Markowitz's mean-variance optimization with investment and constrained reinsurance

This paper deals with the optimal investment-reinsurance strategy for an insurer under the criterion of mean-variance. The risk process is the diffusion approximation of a compound Poisson process and the insurer can invest its wealth into a financial market consisting of one risk-free asset and one risky asset, while short-selling of the risky asset is prohibited. On the side of reinsurance, we require that the proportion of insurer's retained risk belong to $[0, 1]$, is adopted. According to the dynamic programming in stochastic optimal control, the resulting Hamilton-Jacobi-Bellman (HJB) equation may not admit a classical solution. In this paper, we construct a viscosity solution for the HJB equation, and based on this solution we find closed form expressions of efficient strategy and efficient frontier when the expected terminal wealth is greater than a certain level. For other possible expected returns, we apply numerical methods to analyse the efficient frontier. Several numerical examples and comparisons between models with constrained and unconstrained proportional reinsurance are provided to illustrate our results.


1.
Introduction. Reinsurance is a process whereby one entity (the "reinsurer") takes on all or part of the risk covered under a policy issued by an insurance company (the "cedent" or "insurer"). It is an important mechanism of risk management for a cedent to spread its underlying risk by paying some premiums to the reinsurer. The research on optimal reinsurance design dated back to the 1960s. As a sound and prudent risk management tool that permits insurance companies to be protected against adverse fluctuations, optimal reinsurance design has remained an active subject. Some typical reinsurance strategies include stop loss, proportional, excessof-loss, loss-occurring coverage and risk-attaching reinsurance. The proportional (or quota-share) and the excess-of-loss reinsurance have received great attention from the academia and practitioners. Some literatures on the proportional reinsurance include Choulli et al. [7], Højgaard and Taksar [10,11,12], Schmidli [17,18] and Taksar [22]. Some recent works on the excess-of-loss reinsurance are Asmussen et al. [1], Choulli et al. [6], Irgens and Paulsen [13] and Zhang et al. [27]. Specifically, Asmussen et al. [1] study the excess of loss reinsurance through re-parameterizing the controlled risk process by taking the drift as a basic control parameter, then the resulting process has a similar form with the dynamics under proportional reinsurance setting. Thus after the re-parametrization, one can solve the stochastic control problem under the excess-of-loss policy by using techniques in proportional settings.
Besides transferring part of risks from insurance claims by purchasing reinsurance, insurance companies may invest their surpluses in financial markets. Hence the investigation of optimal investment-reinsurance problems of insurance companies by applying stochastic optimal control theory has been one of the central research topics in actuarial science and a great attention has been drawn into this area. Browne [4] firstly introduces Brownian motion with drift to describe the surplus of the insurance company and finds the optimal investment strategy to maximize the expected exponential utility of terminal wealth. After this, Irgens and Paulsen [13] incorporate a proportional reinsurance to an optimal investment problem and derive the optimal reinsurance-investment strategy in a diffusion model from an insurer's perspective. Instead of using diffusion models for insurer's surpluses, Hipp and Plum [9] employ the Cramér Lundberg model to describe the risk process of an insurance company and suppose that the surplus of insurance company is invested in a risky asset whose price is described by a Geometric Brownian motion. Yang and Zhang [24] study the optimal investment policies of an insurer with jump-diffusion process under three criteria, i.e., maximizing exponential utility at a given terminal time; maximizing the survival probability and a general objective function. Bai and Guo [2] consider a financial market with multiple risky assets and obtain the optimal strategy under the criterion of maximizing expected exponential utility of terminal wealth. Zhang and Siu [26] model the optimal investment-reinsurance problem with uncertainty as a two-player, zero-sum, stochastic differential game between the insurance company and the market.
