Premium Principle Research Articles

On penalized goal-reaching probability minimization under borrowing and short-selling constraints

Abstract We consider a robust optimal investment–reinsurance problem to minimize the goal-reaching probability that the value of the wealth process reaches a low barrier before a high goal for an ambiguity-averse insurer. The insurer invests its surplus in a constrained financial market, where the proportion of borrowed amount to the current wealth level is no more than a given constant, and short-selling is prohibited. We assume that the insurer purchases per-claim reinsurance to transfer its risk exposure to a reinsurer whose premium is computed according to the mean–variance premium principle. Using the stochastic control approach based on the Hamilton–Jacobi–Bellman equation, we derive robust optimal investment–reinsurance strategies and the corresponding value functions. We conclude that the behavior of borrowing typically occurs with a lower wealth level. Finally, numerical examples are given to illustrate our results.

Worst-case distortion riskmetrics and weighted entropy with partial information

In this paper, we discuss the worst-case distortion riskmetrics for general distributions when only partial information (mean and variance) is known. This result is applicable to a general class of distortion risk measures and variability measures. Furthermore, we also consider the worst-case weighted entropy for general distributions when only partial information is available. Specifically, we provide some applications for entropies, weighted entropies and risk measures. The commonly used entropies include Gini functional, cumulative residual entropy, tail-Gini functional, cumulative Tsallis past entropy, extended Gini coefficient, among others. The risk measures contain some premium principles and shortfalls based on entropy. The shortfalls include the Gini shortfall, extended Gini shortfall, shortfall of cumulative residual entropy and shortfall of cumulative residual Tsallis entropy with order α.

A two-layer stochastic game approach to reinsurance contracting and competition

We propose a two-layer stochastic game model to study reinsurance contracting and competition in a market with one insurer and two competing reinsurers. The insurer negotiates with both reinsurers simultaneously for proportional reinsurance contracts that are priced using the variance premium principle. The reinsurance contracting between the insurer and each reinsurer is modeled as a Stackelberg game. The two reinsurers compete for business from the insurer and optimize the so-called relative performance, instead of their own surplus, and their competition is settled by a noncooperative Nash game. We obtain a sufficient and necessary condition, related to the competition degrees of the two reinsurers, for the existence of an equilibrium. We show that the equilibrium, if exists, is unique, and the equilibrium strategy of each player is constant, fully characterized in semiclosed form. Furthermore, we obtain interesting sensitivity results for the equilibrium strategies through both analytical and numerical studies.

Optimal reinsurance with multivariate risks and dependence uncertainty

In this paper, we study the optimal reinsurance design from the perspective of an insurer with multiple lines of business, where the reinsurance is purchased by the insurer for each line of business, respectively. For the risk vector generated by the multiple lines of business, we suppose that the marginal distributions are fixed, but the dependence structure between these risks is unknown. Due to the unknown dependence structure, the optimal strategy is investigated for the worst-case scenario. We consider two types of risk measures: Value-at-Risk (VaR) and Range-Value-at-Risk (RVaR) including Expected Shortfall (ES) as a special case, and general premium principles satisfying certain conditions. To be more practical, the minimization of the total risk is conducted under some budget constraint. For the VaR-based model with only two risks, it turns out that the limited stop-loss reinsurance treaty is optimal for each line of business. For the model with more than two risks, we obtain two types of optimal reinsurance strategies if the marginals have convex or concave distributions on their tail parts by constraining the ceded loss functions to be convex or concave. Moreover, as a special case, the optimal quota-share reinsurance with dependence uncertainty has been studied. Finally, after applying our findings to two risks, some studies have been implemented to obtain both the analytical and numerical optimal reinsurance policies.

Hybrid life insurance valuation based on a new standard deviation premium principle in a stochastic interest rate framework

Hybrid life insurance valuation based on a new standard deviation premium principle in a stochastic interest rate framework

Optimal reinsurance design under distortion risk measures and reinsurer’s default risk with partial recovery

AbstractReinsurers may default when they have to pay large claims to insurers but are unable to fulfill their obligations due to various reasons such as catastrophic events, underwriting losses, inadequate capitalization, or financial mismanagement. This paper studies the problem of optimal reinsurance design from the perspectives of both the insurer and reinsurer when the insurer faces the potential default risk of the reinsurer. If the insurer aims to minimize the convex distortion risk measure of his retained loss, we prove the optimality of a stop-loss treaty when the promised ceded loss function is charged by the expected value premium principle and the reinsurer offers partial recovery in the event of default. For any fixed premium loading set by the reinsurer, we then derive the explicit expressions of optimal deductible levels for three special distortion functions, including the TVaR, Gini, and PH transform distortion functions. Under these three explicit distortion risk measures adopted by the insurer, we seek the optimal safety loading for the reinsurer by maximizing her net profit where the reserve capital is determined by the TVaR measure and the cost is governed by the expectation. This procedure ultimately leads to the Bowley solution between the insurer and the reinsurer. We provide several numerical examples to illustrate the theoretical findings. Sensitivity analyses demonstrate how different settings of default probability, recovery rate, and safety loading affect the optimal deductible values. Simulation studies are also implemented to analyze the effects induced by the default probability and recovery rate on the Bowley solution.

