Reinsurance Treaties Research Articles

Stable-1/2 bridges and insurance

We develop a class of non-life reserving models using a stable-1/2 random bridge to to simulate the accumulation of paid claims, allowing for an essentially arbitrary choice of a priori distribution for the ultimate loss. Taking an information-based approach to the reserving problem, we derive the process of the conditional distribution of the ultimate loss. The “best-estimate ultimate loss process” is given by the conditional expectation of the ultimate loss. We derive explicit expressions for the best-estimate ultimate loss process, and for expected recoveries arising from aggregate excess-of-loss reinsurance treaties. Use of a deterministic time change allows for the matching of any initial (increasing) development pattern for the paid claims. We show that these methods are well-suited to the modelling of claims where there is a non-trivial probability of catastrophic loss. The generalized inverse-Gaussian (GIG) distribution is shown to be a natural choice for the a priori ultimate loss distribution. For particular GIG parameter choices, the best-estimate ultimate loss process can be written as a rational function of the paid-claims process. We extend the model to include a second paid-claims process, and allow the two pro- cesses to be dependent. The results obtained can be applied to the modelling of multiple lines of business or multiple origin years. The multi-dimensional model has the property that the dimensionality of calculations remains low, regardless of the number of paid-claims processes. An algorithm is provided for the simulation of the paid-claims processes.

Asymptotic results for conditional measures of association of a random sum

Asymptotic results are obtained for several conditional measures of association. The chosen random variables are the first two order statistics and the total sum within a random sum. Many of the results have confirmed the “one-jump” property of the risk model. Non-trivial limits are obtained when the dependence among the first two order statistics is considered. Our results help in understanding the extreme behaviour of well-known reinsurance treaties that involve only few large claims. Interestingly, the Pearson product-moment correlation coefficient between the first two order statistics provides an alternative procedure to estimate the tail index of the underlying distribution.

Joint tail of ECOMOR and LCR reinsurance treaties

Researchers in actuarial sciences have investigated the tail behavior of the LCR and ECOMOR reinsurance treaties separately for managing extreme risks in reinsurance business. In practice, a reinsurance company may possess these two treaties simultaneously. Therefore, investigating the joint tail behavior of these two treaties is practically useful in risk management. This paper derives the asymptotic limit of the joint tail of these two reinsurance treaties under the setup of Jiang and Tang (2008).

Optimal Reinsurance Under VaR and CTE Risk Measures When Ceded Loss Function is Concave

Cai et al. (2008) explored the optimal reinsurance designs among the class of increasing convex reinsurance treaties under VaR and CTE risk measures. However, reinsurance contracts always involve a limit on the ceded loss function in practice, and thus it may not be enough to confine the analysis to the class of convex functions only. The object of this article is to present an optimal reinsurance policy under VaR and CTE optimization criteria when the ceded loss function is in a class of increasing concave functions and the reinsurance premium is determined by the expected value principle. The outcomes reveal that the optimal form and amount of reinsurance depend on the confidence level p for the risk measure and the safety loading θ for the reinsurance premium. It is shown that under the VaR optimization criterion, the quota-share reinsurance with a policy limit is optima, while the full reinsurance with a policy limit is optima under CTE optimization criterion. Some illustrative examples are provided.

Time-Consistent Investment-Reinsurance Strategies Towards Joint Interests of the Insurer and the Reinsurer Under CEV Models

The present paper studies the time-consistent solutions to an investment-reinsurance problem under a mean-variance framework. The paper is distinguished from other literature by taking into account the interests of both the insurer and reinsurer jointly. The claim process of the insurer is governed by a Brownian motion with a drift. A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle. Both the insurer and the reinsurer are assumed to invest in a risky asset, which is distinct for each other and driven by a constant elasticity of variance model. The optimal decision is formulated on a weighted sum of the insurer’s and reinsurer’s surplus processes, where the weight can be viewed as a regularization parameter to balance the importance of each party in the decision. Upon a verification theorem, which is established with a formal proof for a more general problem, explicit solutions are obtained for the proposed investment-reinsurance model. Moreover, numerous mathematical analysis and numerical examples are given to demonstrate those derived results as well as the economic meanings behind.

