A posteriori ratemaking using bivariate Poisson models

Recently, different bivariate Poisson regression models have been used in the actuarial literature to make an a priori ratemaking taking into account the dependence between two types of claims. A natural extension for these models is to consider a posteriori ratemaking (i.e. experience rating models) that also relaxes the independence assumption. We introduce here two bivariate experience rating models that integrate the a priori ratemaking based on the bivariate Poisson regression models, extending the existing literature for the univariate case to the bivariate case. These bivariate experience rating models are applied to an automobile insurance claims data-set to analyse the consequences for posterior premiums when the independence assumption is relaxed. The main finding is that the a posteriori risk factors obtained with the bivariate experience rating models are significantly lower than those factors derived under the independence assumption.     

An Empirical Analysis of the Effects of Increasing Deductibles on Moral Hazard

Using information on timing and number of claims in a unique data set pertaining to comprehensive automobile insurance with the increasing deductible provision in Taiwan, the authors provide new evidence for moral hazard. Time-varying correlations between the choice of the insurance coverage and claim occurrence are significantly positive and exhibit a smirk pattern across policy months. This empirical finding supports the existence of asymmetric information. A subsample estimation depicts insured drivers' significant responses to increasing deductibles, which implies the existence of moral hazard. According to the probit regression results, the increasing deductible makes policyholders who have ever filed claims less likely to file additional claims later in the policy year. The empirical findings strongly support the notion that the increasing deductible provision helps control moral hazard.     

On Compulsory Per-Claim Deductibles in Automobile Insurance

The purposes of this paper are to analyze the theoretical characteristics of the compulsory deductible system and to verify the rationality of an increasing per-claim deductible in automobile insurance. We derive the optimal variable per-claim deductible by assuming the insurers are financially balanced and the expected utility of the insured is maximized in the absence of moral hazard. Our result suggests that a variable per-claim deductible increasing with the number of claims per year is not optimal. Instead, deductibles should be charged in a decreasing rate forming a second-best solution.     

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.     

The Incentive Effects of Increasing Per-Claim Deductible Contracts in Automobile Insurance

A new rating system of automobile insurance for vehicle damage in Taiwan was launched in 1996, introducing a deductible that increases with the number of claims. In this article, we provide a theoretical rationale for the existence of an increasing per‐claim deductible system and show that the new system is most likely an optimal choice for those insured who tend to have lower claims probability when incentives are present. Using a unique dynamic data set, we are able to conduct a natural experiment to examine the incentive effects (both positive and negative) by looking at the change in claim tendency before and after switching between two deductible plans: an increasing per‐claim deductible and a zero deductible. Our results provide direct evidence of the effects of deductible structures on claim behavior.     

Come Rain or Shine: Evidence on Flood Insurance Purchases in Florida

This article provides a detailed analysis of the operation of the National Flood Insurance Program (NFIP) in Florida, which accounts for 40 percent of the NFIP portfolio. We study the demand for flood insurance with a data set of more than 7.5 million NFIP policies‐in‐force (the largest ever studied) for the years 2000–2005, as well as all NFIP claims filed in Florida. We answer four questions: What are the characteristics of the buyers of flood insurance? What types of contracts (deductibles and coverage levels) are purchased? What are the determinants of claims payments? How are prices determined and how much does NFIP insurance cost?     

Fat-Tailed Regression Modeling with Spliced Distributions

Insurance claims data usually contain a large number of zeros and exhibits fat-tail behavior. Misestimation of one end of the tail impacts the other end of the tail of the claims distribution and can affect both the adequacy of premiums and needed reserves to hold. In addition, insured policyholders in a portfolio are naturally non-homogeneous. It is an ongoing challenge for actuaries to be able to build a predictive model that will simultaneously capture these peculiar characteristics of claims data and policyholder heterogeneity. Such models can help make improved predictions and thereby ease the decision-making process. This article proposes the use of spliced regression models for fitting insurance loss data. A primary advantage of spliced distributions is their flexibility to accommodate modeling different segments of the claims distribution with different parametric models. The threshold that breaks the segments is assumed to be a parameter, and this presents an additional challenge in the estimation. Our simulation study demonstrates the effectiveness of using multistage optimization for likelihood inference and at the same time the repercussions of model misspecification. For purposes of illustration, we consider three-component spliced regression models: the first component contains zeros, the second component models the middle segment of the loss data, and the third component models the tail segment of the loss data. We calibrate these proposed models and evaluate their performance using a Singapore auto insurance claims dataset. The estimation results show that the spliced regression model performs better than the Tweedie regression model in terms of tail fitting and prediction accuracy.     

