Reinsurance Pricing Research Articles

An equilibrium model of reinsurance pricing

Reinsurance is the insurance of insurance companies. We develop a dynamic equilibrium model for reinsurance pricing based on the supply and demand of reinsurance. In the model, an insurance company and a reinsurance company make decisions on the optimal reinsurance policy as a demander and a supplier of reinsurance, respectively. The reinsurance price is determined when the demand matches the supply. We incorporate the proportional reinsurance policy as well as investment opportunities into the framework. The high flexibility of this equilibrium pricing model paves a new way for studies of optimal reinsurance and reinsurance pricing.

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.

Reinsurance, investment and the rationality with a diffusion model approximating a jump model

This article investigates the optimal excess-loss reinsurance, investment, and the rationality of using a diffusion model to approximate a jump model. We assume that the instantaneous rate of return of the risky asset is modelled by an Ornstein-Uhlenbeck (O-U) process, and the insurance claims are modeled by a compound Poisson (CP) process and a diffusion approximation (DA) model. By using the stochastic dynamic programming method, the closed-form expressions for the optimal reinsurance and investment strategies and the corresponding value functions are derived. We find that the insurer with the CP claim model always has higher reinsurance demand than the insurer with the DA claim model. Moreover, the error of using the DA model to approximate the CP model is very much dependent on the reinsurance price. Numerical analysis shows that the insurer adjusts the investment strategy as the state of the financial market changes.

Optimal robust reinsurance contracts with investment strategy under variance premium principle

This paper investigates the optimal robust proportional reinsurance contracts with investment in a liquid financial market under variance premium principle in a principal-agent framework. The surplus process of the insurer (agent) is assumed to follow an approximating compound Poisson risk process. Both the insurer and reinsurer (principal) are allowed to invest in a risk-free asset and a risky asset whose price process is governed by the constant elasticity of variance model. The insurer and reinsurer aim to maximize the expected exponential utility of terminal wealth. The reinsurer is ambiguity-averse and has deterministic ambiguity aversion preferences against the diffusion risk caused by the financial market and the approximated diffusion risk which comes from the claim process. The reinsurance price is described by the reinsurer's safety loading which can be decided by the optimal strategies of the insurer and the reinsurer. By utilizing stochastic optimal control principle and HJB (or HJBI) equations, the optimal (robust) proportional reinsurance-investment strategies and the corresponding value functions are obtained explicitly. Numerical examples are provided.

Optimal risk management with reinsurance and its counterparty risk hedging

In this paper, we revisit the study of an optimal risk management strategy for an insurer who wants to maximize the expected utility by purchasing reinsurance and managing reinsurance counterparty risk with a default-free hedging instrument, where the reinsurance premium is calculated by the expected value principle and the price of the hedging instrument equals the expected payoff plus a proportional loading. Different to previous studies, we exclude ex post moral hazard by imposing the no-sabotage condition on reinsurance contracts and derive the optimal strategy analytically. We find that the stop-loss reinsurance is always optimal, but the form of the optimal hedging payoff depends on the cost difference between reinsurance and hedging instrument. We further show that full risk transfer is optimal if and only if both reinsurance pricing and the hedging price are fair. Finally, numerical analyses are conducted to illustrate the effects of some interesting factors on the optimal risk management strategy.

Optimal reinsurance contract and investment strategy for multiple competitive-cooperative insurers and a reinsurer

Abstract Accepted by: Giorgio Consigli In this article, we consider a reinsurance contract design by taking into account the joint interests of multiple insurers and a reinsurer. The reinsurance contract consists of optimal reinsurance strategies and reinsurance prices. The former are chosen by the competitive–cooperative insurers and the latter are determined by a reinsurer. Both insurers and the reinsurer can invest in the common risk-free asset and one different risky asset. The optimal time-consistent reinsurance–investment strategies of the insurers and the optimal reinsurance prices and investment strategy of the reinsurer are derived analytically. Numerical experiments are carried out to illustrate the influences of model parameters on the optimal reinsurance contract and optimal investment strategies. We find that the establishment of the reinsurance contract is affected by the correlation between the claim sizes of insurers as well as that of claim numbers. The results reveal that competition and cooperation will lead to a decrease and increase of the reinsurance price, respectively, showing the importance of opting for cooperation among insurers. Both competition and cooperation are beneficial to insurers, especially for those with high risk aversion.