Optimal VaR-based risk management with reinsurance

It is well-known that reinsurance can be an effective risk management solution for financial institutions such as the insurance companies. The optimal reinsurance solution depends on a number of factors including the criterion of optimization and the premium principle adopted by the reinsurer. In this paper, we analyze the Value-at-Risk based optimal risk management solution using reinsurance under a class of premium principles that is monotonic and piecewise. The monotonic piecewise premium principles include not only those which preserve stop-loss ordering, but also the piecewise premium principles which are monotonic and constructed by concatenating a series of premium principles. By adopting the monotonic piecewise premium principle, our proposed optimal reinsurance model has a number of advantages. In particular, our model has the flexibility of allowing the reinsurer to use different risk loading factors for a given premium principle or use entirely different premium principles depending on the layers of risk. Our proposed model can also analyze the optimal reinsurance strategy in the context of multiple reinsurers that may use different premium principles (as attributed to the difference in risk attitude and/or imperfect information). Furthermore, by artfully imposing certain constraints on the ceded loss functions, the resulting model can be used to capture the reinsurer’s willingness and/or capacity to accept risk or to control counterparty risk from the perspective of the insurer. Under some technical assumptions, we derive explicitly the optimal form of the reinsurance strategies in all the above cases. In particular, we show that a truncated stop-loss reinsurance treaty or a limited stop-loss reinsurance treaty can be optimal depending on the constraint imposed on the retained and/or ceded loss functions. Some numerical examples are provided to further compare and contrast our proposed models to the existing models.

Asymptotic Results for Conditional Measures of Association of a Random Sum

Asymptotic results are obtained for several conditional measures of association. The chosen random variables are the first two order statistics and the total sum within a random sum. Many of the results have confirmed the “one-jump” property of the risk model. Non-trivial limits are obtained when the dependence among the first two order statistics is considered. Our results help in understanding the extreme behaviour of well-known reinsurance treaties that involve only few large claims. Interestingly, the Pearson product-moment correlation coefficient between the first two order statistics provides an alternative procedure to estimate tail index of the underlying distribution.

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.

Rational and Mechanics of Peak Risk Variance Swap for a Property Insurance Portfolio

In this technical report we explore the motivation, structuring and detailed mechanics of a variance swap contract adapted for a property insurance portfolio. We structure, price and test sensitivities of the swap contract using real event historical and modeled natural catastrophe loss data. Our key motivation is to propose an element of financial engineering innovation to insurance portfolio risk management to allow for constructing hedging strategies that may not be possible to achieve with traditional reinsurance treaties and contracts.

Re-Insurance Treaty Portfolio Tail Risk Aggregation and Metrics

We examine techniques of various tail risk modeling and aggregation methodologies and their impact on risk metrics and overall convergence for a notopnal RE/Insurance treaty portfolio.

Optimal impulse control with variance premium principle

In this paper, we study the optimal impulse control problem with reinsurance strategy. We assume that two reinsurers participate in a reinsurance treaty for some loss, and the insurer and two reinsurers adopt the variance premium principle with different parameter respectively. Further, we assume that the dividends are paid to shareholder of an insurance company, and every dividend payout incurs some fixed transaction costs and proportional taxes, i.e., impulse dividends. In diffusion model, we study the maximum of the expected discounted dividend payout until the time of ruin by dynamic programming approach, and closed-form solutions for the value function and optimal dividend-reinsurance strategy are obtained.

Optimal reinsurance analysis from a crop insurer's perspective

PurposeThe purpose of this paper is to analyze the optimal reinsurance contract structure from the crop insurer's perspective.Design/methodology/approachA very powerful and flexible empirical‐based reinsurance model is used to analyze the optimal form of the reinsurance treaty. The reinsurance model is calibrated to unique data sets, including private reinsurance experience for Manitoba, and loss cost ratio (LCR) experience for all of Canada, under the assumption of the standard deviation premium principle and conditional tail expectation risk measure.FindingsThe Vasicek distribution is found to provide the best statistical fit for the Canadian LCR data, and the empirical reinsurance model stipulates that a layer reinsurance contract structure is optimal, which is consistent with market practice.Research limitations/implicationsWhile the empirical reinsurance model is able to reproduce the optimal shape of the reinsurance treaty, the model produces some inconsistencies between the implied and observed attachment points. Future research will continue to explore the reinsurance model that will best recover the observed market practice.Practical implicationsPrivate reinsurance premiums can account for a significant portion of a crop insurer's budget, therefore, this study should be useful for crop insurance companies to achieve efficiencies and improve their risk management.Originality/valueTo the best of the authors' knowledge, this is the first paper to show how a crop insurance firm can optimally select a reinsurance contract structure that minimizes its total risk exposure, considering the total losses retained by the insurer, as well as the reinsurance premium paid to private reinsurers.

The Market for Reinsurance

Using a unique proprietary data set of primary insurers and reinsurers, we analyze the structure of the reinsurance market. The data set, which spans six years, contains quotes for different reinsurance layers, for different treaties, for different clients, and for different lines of business. There are in fact close to 20,000 quotes for close to 1,000 reinsurance treaties, 70% of which have more than one layer. This is the first study that documents the actual structure of the global reinsurance market using actual quotes, and not only the wining quote, for a large number of layers of a large number of reinsurance treaties. We find that reinsurance treaties have generally become more complex over the years. We also find an interesting linear pattern is the distribution of the quotes by line of insurance.

