Reinsurance Arrangements Research Articles

Optimal reinsurance under multiple attribute decision making

AbstractWe apply methods from multiple attribute decision making (MADM) to the problem of selecting an optimal reinsurance level. In particular, we apply the Technique for Order of Preference by Similarity to Ideal Solution method with Mahalanobis distance. We consider the classical risk model under a reinsurance arrangement – either excess of loss or proportional – and we consider scenarios that have the same finite time ruin probability. For each of these scenarios we calculate three quantities: released capital, expected profit and expected utility of resulting wealth. Using these inputs, we apply MADM to find optimal retention levels. We compare and contrast our findings with those when decisions are based on a single attribute.

Optimal retention for a stop-loss reinsurance with incomplete information

This paper considers the determination of optimal retention in a stop-loss reinsurance. Assume that we only have incomplete information on a risk X for an insurer, we use an upper bound for the value at risk (VaR) of the total loss of an insurer after stop-loss reinsurance arrangement as a risk measure. The adopted method is a distribution-free approximation which allows to construct the extremal random variables with respect to the stochastic dominance order and the stop-loss order. We derive the optimal retention such that the risk measure used in this paper attains the minimum. We establish the sufficient and necessary conditions for the existence of the nontrivial optimal stop-loss reinsurance. For illustration purpose, some numerical examples are included and compared with the results yielded in Theorem 2.1 of Cai and Tan (2007).

Ruin Probability in a Correlated Aggregate Claims Model with Common Poisson Shocks: Application to Reinsurance

This paper considers a correlated aggregate claims model with common Poisson shocks, which allows for dependence in n (n ≥ 2) classes of business across m (m ≥ 1) different types of stochastic events. The dependence structure between different claim numbers is connected with the thinning procedure. Under combination of quota-share and excess of loss reinsurance arrangements, we examine the properties of the proposed risk model. An upper bound for the ruin probability determined by the adjustment coefficient is established through martingale approach. We reduce the problem of optimal reinsurance strategy for maximizing the insurer’s adjustment coefficient and illustrate the results by numerical examples.

The Economics and Regulation of Captive Reinsurance in Life Insurance

The use of captive reinsurance arrangements in life insurance has generated significant debate and led to recent adoption of new regulatory requirements by the National Association of Insurance Commissioners (NAIC). This paper provides an overview of the regulatory reserve requirements that spurred growth in captive reinsurance and how captive arrangements are used, including a summary of data on the use of captives made possible by new NAIC reporting requirements for captive reinsurance in 2013. It elaborates potential efficiencies and risks from the arrangements, and how insurers’ financial incentives, previous regulation, rating agency monitoring, and monitoring by non-insurance creditors mitigate those risks. A.M. Best ratings for life insurers with and without captive arrangements are summarized, documenting that most entities using captive reinsurance have relatively high ratings. The new NAIC regulatory framework for captive reinsurance arrangements and specific requirements for the amounts and types of assets permitted to back the arrangements are discussed.

Reinsurance decisions in life insurance: An empirical test of the risk–return criterion

Recent studies have analyzed optimal reinsurance contracts within the framework of profit maximization and/or risk minimization. This type of framework, however, does not consider reinsurance as a tool for capital management and financing. In the present paper, we consider different proportional reinsurance contracts used in life insurance (viz., quota-share, surplus, and combinations of quota-share and surplus) while taking into account the insurer's capital constraints. The objective is to determine how different reinsurance transactions affect the risk/reward profile of the insurer and whether factors, such as claims severity, premiums, and insurer's risk appetite, influence the choice of a proportional reinsurance coverage. We compare each reinsurance structure based on actual insurance company data, using the risk–return criterion. This criterion determines the type of reinsurance that enables insurer to retain the largest underwriting profits and/or minimize the risk of the retained claims while keeping the insurer's risk appetite constant, assuming a given capital constraint. The results of this study confirm that the choice of reinsurance arrangement depends on many factors, including risk retention levels, premiums, and the variance of the sum insured values (and therefore claims). As such, under heterogeneous insurance portfolio single type of reinsurance arrangement cannot maximize insurer's returns and/or minimize the risk, therefore a combination of different reinsurance coverages should be employed. Hence, future research on optimal risk management choices should consider heterogeneous portfolios while determining the effects of different financial and risk management tools on companies' risk–return profiles.