Reinsurance contracts under Stackelberg game and market equilibrium

In this paper, we investigate the robust reinsurance contracts under Stackelberg game and market equilibrium. Each reinsurance contract contains two decision makers, one insurer and one reinsurer. The insurer is ambiguity-neutral and adopts a loss-dependent premium principle to collect premium. The reinsurer is ambiguity-averse and is a Bayesian learner. By using the stochastic dynamic programming method and the inverse method, the analytical expressions of the optimal risk allocation proportion and reinsurance price are derived for the two types of reinsurance contracts. We shows that the loss-dependent premium principle has the penalty-reward nature. Both the reinsurance price and demand decrease as the extrapolative intensity increases. Learning has important significance and always puts down the reinsurance price and puts up the reinsurance demand. On the contrary, the reinsurer’s ambiguity aversion raises the reinsurance price and decreases the reinsurance demand. Finally, numerical analysis reveals that the reinsurance price is greater under the Stackelberg game than that under the market equilibrium.

Calculating Premiums for Extreme-Tail Risks with Liability Claims and Nuclear Verdicts

Purpose: When insurance claims, particularly liability claims and nuclear verdicts, are governed by fat-tailed distributions, considerable uncertainty exists about the value of the tail index. Material and Methods: Using the theory of risk aversion, this paper establishes a new premium principle (the power principle – analogous to the exponential principle for thin-tailed claims) and investigates its properties. Findings: Applied to claims arising from generalized Pareto distributions, the resultant premium is shown to be the ratio of the two largest expected claims. This structure provides a natural model for incorporating tail-index uncertainty into premiums. The theory is illustrated through possible ‘premiums’ for liability claims and nuclear verdicts, utilizing the consistent pattern of observed extremes. Implications to Theory, Practice and Policy: By integrating statistical methods for tail index estimation and addressing the inherent uncertainty, the power principle offers a robust framework for determining premiums in high-risk environments. The paper concludes with practical implications for insurers, highlighting the need for advanced risk management strategies and regulatory considerations in dealing with extreme liability claims.

The optimal reinsurance strategy with price-competition between two reinsurers

We study optimal reinsurance for an insurer and two reinsurers in the market through stochastic game theory. The relationship between the insurer and reinsurers is described by a Stackelberg model, where reinsurers, as market leaders, set prices for reinsurance treaties, and the insurer, as a price taker, determines reinsurance demand. Furthermore, we employ a Nash game to model the price competition between the two reinsurers who adopt different premium principles: the variance premium principle and the expected value premium principle. Both the insurer and reinsurers aim to maximize their respective mean-variance cost functions, leading to a time inconsistency control problem. This issue is resolved using a corresponding extended Hamilton-Jacobi-Bellman equation in the game-theoretic framework. We find that the insurer will adopt propositional and excess-of-loss reinsurance strategies with two reinsurers, respectively. Moreover, under an exponential claim size distribution, there exists a unique equilibrium reinsurance premium strategy. Our numerical analysis illuminates the effects of claim size, risk aversion, and the interest rates of the insurer and reinsurers on the equilibrium reinsurance and premium strategies, enhancing the understanding of competition in the reinsurance market.

Aggregate Loss Models to Calculate Risk Measures

The concept of aggregate loss models pertains to a stochastic variable representing the total sum of all losses encountered within a set of insurance policies. In the non-life insurance sector, it is employed to assess the potential losses that an insurance company may face when claims made by policyholders exceed the allocated claim reserves. The purpose of studying aggregate loss models is to ascertain risk measures such as standard deviation of premium principles, value at risk (VaR), and conditional tail expectation (CTE). These steps aid insurance companies in the management and quantification of risks associated with aggregate losses. The standard deviation of premium principles is calculated analytically by substituting expected values and variances, while VaR is estimated using the Monte Carlo method to determine quantile values and confidence intervals. CTE is evaluated by computing the average losses that surpass the VaR threshold. These distributions and parameters require the Pareto distribution, which characterizes claim sizes, and the Poisson or Negative Binomial distribution, which factors in the number of claims. It is crucial to carefully consider the selection of the appropriate distribution, as it plays a significant role in determining the accuracy and reliability of the model. Furthermore, other influencing factors, such as loading factors and confidence intervals, should also be taken into account. These factors have the potential to significantly impact the quantification of risk arising from the model.

