Multivariate Tweedie lifetimes: the impact of dependence

Systematic longevity risk is increasingly relevant for public pension schemes and insurance companies that provide life benefits. In view of this, mortality models should incorporate dependence between lives. However, the independent lifetime assumption is still heavily relied upon in the risk management of life insurance and annuity portfolios. This paper applies a multivariate Tweedie distribution to incorporate dependence, which it induces through a common shock component. Model parameter estimation is developed based on the method of moments and generalized to allow for truncated observations. The estimation procedure is explicitly developed for various important distributions belonging to the Tweedie family, and finally assessed using simulation.     

Improving risk classification and ratemaking using mixture-of-experts models with random effects

In the underwriting and pricing of nonlife insurance products, it is essential for the insurer to utilize both policyholder information and claim history to ensure profitability and proper risk management. In this paper, we apply a flexible regression model with random effects, called the Mixed Logit-weighted Reduced Mixture-of-Experts, which leverages both policyholder information and their claim history, to categorize policyholders into groups with similar risk profiles, and to determine a premium that accurately captures the unobserved risks. Estimates of model parameters and the posterior distribution of random effects can be obtained by a stochastic variational algorithm, which is numerically efficient and scalable to large insurance portfolios. Our proposed framework is shown to outperform the classical benchmark models (Logistic and Lognormal GL(M)M) in terms of goodness-of-fit to data, while offering intuitive and interpretable characterization of policyholders' risk profiles to adequately reflect their claim history.     

Intrinsic objective Bayesian estimation of the mean of the Tweedie family

ABSTRACT

The Tweedie family, which is classified by the choice of power unit variance function, includes heavy tailed distributions, and as such could be of significant relevance to actuarial science. The class includes the Normal, Poisson, Gamma, Inverse Gaussian, Stable and Compound Poisson distributions. In this study, we explore the intrinsic objective Bayesian point estimator for the mean value of the Tweedie family based on the intrinsic discrepancy loss function – which is an inherent loss function arising only from the underlying distribution or model, without any subjective considerations – and the Jeffreys prior distribution, which is designed to express absence of information about the quantity of interest. We compare the proposed point estimator with the Bayes estimator, which is the posterior mean based on quadratic loss function and the Jeffreys prior distribution. We carry a numerical study to illustrate the methodology in the context of the Inverse Gaussian model, which is fully unexplored in this novel context, and which is useful to insurance contracts.     

Credibility ratemaking using collateral information

Credibility ratemaking is a technique used in pricing health care, property and casualty, workers’ compensation, and group life coverages. It has been a part of actuarial practice since the time of Mowbray's (1914) contribution. In earlier work, we showed how many types of credibility models could be expressed as special cases of mixed linear models. This article extends this approach to credibility by formally introducing collateral information through the use of Bayesian methods.

Specifically, we derive credibility estimators and mean square errors for normal hierarchical linear models. We provide intuition for the credibility estimators by establishing the link between these estimators and homogeneous and inhomogeneous estimators that appear in non-Bayesian credibility theory.     

Remarks on a copula-based conditional value at risk for the portfolio problem

We deal with a multivariate conditional value at risk. Compared with the usual notion for the single random variable, a multivariate value at risk is concerned with several variables, and thus, the relation between each risk factor should be considered. We here introduce a new definition of copula-based conditional value at risk, which is real valued and ready to be computed. Copulas are known to provide a flexible method for handling a possible nonlinear structure; therefore, copulas may be naturally involved in the theory of value at risk. We derive a formula of our copula-based conditional value at risk in the case of Archimedean copulas, whose effectiveness is shown by examples. Numerical studies are also carried out with real data, which can be verified with analytical results.     

A multivariate Markov chain stock model

ABSTRACT

We propose a dividend stock valuation model where multiple dividend growth series and their dependencies are modelled using a multivariate Markov chain. Our model advances existing Markov chain stock models. First, we determine assumptions that guarantee the finiteness of the price and risk as well as the fulfilment of transversality conditions. Then, we compute the first- and second-order price-dividend ratios by solving corresponding linear systems of equations and show that a different price-dividend ratio is attached to each combination of states of the dividend growth process of each stock. Subsequently, we provide a formula for the computation of the variances and covariances between stocks in a portfolio. Finally, we apply the theoretical model to the dividend series of three US stocks and perform comparisons with existing models. The results could also be applied for actuarial purposes as a general stochastic investment model and for calculating the initial endowment to fund a portfolio of dependent perpetuities.     

