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.     

面板数据下的线性混合模型及其在车险费率厘定中的应用

精算师在进行车险净保费信度厘定时可采用关于面板数据的线性混合模型,本文采用每次交通事故平均损失额和事故发生频率作为车险净保费的计算指标。利用2008~2012年31个省、市、自治区5年的数据,建立面板数据下的线性混合模型,选取人均地区生产总值、每平方公里人口数、民用汽车拥有量作为解释变量,得到每次交通事故平均损失额和事故发生频率的估计模型,进而得到纯保费估计。这一研究可为车险费率市场化提供一定的理论支持和参考。     

A hierarchical credibility regression model

Abstract

The paper develops a hierarchical credibility regression model with random parameters on two levels. The common credibility regression models appear as special cases by assuming one of the parameter levels degenerate. By specializing into another direction we obtain Jewell's (1975c) hierarchical model as a special case.     

The linear growth credibility model

Abstract

The investigation of evolutionary models, i.e. models allowing the risk parameter to change in time, has been one of the main topics of research in credibility theory in the last few years. In the present paper a very special (but rather practicable) evolutionary model is defined and recursions for the credibility estimator are stated.     

Evolutionary Credibility Theory

Abstract

The conventional approach to evolutionary credibility theory assumes a linear state-space model for the longitudinal claims data so that Kalman filters can be used to estimate the claims’ expected values, which are assumed to form an autoregressive time series. We propose a class of linear mixed models as an alternative to linear state-space models for evolutionary credibility and show that the predictive performance is comparable to that of the Kalman filter when the claims are generated by a linear state-space model. More importantly, this approach can be readily extended to generalized linear mixed models for the longitudinal claims data. We illustrate its applications by addressing the “excess zeros” issue that a substantial fraction of policies does not have claims at various times in the period under consideration.     

Abstract credibility

Abstract

This paper falls into three parts. The first part (Sections 1 to 4) consists of an attempt to consolidate the existing credibility models, which are rapidly becoming more numerous, by constructing one large all-embracing model. In order to do this, it is necessary to abstract from the usual setting of Credibility Theory, the insurance world, and work in more abstract Hilbert spaces. This increases the complexity of the theory, but the results appear nevertheless in a form familiar to credibility theorists.

The second part (Section 5) shows that, in the so-called inhomogeneous case of credibility estimation, best estimates are obtained if one uses sufficient statistics, a result which has previously been known under much more restricted circumstances. In the same section it is pointed out that the validity of the analogous result in the homogeneous case is by no means obvious.

The third part (Section 6) demonstrates that a number of existing credibility models can be derived as special cases of the model developed here, and goes on to deal briefly with a couple of new models which arise naturally out of the abstract credibility formulation.     

Is the Magic of the Diaspora Fact or Fiction? A Study of Taiwan’s Trade Performance in the Bamboo Network

Abstract

In this study, we examine the relationship between the ethnic Chinese diaspora network (i.e., bamboo network) and Taiwan’s trade performance. The models are based on theories of international trade to examine hypotheses using panel data from the members of the World Trade Organization (WTO) from 1996 to 2012. The results indicate that the bamboo network has limited effects on Taiwan’s trade performance. Hence, the “magic” effect of the Chinese diaspora is debatable.     

Credibility Using Copulas

Abstract

Credibility is a form of insurance pricing that is widely used, particularly in North America. The theory of credibility has been called a “cornerstone” in the field of actuarial science. Students of the North American actuarial bodies also study loss distributions, the process of statistical inference of relating a set of data to a theoretical (loss) distribution. In this work, we develop a direct link between credibility and loss distributions through the notion of a copula, a tool for understanding relationships among multivariate outcomes.

This paper develops credibility using a longitudinal data framework. In a longitudinal data framework, one might encounter data from a cross section of risk classes (towns) with a history of insurance claims available for each risk class. For the marginal claims distributions, we use generalized linear models, an extension of linear regression that also encompasses Weibull and Gamma regressions. Copulas are used to model the dependencies over time; specifically, this paper is the first to propose using a t-copula in the context of generalized linear models. The t-copula is the copula associated with the multivariate t-distribution; like the univariate tdistributions, it seems especially suitable for empirical work. Moreover, we show that the t-copula gives rise to easily computable predictive distributions that we use to generate credibility predictors. Like Bayesian methods, our copula credibility prediction methods allow us to provide an entire distribution of predicted claims, not just a point prediction.

We present an illustrative example of Massachusetts automobile claims, and compare our new credibility estimates with those currently existing in the literature.     

E. J. Moorhead (1910 – 2004)

Abstract

As is well known in actuarial practice, excess claims (outliers) have a disturbing effect on the ratemaking process. To obtain better estimators of premiums, which are based on credibility theory, Künsch and Gisler and Reinhard suggested using robust methods. The estimators proposed by these authors are indeed resistant to outliers and serve as an excellent example of how useful robust models can be for insurance pricing. In this article we further refine these procedures by reducing the degree of heuristic arguments they involve. Specifically we develop a class of robust estimators for the credibility premium when claims are approximately gamma-distributed and thoroughly study their robustness-efficiency trade-offs in large and small samples. Under specific datagenerating scenarios, this approach yields quantitative indices of estimators’ strength and weakness, and it allows the actuary (who is typically equipped with information beyond the statistical model) to choose a procedure from a full menu of possibilities. Practical performance of our methods is illustrated under several simulated scenarios and by employing expert judgment.     