Credibility Theory for Variance Premium Principle

The method of credibility theory plays a critical role in various research areas of actuarial science. Among others, the hypothetical mean and process variance are two quantities that convey crucial information to insurance companies when determining premiums for the insureds. The classical credibility model charges premiums by applying a linear combination of the mean of past claims and the population mean, and the corresponding estimator is proved to be the best estimator of the hypothetical mean under the mean squared loss criterion. Enlightened by the prestigious variance premium principle, we propose a credibility approach to estimate the linear combination of hypothetical mean and process variance under the quadratic loss function. Our proposed estimator consists of the linear form of observations and their quadratic terms, as well as some quantities representing population information. Meanwhile, a spin-off result is found and utilized to compare with the classical credibility model and the q-credibility model. The nonparametric estimators of structural quantities are also provided for ease of practical usage. Several numerical illustrations are carried out to demonstrate the performance of the estimator. A real dataset from a Swiss insurance company is also analyzed for the practical application of our results, where a data-driven procedure is proposed for determining the safety loading.

Optimal reinsurance under a new design: two layers and multiple reinsurers

We investigate the optimal reinsurance problem considering a more realistic two-layer design involving excess-of-loss reinsurance and multiple reinsurers. Unlike the commonly assumed single-layer design in the existing literature, our approach involves a top layer consisting of excess-of-loss reinsurance and the involvement of multiple reinsurers. Our focus is on the insurer's management of risk exposure and costly capital reinvestment strategies to maximize the company value. We formulate this problem as an optimization problem with constrained multivariate reinsurance exposure variables, and capital injection control. Our findings show that, from a profitability standpoint, when the surplus is low and the excess-of-loss reinsurance is not too expensive, the insurer should obtain a two-layer reinsurance with an excess-of-loss reinsurance at the top layer and a combination of proportional reinsurance from multiple reinsurers at the bottom layer. In contrast, when the surplus is relatively large or the excess-of-loss reinsurance is too expensive, the insurer should purchase a single-layer proportional reinsurance from multiple reinsurers. When the surplus is large enough, the insurer should not acquire any reinsurance. Our results suggest that stand-alone excess-of-loss reinsurance is never optimal when there is also proportional reinsurance with a variance premium principle available for losses below the excess threshold, and we theoretically confirm that a multi-layer reinsurance design is preferable over a single-layer design. Furthermore, our results demonstrate that in order to maximize the company's value, dividends should be paid according to the lump-sum barrier strategy, and capital injections should be considered only if the surplus is null and the transaction costs on capital injections are not too high.

Optimal insurance with mean-deviation measures

This paper studies an optimal insurance contracting problem in which the preferences of the decision maker are given by the sum of the expected loss and a convex, increasing function of a deviation measure. As for the deviation measure, our focus is on convex signed Choquet integrals (such as the Gini coefficient and a convex distortion risk measure minus the expected value) and on the standard deviation. We find that if the expected value premium principle is used, then stop-loss indemnities are optimal, and we provide a precise characterization of the corresponding deductible. Moreover, if the premium principle is based on Value-at-Risk or Expected Shortfall, then a particular layer-type indemnity is optimal, in which there is coverage for small losses up to a limit, and additionally for losses beyond another deductible. The structure of these optimal indemnities remains unchanged if there is a limit on the insurance premium budget. If the unconstrained solution is not feasible, then the deductible is increased to make the budget constraint binding. We provide several examples of these results based on the Gini coefficient and the standard deviation.

Calculating premium principles from the mode of a unimodal weighted distribution

Abstract The theory of utility is a well-known method of constructing insurance premiums (see e.g., Newton et al. (1986) Actuarial Mathematics. Itasca, Illinois: The Society of Actuaries.). Furman and Zitikis ((2008) Insurance: Mathematics and Economics, 42, 459–465.) proposed an alternative method using the mean value of a weighted random variable. According to this approach, for various choices of weighting, popular premiums such as net premium, modified variance premium, Esscher premium, and Kamps premium are obtained. On the other hand, some premiums cannot be obtained with this method, such as the premium of the exponential principle. In this paper, we provide a complementary theory by introducing a family of unimodal weighted distributions for which the mode is a premium principle.

