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.     

完全市场中的资产定价——有限离散时间情形

研究完全市场中有限离散时间情形下的资产定价问题。首先,给出了无风险收益的概念,借助无风险收益定义了一种风险中性概率。基于这个概率,得到了资产的价格等于随机现金流与随机贴现因子乘积的期望,而且资产的价格还等于资产支付关于q的期望对无风险收益的贴现值。其次,借助无风险概率考虑了资产在多期情形下的资产定价,得出了相应的股票期权公式,尤其作为推论给出了欧式看涨期权的定价公式,并对资产价格过程的鞅性作了讨论。     

An analysis of option pricing under systematic consumption risk using GARCH

An issue in the pricing of contingent claims is whether to account for consumption risk. This is relevant for contingent claims on stock indices, such as the FTSE 100 share price index, as investor’s desire for smooth consumption is often used to explain risk premiums on stock market portfolios, but is not used to explain risk premiums on contingent claims themselves. This paper addresses this fundamental question by allowing for consumption in an economy to be correlated with returns. Daily data on the FTSE 100 share price index are used to compare three option pricing models: the Black–Scholes option pricing model, a GARCH (1, 1) model priced under a risk-neutral framework, and a GARCH (1, 1) model priced under systematic consumption risk. The findings are that accounting for systematic consumption risk only provides improved accuracy for in-the-money call options. When the correlation between consumption and returns increases, the model that accounts for consumption risk will produce lower call option prices than observed prices for in-the-money call options. These results combined imply that the potential consumption-related premium in the market for contingent claims is constant in the case of FTSE 100 index options.     

Risk Management, Capital Budgeting, and Capital Structure Policy for Insurers and Reinsurers

This article builds on Froot and Stein in developing a framework for analyzing the risk allocation, capital budgeting, and capital structure decisions facing insurers and reinsurers. The model incorporates three key features: (i) value‐maximizing insurers and reinsurers face product‐market as well as capital‐market imperfections that give rise to well‐founded concerns with risk management and capital allocation; (ii) some, but not all, of the risks they face can be frictionlessly hedged in the capital market; and (iii) the distribution of their cash flows may be asymmetric, which alters the demand for underwriting and hedging. We show these features result in a three‐factor model that determines the optimal pricing and allocation of risk and capital structure of the firm. This approach allows us to integrate these features into: (i) the pricing of risky investment, underwriting, reinsurance, and hedging; and (ii) the allocation of risk across all of these opportunities, and the optimal amount of surplus capital held by the firm.     

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.     

The Design of Private Reinsurance Contracts

This article examines the effect of asymmetric information on the trading of underwriting risk between insurers and reinsurers and how it is mitigated in a context of long-term relationships. It begins by explaining how information problems affect the efficiency of the allocation of risk between insurers and reinsurers and how long-term implicit contracts allow the inclusion of new information in the pricing of reinsurance coverage. A key feature of these relationships is the reliance on loss-contingent rebates and commissions in the pricing of reinsurance coverage. We argue that when information is revealed only over time, long-term implicit contracts between insurers and reinsurers allow the inclusion of new information into reinsurance pricing. Because of this feature, the allocation of risk between insurers and reinsurers is more efficient. Specifically, such arrangements lead to more reinsurance coverage, higher insurer profits, and lower expected distress in the industry. Journal of Economic Literature Classification Numbers: G22, G13, L15, D81.     

Capital standard,forbearance and deposit insurance pricing under GARCH

We propose a multiperiod deposit insurance pricing model that simultaneously incorporates the capital standard and the possibility of forbearance. The model employs the recently developed GARCH option pricing technique in determining the deposit insurance value. Our model offers two distinctive advantages. First, it explicitly considers the implications of the strict enforcement on capital standard as stipulated in FDIC Improvement Act of 1991. Second, the use of the GARCH model allows us to capture many robust features exhibited by financial asset returns. By the GARCH option pricing theory, the value of a contingent claim is a function of the asset risk premium. This unique feature is found to be prominent in determining the bank's deposit insurance value. We also examine the effects of capital forbearance and moral hazard behavior in this multiperiod deposit insurance setting.     

An Empirical Analysis of the Pricing of Bank Issued Options versus Options Exchange Options

Since 1998, large investment banks have become active as issuers of options, generally referred to as call warrants or bank‐issued options. This has led to an interesting situation in the Netherlands, where simultaneously call warrants are traded on the stock exchange, and long‐term call options are traded on the options exchange. Both entitle their holders to buy shares of common stock. We start with a direct comparison between call warrants and call options, written on the same stock and with the same exercise price, but where the call option has a longer time to maturity. In 13 out of 16 cases we find that the call warrants are priced higher, which is a clear violation of basic option pricing rules. In the second part of the analysis we use option pricing models to compare the pricing of call warrants and call options. If implied standard deviations from options are used to price the call warrants, we find that the call warrants are strongly overpriced during the first five trading days. The average overpricing is between 25 and 30%. Only a small part of the overpricing can be explained by rational arguments such as transaction costs. We suggest that the overvaluation can be explained by a combination of an active financial marketing by the banks and the framing effect.     

