Optimal insurance in the presence of insurer’s loss limit

In this paper we consider the optimal insurance problem when the insurer has a loss limit constraint. Under the assumptions that the insurance price depends only on the policy’s actuarial value, and the insured seeks to maximize the expected utility of his terminal wealth, we show that coverage above a deductible up to a cap is the optimal contract, and the relaxation of insurer’s loss limit will increase the insured’s expected utility.When the insurance price is given by the expected value principle, we show that a positive loading factor is a sufficient and necessary condition for the deductible to be positive. Moreover, with the expected value principle, we show that the optimal deductible derived in our model is not greater (lower) than that derived in Arrow’s model if the insured’s preference displays increasing (decreasing) absolute risk aversion. Therefore, when the insured has an IARA (DARA) utility function, compared to Arrow model, the insurance policy derived in our model provides more (less) coverage for small losses, and less coverage for large losses.Furthermore, we prove that the optimal insurance derived in our model is an inferior (normal) good for the insured with a DARA (IARA) utility function, consistent with the finding in the previous literature. Being inferior, the insurance can also be a Giffen good. Under the assumption that the insured’s initial wealth is greater than a certain level, we show that the insurance is not a Giffen good if the coefficient of the insured’s relative risk aversion is lower than 1.     

Analyzing surplus appropriation schemes in participating life insurance from the insurer’s and the policyholder’s perspective

This paper examines the impact of three surplus appropriation schemes often inherent in participating life insurance contracts on the insurer’s shortfall risk and the net present value from an insured’s viewpoint. (1) In case of the bonus system, surplus is used to increase the guaranteed death and survival benefit, leading to higher reserves; (2) the interest-bearing accumulation increases only the survival benefit by accumulating the surplus on a separate account; and (3) surplus can also be used to shorten the contract term, which results in an earlier payment of the survival benefit and a reduced sum of premium payments. The pool of participating life insurance contracts with death and survival benefit is modeled actuarially with annual premium payments; mortality rates are generated based on an extension of the Lee-Carter (1992) model, and the asset process follows a geometric Brownian motion. In a simulation analysis, we then compare the influence of different asset portfolios and shocks to mortality on the insurer’s risk situation and the policyholder’s net present value for the three surplus schemes. Our findings demonstrate that, even though the surplus distribution and thus the amount of surplus is calculated the same way, the type of surplus appropriation scheme has a substantial impact on the insurer’s risk exposure and the policyholder’s net present value.     

Adverse selection or advantageous selection? Risk and underwriting in China’s health-insurance market

Using data from China’s individual health-insurance market, we study the problem of information asymmetry. Our preliminary results appear to contradict standard-model predictions, showing that higher-risk buyers are more likely to purchase “additional” insurance than lower-risk buyers, but that they also tend to purchase lower limits of “basic” insurance coverage. We therefore develop a theoretical model to capture the effects of buyers’ wealth levels and loss amounts, and show empirically that these effects, in the context of asymmetric information, lead to the coexistence of adverse selection and advantageous selection in China’s health-insurance market.     

Long-tail longitudinal modeling of insurance company expenses

The insurance industry is known to have high operating expenses in the financial services sector. Insurers, investors and regulators are interested in models to understand the behavior of expenses. However, the current practice ignores skewness, occasional negative values as well as their temporal dependence.Addressing these three features, this paper develops a longitudinal model of insurance company expenses that can be used for prediction, to identify unusual behavior, and to measure firm efficiency. Specifically, we use a three-parameter asymmetric Laplace density for the marginal distribution of insurers’ expenses in each year. Copula functions are employed to accommodate their temporal dependence. As a function of explanatory variables, the location parameter allows us to analyze an insurer’s expenses in light of the firm’s characteristics. Our model can be interpreted as a longitudinal quantile regression.The analysis is performed using property-casualty insurance company data from the National Association of Insurance Commissioners of years 2001-2006. Due to the long-tailed nature of insurers’ expenses, two alternative approaches are proposed to improve the performance of the longitudinal quantile regression model: rescaling and transformation. Predictive densities are derived that allow one to compare the predictions for individual insurers in a hold-out-sample. Both predictive models are shown to be reasonable with the rescaling method outperforming the transformation method. Compared with standard longitudinal models, our model is shown to be superior in identifying insurers’ unusual behavior.     

