Net Premium Principle Research Articles

Reliability contract in hydrogen networks: Another step towards sustainable transportation

While hydrogen energy is increasingly incorporated into energy hubs supplying the fuel cell electric vehicles, the question arose as to whether the inflexible hydrogen networks can provide a reliable supply for sustainable transportation. This study proposes a reliability contract method to capture the economic benefits of differentiated reliability according to the preferences of energy hubs for hydrogen outages. The innovative method divides the hydrogen network into multiple zones to construct the premiums and reimbursements for the reliability contract. According to the key factors of uncertainty in the energy hub operation problem, risk management and net premium principle, this research identifies the effective application of the zone-based reliability contract to hydrogen networks by using a stochastic programming model. Simulations implemented on a typical structure of hydrogen network with 5.2 h outage duration per year shows that the zone division of the proposed reliability contract effectively reduce the outage to 3.1 h.

Premium calculation on health insurance implementing deductible

Implementation of deductibles on insurance contracts is one of many ways to solve the adverse selection and moral hazard problems that often arise in health insurance. Deductibles will make the amount of expected loss for insurance companies lower. However, the deductible will also affect the amount of net premium. The higher the deductible, the lower the net premium that insurance company will get. If the claims amount are very large, then the insurance company would not be able to pay the claims because the amount of net premium that have been earned is small. Thus, the net premium principle is less suitable to use when the insurance companies. Then, they must find more suitable premium calculation method and in this paper, Proportional Hazard (PH) transform principle proposed by Wang being analyzed as an alternative premium when the insurer applies deductibles. The amount of PH transform premium will be higher than net premium and it has more slightly decrease when deductibles are applied compared to the amount of net premium.

Pareto-Optimal Reinsurance Revisited: A Two-Stage Optimisation Procedure Approach

In this paper, based on the Tail-Value-at-Risk (TVaR) measure, we revisit the Pareto-optimal reinsurance policies for the insurer and the reinsurer via a two-stage optimisation procedure. To reduce ex-post moral hazard, we assume that reinsurance contracts satisfy the principle of indemnity and the incentive compatible constraint which have been advocated by Huberman et al. (1983). We show that the Pareto-optimal reinsurance policy exists if the reinsurance premiums can be expressed as an integral form. The proposed class of premium principles encompasses the net premium principle, expected value premium principle, TVaR premium principle, generalized percentile premium principle, and so on. We further use the TVaR premium principle and the expected value premium principle as examples to illustrate the two-stage optimisation procedure by deriving explicitly the Pareto-optimal reinsurance policies. We extend the results by Cai et al. (2017) when the expected value premium principle is replaced by the TVaR premium principle.

An Alternative Pricing System through Bayesian Estimates and Method of Moments in a Bonus-Malus Framework for the Ghanaian Auto Insurance Market

This paper examines the current No-Claim Discount (NCD) system used in Ghana’s auto insurance market as inefficient and outmoded and, therefore, proposes an alternative optimal Bonus-Malus System (BMS) intended to meet the present market conditions and demand. It appears that the existing BMS fails to acknowledge the frequency and severity of policyholders’ claims in its design. We minimized the auto insurance portfolios’ risk through Bayesian estimation and found that the risk is well fitted by gamma, with the claim distribution modeled by the negative binomial law with the expected number of claims (a priori) as 14%. The models presented in this paper recognize the longevity of accident-free driving and fully reward higher discounts to policyholders from the second year when the true characteristics of the hidden risks posed to the pool have been ascertained. The BMS finally constructed using the net premium principle is very optimal and has reasonable punishment and rewards for both good and bad drivers, which could also be useful in other developing economies.

Insurance ratemaking using the Exponential-Lognormal regression model

AbstractThis paper is concerned with presenting the Exponential-Lognormal (ELN) regression model as a competitive alternative to the Pareto, or Exponential-Inverse Gamma, regression model that has been used in a wide range of areas, including insurance ratemaking. This is the first time that the ELN regression model is used in a statistical or actuarial context. The main contribution of the study is that we illustrate how maximum likelihood estimation of the ELN regression model, which does not have a density in closed form, can be accomplished relatively easily via an Expectation-Maximisation type algorithm. A real data application based on motor insurance data is examined in order to emphasise the versatility of the proposed algorithm. Finally, assuming that the number of claims is distributed according to the classic Negative Binomial and Poisson-Inverse Gaussian regression models, both the a priori and a posteriori, or Bonus–Malus, premium rates resulting from the ELN regression model are calculated via the net premium principle and compared to those determined by the Pareto regression model that has been traditionally used for modelling claim sizes.

The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking

This paper presents the Negative Binomial-Inverse Gaussian regression model for approximating the number of claims as an alternative to mixed Poisson regression models that have been widely used in various disciplines including actuarial applications. The Negative Binomial-Inverse Gaussian regression model can be considered as a plausible model for highly dispersed claim count data and this is the first time that it is used in a statistical or actuarial context. The main achievement is that we propose a quite simple Expectation-Maximization type algorithm for maximum likelihood estimation of the model. Finally, a real data application using motor insurance data is examined and both the a priori and a posteriori, or Bonus-Malus, premium rates resulting from the Negative Binomial-Inverse Gaussian model are calculated via the net premium principle and compared to those determined by the Negative Binomial Type I and the Poisson-Inverse Gaussian regression models that have been traditionally used for a priori and a posteriori ratemaking.