Longitudinal modeling of insurance claim counts using jitters

Modeling insurance claim counts is a critical component in the ratemaking process for property and casualty insurance. This article explores the usefulness of copulas to model the number of insurance claims for an individual policyholder within a longitudinal context. To address the limitations of copulas commonly attributed to multivariate discrete data, we adopt a ‘jittering’ method to the claim counts which has the effect of continuitizing the data. Elliptical copulas are proposed to accommodate the intertemporal nature of the ‘jittered’ claim counts and the unobservable subject-specific heterogeneity on the frequency of claims. Observable subject-specific effects are accounted in the model by using available covariate information through a regression model. The predictive distribution together with the corresponding credibility of claim frequency can be derived from the model for ratemaking and risk classification purposes. For empirical illustration, we analyze an unbalanced longitudinal dataset of claim counts observed from a portfolio of automobile insurance policies of a general insurer in Singapore. We further establish the validity of the calibrated copula model, and demonstrate that the copula with ‘jittering’ method outperforms standard count regression models.     

基于GEM模型的上海航运保险产业竞争力研究

本文通过GEM模型,根据最新数据计算指标值,测量产业集群的竞争力的芡展情况,对影响航运保险产业竞争力的几大因素进行了分析。根据指标计算结果．目前航运保险产业的薄弱环节在于其产业基础不稳,人才匮乏且产品严重缺乏创新．导致集群内恶性竞争严重,行业的发展没有匹配与其相适应的法律环境。为进一步提高上海的航运保险,还需要培养更多专业人才致力于险种创新,政府也需要加大扶持力度。     

Second-order Bayesian revision of a generalised linear model

It is well known that the exponential dispersion family (EDF) of univariate distributions is closed under Bayesian revision in the presence of natural conjugate priors. However, this is not the case for the general multivariate EDF. This paper derives a second-order approximation to the posterior likelihood of a naturally conjugated generalised linear model (GLM), i.e., multivariate EDF subject to a link function (Section 5.5). It is not the same as a normal approximation. It does, however, lead to second-order Bayes estimators of parameters of the posterior. The family of second-order approximations is found to be closed under Bayesian revision. This generates a recursion for repeated Bayesian revision of the GLM with the acquisition of additional data. The recursion simplifies greatly for a canonical link. The resulting structure is easily extended to a filter for estimation of the parameters of a dynamic generalised linear model (DGLM) (Section 6.2). The Kalman filter emerges as a special case. A second type of link function, related to the canonical link, and with similar properties, is identified. This is called here the companion canonical link. For a given GLM with canonical link, the companion to that link generates a companion GLM (Section 4). The recursive form of the Bayesian revision of this GLM is also obtained (Section 5.5.3). There is a perfect parallel between the development of the GLM recursion and its companion. A dictionary for translation between the two is given so that one is readily derived from the other (Table 5.1). The companion canonical link also generates a companion DGLM. A filter for this is obtained (Section 6.3). Section 1.2 provides an indication of how the theory developed here might be applied to loss reserving. A sequel paper, providing numerical illustrations of this, is planned.     

基于拓扑数据模型的车险操作风险度量

以机动车辆保险为研究对象,分析车险核心业务流程中操作风险的表现形式,借助拓扑数据模型及Monte Carlo模拟对其风险进行度量。结果显示:损失强度表现出较强的厚尾性,事件频率具有高频性,但总损失额分布的厚尾特征不明显,这些结果为操作风险进行经济资本配置及管理提供了依据。     

On a multivariate Markov chain model for credit risk measurement

In this paper, we use credibility theory to estimate credit transition matrices in a multivariate Markov chain model for credit rating. A transition matrix is estimated by a linear combination of the prior estimate of the transition matrix and the empirical transition matrix. These estimates can be easily computed by solving a set of linear programming (LP) problems. The estimation procedure can be implemented easily on Excel spreadsheets without requiring much computational effort and time. The number of parameters is O(s² m² ), where s is the dimension of the categorical time series for credit ratings and m is the number of possible credit ratings for a security. Numerical evaluations of credit risk measures based on our model are presented.     

