Nanyang Business School Forum on Risk Management and Insurance
Optimal Investment and Premium Control for an Insurance Company with Ambiguity Aversion
In this paper, our purpose is to find the impacts of model ambiguity on optimal strategies. We consider the optimal investment and premium control problem for insurers who worry about model ambiguity. Different from most of the researches, we assume that insurers’ surplus processes are depicted by non-homogeneous compound Poisson models. With consideration of the existence of ambiguity and the objective of maximizing the expected utility of terminal wealth, we obtain the close-form solutions of the optimal investment and premium control policies by solving the Hamilton-Jacobi-Bellman (HJB) equations.
In recent years, the research fields of actuarial science have been more diversified as we can see the relationship between risk theory and financial mathematics is becoming closer. In addition, optimal control problems are getting larger important and interests. As a result, many researchers paid their attention to optimal investment, optimal reinsurance, optimal dividend and risk control problems. In particular, the investment problem has been extensively studied under mathematical insurance context. In previous studies, the researchers often focused on optimal strategies under a certain probability measure P. A basic assumption behind is that the insurer knows exactly about the true probability measure P. We call this probability measure P the reference probability measure or reference model. However, some researchers argued that the assumption is too strong. We know that the probability measure P is derived from insurers’ limited information, in other words, insurance companies estimate P through from the records of claims, regardless of accurate recognition to underlying risks. Obviously, the probability measure P may prone to misspecification error. Hence, insurers should take model ambiguity into consideration. In order to solve the problems incurred from model ambiguity, insurers need to implement alternative models (which refers to other probability measures instead of P). Many researchers paid much attention to these problems in optimal control problems.
In this paper, we consider an optimal investment and premium control problem for an insurer with a non-homogeneous compound Poisson model who worries about model ambiguity.
Different from the classical risk model, the surplus process of an insurer considered here is a non-homogeneous Poisson process, which means that the claims arrival rate is not a constant rate. In an insurance company, we can observe that the claims arrival rate depends heavily on the premium rate. Naturally, once the premium rate is changed, the claims arrival rate will change. As a result, there is a monotone function describing the relationship between the premium rate and the claim arrival rate. In our work, we use an unspecified monotone function to describe the relationship between the safety loading of premium and the claim arrival rate. Furthermore, the premium rate or the claim arrival rate which depicts the premium income will be served as a control variable. Then, a diffusion model is used to approximate the non-homogeneous compound Poisson process.
For the financial market, it is assumed that the insurer could invest in the risk-free asset or risky asset. It is also assumed that there is only one risk-free asset and one risky asset in a financial market. The interest rate of risk-free asset is denoted as r, and the price process of risky asset is described by the Black–Scholes formula. The correlation between insurance market and financial market is denoted as ρ.
We know that there are different surplus processes and different price processes of risky asset corresponding to different probability measures. We know that the probability measure P in the above may be misspecification. The alternative models (alternative probability measures) considered by insurer should similar to the reference model. We first find the alternative probability measures which are equivalent to P. For the purpose of considering the alternative models, we measure the discrepancy between each alternative model and reference model by using relative entropy. Under the situation of ambiguity aversion, we formulate a robust control problem which is the optimal investment and premium control problem. Our goal is to give solutions of the optimal investment and premium control policies. We derive the Hamilton-Jacobi-Bellman (HJB) equations by the technique of dynamic optimal principle and solve for the solutions with the assumption of that the insurer has a CARA utility.
Under the situation of our research, we find that the impacts of model ambiguity are significant. The optimal investment and premium control policies with considering model ambiguity is exact same as the optimal policies without considering model ambiguity when the insurer’s absolute risk aversion coefficient changed as some larger value. This means that, if the insurer is an ambiguity averse and conservative to the alternative model, he appears more risk averse in the finance market and insurance market. We can see that the more the ambiguity is, the less the insurer is willing to hedge in the financial market. The more the ambiguity is, the higher the premium rate will be. In addition, if the decision maker of an insurance company is ambiguity aversion, his business behavior will be more conservative. Therefore, insurers should not ignore the existence of ambiguity. Particularly, when insurers make a price, they should consider the existence of ambiguity, and should prepare the reserved space for ambiguity.
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