theoretical justification is given and its applicability is illustrated by several experiments, which include estimation of HACs involving up to five different Archimedean families.

Recent Developments in Copula Models

Copula models have become very popular and well studied among the scientific community.[...]

Hierarchical Archimedean copulas through multivariate compound distributions

In this paper, we propose a new hierarchical Archimedean copula construction based on multivariate compound distributions. This new imbrication technique is derived via the construction of a multivariate exponential mixture distribution through compounding. The absence of nesting and marginal conditions, contrarily to the nested Archimedean copulas approach, leads to major advantages, such as a flexible range of possible combinations in the choice of distributions, the existence of explicit formulas for the distribution of the sum, and computational ease in high dimensions. A balance between flexibility and parsimony is targeted. After presenting the construction technique, properties of the proposed copulas are investigated and illustrative examples are given. A detailed comparison with other construction methodologies of hierarchical Archimedean copulas is provided. Risk aggregation under this newly proposed dependence structure is also examined.

The Realized Hierarchical Archimedean Copula in Risk Modelling

This paper introduces the concept of the realized hierarchical Archimedean copula (rHAC). The proposed approach inherits the ability of the copula to capture the dependencies among financial time series, and combines it with additional information contained in high-frequency data. The considered model does not suffer from the curse of dimensionality, and is able to accurately predict high-dimensional distributions. This flexibility is obtained by using a hierarchical structure in the copula. The time variability of the model is provided by daily forecasts of the realized correlation matrix, which is used to estimate the structure and the parameters of the rHAC. Extensive simulation studies show the validity of the estimator based on this realized correlation matrix, and its performance, in comparison to the benchmark models. The application of the estimator to one-day-ahead Value at Risk (VaR) prediction using high-frequency data exhibits good forecasting properties for a multivariate portfolio.

Multivariate statistical modelling of future marine storms

Extreme events, such as wave-storms, need to be characterized for coastal infrastructure design purposes. Such description should contain information on both the univariate behaviour and the joint-dependence of storm-variables. These two aspects have been here addressed through generalized Pareto distributions and hierarchical Archimedean copulas. A non-stationary model has been used to highlight the relationship between these extreme events and non-stationary climate. It has been applied to a Representative Concentration Pathway 8.5 Climate-Change scenario, for a fetch-limited environment (Catalan Coast). In the non-stationary model, all considered variables decrease in time, except for storm-duration at the northern part of the Catalan Coast. The joint distribution of storm variables presents cyclical fluctuations, with a stronger influence of climate dynamics than of climate itself.

Modeling Multicountry Longevity Risk With Mortality Dependence: A Lévy Subordinated Hierarchical Archimedean Copulas Approach

AbstractThis article proposes a new copula model known as the Lévy subordinated hierarchical Archimedean copulas (LSHAC) for multicountry mortality dependence modeling. To the best of our knowledge, this is the first article to apply the LSHAC model to mortality studies. Through an extensive empirical analysis on modeling mortality experiences of 13 countries, we demonstrate that the LSHAC model, which has the advantage of capturing the geographical structure of mortality data, yields better fit, compared to the elliptical copulas. In addition, the proposed LSHAC model generates out‐of‐sample forecasts with smaller standard deviations, when compared to other benchmark copula models. The LSHAC model also confirms that there is an association between geographical locations and dependence of the overall mortality improvement. These results yield new insights into future longevity risk management. Finally, the model is used to price a hypothetical survival index swap written on a weighted mortality index. The results highlight the importance of dependence modeling in managing longevity risk and reducing population basis risk.

Kendall’s tau and agglomerative clustering for structure determination of hierarchical Archimedean copulas

Abstract Several successful approaches to structure determination of hierarchical Archimedean copulas (HACs) proposed in the literature rely on agglomerative clustering and Kendall’s correlation coefficient. However, there has not been presented any theoretical proof justifying such approaches. This work fills this gap and introduces a theorem showing that, given the matrix of the pairwise Kendall correlation coefficients corresponding to a HAC, its structure can be recovered by an agglomerative clustering technique.