Some Optimization and Decision Problems in Proportional Reinsurance

Reinsurance is one of the tools that an insurer can use to mitigate the underwriting risk and then to control its solvency. In this paper, we focus on the proportional reinsurance arrangements and we examine several optimization and decision problems of the insurer with respect to the reinsurance strategy. To this end, we use as decision tools not only the probability of ruin but also the random variable deficit at ruin if ruin occurs. The discounted penalty function (Gerber & Shiu, 1998) is employed to calculate as particular cases the probability of ruin and the moments and the distribution function of the deficit at ruin if ruin occurs.We consider the classical risk theory model assuming a Poisson process and an individual claim amount phase-type distributed, modified with a proportional reinsurance with a retention level that is not constant and depends on the level of the surplus. Depending on whether the initial surplus is below or above a threshold level, the discounted penalty function behaves differently. General expressions for this discounted penalty function are obtained, as well as interesting theoretical results and explicit expressions for phase-type 2 distribution. These results are applied in numerical examples of decision problems based on the probability of ruin and on different risk measures of the deficit at ruin if ruin occurs (the expectation, the Value at Risk and the Tail Value at Risk).

Optimal reinsurance in the presence of counterparty default risk

The optimal reinsurance arrangement is identified whenever the reinsurer counterparty default risk is incorporated in a one-period model. Our default risk model allows the possibility for the reinsurer to fail paying in full the promised indemnity, whenever it exceeds the level of regulatory capital. We also investigate the change in the optimal solution if the reinsurance premium recognises or not the default in payment. Closed form solutions are elaborated when the insurer’s objective function is set via some well-known risk measures. It is also discussed the effect of reinsurance over the policyholder welfare. If the insurer is Value-at-Risk regulated, then the reinsurance does not increase the policyholder’s exposure for any possible reinsurance transfer, even if the reinsurer may default in paying the promised indemnity. Numerical examples are also provided in order to illustrate and conclude our findings. It is found that the optimal reinsurance contract does not usually change if the counterparty default risk is taken into account, but one should consider this effect in order to properly measure the policyholders exposure. In addition, the counterparty default risk may change the insurer’s ideal arrangement if the buyer and seller have very different views on the reinsurer’s recovery rate.

Catastrophic Risk, the Disequilibrium of Agricultural Insurance Markets and Reinsurance - Based on the Data from Hubei

Catastrophic risk existing objectively in agricultural production is the main cause for the imbalance of the agricultural insurance markets, and to promote the sustainable development of the agricultural insurance, it is necessary to improve the agricultural insurance risk transfer mechanisms. Under the condition of farms and insurance companies expected utility maximum, we develop an agricultural insurance market equilibrium model based on the M-V preference function, and then demonstrate that catastrophic risk leads to the imbalance in the agricultural insurance market. Insurance companies involved in reinsurance arrangements helps rectify the imbalances in agricultural insurance market. Based on this analysis we verify the development of Hubeis agricultural insurance market, and then simulate the reinsurance arrangement of Hubei agricultural insurance. The theoretic and empirical analysis has an implication for the design of agricultural insurance catastrophe risk transfer tools.

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 Risk Transfer with Multiple Reinsurers

The classical problem of identifying the optimal reinsurance arrangement for a one-period model is examined under some risk measure criteria. We develop a new methodology via a two-stage optimisation procedure which allows us to not only recover some existing results in the literature, but also makes possible the analysis of high dimensional problems in which the insurance company diversifies its risk with multiple reinsurance counter-parties, where the insurer risk position and the premium charged by the reinsurers are risk quantile functions. Closed form solutions are elaborated for some particular settings, although numerical methods for the second part of our procedure represent viable alternatives for the ease of implementing it in more complex scenarios. Furthermore, we discuss some approaches to obtain more robust results.

Optimal proportional reinsurance under dependent risks

This paper considers a correlated risk model with thinning-dependence structure. The authors investigate the optimal proportional reinsurance that maximizes the adjustment coefficient and the optimal proportional reinsurance under mean variance principle for the proposed model. The authors derive the optimal solutions and the numerical illustrations to show the impact of the dependence among the classes of business on the optimal reinsurance arrangements.