Tail variance allocation, Shapley value, and the majorization problem

AbstractWith a focus on the risk contribution in a portofolio of dependent risks, Colini-Baldeschi et al. (2018) introduced Shapley values for variance and standard deviation games. In this note we extend their results, introducing tail variance as well as tail standard deviation games. We derive closed-form expressions for the Shapley values for the tail variance game and we analyze the vector majorization problem for the two games. In particular, we construct two examples showing that the risk contribution rankings for the two games may be inverted depending on the conditioning threshold and the tail fatness. Motivated by these examples, we formulate a conjecture for general portfolios. Lastly, we discuss risk management implications, including the characterization of tail covariance premiums and reinsurance pricing for peer-to-peer insurance policies.

Pricing reinsurance and determining optimal retention based on the criterion of maximizing social expected utility

PurposeThe authors consider the mutual benefits of the ceding company and reinsurance company in the design of reinsurance contracts. Two objective functions to maximize social expected utilities are established, which are to maximize the sum of the expected utilities of both the ceding company and reinsurance company, and to maximize their products. The first objective function, additive, emphasizes the total gains of both parties, while the second, multiplicative, accounts for the degree of substitution of gains of one party through the loss of the other party. The optimal price and retention of reinsurance are found by a grid search method, and numerical analysis is conducted. The results indicate that the optimal solutions for two objective functions are quite different. However, optimal solutions are sensitive to the change of the means and volatilities of the claim loss for both objective functions. The results are potentially valuable to insurance regulators and government entities acting as reinsurers of last resort.Design/methodology/approachIn this paper, the authors apply relatively simple, but in the view significant, methods and models to discuss the optimization of excess loss reinsurance strategy. The authors only consider the influence of loss distribution on optimal retention and reinsurance price but neglect the investment factor. The authors also consider the benefits of both ceding company and reinsurance company to determine optimal premium and retention of reinsurance jointly based on maximizing social utility: the sum (or the product) of expected utilities of reinsurance company and ceding company. The authors solve for optimal solutions numerically, applying simulation.FindingsThis paper establishes two optimization models of excess-of-loss reinsurance contract against catastrophic losses to determine optimal premium and retention. One model considers the sum of the expected utilities of a ceding company and a reinsurance company's expected utility; another considers the product of them. With an example, the authors find the optimal solutions of premium and retention of excess loss reinsurance. Finally, the authors carry out the sensitivity analysis. The results show that increasing the means and the volatilities of claim loss will increase the optimal retention and premium. For objective function I, increasing the coefficients of risk aversion of or reducing the coefficients of risk aversion of will make the optimal retention reduced but the optimal premium increased, and vice versa. However, for objective function 2, the change of coefficient of risk aversion has no effect on optimal solutions.Research limitations/implicationsUtility of the two partners: The ceding company and the reinsurance company, may have different weights and different significance. The authors have not studied their relative significance. The simulation approach in numerical methods limits us to the probability distributions and stochastic processes the authors use, based on, generally speaking, lognormal models of rates of return. This may need to be generalized to other returns, including possible models of shocks through jump processes.Practical implicationsIn the recent two decades, reinsurance companies have played a great role in hedging mega-catastrophic losses. For example, reinsurance companies (and special loss sharing arrangements) paid as much as two-thirds of the insured losses for the September 11, 2001 tragedy. Furthermore, large catastrophic events have increased the role of governments and regulators as reinsurers of last resort. The authors hope that the authors provide guidance for possible balancing of the needs of two counterparties to reinsurance contracts.Social implicationsNearly all governments around the world are engaged in regulation of insurance and reinsurance, and some are reinsurers themselves. The authors provide guidance for them in these activities.Originality/valueThe authors believe this paper to be a completely new and original contribution in the area, by providing models for balancing the utility to the ceding insurance company and the reinsurance company.

