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.     

Optimal Risk Financing in Large Corporations through Insurance Captives

A captive is an insurance or reinsurance company established by a parent group to finance its own risks. Captives mix internal risk pooling between the business units of the parent group and risk transfer towards the reinsurance market. We analyse captives from an optimal insurance contract perspective. The paper characterises the vertical contractual chain that links firstly business units to insurance captives or to “fronters” through insurance contracts, secondly fronters to reinsurance captives through the cession of risks and thirdly insurance or reinsurance captives to reinsurers through cessions or retrocessions. In particular, the risk cession by fronters to a reinsurance captive trades off the benefits derived from recouped premiums and from the risk-sharing advantage of an “umbrella reinsurance policy”, against the risks that result from the captive liabilities.     

Optimal insurance in the presence of reinsurance

This paper studies an optimal insurance and reinsurance design problem among three agents: policyholder, insurer, and reinsurer. We assume that the preferences of the parties are given by distortion risk measures, which are equivalent to dual utilities. By maximizing the dual utility of the insurer and jointly solving the optimal insurance and reinsurance contracts, it is found that a layering insurance is optimal, with every layer being borne by one of the three agents. We also show that reinsurance encourages more insurance, and is welfare improving for the economy. Furthermore, it is optimal for the insurer to charge the maximum acceptable insurance premium to the policyholder. This paper also considers three other variants of the optimal insurance/reinsurance models. The first two variants impose a limit on the reinsurance premium so as to prevent insurer to reinsure all its risk. An optimal solution is still layering insurance, though the insurer will have to retain higher risk. Finally, we study the effect of competition by permitting the policyholder to insure its risk with an insurer, a reinsurer, or both. The competition from the reinsurer dampens the price at which an insurer could charge to the policyholder, although the optimal indemnities remain the same as the baseline model. The reinsurer will however not trade with the policyholder in this optimal solution.     

VAR and CTE Criteria for Optimal Quota-Share and Stop-Loss Reinsurance

Abstract

It is well known that reinsurance can be an effective risk management tool for an insurer to minimize its exposure to risk. In this paper we provide further analysis on two optimal reinsurance models recently proposed by Cai and Tan. These models have several appealing features including (1) practicality in that the models could be of interest to insurers and reinsurers, (2) simplicity in that optimal solutions can be derived in many cases, and (3) integration between banks and insurance companies in that the models exploit explicitly some of the popular risk measures such as value-at-risk and conditional tail expectation. The objective of the paper is to study and analyze the optimal reinsurance designs associated with two of the most common reinsurance contracts: the quota share and the stop loss. Furthermore, as many as 17 reinsurance premium principles are investigated. This paper also highlights the critical role of the reinsurance premium principles in the sense that, depending on the chosen principles, optimal quota-share and stop-loss reinsurance may or may not exist. For some cases we formally establish the sufficient and necessary (or just sufficient) conditions for the existence of the nontrivial optimal reinsurance. Numerical examples are presented to illustrate our results.     

Pricing Double-Trigger Reinsurance Contracts: Financial Versus Actuarial Approach

This article discusses various approaches to pricing double‐trigger reinsurance contracts—a new type of contract that has emerged in the area of ‘‘alternative risk transfer.’’ The potential coverage from this type of contract depends on both underwriting and financial risk. We determine the reinsurer's reservation price if it wants to retain the firm's same safety level after signing the contract, in which case the contract typically must be backed by large amounts of equity capital (if equity capital is the risk management measure to be taken). We contrast the financial insurance pricing models with an actuarial pricing model that has as its objective no lessening of the reinsurance company's expected profits and no worsening of its safety level. We show that actuarial pricing can lead the reinsurer into a trap that results in the failure to close reinsurance contracts that would have a positive net present value because typical actuarial pricing dictates the type of risk management measure that must be taken, namely, the insertion of additional capital. Additionally, this type of pricing structure forces the reinsurance buyer to provide this safety capital as a debtholder. Finally, we discuss conditions leading to a market for double‐trigger reinsurance contracts.     

Reinsurance for Natural and Man-Made Catastrophes in the United States: Current State of the Market and Regulatory Reforms

U.S. insurers are heavily dependent on global reinsurance markets to enable them to provide adequate primary market insurance coverage. This article reviews the response of the world's reinsurance industry to recent mega-catastrophes and provides recommendations for regulatory reforms that would improve the efficiency of reinsurance markets. The article also considers the supply of insurance and reinsurance for terrorism and makes recommendations for joint public–private responses to insuring terrorism losses. The analysis shows that reinsurance markets responded efficiently to recent catastrophe losses and that substantial amounts of new capital enter the reinsurance industry very quickly following major catastrophic events. Considerable progress has been made in improving risk and exposure management, capital allocation, and rate of return targeting. Insurance price regulation for catastrophe-prone lines of business is a major source of inefficiency in insurance and reinsurance markets. Deregulation of insurance prices would improve the efficiency of insurance markets, enabling markets to deal more effectively with mega-catastrophes. The current inadequacy of the private terrorism reinsurance market suggests that the federal government may need to remain involved in this market, at least for the next several years.     