Economic order quantity model and Taguchi’s cost of poor quality

In this paper we consider defective products and Taguchi’s cost of poor quality in the economic order quantity (EOQ) model. We assume that the product quality performs a normal distribution function, and the Taguchi’s poor quality cost has been involved. From our analysis, it has been found that the annual profit will be decreased if the poor quality of product and Taguchi’s quality cost are involved in the model. It has also been found that economic order quantity in our model is larger than that in a traditional EOQ model.     

Moderate deviations for a risk model based on the customer-arrival process

In this paper, we investigate the moderate deviations for a customer-arrival-based insurance risk model, in which customer’s actual claim sizes are described as independent and identically distributed heavy-tailed random variables multiplying a shot function, and the model can be treated as a Poisson shot noise process.     

Valuation of guaranteed annuity options using a stochastic volatility model for equity prices

Guaranteed annuity options are options providing the right to convert a policyholder’s accumulated funds to a life annuity at a fixed rate when the policy matures. These options were a common feature in UK retirement savings contracts issued in the 1970’s and 1980’s when interest rates were high, but caused problems for insurers as the interest rates began to fall in the 1990’s. Currently, these options are frequently sold in the US and Japan as part of variable annuity products. The last decade the literature on pricing and risk management of these options evolved. Until now, for pricing these options generally a geometric Brownian motion for equity prices is assumed. However, given the long maturities of the insurance contracts a stochastic volatility model for equity prices would be more suitable. In this paper explicit expressions are derived for prices of guaranteed annuity options assuming stochastic volatility for equity prices and either a 1-factor or 2-factor Gaussian interest rate model. The results indicate that the impact of ignoring stochastic volatility can be significant.     

An optimal insurance strategy for an individual under an intertemporal equilibrium

In this paper, we discuss how a risk-averse individual under an intertemporal equilibrium chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. It is shown that the individual’s optimal insurance strategy actually is equivalent to buying a put option, which is written on his/her holding asset with a proper strike price. Since the cost of avoiding risk can be seen as a risk measure, the put option premium can be considered as a reasonable risk measure. Jarrow [Jarrow, R., 2002. Put option premiums and coherent risk measures. Math. Finance 12, 135-142] drew this conclusion with an axiomatic approach, and we verify it by solving the individual’s optimal insurance problem.     

A risk-based model for the valuation of pension insurance

In the US, defined benefit plans are insured by the Pension Benefit Guaranty Corporation (PBGC). Taking account of the fact that the PBGC covers only the residual deficits of the pension fund the sponsoring company is unable to cover and that the plans can be prematurely terminated, we consider a model that accounts for the joint dynamics of the pension fund’s and sponsoring firm’s assets in order to effectively determine the risk-based pension premium for the insurance provided by the PBGC. We obtain a closed-form pricing formula for this risk-based premium. Its magnitude depends highly on the investment portfolio of the pension fund and of the sponsoring company as well as the correlation between these two portfolios.     

Optimal Exercise Strategies for Operational Risk Insurance via Multiple Stopping Times

In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems arising in the setting in which the value function acts on a compound process that is modified by the actions taken at the stopping times. This class of problem is particularly relevant in insurance and risk management settings and we demonstrate this on an important application domain based on insurance strategies in Operational Risk management for financial institutions. In this area of risk management the most prevalent class of loss process models is the Loss Distribution Approach (LDA) framework which involves modelling annual losses via a compound process. Given an LDA model framework, we consider Operational Risk insurance products that mitigate the risk for such loss processes and may reduce capital requirements. In particular, we consider insurance products that grant the policy holder the right to insure k of its annual Operational losses in a horizon of T years. We consider two insurance product structures and two general model settings, the first are families of relevant LDA loss models that we can obtain closed form optimal stopping rules for under each generic insurance mitigation structure and then secondly classes of LDA models for which we can develop closed form approximations of the optimal stopping rules. In particular, for losses following a compound Poisson process with jump size given by an Inverse-Gaussian distribution and two generic types of insurance mitigation, we are able to derive analytic expressions for the loss process modified by the insurance application, as well as closed form solutions for the optimal multiple stopping rules in discrete time (annually). When the combination of insurance mitigation and jump size distribution does not lead to tractable stopping rules we develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.     