Double Risk Catastrophe Reinsurance Premium Based on Houses Damaged and Deaths

The peaks over threshold (POT) model for catastrophe (CAT) reinsurance pricing has been widely used, but has mainly focused on univariate CAT reinsurance pricing. We provide further justification and support for the model by considering the addition of more than one type of CAT risk in the context of extreme value theory. We further extend the applicability of the CAT reinsurance premium model by considering house damage and deaths as CAT risk. Using the proposed model, we present a simulation framework for pricing double risk CAT reinsurance, based on excess-of-loss reinsurance contract. Furthermore, we fit the POT model to the earthquake loss data in Indonesia. Finally, we provide the price of the double risk CAT reinsurance premium under the standard deviation premium principle. The framework results obtained show that the pricing formulas in this study are appropriate for the double risk claim and may be used as a basis for the pricing of double risk CAT excess-of-loss reinsurance contracts.

Određivanje samopridržaja osiguravača u uslovima rastućih cena reosiguranja

Reinsurance provides protection to the Insurer against large and catastrophe claims and mitigates fluctuations in the aggregate amount of claims, thus contributing to the reduction in insolvency risk. However, a transfer of part of liability under the insurance contract to the reinsurer comes at a price, which reduces the Insurer's profitability. The current increase in reinsurance prices, which was caused by a constellation of natural, macroeconomic and political shocks, is one of the largest ever. Under such circumstances, stating the adequate self-retention is of crucial significance for the Insurer's performance. The paper presents a possible approach to defining the level of self-retention based on the revenue and risks. The approach is based on stochastic simulations of the probability distribution of the aggregate amount of claims at the level of the insurance portfolio and the effects of different reinsurance programs on such distribution, in order to select the program that achieves the maximum ratio between the revenue and risk. Such a reinsurance program corresponds to a level of self-retention that is optimal for a given insurer.

Reinsurance And The Determinants Of The Ceding Decision Of Life Insurance Companies In Nigeria: An Empirical Analysis

Purpose: This paper is an empirical analysis of Nigeria's determinants of the ceding decision of life insurance companies. It is birthed from the notion that reinsurance, though highly beneficial to the insurance industry, is rarely undertaken by life insurers. Thus, this study aims at expounding on the determinants of reinsurance decisions of life insurance companies in Nigeria and the relationship of each determinant with ceded reinsurance. Methodology: An ex-post facto research design was adopted, and a sample size of seven (7) core life insurance companies in Nigeria was selected using the purposive sampling technique. Data were sourced from the Nigerian Insurers Digest and the websites of the select insurance companies covering the years 2011 to 2019. Descriptive analysis and unit root tests were conducted on the data to justify its suitability. Data were further analyzed using the panel data regression, and a decision was arrived at using the Hausman and redundant fixed effect tests. Results/Findings: The analysis showed a significant relationship between leverage and ceded reinsurance, firm size and ceded reinsurance, as well as return on assets and ceded reinsurance, while underwriting risk and reinsurance price had insignificant relationships with ceded reinsurance. All the explanatory variables also had positive relationships with the ratio of ceded reinsurance. Thus, it was concluded that leverage, firm size and return on assets are major determinants of the ceding decision of these companies, while underwriting risk and reinsurance price are not. Originality and Practical Implications:This study is an original work of the authors. It practically shows the relationship between the factors considered by life insurers in deciding to cede their risks and ceded reinsurance. With previous studies on reinsurance and its benefit to the insurance industry, this study brings factors that impede the life insurer from deriving these benefits. Therefore, it was recommended that to encourage life insurers to cede more risks, policymakers and regulators should align specific regulations on leverage, firm size and return on assets of life insurers in Nigeria.