In this paper, we apply the mean-variance criterion to proportional reinsurance and investment problem of an insurer whose risk process is driven by the diffusion approximation of a controlled compound Poisson process. The mean-variance criterion is firstly proposed in portfolio selection by Markowitz [16] considering the expected return as well as the variance of the investment in a single period. The continuous-time version is solved in Zhou and Li [28] under the framework of linear-quadratic stochastic control theory. Considering that short-selling of risky assets is always restricted by regulatory, we introduce the no-shorting constraint in our model, then the portfolio is constrained to take nonnegative values hence the corresponding HJB equation has no smooth solution. Li et al. [14] overcome this difficulty by constructing a continuous function via two Riccati equations and show that this function is a viscosity solution to the HJB equation. Recently, Wang et al. [23] point out that the mean-variance problem is also of interest in insurance applications. Then there are increasing interests in adopting the mean-variance criterion in insurance modelling. Bai and Zhang [3] derive the optimal proportional reinsurance and investment strategy in both classical model and its diffusion approximation under the mean-variance criterion.
However, in their model the constraint of proportional reinsurance is not considered, which leads to the situation that the proportion of claim risks that insurance company might take is greater than one. In fact, the amount of claims that the company would have to take can even be more than ten times of the original risk. Such kind of optimal policies can hardly be realistic. Thus in this paper, the proportion of reinsurance is constrained to be in [0, 1], which makes our mean-variance problem challenging because the value function (i.e. the solution of HJB equation) is no longer a quadratic function. As yet, analytical research on the problem is literally nil according to our best knowledge.
The main contribution of this paper is that we construct a viscosity solution, which is not a quadratic function of the company's surplus, of the corresponding HJB equation. Based on this, the explicit expressions of efficient strategy and efficient frontier are derived when the objective expected terminal wealth is greater than a certain level. When the expected return is below that level, it is hard to determine the efficient frontier explicitly since the expression of value function, which coincides with the viscosity solution, is too complicated and involves too many parameters. However, by applying numerical methods and with the assistance of mathematical softwares, we find that the expressions of efficient strategy and efficient frontier vary with the alteration of parameters. When comparing the efficient frontiers of models with constrained and unconstrained proportional reinsurance, we find that if the proportion of retained risk q is only required to be nonnegative, the company will take much lower risks than the constrained reinsurance case. However, the optimal reinsurance proportion might be too high to be realistic, which indicates the importance of reinsurance restriction q ≤ 1.
The rest of this paper is organized as following. In Section 2, we formulate the mean-variance problem under no short-selling constraint and proportional reinsurance setting. We study an auxiliary stochastic control problem in Section 3. A viscosity solution is constructed to the corresponding HJB equation together with the optimal feedback control, where we overcome the difficulties caused by the constrained proportional reinsurance. In Section 4, the efficient strategy and efficient frontier are explicitly derived when the expected return exceeds a certain level and for the remaining objective terminal wealth, numerical analysis is applied. To illustrate the results, two numerical examples and the comparison of efficient frontiers in constrained and unconstrained reinsurance models are presented in Section 5. Finally, some additional remarks are provided in Section 6.
2. Formulation. Let (Ω, F, P) be a probability space with a filtration {F t }. Consider the classical Cramér-Lundberg model where x 0 is the initial surplus, the arrival process N t is a Poisson process with constant intensity λ > 0 and the random variables Y i , i = 1, 2, . . ., are i.i.d claim sizes independent of N t . We let {T i , i = 1, 2, . . .} denote the claim times and G(x) denotes the claim size distribution with finite first and second moments m 1 , m 2 . The premium rate c is assumed to be calculated via the expected value principle, i.e., c = (1 + η)λm 1 , where η > 0 is the relative safety loading factor.
Suppose that the insurer has the choice of both risk management and investment within a finite time horizon [0, T ]. Risk management takes the form of proportional reinsurance, i.e., insurer could transfer a fraction 1 − q(t) of the contingent claims to a reinsurer, where q(t) is F t -measurable and satisfies 0 ≤ q(t) ≤ 1 for all t. For this business, the premium rate payable to reinsurer is (1 + θ)(1 − q(t))λm 1 , where θ (θ ≥ η) represents the loading factor for the reinsurer. The insurer could invest its wealth at hand into a financial market consisting of a risk-free asset (bond) and a risky asset (stock). Particularly, the price process of the risk-free asset follows an ordinary differential equation (ODE) dS 0 (t) = rS 0 (t)dt, r > 0, and the price process of the risky asset is driven by a geometric Brownian motion where {W (t)} is a standard Brownian motion independent of claim process.