Credibility for Severity Revisited

Abstract

It is basic actuarial knowledge that the pure premium of an insurance contract can be written as the product of the expected claim number and the expected claim amount. Actuaries use credibility theory to incorporate the contract’s individual experience into this calculation in a statistically optimal way. For many years, however, the use of credibility was limited to the frequency component. Starting with the paper by Hewitt (1971), there have been various suggestions as to how credibility theory also can be applied to the severity component of the pure premium. The latest such suggestion, Frees (2003), revived the interest in the problem.

In this paper, we review four different formulas incorporating frequency and severity into credibility calculations. We then compare by simulation which one is most accurate at predicting a contract’s next-year outcome. It is found that the classical formula of Bühlmann (1967) is as good as the other ones in many cases. Alternatives, however, may offer easier analysis of the separate effects of frequency and severity on the premium.

We also show that all the formulas reviewed in this paper stem from the same minimization problem, and we present a general, integrated, solution. At the same time, we complete Gerber (1972) by providing a proof to the main result of this paper and by stating required additional assumptions.     

The Financial Modeling of Property-Casualty Insurance Companies

This paper describes a financial model currently being used by a major U.S. multiline property-casualty insurer. The model, which was first developed for solvency monitoring purposes, is now being employed for a variety of internal management purposes as well, including (1) the allocation of equity to corporate units, thereby allowing measurements of profitability by business segment and policy year, as well as analysis of the progression of “free surplus,” (2) the analysis of major risks–such as inflation risks, interest rate risks, and reserving risks–that have heretofore been difficult to quantify, and (3) consideration of varying scenarios on the company’s financial performance, both of macroeconomic conditions as well as of the insurance environment.

Many aspects of financial modeling do not differ significantly between life and property-casualty insurers, and these are not discussed in the paper. Rather, the paper focuses on the following topics:

1. Surplus allocation and profitability: how economic surplus and the returns on this surplus are determined by line of business, separately for new business and for the runoff of existing business, and how the progression of free surplus is viewed.

2. Multifaceted risks: how to model risks that affect multiple components of the insurer’s operations, such as economic risks and financial risks. The multiple effects of macroeconomic conditions and changing inflation rates on workers’ compensation claim frequencies and severities complicate the basic interest rate path modeling of life insurance products and annuity contracts.

3. Scenario building: how to construct scenarios of macroeconomic conditions or industry cyclical movements to test the resilience of the company to changing external conditions.     

Living to 100 and Beyond: Implications of Longer Life Spans

Abstract

Life spans have increased remarkably in the last century. There is substantial disagreement and uncertainty among researchers today about the future course of mortality in the developed countries. Will we continue to live ever-longer lives, or is the human life span headed toward a biological upper limit? The answer to this question has important implications for the elderly, their spouses and children, businesses, and our society as a whole. Continued growth in life expectancy with good health would extend our ability to enjoy all the things we cherish.

Simultaneously, this growth would increase our need to prepare carefully for some unexpected challenges. Among these challenges is a greater need by individuals to save for retirement and to prepare for the possibility of becoming dependent upon family members and others for one’s care. Businesses will enjoy access to experienced workers and expanding markets among the elderly, while they also will try to control the post employment costs that longer life spans will generate. Finally, governments will struggle to manage competing interests as the financial needs of the elderly are weighed against other societal obligations.     

Empirical Bayes credibility

Abstract

A credibility estimator is Bayes in the restricted class of linear estimators and may be viewed as a linear approximation to the (unrestricted) Bayes estimator. When the structural parameters occurring in a credibility formula are replaced by consistent estimators based on data from a collective of similar risks,we obtain an empirical credibility estimator, which is a credibility counterpart of empirical Bayes estimators. Empirical credibility estimators are proposed under various model assumptions, and sufficient conditions for asymptotic optimality are established.     

Loss Reduction Through Worker Satisfaction: The Case of Workers’ Compensation

A prospective study of occupational low back pain (LBP) indicates loss reduction efforts in workers’ compensation that improve workers satisfaction with the treatment of their claim significantly improves levels of recovery (reduces losses) and lowers workers’ compensation insurance costs. The improved outcomes associated with greater worker satisfaction with the firm's treatment of their injury claim, as well as with the treatment from their health care provider, are robust to five alternative measures of back problems, including leg pain and back pain scales, measures of functional limitation, and quality of life scales. Satisfaction with effectiveness of the health care is more important in recovery than satisfaction with the provider's bedside manner. While satisfaction with health care provider significantly improves back pain and functionality at 6 months, satisfaction with the employer's treatment of the claim is equally important at 6 months and grows in quantitative importance at 1 year. Overall, higher satisfaction with claim treatment reduces the likelihood that an injury becomes an indemnity claim and results in almost a 30 percent reduction in claim costs.     