Optimal reinsurance pricing, risk sharing and investment strategies in a joint reinsurer-insurer framework

Abstract We investigate the reinsurance contract and investment strategy problem between an insurer and a reinsurer under the continuous-time framework. For the reinsurance contract design, the joint interests of the insurer and the reinsurer are considered. That is, the insurer determines the claim risk sharing strategy, and the reinsurer determines the reinsurance price. The insurer and the reinsurer can invest in the common risk-free asset and different risky assets to increase their respective wealths. The competition between the insurer and the reinsurer is quantified through the relative performance. Both the insurer and the reinsurer aim at maximizing the expected value of the terminal pay-off while minimizing its variance. By using the stochastic optimal control technique, we derive analytically the optimal time-consistent investment strategy and obtain the optimal reinsurance contract. Finally, the influences of model parameters on the optimal reinsurance contract and time-consistent investment strategy are examined through numerical experiments.

Feature extraction of auto insurance size of loss data using functional principal component analysis

To better understand the major pattern and natural variation of statistical reporting data, visualization and interpretability have been an ongoing challenging issue, particularly in complex statistical data analysis. In auto insurance rate regulation, the study of the size-of-loss distribution is critical because it serves as a benchmark for rate filings conducted by insurance companies. In this work, we propose dimension reduction and feature exaction approaches via functional principal component analysis to analyse auto insurance statistical reporting data. Dimension reduction aims to estimate the overall functional pattern of the size-of-loss, while feature extraction is to study the similarity and variability of extracted features in the feature subspace. We also investigate the functionality of relative claim frequency, size-of-loss, and the pattern and variability of extracted features, using loss data at the industry level. The proposed method helps improve data explainability and understanding of the overall auto insurance size-of-loss distributions pattern. Also, we found that the proposed method can capture the tail behaviour of the size-of-loss distribution. It is also applicable to similar data analysis in many applications, including economics and finance. The impact of this research is to help develop strategies to control and manage the proposed rate changes by insurance companies for rate regulation purposes in the aspect related to large loss loading for reinsurance pricing.

Optimal reinsurance pricing with ambiguity aversion and relative performance concerns in the principal-agent model

This paper first studies the optimal reinsurance problems for two competitive insurers and then studies the optimal reinsurance premium pricing problem for their common reinsurer by using the dynamic programming technique. The two insurers are subject to common insurance systematic risk. Each purchases proportional or excess-of-loss reinsurance for risk control. They aim to maximize the expected utilities of their relative terminal wealth. With the insurers' optimal reinsurance strategies, the reinsurer decides the reinsurance premiums for each insurer, also aiming to maximize the expected utility of her terminal wealth. Thus, the optimal reinsurance pricing problem is formulated as a Stackelberg game between two competitive insurers and a reinsurer, where the reinsurer is the leader, and the insurers are followers. Besides, all three players take model ambiguity into account. We characterize the optimal strategies for the insurers and the reinsurer and provide some numerical examples to show the impact of competition and model ambiguity on the pricing of reinsurance contracts.

A Multi-Agent Incomplete Equilibrium Model and Its Applications to Reinsurance Pricing and Life-Cycle Investment

A Multi-Agent Incomplete Equilibrium Model and Its Applications to Reinsurance Pricing and Life-Cycle Investment

Reinsurance contract design with heterogeneous beliefs and learning

We examine a continuous-time reinsurance contracting problem, where the reinsurer (principal) and the insurer (agent) disagree about the resolution of claim process. We illustrate the reinsurance contract design in an application with Bayesian learning. In the principal–agent framework, the reinsurer dynamically decreases the equilibrium reinsurance price instead of keeping it a constant. However, the reinsurance price increases about the belief differences. Both of these two opposite effects on the reinsurance price induce the insurer’s demand for reinsurance to first increase and then decrease with respect to decision time as the belief differences increase. Moreover, disagreement results in the relatively pessimistic insurer’s risk-taking behavior by raising risk retention as claim volatility increases. Furthermore, we find that when the belief differences are not too large, the reinsurer with belief differences and learning obtains greater satisfaction than the standard one.

