Optimal Investment Problem for Life Insurance Company by Considering Health-Level

In this paper, we study the optimal investment strategy for a life insurance company in a health-level framework. The income-levels of residents in different regions are different and this leads to different health-levels for various regions. We present a new framework to study the risk caused by different health-levels. The surplus process of the insurance company is described by the classical Cramér-Lundberg Model. The company is allowed to invest in a risk-free asset and a risky asset. For mean-variance criterion, we establish the corresponding Hamilton-Jacobi-Bellmen (HJB) equations and derive the time-consistent investment strategy. Finally, we provide numerical simula-tions to analyze the effects of the health-level on the insurer’s value function.


Introduction
Recently, optimal investment problem for insurers has attracted more and more attention. For example, Browne [1] considered the optimal investment problem for an insurance company with diffusion risk model and the stock price was described by a geometric Brownian motion. Hipp and Plum [2] used the classical Cramér-Lundberg risk model and obtained the optimal investment strategy for an insurance company. For other literatures on this field, please refer to [3] [4] [5] [6]. Although many researchers study optimal investment problem for general insurers, there are few literatures studying investment problem for concrete life insurance companies.
In this paper, we consider the optimal investment problem for a life insurance company. Life insurance is a contract between an insurance policy holder and an insurer. The insurers promise to pay a benefit in exchange for a premium, upon the death of a policyholder. Depending on the contract, other events such as terminal illness or critical illness can also trigger payment. At the same time as the rapid economic development, the problem of unbalanced development among different regions emerges. This disparity in development results in differences in the health level and life expectancy of residents between different regions. Thus, insurance companies should consider these differences between different regions when pricing life insurance products. We use the mortality of critical diseases to describe the differences between the different regions. Biagini et al. [7] studied the pricing and hedging of a life insurance portfolio with dependent mortality risk. They introduced the Gaussian random fields to describe the mortality intensities and obtained the analytically results of the hedging strategy. According to the analysis of the mortality of critical diseases in different regions, we introduce the concept of health-levels for different regions. We then analyze the impact of health-level on the amounts of claims and adjust premiums for life insurance in various regions.
Suppose the life insurance company choose the mean-variance criterion.
There are two main approaches to solve the mean-variance problem. One is to obtain the precommitment strategy and the other is to study the time-consistent strategy. Bi and Guo [8] derived the optimal precommitment strategy for insurers under mean-variance criterion for various cases. Li et al. [9] considered the time-consistent investment and reinsurance strategies for an insurer under Hestons stochastic volatility model. For more detailed discussion, see [10] [11] [12]. In this paper, we consider the time-consistent strategy for this optimal investment problem.
This paper according to the regional differences of the health level studies the optimal investment strategy for a life insurance company. The surplus process of the company is described by the classical Cramér-Lundberg Model and the insurer can invest in a risk-free asset and a risky asset. To maximize the profits and minimize the risk, we take the mean-variance criterion into account. Furthermore, we establish the corresponding Hamilton-Jacobi-Bellmen(HJB) equations and obtain the time-consistent optimal investment strategy. Finally, we study the effect of the health-level on the value function and the extra premium by numerical stimulations. This paper is organized as follows: Section 2 formulates the model. In Section 3, we derive the optimal investment strategy and the expectation of the terminal wealth explicitly. Section 4 gives the numerical stimulations. Section 5 concludes the paper.

Model and Optimization Problem
In this section, let ) is the information of the financial and insurance market until time t.

Surplus Process for a Life Insurance Company
In this section, we will model the heath-level by using the mortality of the critical diseases and describe the surplus process of the life insurance company.
Firstly, the health-level is a defined by the mortality of the critical diseases in different regions. Suppose that there are k regions and M critical diseases. So the health-level of the kth region k y is defined as: And the surplus process of the life insurance company is described by the classical Cramér-Lundberg model: where x is the initial capital of the life insurance company, y is the health-level of the location of the company, and ( ) P y is the premium rate of the company, Substituting Equations (3) and (4) into (2) to get a new form of surplus process: According to the approximation of Cramér-Lundberg model,

Financial Model
This section will model the financial assets and the wealth process of the insurance company.
In this paper, the financial market consists of a risk-free asset and a risk asset.
The price of the risk-free (i.e., cash) asset ( ) 0 S t is the following: where 0 0 S > is the initial price of risk-free asset and r is the risk-free interest rate.
The second asset in the market is risk asset which is described by a standard Brownian motion where s µ is the expect return rate of the risk asset; s σ is the volatility of this asset. 2 W is a standard Brownian motion on the space And the correlation coefficient between Suppose that there are no transaction costs and trading is continuous.
Moreover, donate ( ) t π as the money which investment in the risk asset ( ) π π ∈ Π = ∈ ∞ , which means that a short sell of the bonds is not permitted.
Then the wealth process of the life insurance company ( ) Finally, we defined the optimization problem for the continuous-time model (9). We want to maximize the fund size and to minimize the volatility of the terminal wealth. So we choose the mean-variance utility as our main criterion.
And the optimization problem under this criterion can be described as follow: Var (10) where ( ) t π is the investment strategy of the insurance company.