Incorporating the Bühlmann credibility into mortality models to improve forecasting performances

In this paper, we incorporate the Bühlmann credibility into three mortality models (the Lee–Carter model, the Cairns–Blake–Dowd model, and a linear relational model) to improve their forecasting performances, as measured by the MAPE (mean absolute percentage error), using mortality data for the UK. The results show that the MAPE reduction ratios for the three mortality models with the Bühlmann credibility are all significant. More importantly, the MAPEs under the three mortality models with the Bühlmann credibility are very close to each other for each age and forecast year. Thus, by incorporating the Bühlmann credibility we are able to converge the forecasting MAPEs resulting from the three different mortality models to a lower and more consistent level. Moreover, we provide a credibility interpretation with an individual time trend for age x and a group time trend for all ages. Finally, we apply the forecasted mortality rates both with and without the Bühlmann credibility to the net single premiums of life insurance products, and compare the corresponding MAPEs.     

The Actuary’s Role in Managed Care

Abstract

Financial risk is moving to center stage in the $1-trillion U.S. health-care market. The growth of managed care has created new forms of risk and has shifted this risk from insurance companies, which have long dealt with it successfully, to health-care providers and other organizations that have not traditionally accepted the same type and amount of risk. Health-care actuaries have the expertise to help these institutions, and the nation, protect their financial well-being.

Actuaries specialize in the evaluation, quantification, and management of risk. Actuarial models of health-care costs, which help evaluate risk, offer management a window to the managed care world. With these models and other tools, health-care actuaries help organizations succeed in today’s health-care environment by showing how the financial and functional elements of an organization relate to risk.

This report discusses the evolution of the health-care industry and the role that the healthcare actuary has played in that evolution. Eight case studies outline actuarial approaches to assessing risk in the era of managed care by discussing situations affecting five groups: providers, employers, regulators, public policy organizations, and HMOs. Built on experience gained in hundreds of cases, these studies show the range of tasks encountered by managed care actuaries and outline approaches that can help balance risks in today’s health-care system.     

The prophecy of Ulrich Beck: signposts for the social sciences

Abstract

This special issue on the legacy of Ulrich Beck is aimed to stimulate reflection both on the specific uses to which Beck’s conceptual and theoretical apparatus can be put within risk studies and the wider significance of his academic project for the social sciences. In this end-piece, we draw out the key themes which surface in the different contributions relating to five particular areas: the nature of risk; advancements in methods; issues of non-knowledge and uncertainty; the development of cosmopolitan risk communities; and the situated character of individualization. We discuss the implications of the accounts contained in this special issue and reflect on the impact and influence of Beck’s sustained engagement with colleagues around the globe, concluding that the concepts and methods that Beck bequeathed the social sciences are set to live on and thrive.     

农民工职业病保险补偿制度评价与优化研究

工伤保险是当前农民工职业病最主要的保险补偿方式,然而制度设计上偏重普通职业伤害、忽视职业病患者特殊需求的现实,导致大部分农民工职业病患者都无法从工伤保险中得到足额的赔偿金来维持自身及家庭的正常生活。为保障农民工职业病患者的合法权益、缓解因职业病而导致的致贫现象,建议根据当前外出务工农民的现实保险需求,完善这一群体的医疗和养老保险制度,引入商业保险参与农民工职业病患者的保险补偿,构建以工伤保险等社会保险为主体、商业雇主责任保险为补充的多元保险补偿模式。     

Converting Clinical Literature to an Insured Population: A Comparison of Models Using Nhanes

Abstract

The use of clinical literature to set risk classification standards for life insurance underwriting stems from the need to set the most accurate standards using the best available information. A necessary hurdle in this process is converting any excess mortality observed in a clinical study to the appropriate rating for use in underwriting. A widely accepted model in the insurance industry, the Excess Death Rate model, treats the excess as additive to the conditional probability of death for an insurance company’s unimpaired class.

In this paper we test the validity of that model versus other common predictive models of excess mortality in an insured population. Applying these models to National Health and Nutrition Examination Survey (NHANES) data, we derive estimates for excess mortality from three commonly seen underwriting impairments in what could be considered a clinical population. These estimates are added to an estimate of an insurance company’s unimpaired mortality class and then used to predict deaths in an “insurable” subset of that clinical population.

The Excess Death Rate model performed the best of all models, having the smallest cumulative difference of actual to predicted deaths. The use of publicly available data, such as that in NHANES, could help bridge the gap between clinical literature and its application in insurance underwriting if insurable cohorts can be reliably identified from these generally healthy, ambulatory groups.     

Equity and Credibility

We build on previous work concerned with measuring equity and consider the problem of using observed claim data or other information to calculate premiums which maximize equity. When these optimal premiums are used, we show that gathering more information or refining the risk classification always increases equity. We study the case for which the premium is constrained to be an affine function of the claim data and obtain results analogous to classical credibility theory, including the inhomogeneous and homogeneous cases of the Bu¨hlmann-Straub model. We derive formulas for the credibility weights in certain cases.