Robust reinsurance contract with asymmetric information in a stochastic Stackelberg differential game

In this paper, we determine a robust reinsurance contract from joint interests of the insurer and reinsurer under the framework of Stackelberg differential game. More specifically, the reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal proportional reinsurance to purchase. In order to defend the large shocks of wealth process, a loss-dependent premium principle is applied to the insurer. Meanwhile, we incorporate model uncertainty into the reinsurer's controlled surplus due to the asymmetric information. Under the time-consistent mean-variance criterion, we derive the robust reinsurance contract explicitly by solving the coupled extended Hamilton–Jacobi–Bellman systems. It is interesting to prove that the optimal premium control for the reinsurer is determined by a time-adjusted variance principle. In addition, we find that the reinsurer would like to raise the reinsurance price to guard against the model uncertainty, which consequently decreases the insurer's reinsurance demand. Finally, further analyses are provided to show the necessity of considering the model uncertainty; otherwise, the reinsurance company will suffer a great loss of utility.

Enhancing an insurer's expected value by reinsurance and external financing

In this paper, we analyze a decision-making problem for an insurer with limited liability, who is subject to a solvency constraint and wants to maximize the expected value through reinsurance purchase and external financing. We impose mild conditions on the reinsurance premium principle, which are the axioms of law invariance, risk loading and preserving the convex order. These three axioms are satisfied by all the widely used premium principles, except the Esscher principle, listed in Young (2004). If value at risk (VaR) or conditional value at risk (CVaR) is adopted to calculate the insurer's regulatory capital, we show that the optimal reinsurance can be in the form of two layers. The optimal reinsurance form is reduced to one layer once the premium principle further satisfies a weak condition. In particular, under Wang's, expected value, variance and Dutch premium principles, we derive the insurer's optimal strategies of reinsurance and financing, which are obtained explicitly under the VaR risk measure but have to be solved numerically for the CVaR risk measure. Results indicate that the insurer has three types of optimal strategies: the reinsurance only strategy, external financing only strategy, and mixed strategy. Moreover, the insurer's attitude toward ceding the risk through reinsurance and increasing capital through external financing is often greatly affected by the regulatory regime, the financing cost and the reinsurance price.

Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model

In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and the reinsurance price is modelled by a continuous-time Markov chain with two states. The presence of regime-switching substantially complicates the optimal reinsurance problem, as the surplus-independent strategies turn out to be suboptimal. We develop a recursive approach that allows to represent a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation and the corresponding reinsurance strategy as the unique limits of the sequence of solutions to ordinary differential equations and their first- and second-order derivatives. Via Ito’s formula, we prove the constructed function to be the value function. Two examples illustrate the recursive procedure along with a numerical approach yielding the direct solution to the HJB equation.

Quantifying the Role of Occurrence Losses in Catastrophe Excess of Loss Reinsurance Pricing

The aim of this paper is to merge order statistics with natural catastrophe reinsurance pricing to develop new theoretical and practical insights relevant to market practice and model development. We present a novel framework to quantify the role that occurrence losses (order statistics) play in pricing of catastrophe excess of loss (catXL) contracts. Our framework enables one to analytically quantify the contribution of a given occurrence loss to the mean and covariance structure, before and after the application of a catXL contract. We demonstrate the utility of our framework with an application to idealized catastrophe models for a multi-peril and a hurricane-only case. For the multi-peril case, we show precisely how contributions to so-called lower layers are dominated by high frequency perils, whereas higher layers are dominated by low-frequency high severity perils. Our framework enables market practitioners and model developers to assess and understand the impact of altered model assumptions on the role of occurrence losses in catXL pricing.