Insurance ratemaking using a copula-based multivariate Tweedie model

The Tweedie distribution, featured with a mass probability at zero, is a convenient tool for insurance claims modeling and pure premium determination in general insurance. Motivated by the fact that an insurance policy typically provides multiple types of coverage, we propose a copula-based multivariate Tweedie regression for modeling the semi-continuous claims while accommodating the association among different types. The proposed approach also allows for dispersion modeling, resulting in a multivariate version of the double generalized linear model. We demonstrate the application in insurance ratemaking using a portfolio of policyholders of automobile insurance from the state of Massachusetts in the United States.     

The comparative statics of deductible insurance in expected- and non-expected-utility theories

This paper identifies comparative statics results for insurance contracts that distinguish between various models of decision making under risk—specifically, expected utility, rank-dependent expected utility, and weighted utility. Insurance contracts offer full coverage above a deductible. Firms offer premium schedules that give the premium charged as a function of the deductible; households choose both an insurance company and a deductible to maximize utility. A competitive equilibrium requires zero expected profit for firms. We identify changes in the distribution of losses such that the optimal deductible increases for utility representations in a particular class but decreases for some representations outside that class. We give results both for the demand for insurance, as well as for the equilibrium contract.     

Modeling Severity and Measuring Tail Risk of Norwegian Fire Claims

The probabilistic behavior of the claim severity variable plays a fundamental role in calculation of deductibles, layers, loss elimination ratios, effects of inflation, and other quantities arising in insurance. Among several alternatives for modeling severity, the parametric approach continues to maintain the leading position, which is primarily due to its parsimony and flexibility. In this article, several parametric families are employed to model severity of Norwegian fire claims for the years 1981 through 1992. The probability distributions we consider include generalized Pareto, lognormal-Pareto (two versions), Weibull-Pareto (two versions), and folded-t. Except for the generalized Pareto distribution, the other five models are fairly new proposals that recently appeared in the actuarial literature. We use the maximum likelihood procedure to fit the models and assess the quality of their fits using basic graphical tools (quantile-quantile plots), two goodness-of-fit statistics (Kolmogorov-Smirnov and Anderson-Darling), and two information criteria (AIC and BIC). In addition, we estimate the tail risk of “ground up” Norwegian fire claims using the value-at-risk and tail-conditional median measures. We monitor the tail risk levels over time, for the period 1981 to 1992, and analyze predictive performances of the six probability models. In particular, we compute the next-year probability for a few upper tail events using the fitted models and compare them with the actual probabilities.     

Asymmetric Information Problems in Taiwan's Automobile Insurance Market: The Effect of Policy Design on Loss Characteristics

This article investigates asymmetric information problems for the automobile insurance market in Taiwan. Using panel data for the comprehensive automobile insurance coverage from 1995 to 1999, this article analyzes how types of coverage, deductible amounts, and experience ratings have affected the adverse selection and moral hazard problems in Taiwan's automobile insurance market. The empirical results provide partial evidence to demonstrate that the loss frequency and loss ratio were reduced by the addition of self-selection mechanisms in policies with different levels of coverage. In addition, the deductible amounts, experience ratings, and better control of underwriting and claims processing were shown possibly to have decreased potential losses from adverse selection and moral hazard problems.     

The Impact of Regret on the Demand for Insurance

We examine optimal insurance purchase decisions of individuals that exhibit behavior consistent with Regret Theory. Our model incorporates a utility function that assigns a disutility to outcomes that are ex post suboptimal, and predicts that individuals with regret‐theoretical preferences adjust away from the extremes of full insurance and no insurance coverage. This prediction holds for both coinsurance and deductible contracts, and can explain the frequently observed preferences for low deductibles in markets for personal insurance.     

Accident Year Development, Bonus Banks, and Insurance Incentive Compensation

Insurance claims can take years to resolve, which makes insurance performance measurement—and incentive compensation based on such measurement— challenging. The insurance industry utilizes a method of analysis called accident year analysis to manage the temporal challenge inherent in insurance claims. Despite the managerial and economic utility of this method of analysis, it has generally not been applied to incentive compensation programs for insurance company executives and employees. This article will explain accident year analysis, and will show how it can be merged with the bonus bank concept and the Insurance Performance Measure, which is an insurance economic profit metric, to construct an economically consistent insurance incentive compensation program.     

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.     