Estimation of Hierarchical Archimedean Copulas as a Shortest Path Problem

We formulate the problem of finding and estimating the optimal hierarchical Archimedean copula as an amended shortest path problem. The standard network flow problem is amended by certain constraints specific to copulas, which limit scalability of the problem. However, we show in dimensions as high as twenty that the new approach dominates the alternatives which usually require recursive estimation or full enumeration.

A multivariate statistical model of extreme events: An application to the Catalan coast

Wave extreme events can be understood as the combination of Storm-intensity, Directionality and Intra-time distribution. However, the dependence structure among these factors is still unclear. A methodology has been developed to model wave-storms whose components are linked together. The model is composed by three parts: an intensity module, a wave directionality module, and an intra-time distribution module. In the Storm-intensity sub-model, generalized Pareto distributions and hierarchical Archimedean copulas have been used to characterize the storm energy, unitary energy, peak wave-period and duration. In the Directionality and Intra-time sub-models, the wave direction (at the peak of the storm) and the storm growth–decay rates are linked to the variables from the intensity model, respectively. The model is applied to the Catalan coast (NW Mediterranean). The outcomes denote spatial patterns that coincide with the state of knowledge. The proposed methodology is able to provide boundary conditions for wave and near-shore studies, saving computational time and establishing the dependence of the proposed variables. Such synthetic storms reproduce the inter-variable co-dependence of the original data.

Structure and estimation of Lévy subordinated hierarchical Archimedean copulas (LSHAC): Theory and empirical tests

Abstract Levy subordinated hierarchical Archimedean copulas (LSHAC) are flexible models in high dimensional modeling. However, there is limited literature discussing their applications, largely due to the challenges in estimating their structures and their parameters. In this paper, we propose a three-stage estimation procedure to determine the hierarchical structure and the parameters of a LSHAC. This is the first paper to empirically examine the modeling performances of LSHAC models using exchange traded funds. Simulation study demonstrates the reliability and robustness of the proposed estimation method in determining the optimal structure. Empirical analysis further shows that, compared to elliptical copulas, LSHACs have better fitting abilities as well as more accurate out-of-sample Value-at-Risk estimates with less parameters. In addition, from a financial risk management point of view, the LSHACs have the advantage of being very flexible in modeling the asymmetric tail dependence, providing more conservative estimations of the probabilities of extreme downward co-movements in the financial market.

Distorted Copula-Based Probability Distribution of a Counting Hierarchical Variable: A Credit Risk Application

In this paper, we propose a novel approach for the computation of the probability distribution of a counting variable linked to a multivariate hierarchical Archimedean copula function. The hierarchy has a twofold impact: it acts on the aggregation step but also it determines the arrival policy of the random event. The novelty of this work is to introduce this policy, formalized as an arrival matrix, i.e., a random matrix of dependent 0–1 random variables, into the model. This arrival matrix represents the set of distorted (by the policy itself) combinatorial distributions of the event, i.e., of the most probable scenarios. To this distorted version of the [Formula: see text] approach [see Ref. 7 and Ref. 27], we are now able to apply a pure hierarchical Archimedean dependence structure among variables. As an empirical application, we study the problem of evaluating the probability distribution of losses related to the default of various type of counterparts in a structured portfolio exposed to the credit risk of a selected set among the major banks of European area and to the correlations among these risks.

COPULA MODELS OF THE JOINT DISTRIBUTION OF EXCHANGE RATES

The paper aims at investigating the joint distribution of currency rates using HAC, HKC and Vine copulas in several time periods. Models were constructed using Archimedean copulas including Gumbel-Hougaard, Joe BB1 and Frank copulas, and their parameters were estimated by maximum likelihood. The best models were built using hierarchical Archimedean copulas and the worst were obtained with vine copulas. In comparison with the HAC, the main advantage of hierarchical Kendall copulas is the possibility to use a two-parameter copula Joe BB1. The best models were obtained with Frank copula, while Gumbel-Hougaard copula has shown a decent result only in the third period. Additionally, in this paper was made an attempt to get the forecast of exchange rates using Kendall’s and Marshall-Olkin’s algorithms. The most accurate forecast was obtained with Gumbel-Hougaard copula for euro and franc.