A strategy α is described by a pair of stochastic processes (π(t), q(t)), where π(t) represents the amount of wealth invested in the risky asset at time t and q(t) represents the retention proportion of claims at time t. A restriction considered here is the prohibition of short selling the risky asset, i.e., π(t) ≥ 0. While borrowing from the money market (at interest rate r) is still allowed. A strategy α is said to be admissible if (π(t), q(t)) is F t -progressively measurable, and satisfies 0 ≤ q(t) ≤ 1, π(t) ≥ 0 and E[ T 0 π 2 (s)ds] < ∞. We denote the set of all admissible strategies by α S .
Let X α t be the resulting surplus process after incorporating strategy α into (2.1), the dynamics of X α t can be preserved as follows Similar to (1.5) in Højgaard and Taksar [12], we approximate this controlled surplus process by a diffusion process with identical mean and variance, i.e.
. The independence between {W (t)} and {W 0 (t)} comes from the fact that claim process and investment return are uncorrelated.
In this paper, we apply the mean-variance principle, under which our aim is to find an admissible strategy such that the expected terminal wealth satisfies E[X α T ] = d, while the risk measured by the variance of the terminal wealth, i.e., is the terminal wealth at time T if insurance company invests all of its wealth at hand into the risk-free asset and transfers all forthcoming risks to the reinsurer.
Remark 2. The mean-variance principle considers the tradeoff between the expected return and variance of insurer's surplus at the end of time horizon T , regardless of the ruin issue during the period (0, T ). As formulated in many papers, for example, Chen et al. [5], we assume that the insurance company can continue its operation even if its surplus is negative and then the ruin problem is ignored in our model setting.
The above problem can be formulated as the following optimization problem parameterized by d: The optimal strategy of (2.4) is called an efficient strategy, and (V arX α T , d), where V arX α T is the optimal value of (2.4) corresponding to d, is called an efficient point. The set of all efficient points, when the parameter d runs over [d 0 , +∞), is called the efficient frontier.
Since (2.4) is a convex optimization problem, the equality constraint EX α T = d can be dealt with by introducing a Lagrange multiplier β ∈ R. In this way the problem (2.4) can be solved via its dual problem (for every fixed β): (2.6) 3. Value function for the auxiliary problem. For the auxiliary problem (2.6), we define the associated optimal value function by Then we can use dynamic programming approach to find the optimal control for the auxiliary problem (2.6). From standard arguments, we see that if the optimal value function V (t, x) is twice continuously differentiable (i.e., V ∈ C 1,2 ), then it satisfies the following Hamilton-Jacobi-Bellman (HJB) equation For calculation convenience, we rewrite the HJB equation (3.8) as (3.10) Since V ∈ C 1,2 is not be satisfied in most of the cases, we study the viscosity solution of this HJB equation. The notion of viscosity solution was introduced by Crandell and Lions [8] for first-order Hamilton-Jacobi equations and by Lions [15] for second-order partial differential equations. Nowadays, it is a standard tool for studying HJB equations. Adopting the notion of viscosity solution introduced by Crandell and Lions [8] and by Soner [20,21], we define Finally, a continuous function v : [0, T ] × R → R + is said to be a viscosity solution of (3.9)-(3.10) if it is both a viscosity subsolution and a viscosity supersolution at The following theorem shows that (3.9)-(3.10) has a continuous viscosity solution.
Considering the assumption v x ≥ 0, we have in the region: and the minimum is attained at (π * , q * ) = (0, 0).
Assume that the minimum of (3.10) is attained in the interior of the control region, i.e., the optimal π * (t, x) is non-negative and q * (t, Inserting the trivial solution (3.14) and α * = (π * , q * ) into (3.17), we obtain , and g 1 (t) is given by (3.13).
