Optimal Proportional Reinsurance And Investment Strategy With Credit Risk And Model Uncertainty

This paper investigates two types of optimal reinsurance and investment problems using the insurance actuarial theory,the credit risk theory,and using the methods of stochastic control and robust control.One is the optimal reinsurance and investmen-t problem when an insurance company invests in a defaultable corporate bond with credit risk,the other is the optimal reinsurance and investment strategy considering model uncertaintyCredit risk refers to the risk that the debtor fails to fulfill the obligation in the con-tract and causes losses to other economic subjects,among which the most important risk is the default risk.As a typical fixed income investment product,corporate bonds have relatively low investment risk and relatively stable income compared with other investment products such as stocks.Compared with Treasury bills,bank deposits,etc.,the yield is relatively high,which can largely meet the basic requirements of insurance funds on safety,liquidity and profitability,and is held in large proportion by insurance companies.However,this kind of investment object contains default risk.Since the outbreak of the US financial crisis in 2008,academia and the industry have paid much attention to the default risk implicit in financial assets such as corporate bonds.Cor-porate bonds are a typical kind of fixed income investment products.Compared with other investment products,such as stocks,they have relatively lower investment risk,and more stable profit.Compared with Treasury bills and bank accounts,they have relatively high yield,thus they can largely meet the basic requirements of insurance funds on security,liquidity and profitability.They are the main investment choice for an insurance company.As investors,insurance companies are also paying close atten-tion to the default risk in the bond market and adjusting their investment strategies accordingly.Based on this,the first part of this paper studies the optimal reinsur-ance and investment problem of an insurance company when its investment asset pool containing a defaultable corporate bond,corresponding to the content of chapter 3 and chapter 4.Suppose an insurance company can buy proportional reinsurance,and invest in three financial assets at the same time:a bank account,a stock and a de-faultable corporate bond.In chapter 3,the surplus process of the insurance company is described by CEV stochastic volatility model,the stock price is described by the constant elastic variance model,and the default risk of the defaultable corporate bond is described by the reduced method in credit risk.When the default occurs,the bond is recovered according to the principle of market value,and the recovered wealth can be reinvested.The goal is to maximize the exponential expected utility of the insurance company's wealth at the end of investment.Using the stochastic control theory,we ob-tain the explicit solutions of the optimal reinsurance and investment strategy and the value function.In chapter 4,we use the classic Cramer-Lundberg model to describe the surplus process of the insurance company,use a stochastic premium process whose premium is described by an Ornstein-Uhlenbeck process to describe dynamics of the stock price,and use a same dynamic as in chapter 3 to describe the price dynamic of the defaultable corporate bond.The objective is to maximize the exponential utility of the terminal wealth.Using dynamic programming principle,we derive the closed-form solutions for the optimal reinsurance and investment strategies and the corresponding value functions.In the last section of the two chapters,some numerical examples are given to demonstrate the influence of model parameters on the value function and the optimal reinsurance and investment strategy,and the result shows that insurance com-panies investing in defaultable bonds can bring greater utility.On the other hand,there are various uncertainties in the real financial market,and it is hard for the insurance company to describe various risk factors using a single prob-ability measure.It is called model uncertainty or ambiguity problem in the research area.In the second part of this paper,we consider an optimal reinsurance and invest problem involving model uncertainty,corresponding to the content of chapter 5.Sup-pose the insurer is an ambiguity-averse insurer.He can buy proportional reinsurance.meanwhile invest in a money account and a stock with a stochastic premium which is described by an Ornstein-Uhlenbeck process.He aims to maximize the expected exponential utility of terminal wealth.In order to seek more robust reinsurance and invest strategy,the insurer will build a relatively good probability model as a reference model according to his subject probability measure P,and then establish a group of alternative models under a group of probability measures which are equivalent to P The relationship of the reference model and alternative models can be described by the corresponding measure transformation,and the distance between two models can be measured by their relative entropy.Applying the theory of robust optimal control,the optimization objective transforms to a multiplier utility maximize-minimize problem,and the optimal strategies under the estimated worst case are obtained.In the last section of Chapter 5,some numerical examples are given to illustrate the influence of model parameters on the robust optimal strategies and the utility loss caused by considering model uncertainty.