Insurance Market Effects of Risk Management Metrics

We extend the classical analysis on optimal insurance design to the case when the insurer implements regulatory requirements (Value-at-Risk). Presumably, regulators impose some risk management requirement such as VaR to reduce the insurers’ insolvency risk, as well as to improve the insurance market stability. We show that VaR requirements may better protect the insured and improve economic efficiency, but have stringent negative effects on the insurance market. Our analysis reveals that the insured are better protected in the event of greater loss irrespective of the optimal design from either the insured or the insurer perspective. However, in the presence of the VaR requirement on the insurer, the insurer's insolvency risk might be increased and there are moral hazard issues in the insurance market because the optimal contract is discontinuous.     

Optimal insurance and reinsurance policies chosen jointly in the individual risk model

The paper studies the so-called individual risk model where both a policy of per-claim insurance and a policy of reinsurance are chosen jointly by the insurer in order to maximize his/her expected utility. The insurance and reinsurance premiums are defined by the expected value principle. The problem is solved under additional constraints on the reinsurer’s risk and the residual risk of the insured. It is shown that the solution to the problem is the following: The optimal reinsurance is a modification of stop-loss reinsurance policy, so-called stop-loss reinsurance with an upper limit; the optimal insurer’s indemnity is a combination of stop-loss- and deductible policies. The results are illustrated by a numerical example for the case of exponential utility function. The effects of changing model parameters on optimal insurance and reinsurance policies are considered.     

Catastrophe Bonds and Reinsurance: The Competitive Effect of Information-Insensitive Triggers

We identify a new benefit of index or parametric triggers. Asymmetric information between reinsurers on an insurer's risk affects competition in the reinsurance market: reinsurers are subject to adverse selection, since only high-risk insurers may find it optimal to change reinsurers. The result is high reinsurance premiums and cross-subsidization of high-risk insurers by low-risk insurers. A contract with a parametric or index trigger (such as a catastrophe bond) is insensitive to information asymmetry and therefore alters the equilibrium in the reinsurance market. Provided that basis risk is not too high, the introduction of contracts with parametric or index triggers provides low-risk insurers with an alternative to reinsurance contracts, and therefore leads to less cross-subsidization in the reinsurance market.     

Optimal reinsurance under general law-invariant risk measures

In recent years, general risk measures play an important role in risk management in both finance and insurance industry. As a consequence, there is an increasing number of research on optimal reinsurance decision problems using risk measures beyond the classical expected utility framework. In this paper, we first show that the stop-loss reinsurance is an optimal contract under law-invariant convex risk measures via a new simple geometric argument. A similar approach is then used to tackle the same optimal reinsurance problem under Value at Risk and Conditional Tail Expectation; it is interesting to note that, instead of stop-loss reinsurances, insurance layers serve as the optimal solution. These two results highlight that law-invariant convex risk measure is better and more robust, in the sense that the corresponding optimal reinsurance still provides the protection coverage against extreme loss irrespective to the potential increment of its probability of occurrence, to expected larger claim than Value at Risk and Conditional Tail Expectation which are more commonly used. Several illustrative examples will be provided.     

Insurance Derivatives: A New Asset Class for the Capital Markets and a New Hedging Tool for the Insurance Industry

The number and severity of natural catastrophes has increased dramatically over the last decade. As a result, there is now a shortage of capacity in the property catastrophe insurance industry in the U.S. This article discusses how insurance derivatives, particularly the Chicago Board of Trade's catastrophe options contracts, represent a possible solution to this problem. These new financial instruments enable the capital markets to provide the insurance industry with the reinsurance capacity it needs. The capital markets are willing to perform this role because of the new asset class characteristics of securitized insurance risk: positive excess returns and diversification benefits.
The article also demonstrates how insurance companies can use insurance derivatives such as catastrophe options and catastrophe-linked bonds as effective, low-cost risk management tools. In reviewing the performance of the catastrophe contracts to date, the authors report promising signs of growth and liquidity in these markets.     