Considering the condition v x < 0 and q 0 (t, in the region and the minimum is attained at α * = (π * , q * ) = − µ − r σ 2 x − g 0 (t)e r(t−T ) , 1 . If we draw a graph of regions where we have already obtained the solution (see Figure 1), it is interesting to notice that there is still one region where we have no solution yet. Considering that q 0 (t, x) ≤ 1 in A 2 and q 0 (t, x) > 1 in A 3 , in the meanwhile, on the lower boundary of A 2 and upper boundary of A 3 , we have q 0 (t, x) = 1. So it is not surprising to expect that q 0 (t, x) ≡ 1 in the remaining region For this remaining region, we cannot get the solution by the previous standard method. This is because the solution of the HJB equation corresponding to the mean-variance problem we considered is no longer a quadratic function, which makes it difficult to solve the HJB equation analytically. Next we will focus on the construction of solutions in A 4 . By comparing the expression of v(t, x) on upper and lower boundaries of A 4 , we guess v has the form Plugging (3.21) into q 0 (t, x), we obtain q 0 (t, x) ≡ 1 in A 4 , which is exactly what we expect, thus q * = 1. Inserting (3.21) into the HJB equation and rearranging the terms, we obtain , and the minimum is attained at Now we have constructed a solution v(t, x) to HJB equation (3.10) with boundary condition (3.9), which is given in Theorem 3.2.
Remark 3. In the inner region of A i (i = 1, 2, 3, 4), v(t, x) ∈ C 1,2 thus it is a classical solution inside these regiones. From the method used in Bai and Zhang [3], we can prove v(t, x) is a viscosity solution on the switching curve (t, x) ∈ [0, T ] × R : x − g 1 (t)e r(t−T ) = 0 by Definition 3.1. On switching curves C 0 and C 1 , the second order derivative is too complicated to compare so the proof of viscosity needs further study.
Remark 4. From the verification theorem in Yong and Zhou [25], we know that the value function is a viscosity solution of HJB equation and the HJB equation admits at most one viscosity solution. Here, v(t, x) we constructed is a viscosity solution, then we can draw the conclusion: The value function V (t, x) for the auxiliary problem (2.6) coincides with the solution v(t, x) constructed in Theorem 3.2.
And the optimal feedback control is given by the (π * t (x), q * t (x)) in Theorem 3.2, where x = X α * t is the corresponding dynamic process. 4. Efficient strategy and efficient frontier. For notational convenience, in this section we define: According to Theorem 2.5.2 (Fritz John Conditions) in Shi [19], we know that the Lagrange multiplier β in our model exists. Then we could apply the results in Section 3 to the mean-variance problem to find the efficient strategy. Let t = 0 and The value of V β (0, X 0 ) depends on the Lagrange multiplier β. To obtain the optimal value (i.e. the minimum variance of X T ) and optimal strategy for original problem, we only need to maximize the value of V β (0, X 0 ) where β runs over R.

NAN ZHANG, PING CHEN, ZHUO JIN AND SHUANMING LI
Theorem 4.1. For d ≥ X 0 e rT + λm1η r (e rT − 1) + m2 m1θ e rT , the efficient strategy of the problem (2.4) corresponding to the expected terminal wealth EX * T = d is The efficient frontier is where A(0) = (µ−r) 2 σ 2 . Proof. We can find the piecewise maximum first and then compare them to obtain the maximum of V β (0, X 0 ) for β running over R.
It is easy to prove that f (α) = 1−e −αT α is a decreasing function for α > 0 and is always positive.
To obtain the maximum of V β (0, X 0 ) for β ∈ R, we only need to compare V β1 (0, X 0 ) and V β * (0, X 0 ) (since d ≥ d and d > d i , i = 1, 2): and the efficient frontier is given by (4.23).

4.2.
The case of d < X 0 e rT + λm1η r (e rT − 1) + m2 m1θ e rT . For d 0 ≤ d ≤ d, it is hard to obtain the explicit expression of efficient frontier since the expression of V β (0, X 0 ) is too complicated to ascertain its maximum. However, from the numerical results obtained by MatLab, we draw the following conclusion: If d 1 ≤ d 2 , we have: (4.24) • For d 1 < d < d 2 , the maximum of V β (0, X 0 ) when β runs over R is attained atβ * ∈ (β 0 , β 1 ) and the efficient frontier is given by the corresponding Vβ * (0, X 0 ). In this case, explicit expressions for the efficient strategy and efficient frontier are hard to determine; • For d ≥ d 2 , max β∈R V β (0, X 0 ) = V β * (0, X 0 ) and the efficient frontier is given by (4.23).
