Biometric worst-case scenarios for multi-state life insurance policies

It is common actuarial practice to calculate premiums and reserves under a set of biometric assumptions that represent a worst-case scenario for the insurer. The new solvency regime of the European Union (Solvency II) also uses worst-case scenarios for the calculation of solvency capital requirements for life insurance business. Surprisingly, the actuarial literature so far offers no exact method for the construction of biometric scenarios that let premiums and reserves be always on the safe side with respect to a given confidence band for the biometric second-order basis. The present paper partly fills this gap by introducing a general method that allows one to construct such scenarios for homogenous portfolios of life insurance policies. The results are especially informative for life insurance policies with mixed character (e.g. survival and occurrence character). Two examples are given that illustrate the new method, demonstrate its usefulness for the calculation of premiums and reserves, and show how the new approach could improve the calculation of biometric solvency reserves for Solvency II.     

Log-supermodularity of weight functions, ordering weighted losses, and the loading monotonicity of weighted premiums

The paper is motivated by a problem concerning the monotonicity of insurance premiums with respect to their loading parameter: the larger the parameter, the larger the insurance premium is expected to be. This property, usually called the loading monotonicity, is satisfied by premiums that appear in the literature. The increased interest in constructing new insurance premiums has raised a question as to what weight functions would produce loading-monotonic premiums. In this paper, we demonstrate a decisive role of log-supermodularity or, equivalently, of total positivity of order 2 (TP₂) in answering this question. As a consequence, we establish-at a stroke-the loading monotonicity of a number of well-known insurance premiums, and offer a host of further weight functions, and consequently of premiums, thus illustrating the power of the herein suggested methodology for constructing loading-monotonic insurance premiums.     

Ruin probabilities for an insurance company based on some stochastic risk models

The paper is concerned with a stochastic risk model with independent random claims and premiums. Recurrence formulas for the ruin probabilities of an insurance company at times of claim payments are obtained. Both the random premiums and the insurance damages are assumed to be independent and identically distributed. The number of claims and premiums are independent Poisson processes, both of which are independent of the size of premiums and claims. We consider the case when the random premiums and insurance damages are exponentially distributed and the more general case when they are gamma distributed with integer parameters. Based on the probabilities obtained in this paper, it is possible to calculate the ruin probabilities on infinite and finite time intervals. Examples are given.     

综合人寿保险精算模型

保险是金融的重要组成部分，国际保险业发展迅速，我国保险业务较晚，资料匮乏，迫切需要引进国外先进的保险经验和保险技术，并结合我国的实际情况加以运用。本文建立了一个综合的人寿保险精算模型，其中包括生存年金，终身寿险和还本部分。通过适当的调整参数进行组合，可以获得不同的保险产品。     

The Pricing of Deposit Insurance Considering Regulatory Capital and Bankruptcy Cost

We develop a deposit insurance pricing model that explicitly considers regulatory capital and bankruptcy costs. Based on the pricing deposit insurance model, we calculate the deposit insurance premiums of China's 16 listed banks with time span of 2011 to 2017 in this paper. The results demonstrate that the deposit insurance premiums of state-owned banks is lower than joint-stock commercial banks and city commercial banks, however, the deposit insurance premiums of joint-stock commercial banks is higher than city commercial banks. Numerical simulation shows that, ceteris paribus, the value of deposit insurance decreases with regulatory capital ratios and the insured deposits ratios, but it increases with interest rate and bankruptcy costs.     

Dual representation of superhedging costs in illiquid markets

This paper studies superhedging of contingent claims in illiquid markets where trading costs may depend nonlinearly on the traded amounts and portfolios may be subject to constraints. We give dual expressions for superhedging costs of financial contracts where claims and premiums are paid possibly at multiple points in time. Besides classical pricing problems, this setup covers various swap and insurance contracts where premiums are paid in sequences. Validity of the dual expressions is proved under new relaxed conditions related to the classical no-arbitrage condition. A new version of the fundamental theorem of asset pricing is given for unconstrained models with nonlinear trading costs.     

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??We develop a deposit insurance pricing model that explicitly considers regulatory capital and bankruptcy costs. Based on the pricing deposit insurance model, we calculate the deposit insurance premiums of China's 16 listed banks with time span of 2011 to 2017 in this paper. The results demonstrate that the deposit insurance premiums of state-owned banks is lower than joint-stock commercial banks and city commercial banks, however, the deposit insurance premiums of joint-stock commercial banks is higher than city commercial banks. Numerical simulation shows that, ceteris paribus, the value of deposit insurance decreases with regulatory capital ratios and the insured deposits ratios, but it increases with interest rate and bankruptcy costs.     