投资者是非理性的吗——卖空限制下我国权证价格偏离探析

本文研究了影响我国权证价格偏离的主要因素。以往文献的重售动机理论和便利收益理论不能诠释权证定价偏差的不对称性。我们的研究提出理性避险动机是权证价格偏离的主导因素,即我国投资者并不是非理性的进行重售投机,也并没有忽视权证的避险性能。本文避免了以往文献的Black-Scholes模型依赖,在平价准则的基础上构造了非模型的误差度量方法,从而剔除了模型假设导致的价格偏差,保证了研究结果的有效性和准确性。     

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.     

基于DFA方法的中国洪水保险定价研究

对洪水保险进行精确有效的定价十分困难.动态财务分析模型(DFA)能够有效的将多种因素综合在一起进行分析,弥补了以往洪水保险定价方法以单一时点统计数据作为定价基础的缺陷;再保险安排是洪水保险计划中的一个重要组成部分,购买60%比例再保险是最适当的;购买再保险能够极大的降低洪水保险计划的破产概率.     

VOLATILITY AND LIQUIDITY AT NYSE OPENING CALLS: A CLOSER LOOK

The literature suggests that the bid-ask spread is responsible, at least in part, for greater price volatility and more negative autocorrelation at the open than at the close. In this study, we find that these phenomena are not related to the bid-ask spread, but are related instead to pricing errors by specialists or limit-order traders around the open. We use George, Kaul, and Nimalendran's (1991) model, which is less biased than Roll's (1984) model, to estimate the implied spread. The results show that, on average, the implied spread earned by liquidity suppliers is lower at the open than at the close. These results refute the contention that specialists exploit their monopoly position and earn a higher profit at the opening call. The evidence is consistent with the hypothesis that specialists set a lower cost of immediacy to encourage trading and the release of more information at the opening call. This could reduce information asymmetry and make subsequent trades in the continuous market more profitable.     

Municipal Bond Liquidity and Default Risk

This paper examines the pricing of municipal bonds. I use three distinct, complementary approaches to decompose municipal bond spreads into default and liquidity components, and find that default risk accounts for 74% to 84% of the average spread after adjusting for tax‐exempt status. The first approach estimates the liquidity component using transaction data, the second measures the default component with credit default swap data, and the third is a quasi‐natural experiment that estimates changes in default risk around pre‐refunding events. The price of default risk is high given the rare incidence of municipal default and implies a high risk premium.     

Index arbitrage and the pricing relationship between Australian stock index futures and their underlying shares

This paper examines the mispricing of Australian stock index futures. Exogenous and endogenous price volatility is confirmed to have a positive impact on the mispricing spread, after filtering out predictable time series components. More accurate pricing associated with surprise trading volume in the underlying stocks is consistent with arbitrageurs acting to narrow price disparities relative to the futures market. Ex‐ante interest rate volatility is the primary source of risk faced by arbitrageurs and fluctuations in the transaction cost of opening index arbitrage positions influence the extent to which they drive prices towards theoretical fair values.     

巨灾风险厚尾分布：POT模型及其应用

针对Gamma,LognormM和Weibull等传统厚尾分布拟合巨灾风险的不足,本文一方面从理论上分析探讨了POT模型及其拟合巨灾厚尾风险的相对优势,另一方面应用POT模型和GPD分布,对我国1952年～2008年间地震直接经济损失数据进行拟合,发现了POT模型拟合巨灾风险厚尾部分的效果比Gamma、LognormM...     

Asset pricing theory for two price economies

We show that nonlinearly discounted nonlinear martingales are related to no arbitrage in two price economies as linearly discounted martingales were related to no arbitrage in economies satisfying the law of one price. Furthermore, assuming risk acceptability requires a positive physical expectation, we demonstrate that expected rates of return on ask prices should be dominated by expected rates of return on bid prices. A preliminary investigation conducted here, supports this hypothesis. In general we observe that asset pricing theory in two price economies leads to asset pricing inequalities. A model incorporating both nonlinear discounting and nonlinear martingales is developed for the valuation of contingent claims in two price economies. Examples illustrate the interactions present between the severity of measure changes and their associated discount rates. As a consequence arbitrage free two price economies can involve unique discount curves and measure changes that are however specific to both the product being priced and the trade direction. Furthermore the developed valuation operators call into question the current practice of Debt Valuation Adjustments.     

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.     

Market Segmentation and Foreign Share Discount in China

In this paper, I formulate and test a one-period capital asset pricing model under ownership restrictions to explain the price differentials between the classes of shares that can be bought by Chinese citizens and by foreign investors, respectively. I find that time-series variability in the spread between domestic and foreign share returns is consistent with differences in risk exposures and expected risk premium, thus supporting the hypothesis of effective market segmentation and price discrimination. I also find that cross-sectional differences between domestic and foreign share returns are correlated with individual shares'; market betas. The result further supports the price discrimination hypothesis.     

Oil Risk Exposure: The Case of the U.S. Oil and Gas Sector

We estimate oil price risk exposures of the U.S. oil and gas sector using the Fama‐French‐Carhart's four‐factor asset pricing model augmented with oil price and interest rate factors. Results show that the market, book‐to‐market, and size factors, as well as momentum characteristics of stocks and changes in oil prices are significant determinants of returns for the sector. Oil price risk exposures of U.S. oil and gas companies in the oil and gas sector are generally positive and significant. Our study also finds that oil price risk exposures vary considerably over time, and across firms and industry subsectors.