Reinsurance Arrangements Minimizing the Risk-Adjusted Value of an Insurer's Liability

In this paper, we investigate the problem of purchasing a reinsurance policy that minimizes the risk-adjusted value of an insurer's liability, where the valuation is carried out using a cost-of-capital approach. In order to exclude the moral hazard, we assume that both the insurer and reinsurer are obligated to pay more for larger loss in a typical reinsurance treaty. Moreover, the reinsurance premium principle is assumed to satisfy three axioms: law invariance, risk loading and preserving convex order. The proposed class of premium principles is quite general in the sense that it contains all the widely used premium principles except Esscher principle listed in Young(2004). When capital at risk is measured by the value at risk(VaR) or conditional value at risk(CVaR), we find it is optimal for the insurer to cede two separate layers over the prescribed premium principles. By imposing an additional weak constraint on the premium principle, we further get that the reinsurance in the form of a layer is optimal. Finally, to illustrate the applicability of our results, we derive explicitly the optimal one-layer reinsurance for expected value principle and Wang's premium principle, and show that two-layer reinsurance may be optimal for Dutch 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 for the second reinsurer.

An Analysis of Reinsurance Optimisation in Life Insurance

This paper considers optimal reinsurance based on an assessment of the reinsurance arrangements for a large life insurer. The objective is to determine the reinsurance structure, based on actual insurer data, using a modified mean-variance criteria that maximises the retained premiums and minimizes the variance of retained claims while keeping the retained risk exposure constant, assuming a given level of risk appetite. The portfolio of life and disability policies use quota-share, surplus and a combination of both quota-share and surplus reinsurance. Alternative reinsurance arrangements are compared using the modified mean-variance criteria to assess the optimal reinsurance strategy. The analysis takes into account recent claims experience as well as actual premiums paid by insured lives and to the reinsurers. Optimal reinsurance cover depends on many factors including retention levels, premiums and the variance of sum insured values (and therefore claims), as a result an insurer should assess the tradeoff between retained premiums and the variance of retained claims based on its own experience and risk appetite.

AIG – a Greek tragedy 'Wickedness' of the wicked and the 'mischief' of the virtuous

Maurice Greenberg is seen as 'wicked', at least by the Attorney General of the State of New York because Greenberg is accused of being responsible for ancextensive array of accounting distortions providing false and misleading financial statements of American International Group, the insurance enterprise that Greenberg helped create and headed for some 40 years. This essay sees the Greenberg saga as much like a Greek tragedy, along the lines of Oedipus. The essay is less to probe the wickedness of Greenberg than to probe the mischief, which the very virtuous are responsible for, in the years of false and misleading financial statements by AIG. It considers principally the important avenue of the alleged accounting distortions, i.e., related to the use and abuse of reinsurance arrangements.

INSOLVENCY PROBABILITY IN REINSURANCE TREATY : A CASE STUDY IN MALAYSIA

In developing countries such as Malaysia, the avai lability of reinsurance arrangements provides sever al advantages to the primary insurers such as keeping their risk exposures at prudent levels by having th eir large risk exposures reinsured by another company, meeting client requests for larger insurance coverage by having their li mited financial sources supported by another company, and acquiring underwriting skills, experience and ability of handling co mplex claims by depending on another company for such services. This paper aims to model insurance claims and assess t he insolvency probability of reinsurance treaties. Claims data wa s obtained from one of the leading insurers in Mala ysia and R programming with actuar package is used to compute the probabil ity of insolvency. ISSN: 1804-0527 (online) 1804-0519 (print) PP. 62-64

Modeling and detecting potentially ruinous streaks in health expenditures

The mean of a distribution of medical expenditures in an insured population can be affected significantly by the occurrence of a few high cost cases. This fact leads some organizations that hold the primary risk for the population (e.g., health plans or self-insured employers) to seek reinsurance arrangements that spread the risk of high cost cases across a broader pool. Recently, the private reinsurance market has experienced some difficulties, attributable to information asymmetries between primary risk holders and reinsurers. The disproportionate effect of a few high cost cases also has generated interest in the development of "risk-adjustment" systems that attempt to reduce the difference in health plans' unreimbursed costs either to endogenous management decisions or random chance. We discuss these issues in light of a well-known statistical result regarding the probability of "streaks" in random data. We illustrate problems that can arise and suggest methods to distinguish random streaks from systematic trends.

