基于Copula的道指与恒指的尾部相关性分析

Copula函数能有效地刻画随机变量间的非对称性和尾部相关性。在研究当道琼斯工业指数与恒生指数的相关性时,采用非参数核密度估计拟合边缘分布。并应用Kol mogorov-Smirrnov检验三种Archi medean Copula函数。研究发现参数值θ=0.2221的Clayton Copula可以用于描述下尾相关性,而θ=1.1105的Gumbel Copula可用于描述上尾相关性。     

Modeling the dependence structure between default risk premium,equity return volatility and the jump risk: Evidence from a financial crisis

This paper investigates the dependence structure between default risk premium, equity return volatility and jump risk in the equity market before and during the subprime crisis. Using iTraxx CDS index spreads from Japanese and Australian markets, the paper models the different relationships that can exist in different ranges of behavior. We consider several Archimedean copula models with different tail dependence structures, namely, Gumbel, Clayton, Frank, AMH and Joe copulas. Although the dramatic change in the levels of the iTraxx CDS index, we find strong evidence that the dependence structure between CDS and stock market conditions is asymmetric and orienting toward the upper side. In addition, we find that the Japanese CDS market is more sensitive to the stock return volatility than the jump risk and the magnitude of this sensitivity is related to the market circumstances. However, Australian CDS market is more sensitive to the jump risk than stock return volatility before and during the financial crisis. This result has important implications for both global financial stability and default risk management. Specifically, the heterogeneity of markets, coupled with the diversity in the risk exposures cause the default risk premium and equity markets to exhibit different levels of sensitivity.     

On the structure and estimation of hierarchical Archimedean copulas

In this paper we provide a method for estimating multivariate distributions defined through hierarchical Archimedean copulas. In general, the true structure of the hierarchy is unknown, but we develop a computationally efficient technique to determine it from the data. For this purpose we introduce a hierarchical estimation procedure for the parameters and provide an asymptotic analysis. We consider both parametric and nonparametric estimation of the marginal distributions. A simulation study and an empirical application show the effectiveness of the grouping procedure in the sense of structure selection.     

Hierarchical structures in the aggregation of premium risk for insurance underwriting

In the valuation of the Solvency II capital requirement, the correct appraisal of risk dependencies acquires particular relevance. These dependencies refer to the recognition of risk diversification in the aggregation process and there are different levels of aggregation and hence different types of diversification. For instance, for a non-life company at the first level the risk components of each single line of business (e.g. premium, reserve, and CAT risks) need to be combined in the overall portfolio, the second level regards the aggregation of different kind of risks as, for example, market and underwriting risk, and finally various solo legal entities could be joined together in a group.

Solvency II allows companies to capture these diversification effects in capital requirement assessment, but the identification of a proper methodology can represent a delicate issue. Indeed, while internal models by simulation approaches permit usually to obtain the portfolio multivariate distribution only in the independence case, generally the use of copula functions can consent to have the multivariate distribution under dependence assumptions too.

However, the choice of the copula and the parameter estimation could be very problematic when only few data are available. So it could be useful to find a closed formula based on Internal Models independence results with the aim to obtain the capital requirement under dependence assumption.

A simple technique, to measure the diversification effect in capital requirement assessment, is the formula, proposed by Solvency II quantitative impact studies, focused on the aggregation of capital charges, the latter equal to percentile minus average of total claims amount distribution of single line of business (LoB), using a linear correlation matrix.

On the other hand, this formula produces the correct result only for a restricted class of distributions, while it may underestimate the diversification effect.

In this paper we present an alternative method, based on the idea to adjust that formula with proper calibration factors (proposed by Sandström (2007)) and appropriately extended with the aim to consider very skewed distribution too.

In the last part considering different non-life multi-line insurers, we compare the capital requirements obtained, for only premium risk, applying the aggregation formula to the results derived by elliptical copulas and hierarchical Archimedean copulas.     

Types of dependence and time-dependent association between two lifetimes in single parameter copula models

Spreeuw, J. Types of dependence and time-dependent association between two lifetimes in single parameter copula models. Scandinavian Actuarial Journal. Most publications on modeling insurance contracts on two lives, assuming dependence of the two lifetimes involved, focus on the time of inception of the contract. The dependence between the lifetimes is usually modeled through a copula and the effect of this dependence on the pricing of a joint life policy is measured. This paper investigates the effect of association at the outset on the mortality in the future. The conditional law of mortality of an individual, given his survival and given the life status of the partner is derived. The conditional joint survival distribution of a couple at any duration, given that the two lives are then alive, is also derived. We analyze how the degree of dependence between the two members of a couple varies throughout the duration of a contract. We have done that for (mainly Archimedean) copula models, with one parameter for the degree of dependence. The conditional distributions hence derived provide the basis for the calculation of prospective provisions.     

Modelling the dependence structures of Australian iTraxx CDS index

In contrast to market expectations, the correlation between credit default swap (CDS) spreads and their respective stock prices in Australia was found to be positive. The global financial crisis (GFC) affected the nonlinear association between the two asset classes with firms experiencing financial distress and stock prices plummeting. CDSs issuers reacted to such exogenous shocks by increasing their risk premiums on their spreads, reflecting the increased inherent risk. By splitting the data into pre- and post-GFC contexts and by employing the use of Archimedean copulas, we observe a negative co-movement in the post-GFC period. This finding is robust to several equity indices. Overall, such result is critical for investors engaging in arbitrageur activities.     

Expected utility theory on mixture spaces without the completeness axiom

A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.     

CDO pricing with nested Archimedean copulas

Companies in the same industry sector are usually more correlated than firms in different sectors, as they are similarly affected by macroeconomic effects, political decisions, and consumer trends. Despite the many stock return models taking this fact into account, there are only a few credit default models that take it into consideration. In this paper we present a default model based on nested Archimedean copulas that is able to capture hierarchical dependence structures among the obligors in a credit portfolio. Nested Archimedean copulas have a surprisingly simple and intuitive interpretation. The dependence among all companies in the same sector is described by an inner copula and the sectors are then coupled via an outer copula. Consequently, our model implies a larger default correlation for companies in the same industry sector than for companies in different sectors. A calibration to CDO tranche spreads of the European iTraxx portfolio is performed to demonstrate the fitting capability of the model. This portfolio consists of CDS on 125 companies from six different industry sectors and is therefore an excellent portfolio for a comparison of our generalized model with a traditional copula model of the same family that does not take different sectors into account.     

The Use of Archimedean Copulas to Model Portfolio Allocations

A copula is a means of generating an n -variate distribution function from an arbitrary set of n univariate distributions. For the class of portfolio allocators that are risk averse, we use the copula approach to identify a large set of n -variate asset return distributions such that the relative magnitudes of portfolio shares can be ordered according to the reversed hazard rate ordering of the n underlying univariate distributions. We also establish conditions under which first- and second-degree dominating shifts in one of the n underlying univariate distributions increase allocation to that asset. Our findings exploit separability properties possessed by the Archimedean family of copulas.     

Lifetime dependence models generated by multiply monotone functions

We study a family of distributions generated from multiply monotone functions that includes a multivariate Pareto and, previously unidentified, exponential-Pareto distribution. We utilize an established link with Archimedean survival copulas to provide further examples, including a multivariate Weibull distribution, that may be used to fit light, or heavy-tailed phenomena, and which exhibit various forms of dependence, ranging from positive to negative. Because the model is intended for the study of joint lifetimes, we consider the effect of truncation and formulate properties required for a number of parameter estimation procedures based on moments and quantiles. For the quantile-based estimation procedure applied to the multivariate Weibull distribution, we also address the problem of optimal quantile selection.