Determination of Insurance Premium Rates with Aggregation Claims at BPJS with Exponential and Gamma Distributions

Badan Penyelenggara Jaminan Sosial (BPJS) is a legal entity that has been provided by the government for the community with the aim of providing protection for all workers in Indonesia from certain socio-economic risks. National development, marked by planned and continuous strides, embodies a commitment to engage all societal, national, and state levels in fostering progress. Encompassing political, economic, socio-cultural, and defense and security realms, the development is meticulously designed to be comprehensive, targeted, integrated, gradual, and sustainable. The overarching objective is to catalyze an augmentation of national capabilities, align the standard of living for the Indonesian people with developed nations, and elevate overall welfare. To establish premium rates, a method involves multiplying the conditional expected value of claim frequency by the size of the claim, considering observed risk characteristics. A claim, in this context, constitutes a formal request to the insurance company, seeking payment in accordance with the terms of the agreement. The primary objective of this study is to establish the insurance premium rates applicable to policyholders (the insured) through the estimation of parameters in the distribution governing aggregate claims. This involves the distribution of both the number of claims and the size of the claims, and the estimation is performed using the moment method. Premium computations are executed based on two key principles: the pure premium principle and the expected value principle. This research produces the conclusion that the Poisson-Gamma aggregate claims distribution has a premium amount of 3.61 times greater than Poisson-Exponential due to the application of the anticipated value principle, namely IDR 4,403,542.94 per month and IDR 1,219,878.45 per month, respectively.

Multidimensional credibility: A new approach based on joint distribution function

Abstract In the traditional multidimensional credibility models developed by Jewell ((1973) Operations Research Center, pp. 73–77.), the estimation of the hypothetical mean vector involves complex matrix manipulations, which can be challenging to implement in practice. Additionally, the estimation of hyperparameters becomes even more difficult in high-dimensional risk variable scenarios. To address these issues, this paper proposes a new multidimensional credibility model based on the conditional joint distribution function for predicting future premiums. First, we develop an estimator of the joint distribution function of a vector of claims using linear combinations of indicator functions based on past observations. By minimizing the integral of the expected quadratic distance function between the proposed estimator and the true joint distribution function, we obtain the optimal linear Bayesian estimator of the joint distribution function. Using the plug-in method, we obtain an explicit formula for the multidimensional credibility estimator of the hypothetical mean vector. In contrast to the traditional multidimensional credibility approach, our newly proposed estimator does not involve a matrix as the credibility factor, but rather a scalar. This scalar is composed of both population information and sample information, and it still maintains the essential property of increasingness with respect to the sample size. Furthermore, the new estimator based on the joint distribution function can be naturally extended and applied to estimate the process covariance matrix and risk premiums under various premium principles. We further illustrate the performance of the new estimator by comparing it with the traditional multidimensional credibility model using bivariate exponential-gamma and multivariate normal distributions. Finally, we present two real examples to demonstrate the findings of our study.

On optimal reinsurance in the presence of premium budget constraint and reinsurer’s risk limit

Abstract In this paper, we propose two new optimal reinsurance models in which both premium budget constraints and the reinsurer’s risk limits are taken into account. To be precise, we assume that the reinsurance premium has an upper bound, and that the admissible ceded loss functions have a pre-specified upper limit. Moreover, we assume that the reinsurance premium principle is calculated by the expected value premium principle. Under the optimality criteria of minimizing the value at risk and conditional value at risk of the insurer’s total risk exposure, we derive the explicit optimal reinsurance treaties, which are layer reinsurance treaties. A new approach is developed to construct the optimal reinsurance treaties. Comparisons with existing studies are also made. Finally, we provide a numerical study based on real data and an example to illustrate the proposed models and results. Our work provides a novel generalization of several known achievements in the literature.

Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

Non-zero-sum reinsurance and investment game under thinning dependence structure: mean–variance premium principle

This paper considers a non-zero-sum stochastic differential reinsurance and investment game between two competitive insurers under the expected exponential utility. It is assumed that the surplus process of each insurer is characterized by a compound Poisson model and their claim businesses are correlated through thinning dependence structure. Each insurer can purchase proportional reinsurance and is allowed to invest in a risk-free asset and a risky asset. The insurers compete with each other and are concerned with the relative performance. By using the stochastic control technique, not only the existence and uniqueness of the optimal strategies are proved, but also the specific analyses for the constraint reinsurance strategies are provided. We find that (i) the thinning dependence structure has significant impact on optimal reinsurance strategies for both insurers and can help them reduce risks; (ii) the behavior of non-zero-sum game will reduce the proportion of purchasing reinsurance for each insurer.

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.