ECOMOR and LCR reinsurance with gamma-like claims

Assuming that the claim sizes of an insurance company have a common distribution with gamma-like tail, we study the asymptotic tail behaviour of the reinsured amounts under the ECOMOR and LCR reinsurance treaties, respectively. Our novel results include a precise asymptotic expansion for the tail probability of the reinsured amounts under the ECOMOR treaty and tight asymptotic bounds for the LCR case. As a by-product we derive a precise asymptotic expansion for the tail of the product of independent regularly varying random variables.

VaR Criteria for optimal limited change-loss and truncated change-loss reinsurance

Reinsurance can provide an effective way for insurer to manage its risk exposure. In this paper, we further analyze the optimal reinsurance models recently proposed by J. Cai and K. S. Tan [Astin Bulletin, 2007, 37(1): 93–112]. With the criteria of minimizing the value-at-risk (VaR) risk measure of insurer’s total loss exposure, we derive the optimal values of sharing proportion a, retention d, and layer l of two reinsurance treaties: the limited change-loss f(x) = a{(x − d)+ − (x − l)+} and the truncated change-loss f(x) = a(x−d)+I(x⩽l). Both of the reinsurance plans have been considered to be more realistic and practical in the real business. Our solutions have several appealing features: (i) there is only one condition to verify for the existence of optimal limited change-loss reinsurance while there always exists an optimal truncated change-loss reinsurance, (ii) the resulting optimal parameters have simple analytic forms which depend only on assumed loss distribution, reinsurer’s safety loading, and insurer’s risk tolerance, (iii) the optimal retention d for limited change-loss reinsurance is the same as that by Cai and Tan while the optimal value is smaller for truncated change-loss, (iv) the optimal sharing proportion and layer are always the same for both reinsurance plans, (v) minimized VaR are strictly lower than the values derived by Cai and Tan, (vi) the optimization results reveal possible drawbacks of VaR-based risk management that a heavy tail risk exposure may be expressed by lower VaR.

Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles

Recently the optimal reinsurance strategy concerning the insurer’s risk attitude and the reinsurance premium principle has been an interesting topic. This paper discusses the optimal reinsurance problem with the insurer’s risk measured by distortion risk measure and the reinsurance premium calculated by a general principle including expected premium principle and Wang’s premium principle as its special cases. Explicit solutions of the optimal reinsurance strategy are obtained under the assumption that both the ceded loss and the retained loss are increasing with the initial loss. We present a new method for discussing the optimal problem. Based on our method, one can explain the optimal reinsurance treaty in the view of a balance between the insurer’s risk measure and the reinsurance premium principle.

Optimal reinsurance arrangements in the presence of two reinsurers

In this paper, we investigate the optimal form of reinsurance from the perspective of an insurer when he decides to cede part of the loss to two reinsurers, where the first reinsurer calculates the premium by expected value principle while the premium principle adopted by the second reinsurer satisfies three axioms: distribution invariance, risk loading, and preserving stop-loss order. In order to exclude the moral hazard, a typical reinsurance treaty assumes that both the insurer and reinsurers are obligated to pay more for the larger loss. Under the criterion of minimizing value at risk (VaR) or conditional value at risk (CVaR) of the insurer's total risk exposure, we show that an optimal reinsurance policy is to cede two adjacent layers, where the upper layer is distributed to the first reinsurer. To further illustrate the applicability of our results, we derive explicitly the optimal layer reinsurance by assuming a generalized Wang's premium principle to the second reinsurer.

Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability

Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability

Optimal reinsurance under variance related premium principles

In this paper, we investigate the optimal form of reinsurance when the insurer seeks to minimize the value at risk(VaR) or the conditional value at risk(CVaR) of his/her total risk exposure. In order to exclude the moral hazard from a reinsurance treaty, both the ceded and retained loss functions are constrained to be increasing. Under the additional assumption that the reinsurance premium is calculated by a variance related principle, we show that the layer reinsurance is always optimal over both the VaR and CVaR criteria. Finally, the variance and standard deviation premium principles are applied to illustrate how to derive the optimal deductible and the upper limit of layer reinsurance.

Index clause: analytical properties and the capitalization strategy

In non-life insurance, some bodily liability losses (e.g. loss of income) are paid out as annuity until specified age or death of victim. In case of excess of loss reinsurance, reinsurers often do not accept the inflation risk embedded in such losses and therefore include in the reinsurance treaty so called index clause which increases the priority by the impact of the inflation on the loss. A formula to calculate the expected reinsurance recoveries and its limiting behavior in case of presence of the index clause is derived and illustrated. A capitalization strategy which minimizes the insurer’s loss is derived.