Correlations among cryptocurrencies: Evidence from multivariate factor stochastic volatility model

This paper is the first study to apply the multivariate factor stochastic volatility model (MFSVM) for analyzing the correlations among six cryptocurrencies. We use MFSVM with the Bayesian estimation procedure for the period from August 8, 2015, to January 1, 2020. According to the findings, there is a significant positive correlation between price volatility values of Bitcoin and Litecoin. Besides, the volatility values of Ethereum have a positive correlation with both Ripple and Stellar. There is also a positive correlation between the volatility values of Ripple and Dash. These findings are robust to consider different correlation networks. The evidence implies that Bitcoin is mainly related to Litecoin, but Ethereum is associated with other cryptocurrencies.     

Modelling multivariate skewness in financial returns: a SGARCH approach

Skewness of financial time series is a relevant topic, due to its implications for portfolio theory and for statistical inference. In the univariate case, its default measure is the third cumulant of the standardized random variable. It can be generalized to the third multivariate cumulant that is a matrix containing all centered moments of order three which can be obtained from a random vector. The present paper examines some properties of the third cumulant under the assumptions of the multivariate SGARCH model introduced by De Luca, Genton, and Loperfido [2006. A multivariate skew-GARCH model. Advances in Econometrics 20: 33–57]. In the first place, it allows for parsimonious modelling of multivariate skewness. In the second place, all its elements are either null or negative, consistently with previous empirical and theoretical findings. A numerical example with financial returns of France, Spain and Netherlands illustrates the theoretical results in the paper.     

The maximum surplus before ruin for dependent risk models through Farlie–Gumbel–Morgenstern copula

We extend the classical compound Poisson risk model to consider the distribution of the maximum surplus before ruin where the claim sizes depend on inter-claim times via the Farlie–Gumbel–Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for this distribution, of which the Laplace transform is provided. We obtain the renewal equation and explicit expressions for this distribution are derived when the claim amounts are exponentially distributed. Finally, we present numerical examples.     

论中国渔业互保协会的运作模式及其完善

中国渔业互保协会在实际运行中采用了"合作组织+行政辅助"运作模式,这种模式缘起在于渔民的自组织成本高于官方组织成本这一现实,两种成本之比决定了"合作组织+行政辅助"这一制度安排动态变动的边界.在这一制度安排中,实现了政府、协会本身和协会会员之间的三方正和博弈状态,这构成了该种制度安排的运作逻辑.随着渔业保险规模的扩大,...     

Option pricing using a binomial model with random time steps (A formal model of gamma hedging)

This paper provides a new option pricing model which justifies the standard industry implementation of the Black-Scholes model. The standard industry implementation of the Black-Scholes model uses an implicit volatility, and it hedges both delta and gamma risk. This industry implementation is inconsistent with the theory underlying the derivation of the Black-Scholes model. We justify this implementation by showing that these adhoc adjustments to the Black-Scholes model provide a reasonable approximation to valuation and delta hedging in our new option pricing model.     

Asymmetric information and imperfect competition in a continuous time multivariate security model

This paper deals with the problem of price formation in a market with asymmetric information and several risky assets. We then extend the multivariate security model of Caballé and Krishnan (1994) to a continuous time framework, and general utility function. Our model enables us to observe some results which are specific to multi security markets such as Giffen effect. An application of the main result will be the non trivial generalizations of the models of Back (1992) and Cho (1997).Mathematics Subject Classification (1991): 49L10, 60G44, 90A15JEL Classification: G11, G12The author would like to thank his supervisor H. Pham, K. Back and an anonymous referee for useful comments and discussions.     

Predicting bankruptcy using the discrete-time semiparametric hazard model

The usual bankruptcy prediction models are based on single-period data from firms. These models ignore the fact that the characteristics of firms change through time, and thus they may suffer from a loss of predictive power. In recent years, a discrete-time parametric hazard model has been proposed for bankruptcy prediction using panel data from firms. This model has been demonstrated by many examples to be more powerful than the traditional models. In this paper, we propose an extension of this approach allowing for a more flexible choice of hazard function. The new method does not require the assumption of a parametric model for the hazard function. In addition, it also provides a tool for checking the adequacy of the parametric model, if necessary. We use real panel datasets to illustrate the proposed method. The empirical results confirm that the new model compares favorably with the well-known discrete-time parametric hazard model.