Insurance pricing under ambiguity

An actuarial model is typically selected by applying statistical methods to empirical data. The actuary employs the selected model then when pricing or reserving an individual insurance contract, as the selected model provides complete knowledge of the distribution of the potential claims. However, the empirical data are random and the model selection process is subject to errors, such that exact knowledge of the underlying distribution is in practice never available. The actuary finds her- or himself in an ambiguous position, where deviating probability measures are justifiable model selections equally well. This paper employs the Wasserstein distance to quantify the deviation from a selected model. The distance is used to justify premiums and reserves, which are based on erroneous model selections. The method applies to the Net Premium Principle, and it extends to the well-established Conditional Tail Expectation and to further, related premium principles. To demonstrate the relations and to simplify the computations, explicit formulas for the Conditional Tail Expectation for standard life insurance contracts are provided.

OPTIMAL REINSURANCE WITH LIMITED CEDED RISK: A STOCHASTIC DOMINANCE APPROACH

AbstractAn optimal reinsurance problem from the perspective of an insurer is studied in this paper, where an upper limit is imposed on a reinsurer's expected loss over a prescribed level. In order to reduce the moral hazard, we assume that both the insurer and the reinsurer are obligated to pay more as the amount of loss increases in a typical reinsurance treaty. We further assume that the optimization criterion preserves the convex order. Such a criterion is very general as most of the criteria for optimal reinsurance problems in the literature preserve the convex order. When the reinsurance premium is calculated as a function of the actuarial value of coverage, we show via a stochastic dominance approach that any admissible reinsurance policy is dominated by a stop-loss reinsurance or a two-layer reinsurance, depending upon the amount of the reinsurance premium. Moreover, we obtain a similar result to Mossin's Theorem and find that it is optimal for the insurer to cede a loss as much as possible under the net premium principle. To further examine the reinsurance premium for the optimal piecewise linear reinsurance policy, we assume the expected value premium principle and derive the optimal reinsurance explicitly under (1) the criterion of minimizing the variance of the insurer's risk exposure, and (2) the criterion of minimizing the risk-adjusted value of the insurer's liability where the liability valuation is carried out using the cost-of-capital approach based on the conditional value at risk.

Premiums and reserves, adjusted by distortions

The net premium principle is considered to be the most genuine and fair premium principle in actuarial applications. However, actuarial due diligence requires additional caution in pricing of insurance contracts to avoid, for example, at least bankruptcy of the insurer. This paper addresses the distorted premium principle from various angles. Distorted premiums are typically computed by underweighting or ignoring low, but overweighting high losses. Dual characterizations, which are elaborated in a first part of the paper, support this interpretation. The main contribution consists in an opposite point of view—an alternative characterization—which leaves the probability measure unchanged, but modifies (increases) the outcomes instead in a consistent way. It turns out that this new point of view is natural in actuarial practice,as it can be used for premium calculations, but equally well to determine the reserve process in subsequent years in a time consistent way.

The uncertain premium principle based on the distortion function

In this paper, we discuss the premium principle in uncertain environment. First, the net premium principle for uncertain risks is presented within the framework of uncertainty theory. With the help of distortion function, a new uncertain premium principle is derived from the uncertain net premium. Some properties of uncertain distortion premium principle are proved. Furthermore, a sufficient and necessary condition that an uncertain premium principle is an uncertain distortion premium principle has been characterized. Finally, some examples are given to illustrate the calculations of the uncertain distortion premium.

Optimal Premium Plans for Reinsurance with Reinstatements

The present paper is concerned with optimal premium plans for a reinsurance contract with reinstatements. In the reinsurance contract considered here, constant reinstatement premiums are due when the reinsurer’s loss exceeds certain bounds. For this reinsurance contract we examine the existence of a premium plan which minimizes the expected squared difference between the loss and the total premium income of the reinsurer. We show that an optimal premium plan exists, that it is unique, and that it satisfies the net premium principle.

Optimal Premium Plans for Reinsurance with Reinstatements

The present paper is concerned with optimal premium plans for a reinsurance contract with reinstatements. In the reinsurance contract considered here, constant reinstatement premiums are due when the reinsurer’s loss exceeds certain bounds. For this reinsurance contract we examine the existence of a premium plan which minimizes the expected squared difference between the loss and the total premium income of the reinsurer. We show that an optimal premium plan exists, that it is unique, and that it satisfies the net premium principle.

Positive homogeneity and multiplicativity of premium principles on positive risks

In this note it is shown that the Swiss premium principle on positive risks is, under a mild regularity condition, positively homogeneous or multiplicative if and only if it is the net premium principle. A similar result is obtained for a generalized form of the Orlicz principle.

Decision theoretic foundations of credibility theory

Most credibility formulas that have been developed till today refer to the net premium, and the problem of an adequate loading has not yet been solved satisfactorily. In the present paper an approach to premium calculation based on loss functions is introduced, and credibility formulas for the resulting premium principles (net premium principle, exponential principle, Esscher principle, e.g.) are derived. These formulas turn out to be Bayes rules in the sense of statistical decision theory. This gives rise to considering minimax rules as well.

Properties of premium calculation principles

All theoretical premium principles, which use a utility function (such as exponential principle, mean value principle, zero utility principle, Swiss premium calculation principle, Orlicz principle, Esscher principle) are analyzed in the light of practical properties such as homogeneity (as usual for quota shares) and sub-additivity. It is proved that a theoretical premium principle, which fulfills only very weakened forms of both practical properties, reduces necessarily to the net premium principle. Therefore it is impossible that the principles and the properties above are reasonable simultaneously.