Multivariate probit models for conditional claim-types

This paper considers statistical modeling of the types of claim in a portfolio of insurance policies. For some classes of insurance contracts, in a particular period, it is possible to have a record of whether or not there is a claim on the policy, the types of claims made on the policy, and the amount of claims arising from each of the types. A typical example is automobile insurance where in the event of a claim, we are able to observe the amounts that arise from say injury to oneself, damage to one’s own property, damage to a third party’s property, and injury to a third party. Modeling the frequency and the severity components of the claims can be handled using traditional actuarial procedures. However, modeling the claim-type component is less known and in this paper, we recommend analyzing the distribution of these claim-types using multivariate probit models, which can be viewed as latent variable threshold models for the analysis of multivariate binary data. A recent article by Valdez and Frees [Valdez, E.A., Frees, E.W., Longitudinal modeling of Singapore motor insurance. University of New South Wales and the University of Wisconsin-Madison. Working Paper. Dated 28 December 2005, available from: http://wwwdocs.fce.unsw.edu.au/actuarial/research/papers/2006/Valdez-Frees-2005.pdf] considered this decomposition to extend the traditional model by including the conditional claim-type component, and proposed the multinomial logit model to empirically estimate this component. However, it is well known in the literature that this type of model assumes independence across the different outcomes. We investigate the appropriateness of fitting a multivariate probit model to the conditional claim-type component in which the outcomes may in fact be correlated, with possible inclusion of important covariates. Our estimation results show that when the outcomes are correlated, the multinomial logit model produces substantially different predictions relative to the true predictions; and second, through a simulation analysis, we find that even in ideal conditions under which the outcomes are independent, multinomial logit is still a poor approximation to the true underlying outcome probabilities relative to the multivariate probit model. The results of this paper serve to highlight the trade-off between tractability and flexibility when choosing the appropriate model.     

Supply chain coordination with insurance contract

We propose an insurance contract under which the supplier shares the risk of overstock and understock with the retailer, improving the efficiency of the supply chain with a newsvendor-type product. We first show that the insurance contract could coordinate the supply chain, and obtain bargaining solution in the supply chain model. Then we investigate the effects of agents’ risk aversion on the supply chain model and acquire the Pareto-optimal solution through the mean–variance approach. After that, we compare the insurance contract with the revenue sharing contract, focusing particularly on their differences. Finally, extensive numerical studies are conducted, and managerial implications are proposed.     

Industry-wide expense standards using random coefficient regression

A large regression model is constructed to convert data from an insurance company's annual statement into an industry-wide standard expected total expense figure. Bayesian methods are used to estimate parameters. A predictive test is used to demonstrate the superiority of the Bayesian approach to ordinary least squares and to compare the performance of several Bayesian estimators.     

A comprehensive extension of an integrated inventory model with ordering cost reduction and permissible delay in payments

Huang (2010) [1] proposed an integrated inventory model with trade credit financing in which the vendor decides its production lot size while the buyer determines its expenditure to minimize the annual integrated total cost for both the vendor and the buyer. In this paper, we extend his integrated supply chain model to reflect the following four facts: (1) generated sales revenue is deposited in an interest-bearing account for the buyer, (2) the buyer’s interest earned is not always less than or equal to its interest charged, (3) the total number of shipments in one lot size is the vendor’s decision variable to minimize the cost, and (4) it is vital to have a discrimination term which can determine whether the buyer’s replenishment cycle time is less than the permissible delay period or not. We then derive the necessary and sufficient conditions to obtain the optimal solution, and establish some theoretical results to characterize the optimal solution. Finally, numerical examples are presented to illustrate the proposed model and its optimal solution.     

On the optimal product mix in life insurance companies using conditional value at risk

This paper proposes a Conditional Value-at-Risk Minimization (CVaRM) approach to optimize an insurer’s product mix. By incorporating the natural hedging strategy of Cox and Lin (2007) and the two-factor stochastic mortality model of Cairns et al. (2006b), we calculate an optimize product mix for insurance companies to hedge against the systematic mortality risk under parameter uncertainty. To reflect the importance of required profit, we further integrate the premium loading of systematic risk. We compare the hedging results to those using the duration match method of Wang et al. (forthcoming), and show that the proposed CVaRM approach has a narrower quantile of loss distribution after hedging—thereby effectively reducing systematic mortality risk for life insurance companies.     