Application of the phase‐type mortality law to life contingencies and risk management

The class of phase‐type distributions has recently gained much popularity in insurance applications due to its mathematical tractability and denseness in the class of distributions defined on positive real line. In this paper, we show how to use the phase‐type mortality law as an efficient risk management tool for various life insurance applications. In particular, pure premiums, benefit reserves, and risk‐loaded premiums using CTE for standard life insurance products are shown to be available in analytic forms, leading to efficient computation and straightforward implementation. A way to explicitly determine provisions for adverse deviation for interest rate and mortality is also proposed. Furthermore, we show how the interest rate risk embedded in life insurance portfolios can be analyzed via interest rate sensitivity index and diversification index which are constructed based on the decomposition of portfolio variance. We also consider the applicability of phase‐type mortality law under a few non‐flat term structures of interest rate. Lastly, we explore how other properties of phase‐type distributions may be applied to joint‐life products as well as subgroup risk ordering and pricing within a given pool of insureds. Copyright © 2017 John Wiley & Sons, Ltd.     

Simulation of a health insurance market with adverse selection

A health insurance market is examined in which individuals with a history of high utilization of health care services tend to select fee-for-service (FFS) insurance when offered a choice between FFS and health maintenance organizations (HMOs). In addition, HMOs are assumed to practice community rating of employee groups. Based on these observations and health plan enrollment and premium data from Minneapolis-St. Paul, a deterministic simulation model is constructed to predict equilibrium market shares and premiums for HMO and FFS insurers within a firm. Despite the fact that favorable selection enhances their ability to compete with FFS insurers, the model predicts that HMOs maximize profits at less than 100% market share, and at a lower share than they could conceivably capture. That is, HMOs would not find it to their advantage to drive FFS insurers from the market even if they could. In all cases, however, the profit-maximizing HMO premium is greater than the experience-rated premium and, thus, the average health insurance premium per employee in firms offering both HMOs and FFS insurance is predicted to be greater than in firms offering one experience-rated plan. The model may be used to simulate the effects of varying the employer's method of contributing to health insurance premiums. Several contribution methods are compared. Employers who offer FFS and HMO insurance and pay the full cost of the lowest-cost plan are predicted to have lower average total premiums (employer plus employee contributions) than employers who pay any level percent of the cost of each plan.     

Optimal Claiming in an Automobile Insurance System with Bonus-Malus Structure

A motorist involved in an accident will have to decide whether to claim from his insurance company or not when he is at fault. An optimal decision rule can only be determined in the light of future developments and future decisions, since the consequences of claiming or not claiming are felt in the subsequent year's premiums. In this paper, optimal no-claim limits are determined for a common Dutch type of insurance policy with bonus-malus structures, using generalized Markovian programming. The computational results are given for various values of the expected number of accidents per year.     

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??In classical credibility theory, the claim amounts of different
insurance policies in a portfolio are assumed to be independent and the premiums are derived
under squared-error loss function. Wen et al. (2012) studied the credibility models with a
dependence structure among the claim amounts of one insurance policy that is called time
changeable effects and obtained the credibility formula. In this paper, we generalized this
dependence structure called time changeable effects to the claim amounts of different
insurance policies in a portfolio. Credibility premiums are obtained for Buhlmann and
Buhlmann-Straub credibility models with dependence structure under balanced loss function.     

分期付费的投资连结保险定价研究

本文在投资基金价格服从几何布朗运动假设下,根据布朗运动的定义和Schwarz不等式得到了具有最小保证金和分期付费的投资连结保险保费的一个上限,并通过Monte Carlo模拟方法检验了它的合理性.这结论对于此保险的定价具有较大的理论与实践意义.     

A priori ratemaking using bivariate Poisson regression models

In automobile insurance, it is useful to achieve a priori ratemaking by resorting to generalized linear models, and here the Poisson regression model constitutes the most widely accepted basis. However, insurance companies distinguish between claims with or without bodily injuries, or claims with full or partial liability of the insured driver. This paper examines an a priori ratemaking procedure when including two different types of claim. When assuming independence between claim types, the premium can be obtained by summing the premiums for each type of guarantee and is dependent on the rating factors chosen. If the independence assumption is relaxed, then it is unclear as to how the tariff system might be affected. In order to answer this question, bivariate Poisson regression models, suitable for paired count data exhibiting correlation, are introduced. It is shown that the usual independence assumption is unrealistic here. These models are applied to an automobile insurance claims database containing 80,994 contracts belonging to a Spanish insurance company. Finally, the consequences for pure and loaded premiums when the independence assumption is relaxed by using a bivariate Poisson regression model are analysed.     

