Pricing catastrophe options with stochastic claim arrival intensity in claim time

We model claim arrival and loss uncertainties jointly in a doubly-binomial framework to price an Asian-style catastrophe (CAT) option with a non-traded underlying loss index using the no-arbitrage martingale pricing methodology. We span these uncertainties by benchmarking to the shadow price of a one-claim bond and the premium of a reinsurance contract. We implement a stochastic time change from calendar time to claim time to more efficiently price the CAT option as a random sum – a binomial sum of claim time binomial Asian option prices. This choice of the operational time dimension allows us to incorporate different patterns of catastrophe arrivals by adjusting the claim arrival probability. We demonstrate this versatility by incorporating a mean-reverting Ornstein-Uhlenbeck intensity arrival process. Simulation results verify our model predictions and demonstrate how the claim arrival probability varies with the expected claim arrival intensity.     

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.     

Valuing catastrophe derivatives under limited diversification: A stochastic dominance approach

We present a new approach to the pricing of catastrophe event (CAT) derivatives that does not assume a fully diversifiable event risk. Instead, we assume that the event occurrence and intensity affect the return of the market portfolio of an agent that trades in the event derivatives. Based on this approach, we derive values for a CAT option and a reinsurance contract on an insurer’s assets using recent results from the option pricing literature. We show that the assumption of unsystematic event risk seriously underprices the CAT option. Last, we present numerical results for our derivatives using real data from hurricane landings in Florida.     

我国巨灾风险管理模式的设计及其运作

商业化、事前补偿的巨灾保险是巨灾风险管理发展的趋势。我国应该逐步构建以政府为主导,涵盖政府、保险公司、再保险公司、资本市场和潜在受灾者五个主体的巨灾风险管理模式。在实际运作中,要考虑巨灾保险承保、保险公司巨灾风险转移和区分潜在客户等。     

Ruin probabilities and aggregrate claims distributions for shot noise Cox processes

We consider a risk process R _t where the claim arrival process is a superposition of a homogeneous Poisson process and a Cox process with a Poisson shot noise intensity process, capturing the effect of sudden increases of the claim intensity due to external events. The distribution of the aggregate claim size is investigated under these assumptions. For both light-tailed and heavy-tailed claim size distributions, asymptotic estimates for infinite-time and finite-time ruin probabilities are derived. Moreover, we discuss an extension of the model to an adaptive premium rule that is dynamically adjusted according to past claims experience.     

Modeling catastrophic deaths using EVT with a microsimulation approach to reinsurance pricing

Recently, a marked Poisson process (MPP) model for life catastrophe risk was proposed in Ekheden & Hössjer (2014). We provide a justification and further support for the model by considering more general Poisson point processes in the context of extreme value theory (EVT), and basing the choice of model on statistical tests and model comparisons. A case study examining accidental deaths in the Finnish population is provided. We further extend the applicability of the catastrophe risk model by considering small and big accidents separately; the resulting combined MPP model can flexibly capture the whole range of accidental death counts. Using the proposed model, we present a simulation framework for pricing (life) catastrophe reinsurance, based on modeling the underlying policies at individual contract level. The accidents are first simulated at population level, and their effect on a specific insurance company is then determined by explicitly simulating the resulting insured deaths. The proposed microsimulation approach can potentially lead to more accurate results than the traditional methods, and to a better view of risk, as it can make use of all the information available to the re/insurer and can explicitly accommodate even complex re/insurance terms and product features. As an example, we price several excess reinsurance contracts. The proposed simulation model is also suitable for solvency assessment.     

巨灾债券与巨灾保险风险分散

巨灾债券,作为一种债权合同,相对于巨灾再保险而言,虽然是一个两极端产品,但在分散风险方面具有其不可比拟的优势。在大额损失时,巨灾债券是巨灾再保险的一种很好的替代产品。另外,巨灾风险债券的发行对巨灾再保险免赔额具有积极影响。     

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.     

Pricing catastrophe risk in life (re)insurance

What is the catastrophe risk a life insurance company faces? What is the correct price of a catastrophe cover? During a review of the current standard model, due to Strickler, we found that this model has some serious shortcomings. We therefore present a new model for the pricing of catastrophe excess of loss cover (Cat XL). The new model for annual claim cost C is based on a compound Poisson process of catastrophe costs. To evaluate the distribution of the cost of each catastrophe, we use the Peaks Over Threshold model for the total number of lost lives in each catastrophe and the beta binomial model for the proportion of these corresponding to customers of the insurance company. To be able to estimate the parameters of the model, international and Swedish data were collected and compiled, listing accidents claiming at least twenty and four lives, respectively. Fitting the new model to data, we find the fit to be good. Finally we give the price of a Cat XL contract and perform a sensitivity analysis of how some of the parameters affect the expected value and standard deviation of the cost and thus the price.     