To illustrate how the pre-given expected terminal wealth level d and the sizing of d 1 and d 2 affect the efficient strategies and efficient frontiers, we list some numerical results in Table 1 and Table 2. In both tables, the potential piecewise maximum value of V β (0, X 0 ) are listed under several commonly used claim size distributions and different expected returns. The global maximum (i.e., V arX * T ) in each case is represented in boldface.
In Table 1, with parameters specified in the table, we have d 1 < d 2 under all distributions. Corresponding to all the three categories of d (i.e., d 0 ≤ d < d 1 , d 1 ≤ d < d 2 and d ≥ d 2 ), we pick one specific value in each case (i.e., d = d0+d1 2 , d = d1+d2 2 and d = d2+d 2 ). By comparing the potential piecewise maximum values, we find that the global maximum are attained at β * ,β * and β * respectively. In Table   2 we always have d 1 > d 2 with the granted parameters. To make the calculation more efficient in mathematical softwares, we choose a particular expected return d = d 2 − 100 instead ofd +d 2 for the category d ∈ (d,d). Results indicate that the global maximum is attained at β * ,β * and β * when the expected terminal wealth is in the range (d 0 ,d), (d,d) and (d, +∞), respectively.
In both tables, we notice that the global maximum value of V β goes up with the increase of targeted terminal wealth d, which indicates that the variance and expected terminal wealth are positively correlated. This is because lower risk investments, while good for peace of mind, will generally provide a lower expected long term return than a high risk investment. Considering that the expected returns d in Table 2 are much larger than those in Table 1, the variances in Table 2 are Table 1. Piecewise and global maximum values of V β (0, X 0 ) under different distributions, if λ = 10, θ = 0.3, η = 0.2, µ = 0.06, r = 0.04, σ = 1, T = 100 and X 0 = 50, which lead to d 1 < d 2 in all the following distributions bigger, which coincides with the mechanism that greater return comes along with more risks.  For a given expected return d, we calculate V β (0, X 0 ), then to determine the global maximum point and global maximum value of it, in this way we can obtain V arX * T thereof the efficient strategy and efficient frontier, the graphs of V β (0, X 0 ) are presented in Figure 2 Figure 2(e) exhibits the three-dimensional graph of V β (0, X 0 ) if we consider it as a function of two variables d and β.
From Figure 2 we notice that V β (0, X 0 ) is a concave function of β in this specific setting. It monotonically increases to the maximum point, which is β * when and is β * when d ≥ d 2 , thenceforth start to shrink. When d is close to d 0 , which is the company's risk free return at time T , the variance under efficient strategy is close to 0. The risk represented by variance gradually goes up as d increases, which manifests that the company should face more risks if it seeks after higher earnings.  Figure 3(a) - Figure 3(d) present the graphs of V β (0, X 0 ) with different values of pre-given expected terminal wealth d. If we deem V β (0, X 0 ) as a function of two variables, d and β, the three-dimensional graph is provided by Figure 3(e).
In this example, for β ∈ (β 0 , β 1 ), V β (0, X 0 ) will firstly increase to the local maximum then monotonically decreases to its local minimum point and then start to bounce back. Therefore, comparing with the previous example, the curve of V β (0, X 0 ) has at least one more inflection point. When d 0 ≤ d < d 2 , β * does not exist. V β (0, X 0 ) keeps going upwards for β ∈ (−∞, β 0 ] and expanding until reaching the locally maximum pointβ * , then gradually shrinks to its local minimum and bounce back after that point. Once hitting the local maximum at β * , it will start to diminish. Thus the global maximum can only be V β * (0, X 0 ) or Vβ * (0, X 0 ), depending on d ≤d or d >d (corresponding to Figure 3(a) and Figure 3(b)). When d ≥ d 2 , we always have Vβ * (0, X 0 ) > max β≥β1 V β (0, X 0 ). So the global maximum will be V β * (0, X 0 ) or V β * (0, X 0 ), depending on d 2 ≤ d ≤d or d >d. From Figure 3(a) - Figure 3(d) we notice that the variance and expected terminal wealth are positively correlated, which coincides with the conclusion in previous example.