Solution of the Optimal Control Problem
In this section, we will find out the optimal solution of the problem (10). Using the methods in [13], the mean-variance optimal control problem is equal to the following Markovian time inconsistent stochastic optimal control problem: where 0 γ > is a coefficient representing the degree of risk aversion of the insurance company. And γ also helps establish the optimal strategy of mean-variance optimal control problem. And the optimal investment strategy * π satisfies ( ) ( ) , , , , where 2 2 then there exist * π ∈ Π is the optimal strategy of problem (11), and Proof The proof of this theorem is similar to the proof in He and Liang [14], so we omit the details here.  After giving the theorem, we will solve the HJB Equations ( (14), (16), (17)).
Firstly, we establish the optimal strategy of problem (11).
From (13), we get Taking (19) and (18) into (14) and differentiating (14) with respect to π , we can obtain ( ) ( ) Substituting (21) into (14) and (16), and G t x have the following form: Differentiating (24) with respect to x and t, we get and taking (25) into (22) and (23), then the equations become Let the coefficient of x and constant term of (26) and (27) And then we get the solution of these ordinary differential equations, the results are as follows: where * X π is the unique solution of the following equation:

Numerical Simulation
In this section, we will study the effects of model parameters on the effect of value functions ( ) , V t x and the extra premium ( ) k y . Throughout the numerical simulation, the initial parameters are given in Table 1 unless otherwise stated. the extra premuim ( ) k y . We find that the extra claims increase with y. This is consistent with intuition. Larger y means lower health-level, thus the extra claims ( ) f y increase as the health-level declines. Moreover, the extra premium ( ) k y increases with y, too. This shows that when the health level of a region is high, the company can appropriately reduce its premium. While the insurance company will charge more for the policyholder from regions with low health-level. And because the insurance company should afford a part of the risk of the health-level y, the extra claim is a little more than the extra premium from beginning to end. (Modify according to comment 2) Figure 2 shows the sensitivity analyzes of the value function of the insurance company. We find that the value functions increase with y first and it will decrease when the value of y is large enough. As y increases, the health-level decreases which means the extra claim is increasing and thus the value function becomes smaller accordingly. We also see that the value function is a increasing The different extra premium can make the value function keep nondecreasing with y and t. (Modify according to comment 3 and 4) Figure 2(b) plots the effect of the initial wealth x on the value function. We see that the value function of the insurance company increases with x. This is consistent with intuition. The more the initial wealth is, the larger the value function is. And the effect of y on the value function is similar to Figure 2 From the form of optimal strategy * π , we find that the health-level y doesn't influence the optimal strategy. So we do not study the effects of parameters on optimal strategy.

Conclusion
In this paper, we study the optimal investment strategy for a life insurance company with considering the differences in the health-level between different regions. The health-level of a region is defined by the incidence rate of the critical diseases. This paper first defines the expression of the health-level y and fits the extra claims ( ) f y according to the actual data. (Modify according to comment 5). The surplus process of the insurance company is described by Cramér-Lundberg model and the insurer is allowed to invest in a risk-free asset and a risky asset. The price process of the risky asset follows the Brownian motion and the insurer considers the mean-variance criterion. By the dynamic programming approach, we establish the HJB equations and derive the optimal investment strategy explicitly. Finally, numerical simulation is provided to analyze the effects of health-level and other parameters on the extra premium and value function. We find that the value function decreases as the corresponding health-level declines. Thus the life insurance company should consider a better expression of the extra premium ( ) k y to reduce the loss from the difference of the health-level in different regions and make the company can operate in a long term. (Modify according to comment 4 and 5). From Figure 2(a), we can find that the value function may decrease with t and y. So the extra premium is not enough to hedge the health-level risk. Thus we need a better definition of extra premium. For future research, we will try to find another way to calculate the extra premium ( ) k y . (Modify according to comment 6).

Fitting of the Extra Claims f(y)
We use the data from the URL: www.cdc.gov, www.acli.com and www.census.gov to fit the extra claim ( ) f y . Some data are given in Tables A1-A3. To fit the extra claims, we give the following assumptions.
• The difference of claims between different regions is only caused by different health-levels.
• The beginning time is 2000.
• The cumulate claims