Optimal health insurance

I formulate expected-utility-maximizing models for health insurance with a single optimal coinsurance (C*) and (separately) a single optimal deductible (D*). While so-doing, I formalize Nyman's challenge to standard welfare-loss models, clarifying when and by how much this alters unadjusted models. Using MEPS-calibrated lognormal distributions and incorporating skewness and kurtosis measures of financial risk, I show how C* shifts as various economic parameters change. For reasonable parameter values, C* < 0.1, much lower than variance-only estimates would conclude. Omitting higher-order risk parameters importantly understates risk and hence understates optimal insurance coverage. I separately develop methods to determine D*, showing that it is approximately a fixed percentage of income that falls as the distribution of financial risks rise. This finding contrasts with existing US public policy regarding high-deductible health plans, which employ fixed deductibles, independent of income.     

Are Territorial Rating Models Outdated in Residential Property Insurance Markets? Evidence From the Florida Property Insurance Market

The fundamental shift in rating methodology from historical loss costs to catastrophe modeling for windstorm coverage calls into question the accuracy of rates developed using rating territories. Using premiums and modeled average annual loss (AAL) estimates from Citizens Property Insurance Corporation (Citizens) in Florida, this article analyzes the use of distance to coast (DtC) as a rating variable in providing coverage for the windstorm peril in homeowners insurance. Catastrophe models used to generate AAL costs do not rely on the same application of the law of large numbers as using historical loss costs and thus allows for more granular pricing of the windstorm peril. The results show that DtC, a rating variable that is property specific, more closely aligns premiums and AALs than territorial rating, and allows more granular pricing of the windstorm peril. More granular risk based pricing provides better incentives for homeowners regarding location and mitigation choices and may help reduce aggregate exposure to windstorm damages in the long run.     

Optimal Insurance Under Random Auditing

We provide a characterization of an optimal insurance contract (coverage schedule and audit policy) when the monitoring procedure is random. When the policyholder exhibits constant absolute risk aversion, the optimal contract involves a positive indemnity payment with a deductible when the magnitude of damages exceeds a threshold. In such a case, marginal damages are fully covered if the claim is verified. Otherwise, there is an additional deductible that disappears when the damages become infinitely large. Under decreasing absolute risk aversion, providing a positive indemnity payment for small claims with a nonmonotonic coverage schedule may be optimal.     

Feasible fraud and auditing probabilities for insurance companies and policyholders

Insurance claims fraud is counted among the major concerns in the insurance industry, the reason being that excess payments due to fraudulent claims account for a large percentage of the total payments each year. We formulate optimization problems from the insurance company as well as the policyholder perspective based on a costly state verification approach. In this setting??while the policyholder observes his losses privately??the insurance company can decide to verify the truthfulness of incoming claims at some cost. We show simulation results illustrating the agreement range which is characterized by all valid fraud and auditing probability combinations both stakeholders are willing to accept. Furthermore, we present the impact of different valid probability combinations on the insurance company??s and the policyholder??s objective quantities and analyze the sensitivity of the agreement range with respect to a relevant input parameter. This contribution summarizes the major findings of a working paper written by Müller et?al. (Working Papers on Risk Management and Insurance (IVW-HSG), No. 92, 2011).     

Selection Bias and Auditing Policies for Insurance Claims

Selection bias results from a discrepancy between the range of estimation of a statistical model and its range of application. This is the case for fraud risk models, which are estimated on audited claims but applied on incoming claims in the design of auditing strategies. Now audited claims are a minority within the parent sample since they are chosen after a severe selection performed by claims adjusters. This article presents a statistical approach that counteracts selection bias without using a random auditing strategy. A two‐equation model on audit and fraud (a bivariate probit model with censoring) is estimated on a sample of claims where the experts are left free to take the audit decision. The expected overestimation of fraud risk derived from a single‐equation model is corrected. Results are close to those obtained with a random auditing strategy, at the expense of some instability with respect to the regression components set. Then we compare auditing policies derived from the different approaches.     

Detection of Automobile Insurance Fraud With Discrete Choice Models and Misclassified Claims

The insurance industry is concerned with the detection of fraudulent behavior. The number of automobile claims involving some kind of suspicious circumstance is high and has become a subject of major interest for companies. This article demonstrates the performance of binary choice models for fraud detection and implements models for misclassification in the response variable. A database from the Spanish insurance market that contains honest and fraudulent claims is used. The estimation of the probability of omission provides an estimate of the percentage of fraudulent claims that are not detected by the logistic regression model.