A sensitivity analysis of typical life insurance contracts with respect to the technical basis

In [Christiansen, M.C., 2007. A sensitivity analysis concept for life insurance with respect to a valuation basis of infinite dimension. Insurance: Math. Econom. doi:10.1016/j.insmatheco.2007.07.005] a sensitivity analysis concept was introduced for the prospective reserve of individual life insurance contracts as functional of the technical basis parameters such as interest rate, mortality probability, disability probability, et cetera. On the basis of that concept, the present paper gives in addition the sensitivities of the premium level.Applying these approaches, an extensive sensitivity analysis is carried out: A study of the basic life insurance contract types ‘pure endowment insurance’, ‘temporary life insurance’, ‘annuity insurance’ and ‘disability insurance’ identifies their diverse characteristics, in particular their weakest points concerning fluctuations of the technical basis. An investigation of combinations of these insurance contract types shows what synergy effects can be expected by creating insurance packages.     

Optimal control and dependence modeling of insurance portfolios with Lévy dynamics

In this paper we are interested in optimizing proportional reinsurance and investment policies in a multidimensional Lévy-driven insurance model. The criterion is that of maximizing exponential utility. Solving the classical Hamilton-Jacobi-Bellman equation yields that the optimal retention level keeps a constant amount of claims regardless of time and the company’s wealth level.A special feature of our construction is to allow for dependencies of the risk reserves in different business lines. Dependence is modeled via an Archimedean Lévy copula. We derive a sufficient and necessary condition for an Archimedean Lévy generator to create a multidimensional positive Lévy copula in arbitrary dimension.Based on these results we identify structure conditions for the generator and the Lévy measure of an Archimedean Lévy copula under which an insurance company reinsures a larger fraction of claims from one business line than from another.     

Integrating long-term care insurance purchase decisions with saving and investment for retirement

Risk related to long-term care (LTC) is high for the elderly. Planning for LTC is now regarded as the ‘third leg’ of retirement planning. In this paper, planning for LTC is integrated with saving and investment decisions for an integrated approach to retirement planning. Optimal LTC insurance purchase decisions are obtained by developing a trade-off between post-retirement LTC costs and LTC insurance premiums paid and coverage received. Integrating insurance purchase with wealth evolution, consisting of saving and investment decisions, allows addressing affordability issues.Two-way branching models are used for the stochastic health events and asset returns. The problem, formulated as a nonlinearly constrained mixed-integer optimization problem, is solved using a heuristic. Sensitivity analyses are performed for initial health and wealth status. Some important aspects of an individual’s behavioral preferences are also addressed in this framework to provide more robust decision support.     

A generalization of sensitivity of the inventory model with partial backorders

In this paper, we use the elementary techniques of differential calculus to investigate the sensitivity analysis of Montgomery et al.’s [Montgomery, D.C., Bazaraa, M.S., Keswani, A.K., 1973. Inventory models with a mixture of backorders and lost sales. Naval Research Logistics Quarterly 20, 225–263] inventory model with a mixture of backorders and lost sales and generalize Chu and Chung’s [Chu, P., Chung, K.J., 2004. The sensitivity of the inventory model with partial backorders. European Journal of Operational Research 152, 289–295] sensitivity analysis. We provide three numerical examples to demonstrate our findings, and remark the interpretation of the global minimum of the average annual cost at which the complete backordering occurs.     

Global loss diversification in the insurance sector

We study the possibility for international diversification of catastrophe risk by the insurance sector. Adopting the argument that large insurance losses may be a ‘globalizing factor’ for the industry, we study the dependence of geographically distant insurance markets via equity returns. In particular, we employ conditional copula theory to model the bivariate dependence of the insurance industry. In contrast to earlier literature on this subject, we disentangle the causes of dependence stemming from the asset side from those from the liability side by conditioning on general market conditions. We find that for both Europe-America and Europe-Asia the dependence is significant. Moreover, we find asymmetric effects: the international dependence is particularly high for losses, even after conditioning for the asset side dependence. Finally, we investigate the time variation in copula parameters and find evidence that dependence in the insurance sector has increased over time, thus reducing the scope for international diversification of large losses in this sector.