Convergence of Insurance and Financial Markets: Hybrid and Securitized Risk-Transfer Solutions

One of the most significant economic developments of the past decade has been the convergence of the financial services industry, particularly the capital markets and (re)insurance sectors. Convergence has been driven by the increase in the frequency and severity of catastrophic risk, market inefficiencies created by (re)insurance underwriting cycles, advances in computing and communications technologies, the emergence of enterprise risk management, and other factors. These developments have led to the development of hybrid insurance/financial instruments that blend elements of financial contracts with traditional reinsurance as well as new financial instruments patterned on asset-backed securities, futures, and options that provide direct access to capital markets. This article provides a survey and overview of the hybrid and pure financial markets instruments and provides new information on the pricing and returns on contracts such as industry loss warranties and Cat bonds.     

A unifying approach to risk-measure-based optimal reinsurance problems with practical constraints

The design of optimal reinsurance treaties in the presence of multifarious practical constraints is a substantive but underdeveloped topic in modern risk management. To examine the influence of these constraints on the contract design systematically, this article formulates a generic constrained reinsurance problem where the objective and constraint functions take the form of Lebesgue integrals whose integrands involve the unit-valued derivative of the ceded loss function to be chosen. Such a formulation provides a unifying framework to tackle a wide body of existing and novel distortion-risk-measure-based optimal reinsurance problems with constraints that reflect diverse practical considerations. Prominent examples include insurers’ budgetary, regulatory and reinsurers’ participation constraints. An elementary and intuitive solution scheme based on an extension of the cost–benefit technique in Cheung and Lo [Cheung, K.C. & Lo, A. (2015, in press). Characterizations of optimal reinsurance treaties: a cost–benefit approach Scandinavian Actuarial Journal. doi:10.1080/03461238.2015.1054303.] is proposed and illuminated by analytically identifying the optimal risk-sharing schemes in several concrete optimal reinsurance models of practical interest. Particular emphasis is placed on the economic implications of the above constraints in terms of stimulating or curtailing the demand for reinsurance, and how these constraints serve to reconcile the possibly conflicting objectives of different parties.     

Catastrophe Bonds,Reinsurance, and the Optimal Collateralization of Risk Transfer

Catastrophe bonds feature full collateralization of the underlying risk transfer and thus abandon the reinsurance principle of economizing on collateral through diversification of risk transfer. Our analysis demonstrates that this feature places limits on catastrophe bond penetration, even if the structure possesses frictional cost advantages over reinsurance. However, we also show that catastrophe bonds have important uses when buyers and reinsurers cannot contract over the division of assets in the event of insolvency and, more generally, cannot write contracts with a full menu of state‐contingent payments. In this environment, segregation of collateral—in the form of multiple reinsurance companies, as well as catastrophe bond vehicles—can ameliorate inefficiencies due to reinsurance contracting constraints by improving welfare for those exposed to default risk. Numerical simulation illustrates how catastrophe bonds improve efficiency in market niches with correlated risks, or with uneven exposure of buyers to reinsurer default.     

保险公司负债公允价值的评估

保险公司的资产与负债按照财务报告或风险管理的要求要用公允价值来表示其价值。保险公司负债主要由各种类型的保险合同组成。尽管金融产品的交易非常活跃,市场价格可以相对容易地获得,但像保险合同这样缺乏交易市场,并且风险是非系统化的金融产品的公允价值的计算却十分困难。通过分层次的方法可以有效地评估金融工具的公允价值,其中评估保险公司负债的公允价值最常用的方法是现值法。现值法中包含了直接法与间接法两种本质上等价的方法。本文对这两种方法分别作了分析,并提出了在评估保险公司负债的公允价值时需要考虑的一些现实问题。     

Optimal Risk Transfer: A Numerical Optimization Approach

Capital efficiency and asset/liability management are part of the Enterprise Risk Management Process of any insurance/reinsurance conglomerate and serve as quantitative methods to fulfill the strategic planning within an insurance organization. A considerable amount of work has been done in this ample research field, but invariably one of the last questions is whether or not, numerically, the method is practically implementable, which is our main interest. The numerical issues are dependent on the traits of the optimization problem, and therefore we plan to focus on the optimal reinsurance design, which has been a very dynamic topic in the last decade. The existing literature is focused on finding closed-form solutions that are usually possible when economic, solvency, and other constraints are not included in the model. Including these constraints, the optimal contract can be found only numerically. The efficiency of these methods is extremely good for some well-behaved convex problems, such as Second-Order Conic Problems. Specific numerical solutions are provided to better explain the advantages of appropriate numerical optimization methods chosen to solve various risk transfer problems. The stability issues are also investigated together with a case study performed for an insurance group that aims capital efficiency across the entire organization.     