Valuation of life insurance products under stochastic interest rates

In this paper, we introduce a consistent pricing method for life insurance products whose benefits are contingent on the level of interest rates. Since these products involve mortality as well as financial risks, we present an approach that introduces stochastic models for insurance products through stochastic interest rate models. Similar to Black et al. [Black, Fisher, Derman, Emanuel, Toy, William, 1990. A one-factor model of interest rates and its application to treasury bond options. Financ. Anal. J. 46 (January-February), 33-39], we assume that the premiums and volatilities of standard insurance products are given exogenously. We then project insurance prices to extract underlying martingale probability structures. Numerical examples on variable annuities are provided to illustrate the implementation of this method.     

Statistical risk evaluation applied to (Belgian) car insurance

Before applying actuarial techniques to determine different subportfolios and adjusted insurance premiums for contracts that belong to a more or less heterogeneous portfolio, e.g. using credibility theory, it is worthwhile performing a statistical analysis on the relevant factors influencing the risk in the portfolio. Also the distributional behaviour of the portfolio should be examined. In this paper such a programme is presented for car insurance data using logistic regression, correspondence analysis, and statistical techniques from survival analysis. The specific mechanisms governing large claims in such portfolios will also be described. This work is based on a representative sample from Belgian car insurance data from 1989.     

Points systems for car insurance

Some data on car insurance premiums charged to 120 categories of customer, arranged in a 5×4×3×2 table, has been analysed. A method similar to analysis of variance was used to find how closely the data could be reproduced by additive combinations of parameters from the four dimensions, the unusual feature was that these parameters were restricted to be small whole numbers. The reason for this is the attractiveness of such a points system in marketability and in ease of comprehension by the customer.     

The utility of deductibles from the insurer's point of view

The relationship between the premiums for deductible cover and for full cover are analyzed with respect to the utility for the insurer. Theorem 1 shows that within natural bounds for the premiums neither form of coverage is generally preferred by all insurers. Theorems 2–4 indicate that in many realistic cases a deductible cover requires a higher risk loading factor by the insurer than a full insurance cover. Theorem 2(b) also contains an exception to this rule.     

Credibility premiums for the zero-inflated Poisson model and new hunger for bonus interpretation

The purpose of this paper is to explore and compare the credibility premiums in generalized zero-inflated count models for panel data. Predictive premiums based on quadratic loss and exponential loss are derived. It is shown that the credibility premiums of the zero-inflated model allow for more flexibility in the prediction. Indeed, the future premiums not only depend on the number of past claims, but also on the number of insured periods with at least one claim. The model also offers another way of analysing the hunger for bonus phenomenon. The accident distribution is obtained from the zero-inflated distribution used to model the claims distribution, which can in turn be used to evaluate the impact of various credibility premiums on the reported accident distribution. This way of analysing the claims data gives another point of view on the research conducted on the development of statistical models for predicting accidents. A numerical illustration supports this discussion.     

Livestock mortality catastrophe insurance using fatal shock process

A major problem facing livestock producers is animal mortality risk. Livestock mortality insurance is still at the initial stages, and premium computation approaches are still relatively new and will require more research. We study multi-peril mortality insurance covering the death of livestock in Canada due to a number of natural causes and animal diseases. The coverage includes diseases that must be reported to the CFIA (Canadian Food Inspection Agency). When a Federal reportable disease (FRD) occurs, the CFIA orders the slaughter of animals. A general model to compute premiums, based on actuarial approaches, has been developed for mortality insurance incorporating FRD. This model can be applied to hogs, cattle, and poultry, and is designed to cover all stages of livestock production.Mortality multi-peril insurance premiums are computed for illustration purposes. Hogs are used as an example, specifically in their final 16 weeks (from the 9th week to the 25th week) when they weigh between 23 kg to 113 kg. This is referred to as the third stage (cycle) or the finishing/grower stage. Premium estimates are generated based on the mortality data. In addition, an additional CFIA reportable disease not seen in the data is assumed. However, it is assumed that producers receiving animal mortality compensation from the CFIA would have their mortality insurance indemnity payouts reduced by the amount of the CFIA animal compensation (no double collection of funds by producers). We introduce fatal shock processes to incorporate the CFIA reportable disease. Having these shocks, all hogs raised in the same farm are facing the same fate with FRD since all hogs will be slaughtered when it occurs. Mortality data is obtained from a North American sample from 1999–2007, covering 139 million hog-months over a number of monthly periods. We calculate premium rate based on per 1,000 hogs raised in the same farm with modifications including deductible and coverage level.     

On the Optimal Dividend Problem in the Dual Model with Surplus-Dependent Premiums

This paper concerns the dual risk model, dual to the risk model for insurance applications, where premiums are surplus-dependent. In such a model, premiums are regarded as costs and claims refer to profits. We calculate the mean of the cumulative discounted dividends paid until the time of ruin, if the barrier strategy is applied. We formulate the associated Hamilton–Jacobi–Bellman equation and identify sufficient conditions for a barrier strategy to be optimal. Numerical examples are provided.