Time-consistent reinsurance and investment strategies for an AAI under smooth ambiguity utility

This paper investigates time-consistent reinsurance(excess-of-loss, proportional) and investment strategies for an ambiguity averse insurer(abbr. AAI). The AAI is ambiguous towards the insurance and financial markets. In the AAI's attitude, the intensity of the insurance claims' number and the market price of risk of a stock can not be estimated accurately. This formulation of ambiguity is similar to the uncertainty of different equivalent probability measures. The AAI can purchase excess-of-loss or proportional reinsurance to hedge the insurance risk and invest in a financial market with cash and an ambiguous stock. We investigate the optimization goal under smooth ambiguity given in Klibanoff, P., Marinacci, M., & Mukerji, S. [(2005). A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892], which aims to search the optimal strategies under average case. The utility function does not satisfy the Bellman's principle and we employ the extended HJB equation proposed in Björk, T. & Murgoci, A. [(2014). A theory of Markovian time-inconsistent stochastic control in discrete time. Finance and Stochastics 18(3), 545–592] to solve this problem. In the end of this paper, we derive the equilibrium reinsurance and investment strategies under smooth ambiguity and present the sensitivity analysis to show the AAI's economic behaviors.     

Optimal proportional reinsurance with a loss-dependent premium principle

ABSTRACT

Empirical studies suggest that many insurance companies recontract with their clients on premiums by extrapolating past losses: a client is offered a decrease in premium if the monetary amounts of his claims do not exceed some prespecified quantities, otherwise, an increase in premium. In this paper, we formulate the empirical studies and investigate optimal reinsurance problems of a risk-averse insurer by introducing a loss-dependent premium principle, which uses a weighted average of history losses and the expectation of future losses to replace the expectation in the expected premium principle. This premium principle satisfies the bonus-malus and smoothes the insurer's wealth. Explicit expressions for the optimal reinsurance strategies and value functions are derived. If the reinsurer applies the loss-dependent premium principle to continuously adjust his premium, we show that the insurer always needs less reinsurance when he also adopts this premium principle than when he adopts the expected premium principle.     

Traditional versus non-traditional reinsurance in a dynamic setting

We consider a stochastic risk reserve process whose risk exposure can be controlled dynamically by applying proportional reinsurance and by issuing CAT Bonds. The CAT Bond payments are only partly correlated with the insurers losses. The aim is to minimize the probability of ruin. Using a two-dimensional diffusion approximation we obtain a controlled diffusion problem which can be solved explicitly with the help of the HJB equation. We present some numerical results and discuss to which extend the proportional reinsurance can be replaced by issuing CAT Bonds.     

Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps

This paper considers a robust optimal excess-of-loss reinsurance-investment problem in a model with jumps for an ambiguity-averse insurer (AAI), who worries about ambiguity and aims to develop a robust optimal reinsurance-investment strategy. The AAI’s surplus process is assumed to follow a diffusion model, which is an approximation of the classical risk model. The AAI is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset whose price is described by a jump-diffusion model. Under the criterion for maximizing the expected exponential utility of terminal wealth, optimal strategy and optimal value function are derived by applying the stochastic dynamic programming approach. Our model and results extend some of the existing results in the literature, and the economic implications of our findings are illustrated. Numerical examples show that considering ambiguity and reinsurance brings utility enhancements.     

On the optimality of proportional reinsurance

Proportional reinsurance is often thought to be a very simple method of covering the portfolio of an insurer. Theoreticians are not really interested in analysing the optimality properties of these types of reinsurance covers. In this paper, we will use a real-life insurance portfolio in order to compare four proportional structures: quota share reinsurance, variable quota share reinsurance, surplus reinsurance and surplus reinsurance with a table of lines.     

国际再保险业战略发展模式及其对我国的借鉴

按照再保险战略发展模式的历史演进顺序,将目前国际再保险业的战略发展模式总结为四种:专业再保险模式,再保与直保一体化模式,金融一体化单元模式和多元化单元模式。结合选取了不同发展模式的典型国际再保险公司,对不同模式进行分析和比较,并进一步提出了我国再保险业战略发展建议。     

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.     

Optimal excess-of-loss reinsurance contract with ambiguity aversion in the principal-agent model

ABSTRACT

We discuss an optimal excess-of-loss reinsurance contract in a continuous-time principal-agent framework where the surplus of the insurer (agent/he) is described by a classical Cramér-Lundberg (C-L) model. In addition to reinsurance, the insurer and the reinsurer (principal/she) are both allowed to invest their surpluses into a financial market containing one risk-free asset (e.g. a short-rate account) and one risky asset (e.g. a market index). In this paper, the insurer and the reinsurer are ambiguity averse and have specific modeling risk aversion preferences for the insurance claims (this relates to the jump term in the stochastic models) and the financial market's risk (this encompasses the models' diffusion term). The reinsurer designs a reinsurance contract that maximizes the exponential utility of her terminal wealth under a worst-case scenario which depends on the retention level of the insurer. By employing the dynamic programming approach, we derive the optimal robust reinsurance contract, and the value functions for the reinsurer and the insurer under this contract. In order to provide a more explicit reinsurance contract and to facilitate our quantitative analysis, we discuss the case when the claims follow an exponential distribution; it is then possible to show explicitly the impact of ambiguity aversion on the optimal reinsurance.     

Optimal reinsurance under adjustment coefficient measure in a discrete risk model based on Poisson MA(1) process

In this paper, we study the retention levels for combinations of quota-share and excess of loss reinsurance by maximizing the insurer’s adjustment coefficient, which in turn minimizes the asymptotic result of ruin probability. Assuming that the premiums are determined by the expected value principle, we consider a discrete risk model, in which a dependence structure is introduced based on Poisson MA(1) process between the claim numbers for each period. The impact of dependence parameter on the adjustment coefficient is discussed and numerical examples are provided to illustrate the results obtained in this paper.     

Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria

In this paper, an ambiguity-averse insurer (AAI) whose surplus process is approximated by a Brownian motion with drift, hopes to manage risk by both investing in a Black–Scholes financial market and transferring some risk to a reinsurer, but worries about uncertainty in model parameters. She chooses to find investment and reinsurance strategies that are robust with respect to this uncertainty, and to optimize her decisions in a mean-variance framework. By the stochastic dynamic programming approach, we derive closed-form expressions for a robust optimal benchmark strategy and its corresponding value function, in the sense of viscosity solutions, which allows us to find a mean-variance efficient strategy and the efficient frontier. Furthermore, economic implications are analyzed via numerical examples. In particular, our conclusion in the mean-variance framework differs qualitatively, for certain parameter ranges, with model-uncertainty robustness conclusions in the framework of utility functions: model uncertainty does not always result in an agent deciding to reduce risk exposure under mean-variance criteria, opposite to the conclusions for utility functions in Maenhout and Liu. Our conclusion can be interpreted as saying that the mean-variance problem for the AAI explains certain counter-intuitive investor behaviors, by which the attitude to risk exposure, for an AAI facing model uncertainty, depends on positive past experience.     

On a continuous time stock price model with regime switching,delay, and threshold

Motivated by the need to describe bear-bull market regime switching in stock prices, we introduce and study a stochastic process in continuous time with two regimes, threshold and delay, given by a stochastic differential equation. When the difference between the regimes is simply given by a different set of real valued parameters for the drift and diffusion coefficients, with changes between regimes depending only on these parameters, we show that if the delay is known there are consistent estimators for the threshold as long we know how to classify a given observation of the process as belonging to one of the two regimes. When the drift and diffusion coefficients are of geometric Brownian motion type we obtain a model with parameters that can be estimated in a satisfactory way, a model that allows differentiating regimes in some of the NYSE 21 stocks analyzed and also, that gives very satisfactory results when compared to the usual Black–Scholes model for pricing call options.     

混合供给模式下巨灾保险市场的激励与约束研究

巨灾保险具有准公共物品属性,既可以由政府提供,也可以由市场提供,单纯依靠政府和私人市场提供都存在较大弊端,政府和私人部门合作,采用混合供给模式可以形成互补优势,提高效率,这已成为国际巨灾保险市场发展的一大趋势。本文以混合供给模式下的巨灾保险市场为对象,分析巨灾保险市场中的委托代理关系,并对如何构建有效的激励与约束机制进行了初步探讨。     

Completion time structures of stock price movements

Summary This paper proposes to model movements in more than a century of daily US stock prices as the outcome of a multi-state marked point process and studies the time it takes for stock prices to complete an up or a down move of a certain size. We present a new econometric specification for a class of dynamic models that account for autoregressive conditional duration effects. We also present a method to account for the effect of time-varying state variables that may change within a duration.We find strong evidence of dynamic dependencies in the direction and speed of stock price movements. Past interest rates are also found to affect the speed and direction of completion times. Out-of-sample prediction results show that forecasts of the direction of moves in stock prices can be greatly improved by including covariates such as interest rates and allowing for dynamics in the econometric specification.We thank an anonymous referee, an associate editor, Rob Engle, Mark Machina and Ruth Williams for helpful conversations. We are grateful to INQUIRE UK for financial support for this research.     

Pricing mortgage insurance contracts under housing price cycles with jump risk: evidence from the U.K. housing market

Previous studies have investigated the determinants of housing price cycles in the housing market; however, we observed the phenomenon of housing price jumps in the 2007 subprime crisis. This paper presents a discussion on the housing price cycle and abnormal price jumps to describe the behavior of housing prices in the United Kingdom. The empirical results show that the impact factors of housing cycles are market risk and the switching factor. Furthermore, the impact factors of jump risks include the bursting of the housing bubble and financial crises. Therefore, in this paper, we employ the Markov switching model with jump risks to value the MI contracts and analyze the influences of housing price cycles, jump risks, risks of market interest rate, and the prepayment risks on MI premiums. The results of sensitivity analysis show that more volatile housing price index returns, as well as longer periods of higher volatility in housing prices, raise MI premiums. Moreover, the MI premium is positively related to the absolute value of the average jump amplitude and the shock frequency of abnormal events. There is the tradeoff between the market interest rate and the prepayment risk. The influences of market interest rate are different on MI premium with/without prepayment risks.     

Changes in subjective risks of hurricanes as time passes: analysis of a sample of Katrina evacuees

Using a quasi‐field experiment, we report on subjects' perceptions of the risks of hurricanes. All experimental subjects were displaced by either Hurricane Katrina or Rita, in New Orleans and other Gulf Coast areas, except for a small control group consisting of people who live in central Texas. We examine their perceptions of risks just after the hurricanes occurred, and over one year later to evaluate the change in subjective risk perceptions over time. A latent risk model is estimated in which subjective probabilities of hurricane strike risk are represented as a function of respondents' demographic characteristics and experiences following the storms.     

Speculative attacks: A laboratory study in continuous time

We examine speculative attacks in a controlled laboratory environment featuring continuous time, size asymmetries, and varying amounts of public information. Attacks succeeded in 233 of 344 possible cases. When speculators have symmetric size and access to information: (a) weaker fundamentals increase the likelihood of successful speculative attacks and hasten their onset, and (b) contrary to some theory, success is enhanced by public access to information about either the net speculative position or the fundamentals. The presence of a larger speculator further enhances success, and experience with large speculators increases small speculators' response to the public information. However, giving the large speculator increased size or better information does not significantly strengthen his impact.