5.2.
Main results from unconstrained reinsurance model. When the proportional reinsurance in our model is not constrained, i.e., we assume that q ≥ 0, the problem coincides with the diffusion model with ρ S = 0 in Bai and Zhang [3]. To 5.3. Impacts of reinsurance constraint. Figure 4(a) presents the efficient frontiers in models with constrained and unconstrained reinsurance in Example 1. From this figure we can find that when proportional reinsurance is unconstrained, the frontier (the red line) is linear for d ≥ d 0 . In fact, the standard deviation is almost 0 even when the expected terminal wealth is very large (7 × 10 −7 when d =d). The corresponding reinsurance strategy is q * = 12.04, which means the insurance company should act as a reinsurer and soar a twelvefold business. Hardly can this be true and the optimal variance is not realistic. If strict proportional reinsurance is considered, i.e., q ∈ [0, 1], the efficient frontier (the blue curve) is linear when d ∈ [d 0 , d 1 ]. When d ∈ (d 1 , d 2 ) or d ∈ [d 2 , +∞), the efficient frontier is part of a hyperbola while the two hyperbolas are different. As for the efficient strategies, when d increases from d 0 to d 1 , the retained proportion of claim risks is moving up from 0 to 1 and the amount invested in risky asset is gradually rising up to m 2 (µ − r)/(m 1 θσ 2 ). In this spectrum, both insurance and investment will lead to more profits to the company, as well as more risks. If d continues to increase, the company should always keep all the claim risks in order to meet its expectation for high yields. The amount invested in risky asset will remain stationary at m 2 (µ − r)/(m 1 θσ 2 ) until the company decides to pursue a wealth greater than d 2 at terminal time T . Thus [d 1 , d 2 ] can be deemed as an inaction region, during which the company do not need to adjust its investment and risk management strategies. In this region, the additional claim risks and profits are weighted equally to the company. If the company prefers a even higher profit, it should restructure its investment strategy to place even more money into risky asset and thus put itself into a more vulnerable situation.
The graph of efficient frontiers for Example 2 is given by Figure 4(b). The red line represents the frontier of unconstrained reinsurance model. Though the minimum variances corresponding to expected return d are not as good as those in Example 1, they are still much smaller than those in constrained reinsurance case. The frontier for constrained reinsurance model (the blue curve) is linear when d ∈ [d 0 ,d] and is hyperbola when d ∈ (d,d) or d ∈ [d, +∞). As for the efficient strategies, as d increases from d 0 , the company's retained proportion of risk gradually goes up from 0 to 1 and then always keeps all the risk. The amounts invested in risky asset will keep increasing when d <d, then stay at m 2 (µ − r)/(m 1 θσ 2 ) in the inaction region (d,d). To hunt for a return higher thand, company should put additional money in risky asset, which leads to even greater risks.
6. Concluding remarks. This paper derives the viscosity solution of HJB equation for mean-variance problem under proportional reinsurance and no-shorting investment. Explicit expressions for efficient strategies and efficient frontiers are derived when d ≥ d. For d 0 ≤ d < d, considering that the solution of HJB equation is not quadratic and the expression of V β (0, X 0 ) involves too many parameters thus is overly complicated, it is hard to compare the size of piecewise maximum thereof to obtain the global maximum depending on their explicit expressions. In this paper, by the assistance of mathematical software, we apply numerical method to obtain the efficient frontiers. However, rigorous proofs are left as an open problem.
In reality, to reduce the risk and improve the profit, insurance company will invest its wealth into multiple risky assets, which lead to another open problem. Suppose the price processes of risky assets are all driven by geometric Brownian motions, then we can obtain viscosity solutions of the corresponding HJB equation by the same procedure applied in this paper. However, the efficient strategies and efficient frontiers are difficult to derive. This is because when determining the optimal strategies, we could only derive the implicit expressions of optimal investment strategies. Without explicit expressions, hardly can we obtain the the maximum value of V β (0, X 0 ) and efficient frontiers.