Optimal Reinsurance Under the Risk-Adjusted Value of an Insurer’s Liability and an Economic Reinsurance Premium Principle

In this article, an optimal reinsurance problem is formulated from the perspective of an insurer, with the objective of minimizing the risk-adjusted value of its liability where the valuation is carried out by a cost-of-capital approach and the capital at risk is calculated by either the value-at-risk (VaR) or conditional value-at-risk (CVaR). In our reinsurance arrangement, we also assume that both insurer and reinsurer are obligated to pay more for a larger realization of loss as a way of reducing ex post moral hazard. A key contribution of this article is to expand the research on optimal reinsurance by deriving explicit optimal reinsurance solutions under an economic premium principle. It is a rather general class of premium principles that includes many weighted premium principles as special cases. The advantage of adopting such a premium principle is that the resulting reinsurance premium depends not only on the risk ceded but also on a market economic factor that reflects the market environment or the risk the reinsurer is facing. This feature appears to be more consistent with the reinsurance market. We show that the optimal reinsurance policies are piecewise linear under both VaR and CVaR risk measures. While the structures of optimal reinsurance solutions are the same for both risk measures, we also formally show that there are some significant differences, particularly on the managing tail risk. Because of the integration of the market factor (via the reinsurance pricing) into the optimal reinsurance model, some new insights on the optimal reinsurance design could be gleaned, which would otherwise be impossible for many of the existing models. For example, the market factor has a nontrivial effect on the optimal reinsurance, which is greatly influenced by the changes of the joint distribution of the market factor and the loss. Finally, under an additional assumption that the market factor and the loss have a copula with quadratic sections, we demonstrate that the optimal reinsurance policies admit relatively simple forms to foster the applicability of our theoretical results, and a numerical example is presented to further highlight our results.     

Översikt av utländska aktuarietidskrifter

Abstract

In this paper a continuous-time model of a reinsurance market is presented, which contains the principal components of uncertainty transparent in such a market: Uncertainty about the time instants at which accidents take place, and uncertainty about claim sizes given that accidents have occurred.

Due to random jumps at random time points of the underlying claims processes, the absence of arbitrage opportunities is not sufficient to give unique premium functionals in general. Market preferences are derived under a necessary condition for a general exchange equilibrium. Information constraints are found under which premiums of risks are determined. It is demonstrated how general reinsurance treaties can be uniquely split into proportional contracts and nonproportional ones.

Several applications to reinsurance markets are given, and the results are compared to the corresponding theory of the classical one-period model of a reinsurance syndicate.

This paper attempts to reach a synthesis between the classical actuarial risk theory of insurance, in which virtually no economic reasoning takes place but where the net reserve is represented by a stochastic process, and the theory of partial equilibrium price formation at the heart of the economics of uncertainty.     

WHERE'S THE CAT GOING? SOME OBSERVATIONS ON CATASTROPHE BONDS

This article provides an assessment of the current state of the market for catastrophe (or "Cat") bonds. Given the changes in insurance markets since September 11th, the demand for Cat bonds is likely to increase. For issuers, Cat bonds have the effect of transferring risks to the capital markets that would normally be underwritten by insurance or reinsurance companies. And as a substitute for insurance, Cat bonds have the potential to help issuers address problems such as lack of capacity and real risk transfer, cyclicality, and credit risk that are commonly associated with insurance and reinsurance markets. Investors value Cat bonds in part because of their low correlations with stocks and conventional bonds. Notable trends in the structuring of the products involve higher levels of risk transfer, longer-term contracts, and linkage to a portfolio of catastrophic risks.     

On the Lack of Participating Policy Usage by Stock Insurance Companies

Stock insurers can reduce or eliminate agency conflicts between policyholders and stockholders by issuing participating insurance. Despite this benefit, most stock companies don't offer participating contracts. This study explains why. We study an equilibrium with both stock and mutual insurers in which stockholders set premiums to provide a fair expected return on their investment, and with a policyholder who chooses the insurance contract that maximizes her expected utility. We demonstrate that stockholders cannot profitably offer fully participating contracts, but can profitably offer partially participating insurance. However, when the policyholder participation fraction is high, the fair‐return premium is so large that the policyholder always prefers fully participating insurance from the mutual company. Policies with lower levels of policyholder participation are optimal for policyholders with relatively high risk aversion, though such policies are usually prohibited by insurance legislation. Thus, the reason stock insurers rarely issue participating contracts isn't because the potential benefits are small or unimportant. Rather, profitability or regulatory constraints simply prevent stock insurers from exercising those benefits in equilibrium.