Spatial Dependence & Aggregation in Weather Risk Hedging

Adverse weather related risk is a main source of crop production loss, and in addition to farmers, this exposure is a major concern and uncertainty for insurers and reinsurers who act as weather risk underwriters. To date, weather hedging has had limited success, largely due to challenges regarding basis risk. Therefore, this paper develops and compares different weather risk hedging strategies for agricultural insurers and reinsurers, through investigating the spatial dependence and aggregation level of systemic weather risks across a country. This paper proposes a flexible time series model that assumes a general hyperbolic (GH) family for the margins to capture the heavy-tail property of the data, together with the Levy subordinated hierarchical Archimedean copula (LSHAC) model to overcome the challenge of high-dimensionality in modeling the dependence of weather risk. Wavelet analysis is employed to study the detailed characteristics within the data from both time and frequency scales. The analysis shows that the LSHAC model proposed in this paper reduces extreme weather downside risk by $3920.89 million, providing an additional $321.61 million risk reduction compared to the independent model assumption. Further, the results reveal that more effective hedging may be achieved as the spatial aggregation level increases.

Modeling Multi-Country Longevity Risk with Mortality Dependence: A LLvy Subordinated Hierarchical Archimedean Copulas (LSHAC) Approach

This paper proposes a new copula model known as the Levy subordinated hierarchical Archimedean copulas (LSHAC) for multi-country mortality dependence modeling. To the best of our knowledge, this is the first paper to apply the LSHAC model to mortality studies. Through an extensive empirical analysis on modelling mortality experiences of 13 countries, we demonstrate that the LSHAC model, which has the advantage of capturing the geographical structure of mortality data, yields better fit, more accurate and robust out-of-sample forecasting, when compared to other benchmark copula models. The LSHAC model also confirms that there is an association between geographical locations and dependence of the overall mortality improvement. These results yield new insights into future longevity risk management. Finally, the model is used to price a hypothetical survival index swap written on a weighted mortality index. The results highlight the importance of dependence modeling in managing longevity risk and reducing population basis risk.

Using Copulas in Data Mining Based on the Observational Calculus

The objective of the paper is a contribution to data mining within the framework of the observational calculus, through introducing ǵeneralized quantifiers related to copulas. Fitting copulas to multidimensional data is an increasingly important method for analyzing dependencies, and the proposed quantifiers of observational calculus assess the results of estimating the structure of joint distributions of continuous variables by means of hierarchical Archimedean copulas. To this end, the existing theory of hierarchical Archimedean copulas has been slightly extended in the paper: It has been proven that sufficient conditions for the function defining a hierarchical Archimedean copula to be indeed a copula, which have so far been rigorously established only for the special case of fully nested Archimedean copulas, hold in general. These conditions allow us to define three new generalized quantifiers, which are then thoroughly validated on four benchmark data sets and one data set from a real-world application. The paper concludes by comparing the proposed quantifiers to a more traditional approach—maximum weight spanning trees.

Structure and Estimation of LLvy Subordinated Hierarchical Archimedean Copulas (LSHAC): Theory and Empirical Tests

Levy subordinated hierarchical Archimedean copulas (LSHAC) are flexible models in high dimensional modeling. However, there is limited literature discussing their applications, largely due to the challenges in estimating their structures and their parameters. In this paper, we propose a three-stage estimation procedure to determine the hierarchical structure and the parameters of a LSHAC. This is the first paper to empirically examine the modeling performances of the LSHAC models using ex- change traded funds. Simulation study demonstrates the reliability and robustness of the proposed estimation method in determining the optimal structure. Empirical analysis further shows that, compared to elliptical copulas, LSHACs have better fitting abilities as well as more accurate out-of-sample Value-at-Risk estimates with less parameters. In addition, from a financial risk management point of view, the LSHACs have the advantage of being very flexible in modeling the asymmetric tail dependence, providing more conservative estimations of the probabilities of extreme downward co-movements in the financial market.