Designing Longevity Risk Transfers: The Point of View of the Cedant

This paper focuses on longevity risk, i.e. the risk coming from the random evolution of mortality at adult ages. Tools to face such risk when managing life annuities are discussed, namely capital allocation and risk transfers. With regard to the latter, taking example from traditional reinsurance arrangements, an Excess of Loss risk transfer, involving the duration of each annuity in the portfolio, and Stop-Loss arrangements, linking the counter party's intervention to either assets backing the portfolio reserve or to annual cash flows, are dealt with. The amounts of capital required in the different situations are then compared, within a simplified probabilistic framework.

Mean-Variance Optimal Reinsurance Arrangements

Reinsurance reduces the risk but it also reduces the potential profit. The aim of the paper is to derive optimal, from the cedent's point of view, reinsurance arrangements balancing the risk measured by variance and expected profits under various mean-variance premium principles of the reinsurer. We find that quota share, excess of loss or combinations of excess of loss with quota share are the optimal rules according to a fixed expected gain of the cedent

A Comparison of Risk-Based Capital Standards under the Expected Policyholder Deficit and the Probability of Ruin Approaches

ABSTRACT Two competing approaches to setting risk-based capital (RBC) parameters are the traditional probability of ruin approach and the more recent expected policyholder deficit (EPD) ratio approach. The probability of ruin approach develops capital standards based on a fixed maximum probability of insolvency regardless of risk. The EPD ratio approach allows tradeoffs between the risk of insolvency and the average cost of insolvency so as to force the product of these two numbers, the EPD, to some fixed value. After an explanation of the underlying mathematics, the author develops risk-based capital parameters for private passenger auto liability reserve risk using both methods. The author shows that capital standards developed using the EPD approach increase the risk of insolvency for larger insurers that pose the greatest potential indirect costs to the insurance market through market disruptions in the wake of a major insolvency. These findings have important public policy implications because the EPD approach cu rrently forms the basis for both the Standard & Poor's capital adequacy model and the A.M. Best Company's Best's Capital Adequacy Ratio (BCAR), and it is also used by the American Academy of Actuaries' Risk-Based Capital Task Force as a basis for recommendations to the National Association of Insurance Commissioners (NAIC) on parameters in the regulatory RBC model. INTRODUCTION In response to concerns about the minimum capital standards applicable to insurance companies, the National Association of Insurance Commissioners (NAIC) adopted a regulatory risk-based capital (RBC) formula for property-liability insurers in 1994. The NAIC regulatory RBC formula computes a minimum capital requirement based on each insurer's risk profile. Most insurers, though, actually have much more than the minimum capital required by the NAIC formula. Many considerations affect the actual level of capital, including franchise value, taxes, reinsurance arrangements, and the level of risk aversion within the company. Insurers use internal RBC models to help develop optimal capital levels, and those optimal levels are almost always higher than the minimum standards under the NAIC RBC formula. In addition, private rating agencies such as A.M. Best and Standard & Poor's use their own internally developed RBC formulas to evaluate each insurer's capitalization as part of the quality rating process. Therefore, th e process of selecting appropriate RBC parameters, whether as part of a regulatory formula, a rating agency formula, or an internal management formula, affects virtually every insurer. Two of the competing approaches to assigning appropriate RBC parameters to the various risk elements in these RBC formulas are the traditional probability of ruin approach and the more recent expected policyholder deficit (EPD) approach (Bustic, 1994). This study will compare and contrast these two approaches. In recent years, the EPD methodology has been adopted as the basis for risk parameters in the capital adequacy models both A.M. Best and Standard & Poor's use. The American Academy of Actuaries' Risk-Based Capital Task Force has also used the EPD approach as the basis for recommendations on risk parameters in the NAIC's regulatory RBC formula (NAIC, 1993). Additionally, internal RBC models used by individual insurance companies may employ either of these approaches or yet another alternative approach. A firm grasp of the underlying mathematics of these approaches is necessary to evaluate the suitability of each approach in developing appropriate capital standards. This article is organized into four sections. The first section gives general background and descriptions of both the probability of ruin and the EPD approaches to setting capital standards. The second section compares the computation of risk parameters using different assumptions about the underlying risk of a particular insurer. …