巨灾风险管理中的行业损失担保

行业损失担保是一种赔付主要由巨灾所造成的整个保险行业损失所触发的保险连结证券。与传统再保险相似,它也要事先确定合约的涵盖地域、灾害种类、责任限额和有效时间等。但它与传统再保险不同在于,赔付取决于两个损失触发条件,即购买者的实际损失和整个保险行业的损失。本文从行业损失担保的市场发展、定义与运行机制、精算定价等角度,对其进行了系统梳理分析,并把它与其他巨灾风险连接证券进行了比较。     

An Equilibrium Model of Catastrophe Insurance Futures and Spreads

This article presents a valuation model of futures contracts and derivatives on such contracts, when the underlying delivery value is an insurance index, which follows a stochastic process containing jumps of random claim sizes at random time points of accident occurrence. Applications are made on insurance futures and spreads, a relatively new class of instruments for risk management launched by the Chicago Board of Trade in 1993, anticipated to start in Europe and perhaps also in other parts of the world in the future. The article treats the problem of pricing catastrophe risk, which is priced in the model and not treated as unsystematic risk. Several closed pricing formulas are derived, both for futures contracts and for futures derivatives, such as caps, call options, and spreads. The framework is that of partial equilibrium theory under uncertainty.     

Valuation of Catastrophe Equity Puts With Markov‐Modulated Poisson Processes

We derive the pricing formula for catastrophe equity put options (CatEPuts) by assuming catastrophic events follow a Markov Modulated Poisson process (MMPP) whose intensity varies according to the change of the Atlantic Multidecadal Oscillation (AMO) signal. U.S. hurricanes events from 1960 to 2007 show that the CatEPuts pricing errors under the MMPP(2) are smaller than the PP by 30 percent to 66 percent. The scenario analysis indicates that the MMPP outperforms the exponential growth pattern (EG) if the hurricane intensity is the AMO signal, whereas the EG may outperform the MMPP if the future climate is warming rapidly.     

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.     

Capital, corporate income taxes, and catastrophe insurance

We provide estimates of the equity capital needed and the resulting tax costs incurred when supplying catastrophe insurance/reinsurance using a partial equilibrium model that incorporates a specific loss distribution for US catastrophe losses. After consideration of insurer investment in tax-exempt securities, tax loss carry-back/forward provisions, and personal taxes, our results imply that the tax costs of equity finance alone have a substantial effect on the cost of supplying catastrophe reinsurance. These results help explain a variety of industry developments that reduce tax costs. Also, when coupled with non-tax costs of capital, these results help explain the limited scope of catastrophe insurance/reinsurance.     

Dynamic equilibrium and the structure of premiums in a reinsurance market

In this paper we present an economic equilibrium analysis of a reinsurance market. The continuous-time model contains the principal components of uncertainty; about the time instants at which accidents take place, and about claim sizes given that accidents have occurred. We give sufficient conditions on preferences for a general equilibrium to exist, with a Pareto optimal allocation, and derive the premium functional via a representative agent pricing theory. The marginal utility process of the reinsurance market is represented by the density process for random measures, which opens up for numerous applications to premium calculations, some of which are presented in the last section. The Hamilton-Jacobi-Bellman equations of individual dynamic optimization are established for proportional treaties, and the term structure of interest rates is found in this reinsurance syndicate. The 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 equilibrium price formation at the heart of the economics of uncertainty.     

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.     

The Optimal Write-Down Coefficients in a Percentage for a Catastrophe Bond

Catastrophe bonds, also known as CAT bonds, are insurance-linked securities that help to transfer catastrophe risks from insurance industry to bond holders. When the aggregate catastrophe loss exceeds a specified amount by the maturity, the CAT bond is triggered and the future bond payments are reduced. This article first presents a general pricing formula for a CAT bond with coupon payments, which can be adapted to various assumptions for a catastrophe loss process. Next, it gives formulas for the optimal write-down coefficients in a percentage, implemented by Monte Carlo simulations, which maximize two measurements of risk reduction, hedge effectiveness rate (HER) and hedge effectiveness (HE), respectively, and examines how the optimal write-down coefficients in a percentage help reinsurance or insurance companies to mitigate extreme catastrophe losses. Last, it demonstrates how the number of coupon payments, loss share, retention level, strike price, maturity, frequency, and severity parameters of the catastrophe loss process and different interest rate models affect the optimal write-down coefficients in a percentage with numerical examples for illustrations.     

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.